Download Determination of Risk Aversion and Moment

Determination of
Risk Aversion and Moment-Preferences:
A Comparison of Econometric models
DISSERTATION
der Universität St.Gallen,
Hochschule für Wirtschafts-,
Rechts- und Sozialwissenschaften (HSG)
zur Erlangung der Würde eines
Doktors der Wirtschaftswissenschaften
vorgelegt von
Fabian Wenner
von
Murten (Freiburg) und aus Deutschland
Genehmigt auf Antrag der Herren
Prof. Dr. Klaus Spremann
und
Prof. Dr. Alex Keel
Dissertation Nr. 2606
Difo-Druck GmbH, Bamberg 2002
Die Universität St. Gallen, Hochschule für Wirtschafts-, Rechts- und Sozialwissenschaften (HSG) gestattet hiermit die Drucklegung der vorliegenden Dissertation, ohne damit zu den darin ausgesprochenen Anschauungen Stellung
zu nehmen.
St. Gallen, den
Der Rektor:
Prof. Dr. Peter Gomez
To
my parents Gerhard Wenner
and Cornelia Wenner-Deloséa
and my wife Beatrice Zellweger
4
Acknowledgements
I would like to thank Professor Takeshi Amemiya and Professor Klaus Spremann for their patient supervision and their ingenious advice. Without their
encouragement, their guidance, confidence and their academic example this
thesis could not have been written. Profound gratitude goes to my coexaminer
Professor Alex Keel who made me familiar with the fundamentals of statistics during my studies and spontaneously and genuinely agreed to support
and supervise my thesis.
Professor Spremann has been very supportive since the very beginning of
my studies. His academic gatherings in Vättis and the accompanying discussions provided fertile ground for the development of new ideas - one of their
products being the thesis on hand.
I am also much indebted to Prof. Takeshi Amemiya for giving me the
opportunity of a stimulating and very hospitable stay at the Economics department at Stanford University. It was an academic experience that will
accompany me for the rest of my life.
Inexpressible is my gratitude for the help and support, the love and care
of my parents. My mother’s and sister’s love for languages not only urged
me to write this thesis abroad, it also cut them out for the ungrateful job
of correcting this manuscript. Many thanks on that score, especially to my
Mum.
For valuable and helpful discussions I am indebted to Kenneth Arrow, Andreas Dische, Luigi Pistaferri, Clemens Sialm, Ed Vytlacil, Alexandre Ziegler.
Valuable and ingenious input during a presentation of my paper also came
from Whitney Newey and James Powell. I’d also like to thank Stephanie
Winhart for helping me with evaluating a survey on risk taking in the early
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ii
stage of this thesis. All errors are mine alone.
Financial support from the Swiss National Fund for my one-year stay at
Stanford University is gratefully acknowledged. It considerably helped to
focus on academic research in Silicon Valley in a time when its real estate
‘bubble’ was at its peak.
Many thanks also go to Camilla and Antonio Cozzio who welcomed us
heartily in our new environment in Palo Alto and made us a part of their
family during our stay in the bay.
Last but definitely not least I would like to thank the person most essential
for the stepwise development of this thesis: Beatrice has walked along with
the ups and downs of its growth path, patient but always urging on, cheering
on the way, listening to partly obscure thoughts and making the time working
on it worth living.
Zurich, Summer 2001.
Fabian Wenner
Table of Abbreviations
Abbreviation
Meaning
AIC
αa , αAP
αi
αr
appe
ARA
BNL
CAPM
CD
CE
CLM
CLT
CPPI
CRRA
DCM
ε
ER
EU
EVD
FS
GBM
Govnmt
IID
Akaike Information Criterion
Pratt-Arrow measure of absolute risk aversion
Measure of relative risk aversion for investor i
Pratt-Arrow measure of relative risk aversion
Average per period exposure
Absolute risk aversion
Binomial Logit Model
Capital Asset Pricing Model
Certificate of Deposit
Certainty Equivalent
Conditional Logit Model
Central Limit Theorem
Constant proportion portfolio insurance
Constant relative risk aversion
Discrete Choice Model
Error term
Expected Return
Expected Utility
Extreme Value Distribution
Fisher skewness = m3 /σ 3
Geometric Brownian Motion
Government
Independently and Identically Distributed
continued on next page
iii
iv
Abbreviation
Meaning
IRA
j
k
KEOGH
Individual Retirement Account
Subscript representing the risk category
Subscript representing the independent factor
Tax-deferred pension account for employees of unincorporated businesses or for persons who are self-employed.
Likelihood function
Log-Likelihood function
Linear Probability Model
Likelihood Ratio Test
Non-normalized third central moment of the
return distribution
Multinomial Logit Model
Subscript representing the observation of the sample
Negative Exponential Utility
(Null hypothesis) cannot be refuted
Ordinary Least Squares
Probability
Probability Distribution Function
Density function of the standard normal variable
Distribution function of the standard normal variable
Risk premium
Skewness preference of investor i,
defined as W 2 · U /U Dividend yield of index option
Simple return
Riskfree rate
Risk Aversion
(Null hypothesis) can be refuted
Skewness or non-normalized third central moment
Standard Deviation of a random variable
Total dollar value
Time invariant portfolio protection
Total market value
continued on next page
L
logL
LPM
LRT
m3
MNL
n
NEU
n.r.
OLS
P
PDF
φ
Φ
π
ψi
q
R
rf
RAV
ref.
S
σ
tdv
TIPP
tmv
Chapter 0. Table of Abbreviations
Abbreviation
Meaning
U
W
WT
WLS
yi
Utility
Wealth
End-of-period Wealth
Weighted Least Squares
Continuous dependent variable representing
the stock ratio of investor i’s portfolio
Discrete dependent variable representing
the stock ratio j of investor i’s portfolio
Yi (j = 1)
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Contents
Acknowledgements
i
Table of Abbreviations
iii
Abstract
xv
1 Introduction
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Earlier assessment methods: overview and critique . . . .
1.2.1 Intuitive approaches . . . . . . . . . . . . . . . . .
1.2.2 Gambles . . . . . . . . . . . . . . . . . . . . . . . .
1.2.3 Derivation of risk aversion in asset pricing models
1.3 Research idea . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.1 Portfolio theory’s old and new facts . . . . . . . .
1.3.2 Overview of models . . . . . . . . . . . . . . . . .
1.3.3 Structuring the Asset Allocation Decision . . . . .
1.3.4 Outline of thesis . . . . . . . . . . . . . . . . . . .
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Two-Moment Risk Preference
2 The CAPM and two moment risk preference
2.1 Assumptions of mean-variance portfolio selection .
2.2 The concept of two-moment risk aversion . . . . .
2.2.1 The Markowitz Premium . . . . . . . . . .
2.2.2 The Pratt-Arrow measure of Risk Aversion
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viii
CONTENTS
2.3
2.4
2.5
2.6
Relevance of assessing the correct asset allocation . . . . .
2.3.1 Calculating Expected Utility loss . . . . . . . . . .
2.3.2 All stocks half the time or half stocks all the time?
Switching and the time horizon controversy . . . . . . . .
Risk aversion and changes in wealth . . . . . . . . . . . .
Bounded rationality and goal of methodology . . . . . . .
2.6.1 Fundamentals of bounded rationality . . . . . . . .
2.6.2 Limits of Expected Utility Maximization . . . . . .
2.6.3 Presentation determines investment choice . . . . .
2.6.4 Goal of questionnaire . . . . . . . . . . . . . . . .
2.6.5 General problem of the approach . . . . . . . . . .
3 Empirical Analysis
3.1 The Data set . . . . . . . . . . . . . . . . . . . . . .
3.2 Selection of factors and hypotheses . . . . . . . . . .
3.3 Testing for bounded rationality . . . . . . . . . . . .
3.3.1 Putting things in perspective . . . . . . . . .
3.3.2 ‘Correctly’ assigned observations . . . . . . .
3.3.3 ‘Wrongly’ assigned observations . . . . . . . .
3.3.4 Stated and observed preferences . . . . . . .
3.4 Structure of Analysis and Nests . . . . . . . . . . . .
3.5 Characterization of econometric models . . . . . . .
3.5.1 The Objective Function . . . . . . . . . . . .
3.5.2 Ordinary and Weighted Least Squares Model
3.5.3 The Tobit Model . . . . . . . . . . . . . . . .
3.5.4 The Ordered Logit Model . . . . . . . . . . .
3.5.5 The Binomial- and Multinomial Logit Model
3.5.6 The Conditional Logit Model (CLM) . . . . .
3.5.7 The Nested Logit Model . . . . . . . . . . . .
3.6 Goodness of fit and hypotheses testing . . . . . . . .
3.6.1 Classification Tables and Error Distance . . .
3.6.2 Akaike Information Criterion (AIC) . . . . .
3.6.3 Likelihood ratio tests . . . . . . . . . . . . . .
3.7 Results of regressions for Setting 1 - “All Categories”
3.7.1 Coefficients . . . . . . . . . . . . . . . . . . .
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CONTENTS
ix
3.7.2
Results of Classification Tables . . . . . . . . . . . . . . 88
3.8
3.9
Results of regressions for Setting 2 - Two-step estimation . . . 89
3.8.1
Setting 2a - “Assetholder or Non-asset holder” . . . . . 89
3.8.2
Setting 2b - “Assetholders only” . . . . . . . . . . . . . 92
Results of regressions for setting 3 - Three-step estimation . . . 93
3.9.1
Setting 3b - “Stock- or Non-stock holder” . . . . . . . . 94
3.9.2
Setting 3c - “Stockholders only” . . . . . . . . . . . . . 95
3.10 Conclusion of the two-moment setting . . . . . . . . . . . . . . 96
3.10.1 The factors and their explanative power . . . . . . . . . 96
3.10.2 Performance of the econometric models . . . . . . . . . 97
3.10.3 Transferability of results . . . . . . . . . . . . . . . . . . 98
4 Joint estimation by gambles and observed stock ratio
II
101
4.1
Determining Two-Moment Risk Aversion by Gambles . . . . . 102
4.2
Joint estimation of econometric choice and gamble . . . . . . . 104
Three-moment risk preference
107
5 Shortcomings of two-moment asset pricing
111
5.1
5.2
Critique of the mean-variance approach . . . . . . . . . . . . . 111
5.1.1
Quadratic utility . . . . . . . . . . . . . . . . . . . . . . 112
5.1.2
Normal distribution . . . . . . . . . . . . . . . . . . . . 113
5.1.3
Performance measurement of optioned portfolios . . . . 115
Summary of critique . . . . . . . . . . . . . . . . . . . . . . . . 116
6 Skewness
117
6.1
Higher moment preferences . . . . . . . . . . . . . . . . . . . . 118
6.2
Prospect theory and adaptive aspiration . . . . . . . . . . . . . 120
6.3
Gambling and insurance habits . . . . . . . . . . . . . . . . . . 123
6.3.1
Cubic utility and skewness
6.3.2
Implications of the cubic utility . . . . . . . . . . . . . . 123
. . . . . . . . . . . . . . . . 123
6.3.3
Insurance and Gambling . . . . . . . . . . . . . . . . . . 125
x
CONTENTS
7 Determining skewness preference through gambles
7.1 Risk premia for three moment approximation . . . .
7.2 Creating gambles for skewness preference . . . . . .
7.2.1 Interpretation of three-moment preferences .
7.2.2 Example for the assessment of risk aversion .
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. 135
. 136
8 Creating portfolio skewness with options
8.1 Portfolio distribution . . . . . . . . . . . . . . . . .
8.2 Truncating the distribution’s lower end with Puts .
8.3 Enhancing the distribution’s upper end with Calls
8.4 Implementation and Limits of Options strategies .
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9 Joint estimation of three moment preference
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10 Summary and Conclusion
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A Asset Allocation and Higher Moment Models
A.1 Optimal Asset Allocation in the 2-Moment Model
A.1.1 Objective Function . . . . . . . . . . . . . .
A.1.2 Solution of the Objective Function . . . . .
A.2 Optimal Asset Allocation in the 3-Moment Model
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B Deriving the MNL model from Utility maximization
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C Likelihood function of the Nested Logit Model
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D Structure of Estimations
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E Empirical Results of Regressions
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E.1 In-sample estimation SCF1998 . . . . . . . . . . . . . . . . . . 179
E.2 Out-of-sample estimation SCF 1995 in SCF1998 . . . . . . . . 215
E.3 Likelihood Ratio Tests . . . . . . . . . . . . . . . . . . . . . . . 247
F Independent Factors
251
G Various Riskrulers
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G.1 Risk Quota by Fidelity Investments . . . . . . . . . . . . . . . . 261
G.2 Allianz Anleger Analyse . . . . . . . . . . . . . . . . . . . . . . 261
CONTENTS
xi
G.3 Union Investment . . . . . . . . . . . . . . . . . . . . . . . . . . 261
H Practical implementation of a Risk Ruler
H.1 Proceeding when developing a Risk Ruler
H.2 Example of an interactive Risk Ruler . . .
H.2.1 Small sample Internet survey . . .
H.2.2 Calculating the predicted choice .
I
Empirical performance of risk classes
J Survey of Consumer Finances: Details
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xii
CONTENTS
List of Figures
1.1
1.2
Structuring the asset allocation decision . . . . . . . . . . . . . 15
Structure and outline of thesis . . . . . . . . . . . . . . . . . . 18
2.1
2.2
2.3
Dividing the CML into 6 funds . . . . . . . . . . . . . . . . . . 22
Typical utility function of a risk averter . . . . . . . . . . . . . 26
Skewed gambles and their Pratt-Arrow risk premia . . . . . . . 31
3.1
3.2
3.3
3.4
The data sets: SCF 1995 and 1998 . . . . . . . .
Factor coefficients of different risk classes, setting
Factor coefficients of different risk classes, setting
Factor coefficients of different risk classes, setting
5.1
Distorted performance of option strategies in the CAPM . . . . 116
8.1
8.2
8.3
Lognormal portfolio hedged with puts, floor 85% . . . . . . . . 144
Lognormal portfolio hedged with puts of different strikes . . . . 145
Moments of hedged portfolios . . . . . . . . . . . . . . . . . . . 149
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1 .
2b
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C.1 Simple 3 response nested logit model . . . . . . . . . . . . . . . 175
D.1 Structure and nests of empirical analyses . . . . . . . . . . . . . 178
G.1
G.2
G.3
G.4
‘Risk Quota by Fidelity Investments’ .
‘Allianz Anleger Analyse’ . . . . . . .
‘Union Investment’ - Answering Sheet
‘Union Investment’ - Evaluation Sheet
xiii
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LIST OF FIGURES
H.1 Internet questionnaire: Small Sample Data . . . . . . . . . . . 276
H.2 Internet Survey: Evaluation Sheet . . . . . . . . . . . . . . . . 281
I.1
I.2
Return distribution of portfolios with low stock ratios . . . . . 285
Return distribution of portfolios with high stock ratios . . . . . 286
Abstract
The assessment of an individual investor’s risk aversion is an obligatory part
of professional investment advice. It is also essential for the financial planning
process in general. The assessment approaches of banks, funds and insurances
however, usually lack structure, scientific methods and empirical testing.
An investor’s total wealth can be conceptually divided into tied assets which are budgeted for specific short-or mid-term projects - and into free
assets - whose purpose is yet undefined. While the shortfall risk approach
seems best suited for the first quantity, allocation according to individual risk
aversion is adequate for the latter part of wealth.
The thesis at hand shows how to determine individual risk aversion with
different discrete choice models, with gambles and jointly with both methods. The methods developed thus aim at allocating the investor’s free part of
wealth.
In the first part empirical estimation of selected socioeconomic factor coefficients that determine risk preference is carried out by OLS, WLS, Tobit,
Ordered Logit and Multinomial Logit models. The Survey of Consumer Finances 1995 and 1998 proves to be the ideal data sample for this purpose.
The preliminary analysis of the first part is carried out assuming the traditional two-moment mean-variance framework. While demographic factors
such as age, gender and marital status will prove insignificant, different saving
reasons, the investment horizon and expectations about the economic development - among others - allow a stepwise out-of-sample assignment precision
of up to 90%.
In the second part the traditional asset pricing model analysis is extended
to incorporate the third moment, skewness. A joint estimation employing obxv
xvi
served and stated preferences assists in determining an individual investor’s
preference pattern regarding the trade-offs between the three moments - mean,
variance and skewness. Investors favor positive skewness and some will accept a lower mean or higher return variance to obtain higher positive skewness.
The study briefly shows how to translate these preference patterns into option
strategies that are added to the portfolio. For the skewness analysis the traditional Pratt-Arrow measure of risk aversion is extended to a three moment
risk premium.
The study exemplifies how to proceed when determining risk aversion on
an individual basis. It shows how to create a questionnaire based on hypotheses, it examines what model to use for what data basis in order to achieve
the highest level of prediction and it illustrates how to include higher moment
preferences within this framework.
Chapter 1
Introduction
“...and they see only their own shadows, or the
shadows of one another, which the fire throws on the
opposite wall of the cave [...]”
– Plato, The Republic, VII.
While the theoretical knowledge about diversification, risk-return tradeoff and portfolio-separation has long been at our disposal, the overwhelming
majority of both investors and investment advisors has not implemented the
concepts introduced by Markowitz (1952), Sharpe (1965), Ross (1976), Tobin
(1958) and others:
Puzzlingly stock-ownership especially in Europe is still limited to a minority among savers, and financial institutions are extremely lax about profoundly educating their clients and show little effort determining their investor’s risk aversion. Banks and brokers usually content themselves with
informing investors about the risk-return trade-off and advise them to adapt
their portfolio to changing needs. For profound investment advice, however,
the conceptual division of one’s financial assets into reserved funds and free
wealth is of paramount importance to the asset allocation decision. Sound
assessment of the investor’s risk aversion is its second most relevant step.
One reason for the lack of more serious analysis of risk aversion and the
development of a profound quantitative assessment method in practice is obviously the difficulty to determine risk aversion (RAV) with confidence. This
1
2
1.1. Motivation
is rather puzzling, as it is such a fundamentally important variable in most
financial and economic models.
While some studies have chosen a cross-sectional approach to quantify it
on an aggregate basis, others have taken interest in the individual investor and
have chosen a more direct approach using gambles or questionnaires whose
significance was not tested empirically. Whatever method used, RAV has been
determined only within a certain range and often not without paradoxical
results. The latter arise due to well-known behavioral anomalies described in
articles by Tversky, Kahneman, Thaler and others.
The study at hand aims at developing a methodological framework for
assessing investors’ risk preferences for their free part of wealth. The analysis
of risk aversion is carried out by discrete choice models1 and at a later stage
by gambles. The framework lays out the structure of the investor’s asset allocation and examines the patterns underlying the choice of his risk level. The
gathered information can be utilized to set up a questionnaire that predicts
an individual’s investment choice by weighing those factors found significant
in explaining the choice of the investment risk level.
The goal is to determine significant behavioral factors that capture as
many aspects of investment decision making as possible and integrate them
into an empirical model.
1.1
Motivation
Several studies as the one by Kritzman (1992) but also many behavioral finance studies2 , prove that single investors are typically not conversant with
the mathematical and theoretical underpinnnings of asset allocation. A necessary condition for good advice is therefore to present the investment choice
and asset allocation in a way that appeals to an individual’s intuition and
that lets the investor easily understand and relate to the basic differences of
concurring investment choices. The study will develop a framework of questions and gambles for individuals whose evaluation yields an estimate for the
1 In the year 2000, the year this thesis was written, D. McFadden and J. Heckman were
awarded the Nobel prize in economics for the development of the models and methods used
in this thesis’ empirical analysis.
2 See for example Shleifer (1999).
Chapter 1. Introduction
3
asset allocation of the free part of wealth. Its major benefit lies in the fact
that the answering of the questionnaire does not require the investor to have
much financial expertise.
Another motivation for the development of more profound assessment
methods is the legal urge for European banks to develop sound practice in
advising investors3 . Sound practice necessitates quantitative methods for
assessing risk aversion. Despite the large body of knowledge related to individual risk-taking and investment allocation, there still remain many open
questions. As public policy makers consider changes to public and private
pension regulation, it is increasingly important that there be a clear understanding of the factors that influence individual investment choices. Financial
behavior of individuals has also increasingly awakened academic interest, as it
is either explicitly or implicitly at the core of traditional predictive models of
asset pricing and aggregate behavior. As a by-product the analysis of factors
which explain investment behavior and the assessment of investment preferences proves valuable to product development in asset management. Surveys
of financial behavior help decide what type of funds to offer to whom.
1.2
1.2.1
Earlier assessment methods: overview and
critique
Intuitive approaches
Most risk assessment methods used in practice lack empirical foundation.
One example is the ‘Risk Quota’ (RQ) Questionnaire of 1991 by Fidelity
Investments as described in Luenberger (1998)4 . Fidelity’s RQ consists of 20
factors that are supposed to be most relevant to the asset allocation decision.
All questions are derived from normative theory and thus concerned with how
people should invest. The answers given result in an overall score that assigns
3 The
insight that the average individual has insufficient understanding of the financial
markets has led most European countries as well as the U.S. to enact laws about the
general duties of care for the protection of individual investors: Art.31, Abs. 2 & 3 WpHG
in Germany, Art. 11 BEHG in Switzerland, Financial Services Act of 1998 (H.R. 10) in the
U.S.
4 please see Appendix G.1
4
1.2. Earlier assessment methods: overview and critique
the investor to one of five investment funds that have distinctly different
stocks ratios or risk levels. The calculation of the factor weights and the
overall score occurs by rule of thumb. These assessment attempts can in
general be identified as intuitive methods that do not analyze the influence
and weight of each independent factor empirically - let alone their interaction
effects. Instead, the weights are assigned arbitrarily or they are derived from
separate normative theories. The most severe drawback of these assessment
attempts, however, is to mix questions about future projects with questions
about one’s personality. The two concepts relating to these issues are to be
treated separately, as will be discussed later.
Many other Risk Rulers used in practice are based on psychological theory
and were implemented without having to pass empirical testing. Examples
are the ‘Allianz Anleger Analyse’5 , ‘Bœrsencoach’ by aixigo6 , ‘riskquestionnaire’ by promistar7 , ‘Psychotrainer’ by Bœrse Online/Dialego8 and Schwab’s
‘Risk Ruler’9 . Major banks provide rather simplified versions of their Risk
Rulers online, as they are afraid of imitation. The ‘Riskruler’ by UBS and the
‘Risk Analyser’ by CreditSuisse10 fit this description. Closely related to Risk
Rulers are so-called Asset-Allocation Tools provided by most banks, insurance groups and Online brokers: Accutrade’s ‘Asset Allocation Worksheet’11 ,
Smartmoney’s ‘Asset Allocator’12 and ‘Allocator’ by Investsearch.
1.2.2
Gambles
Gambles are based on the Bernoulli principle13 which assumes that individuals maximize expected utility when given the choice between different payoffs. These payoffs are typically described by two or more outcomes equalling
monetary amounts that will arise with a given probability. Each gamble and
choice refers to a different point on the investor’s utility curve. By having the
5 please
see Appendix G.2
6 http://www.boersen-coach.com/frames.php3?start=fb
7 http://www.promistar.com/LTCRiskProfileQuestionnaire.pdf
8 http://www.dialego.de/bo11/
9 http://www.schwab.com/SchwabNOW/navigation/mainFrameSet/0,4528,552—829,00.html
10 http://investmentmanager.cspb.com/en/riskanalyzer/index.html
11 http://www.accutrade.com/fhtml/assetallocation.fhtml
12 http://www.smartmoney.com/oneasset/
13 See
the translation of Bernoulli’s original text of 1738 in Sommer (1954).
Chapter 1. Introduction
5
investor choose among gambles several times, an approximate outline of his
utility function can be derived14 . The form of the utility function contains the
investor’s degree of risk aversion. More demanding and less accurate gambles
as the ones by Mosteller and Nogee (1951) present risky and riskless choices
and let the investor quantify his certainty-, outcome- or probability- equivalent. As Schoemaker and Hershey (1992) proved, this assessment method
differs even intra-personally depending on the method chosen and thus yields
very unreliable results. Examples of the avenue of investigating risk taking
behavior as outlined above are: Gertner (1993), Metrick (1995), Altaf (1993)
and Levy (1994). Evidence on those studies is mixed. Most find that people’s
behavior is inconsistent with the predictions of expected utility theory. Another issue that comes up with gambles is whether there is any down-side risk
presented. If there is none then the results may not be generalizable to behavior in the real world. A serious drawback is often the often small sample size
of most of these experiments. Lastly, for average investors gambles generally
seem to be too difficult to understand. This might also explain intrapersonal
inconsistencies and paradoxical outcomes. Gambles and the rigorous belief in
expected utility maximization as the general decision making rule during the
1950s have been criticized for their ambiguous and paradoxical results. Three
critiques seem worthwhile mentioning:
1. Maurice Allais (1953) opposed the American school for using the Bernoulli
principle as its major working hypotheses at the time. He remarked that
the experimental observation of the behavior of men who are considered
rational by public opinion invalidates it. Four facts account for its failure
in reality:
• Very prudent people behave non-according when gambling small
sums.
• The choice of risks bordering on certainty contradicts the independence principle of Savage.
• The choice of risks bordering on certainty contradicts the substitutability principle of Samuelson.
14 This
was shown by Meyer and Pratt (1968)
6
1.2. Earlier assessment methods: overview and critique
• The behavior of entrepreneurs when great losses are possible deviates from expected utility maximization. For Allais the dispersion
as well as the general properties of the form of the probability distribution of psychological values are essential to the theory of risk.
They must be taken into account by every theory of risk if it is to
be realistic. The Bernoulli principle as a description of rationality
has no interest per se. Rationality must be defined either in the
abstract by referring to a general criterion of internal consistency15
or it must be defined experimentally by observing the actions of
people who can be regarded as acting in a rational manner. Allais viewed all the fundamental postulates leading to the Bernoulli
principle as based on false evidence.
2. Machina (1982) showed that the outcome of individuals’ decision making
differs significantly from what Expected Utility Theory would suggest.
3. Kahneman and Tversky (1979) coined the term ‘bounded rationality’16
and proved that individuals do not interpret probabilities according to
expected utility. Tversky and Kahneman (1986) show that preferences
are quite insensitive to small changes of wealth, but highly sensitive to
corresponding changes in the reference income level. Losses and gains
must therefore be considered with respect to this reference income level.
Also, individuals exhibit loss aversion, i.e. they are much more responsive to losses than to gains. A certain income decrease thus results in
a higher loss of utility than the utility gain associated with the same
income increase. Furthermore, the utility function becomes convex in
the loss region which implies local risk loving behavior. In other words,
there exists a range of income levels below the reference income where
bounded rationality reduces risk aversion.17
15 Criterion
implying the coherence of desired goals and the use of appropriate means for
attaining them.
16 See Section 2.6.
17 This revives the discussion of the shape of the utility function. Confer in this context
Friedman and Savage (1948) as well as Kahneman and Tversky (1979) among others.
Chapter 1. Introduction
7
As an alternative to expected utility theory the concept of bounded rationality
demonstrates better compatibility with empirical findings18 . As Brunnermeier
(1996) shows, it can provide theoretical explanation for inconsistent behavior
of individuals towards income lotteries and other anomalies like loss aversion
and status quo bias. In contrast to rational individuals, boundedly rational
decision makers are restricted both by the availability of information and their
ability to learn. However, their persistent search for ‘better’ heuristics leads
to a rational thinking process eliminating systematic errors.
1.2.3
Derivation of risk aversion in asset pricing models
Various empirical studies employing intertemporal equilibrium models derive
risk aversion from option prices and realized asset returns19 . Studies on the
equity-premium puzzle20 and its implications are one example. All of them
choose a cross-sectional approach to determine risk aversion on average. They
infer statements about properties of absolute and relative RAV and the change
of RAV over time which are relevant for asset pricing. As shares are generally
riskier than bonds and as the average investor can be said to show risk averse
behavior, stocks earn a premium on the capital markets. Over the last 20
years this risk premium was about 8% higher than the one for bonds - a
difference inexplicable by the traditional asset pricing models alone. In fact
a difference so significant that relating theories in asset pricing are rendered
incompatible. Given the above equity premium, the measure of absolute risk
aversion implied by the traditional CAPM would come to about 18. This
is significantly higher than the coefficient of proportional risk aversion which
Blume and Friend (1975) estimated to be in excess of two. Either our asset
pricing models do not explain reality convincingly or the empirically estimated
measure of risk aversion is in fact drastically higher than we have assumed so
18 Non-expected
utility and non-utility models as proposed by Fishburn (1988), Machina
(1989) as well as Epstein and Zin (1990) also offer feasible alternative solutions.
19 Cf. Ait-Sahalia and Lo (1999), Jackwerth and Rubinstein (1996) and Jackwerth (1998).
20 Cf. Mehra and Prescott (1985). They state that standard asset pricing models fail
to explain the high average excess return on the US stock market. Their study is based
on a consumption-based asset pricing model with power utility function which causes the
elasticity of intertemporal substitution to be the reciprocal of the coefficient of relative risk
aversion. This relation is an undesirable feature and avoidable if another utility function is
assumed, as done by Epstein and Zin (1991) and Weil (1989).
8
1.2. Earlier assessment methods: overview and critique
far.
Arrow (1971) found constant relative risk aversion to be around 1. The
proposed log utility function, however, does not seem to be an appropriate
assumption for the average investor. Tobin and Dolde (1971) suggest a constant relative risk aversion (CRRA) of 1,5. They supplement their observations by the theory of life cycle patterns and changing risk aversion. Mehra
and Prescott (1985) quote several microeconometric estimates that bound the
risk aversion coefficient from above three. Pindyck (1988) in his research of
the US stock market found relative risk aversion to equal approximately 3 - 4.
Mankiw and Zeldes (1991) continue Mehra and Prescott’s thoughts arguing
that an individual with a coefficient of relative risk aversion above ten would
be willing to pay unrealistically large amounts to avoid bets. Empirically
they find constant relative risk aversion to be about 26, provided the rational
expected utility maximization models hold.
Kandel and Stambaugh (1991) show that even values of relative risk aversion as high as 30 imply quite reasonable behavior when the bet involves a
maximal potential loss of around one percent of the gambler’s wealth. Blake
(1996) contradicts these views by arguing for a CRRA of about 2.
Other works about risk aversion were carried out by Aggarwal (1990) who
showed that distributions with low variance, positive skewness and negative
kurtosis are preferred by investors. Cohn, Lewellen, and Schlarbaum (1975),
King and Leape (1984) as well as Morin and Suarez (1983) reject the CRRAthesis and instead suggest decreasing risk aversion for increasing levels of
wealth.
Kandel and Stambaugh, Mankiw and Zeldes and others, believe it possible that there is no equity premium puzzle. Individuals might be more risk
averse than we thought and this high degree of risk aversion is reflected in the
spread between stocks and bonds. Benartzi and Thaler (1995) and Bonomo
and Garcia (1993) give a different explanation by showing that loss aversion
combined with a short horizon can rationalize investors’ unwillingness to hold
stocks even in the face of a large equity premium. Supporters of the young
field of behavioral finance - Haugen (1995), Shefrin and Statman (1999) among
others - find it unlikely that the observed patterns of excess return predictability can be explained purely by time-varying risk premia generated by highly
risk averse agents in a complete markets economy.
Chapter 1. Introduction
9
Studies about habit-formation by Constantinides (1990) and by Campbell
and Cochrane (1998) disagree and present evidence by explaining a good part
of the excessive stock returns.21 While the first study is able to explain the
intertemporal dynamics of returns, it lacks to account for the differences in
average returns across assets. The latter study by Campbell and Cochrane
suggests that investors seem to fear stocks primarily because they do poorly in
occasional serious recessions - in the times of low surplus consumption ratios
- unrelated to the risks of long-run average consumption growth. Empirical
observations do suggest that equity risk premia are higher at business cycle
troughs than they are at peaks. Risk premia shrink significantly in up-markets
as investors become eager to profit from rising stock prices and increase their
risk exposure. The latter fact sometimes gives rise to speculative bubbles,
as documented by Kindleberger (1996) and Shiller (2000). Excess returns
on common stocks over treasury bills seem forecastable, and many of the
variables that predict excess returns are correlated with or predict business
cycles.
While models of habit formation are able to explain the equity premium
puzzle, their validity seems incompatible with the structure of stock-ownership
as was pointed out by Benartzi and Thaler (1995): most of the assets on the
American stock market and even more on the European stock markets are
owned by three groups of investors: pension funds, endowments and very
wealthy individuals. It is indeed “hard to understand why habit formation
should apply to these investors”. Based on Kahneman and Tversky (1979),
Benartzi and Thaler (1995) propose a model of myopic loss averse investors
that demonstrate a high sensitivity to losses with a prudent tendency to frequently monitor one’s wealth. The domain of the utility function is thus
shifted from consumption to returns whose variability is accepted only in exchange for a large premium. Their results suggest that behavioral anomalies
deserve consideration, but that single phenomena cannot explain the riddle.
21 Habit-models
are an important area of consumption-based pricing models. The Habit
formation describes a positive effect of today’s consumptions on tomorrow’s marginal utility
of consumption. Utility today can be presented as a constant elasticity function of current
consumption and future utility. In this function, the degree of risk aversion of the consumer
as well as the elasticity of intertemporal substitution is governed by different parameters.
Disentangling risk aversion and intertemporal substitution helps explaining various aspects
of asset pricing behavior that appear anomalous in the context of the preferences.
10
1.3. Research idea
Thus, the equity premium puzzle still remains unsolved.
1.3
Research idea
We assume that common behavioral factors relating to risk preferences account for people choosing the same equity ratio in their portfolio. Significant
independent factors can be used for predicting the choice of a particular investor. They can assist in advising customers who do not have sufficient
financial knowledge to make a choice themselves.
The following two subsections briefly summarize the status quo of portfolio
theory and draft the theoretical framework of financial assumptions relevant
for this study. The first subsection revises some old facts about portfolio
theory while the second reviews the categories of asset pricing models. These
two subsections lay the foundation for the structure of the asset allocation
decision that is employed in this study and presented in the third subsection.
1.3.1
Portfolio theory’s old and new facts
In traditional asset pricing models markets were, to a good approximation, assumed to be informationally efficient.22 While the CAPM is per se irrefutible,
the evidence on the predictability of long-term returns reported in empirical
studies is overwhelming. Also, there are strategies with which one can reliably outperform simple indexes and passive portfolio strategies. The assets
they consist of produce high average returns without large betas implying
that beta as the sole risk measure does not suffice and in fact does not do justice to reality. Multifactor models such as the Arbitrage Pricing Theory have
much improved our knowledge about the risk factors that earn a premium on
the market. They also enabled us to better adjust for risk when measuring
investment performance.
Several studies showed that stock and bond returns are predictable at long
horizons. Their negative serial correlation implies that asset returns follow
a mean reverting process23 that seems to be associated with business cycles
22 The
statements in this subsection are partly based upon Cochrane (1999, 2000)
reversion implies stationarity and thus constant unconditional moments. Most
important, however, is the fact that under mean reversion return variance grows less than
23 Mean
Chapter 1. Introduction
11
and financial distress.24 However, annual returns prove much less predictable
and all the more so daily, weekly and monthly returns. Their process comes
closest in resembling a random walk.25
The value-added of most actively managed funds seems questionable. Their
returns prove to be slightly predictable: Past winning funds perform better
than average in the future while past losing funds do worse. Most of this
seemingly persistence however can be explained by fairly mechanical investment styles derived by multifactor models and not by persistent skill at stock
selection.
All of these findings suggest that one can earn a premium for holding
macroeconomic risks associated with the business cycle and for holding assets
that do poorly in times of financial distress, in addition to the risks represented
by overall market movements.
Cochrane (2000) thus advises the investor to hold - in addition to the
market26 and the risk-free asset - a number of passively managed ‘style’ funds
that capture the broad risks common to the majority of investors.
In addition to the overall level of risk aversion, an investor must determine his aversion to some additional risk factors: Each Survey of Consumer
Finances reconfirms that the major proportion of US stockholders largely own
stocks in the company they work for. Clearly, such an allocation runs counter
to diversification of one’s financial risk exposure. In the case of the company’s
distress the investor is in danger of losing both his job, his income and a large
part of his savings.
A wealthy investor, on the other hand, who is less dependent on his income
from work, can afford to invest in value stocks and similar assets that seem
to offer a premium in return for potentially poor performance in times of
financial distress. It is the purpose of the stock market to help transfer risks
from those unable or unwilling to bear them to those who for a risk premium
proportionally with time.
24 Cochrane (1999)
25 While these rather short-term returns are still strikingly unpredictable, it is the returns
at five-year horizons and more that seem highly predictable, confer Fama and French (1988).
A substantial amount of their variation can be explained by variables like the dividend/price
ratio and the term premium.
26 Where the market is a blend of styles (risk factors) identified by an adequate multifactor
model.
12
1.3. Research idea
can and will afford to take them.
Since returns are somewhat predictable, investors can enhance their average returns by moving their assets around among broad categories or ‘styles’
of investments. The question what styles to choose depends on one’s risk
tolerance and other specific circumstances.
An investor desiring more return and willing to take more risk than the
market portfolio is best advised to borrow to invest in the market rather
than to compose a portfolio of riskier stocks. An alternative for adjusting the
risk-return profile is to purchase call or put options on the market portfolio.
The basic decision for the investor is to what proportion he wants to hold
the market portfolio. The empirical evidence of market efficiency and the
poor performance of professional managers of active funds relative to passive
indexation, strongly suggests that stock-selection and timing attempts will
not pay off for most investors. Thus, the standard advice is to hold passively
managed funds that concentrate on minimizing transaction costs and fees.
The two-fund separation theorem leaves open the possibility that the investor’s time horizon matters as well as his risk aversion.
An important question concerns the intertemporal allocation of assets: To
what extent does the time horizon of the investment impact the allocation
decision? - The customary advice that a long-term investor can afford to sit
out all the market’s short-term volatility needs to be qualified. It is true that
a short-term investor should avoid stocks, as he may have to sell at the bottom
rather than wait for the inevitable recovery after a price drop. However, for
the long-term investor the time diversification fallacy lies in the ‘inevitable’
recovery implied by the customary advise. If returns are close to independent
over time, and prices are close to a random walk, a price drop makes it no
more likely that prices will rise more in the future. It may take a long time
until the former, all-time-high value of the portfolio is reached again.27
27 This
implies that stocks are not safer in the long run and the stock/bond allocation
should be independent of the investment horizon. If returns are iid over time then the mean
and variance of continuously compounded returns rises in proportion to the horizon: the
mean and variance of ten-year returns are ten times those of one-year-returns, so the ratio
of mean to variance is the same at all horizons. Merton (1969) and Samuelson (1969) show
that an investor with a utility exhibiting constant relative risk aversion who can continually
rebalance his portfolio between stocks and bonds will always choose the same stock/bond
proportion regardless of investment horizon, when returns are independent over time.
Chapter 1. Introduction
13
Only if returns mean-revert, if they exhibit negative serial correlation,
prices are more likely to come back after a shock. The evidence, in fact, suggests that stock prices do tend to mean-revert. Return variances at horizons
of five years are about 50-60% larger than short-horizon variances.
If long-term returns are predictable, the mean and variance no longer
change proportionally with the time horizon. With negative serial correlation or mean reversion the variance of long-horizon returns is proportionally
lower than the variance of single-period returns. In this case, stocks are more
attractive in the long run.
The most relevant conclusion of the above exposition for this study materializes in the fact that the one input essential to optimal portfolio advice is
risk tolerance. Providers of investment services have at last started to think
about how to measure risk tolerance using a series of questionnaires. This
is the trickiest part of the conventional advice, partly because conventional
measures of risk tolerance often seem incompatible with aggregate risk aversion displayed in asset markets. The basic pragmatic question is whether one
is more or less risk tolerant than the average investor. This question will be
conceptualized and answered in quantitative terms in the study.
1.3.2
Overview of models
The traditional discrete-time, single-period Capital Asset Pricing Model (CAPM)
by Sharpe (1964), Lintner (1965), Mossin (1966) has been criticized extensively by Roll (1977)28 , Ross (1976) and Elton and Gruber (1987)29 . The
28 Roll’s
CAPM critique comprised:
• Because of various actual trading restrictions such as unlimited riskless borrowing
and short-selling, investors may not be able to hold MVE frontier portfolios.
• Due to the fact that security returns are not normally distributed, investors may
have skewness preferences and end up holding inefficient portfolios.
• Since transaction costs and taxes affect security returns, investors who face different
costs may take inefficient portfolio positions gross of cost.
• Investors may suboptimize their portfolio positions when holding indivisible assets
such as their own human capital (i.e., the present value of their future earnings).
29 Elton and Gruber (1987) showed that with homogeneous expectations and meanvariance efficiency, portfolio selection reduces to simple after-tax optimization. Their study
14
1.3. Research idea
CAPM will be briefly dealt with in Chapter 2. Otherwise its fundamental
assumptions, statements and results are presupposed in this study and won’t
be discussed here.
The continuous time models exhibit a normative advantage, as they are
more realistic even though their assumption of continuous rebalancing is not
unproblematic. Continuous rebalancing (and mean reversion) leads to a geometric Brownian motion (GBM) of stock prices (with drift). If continuously
compounded returns are normally distributed as implied by the GBM, then
in the discrete-time case single period returns are lognormally distributed. Investors are thus faced with positively skewed returns due to which they choose
portfolios not efficient in terms of the mean-variance framework. Skewness
cannot be portrayed in the traditional two-moment model. When there are
more than two moments, diversity in tastes e.g. moment preference will ensure that unidentical portfolios result. It is for this reason that Rubinstein
wrote his paper on the Taylor expansion of the utility in 1973. Moment preference models incorporating skewness can approximate optimal strategies more
accurately.
Periodic lognormally distributed returns imply continuous normally distributed returns.30 If one wishes to use the properties of the normal distribution to estimate an investment’s likelihood of loss, one must convert the
investment’s periodic returns to continuous returns. It is then feasible to use
the mean and standard deviation of these values (continuous returns), along
with the normal distribution, to estimate the likelihood of loss.
1.3.3
Structuring the Asset Allocation Decision
According to modern portfolio theory all investors are to allocate their assets between the riskfree asset and the market portfolio. They thus decide
for a specific point on the capital market line to represent their investment
choice. Which point an investor exactly chooses depends on his individual
risk aversion. Risk aversion itself is a concept derived from decision making
also relates to the “clientele effect” first reported by Lewellen, Lease, and Schlarbaum
(1977). This effect claims that investors of different income tax brackets will prefer stocks
that differ in their dividend policy.
30 Kritzman (2000).
Chapter 1. Introduction
15
theory and closely related to expected utility theory. Unfortunately behavior
as predicted by this theory strongly deviates from that observable in reality.
In order to be able to determine risk preference in practice, the problem
of allocating free and reserved wealth must be treated separately31 . The
distinction is crucial to isolate the effect of individual risk aversion on the
allocation decision. Factors such as the investment horizon, the proximity to
retirement, in short all questions concerning risk taking capacity are thereby
taken into consideration (see Figure 1.1). The usual pitfalls can be avoided
Figure 1.1: Structuring the Asset Allocation Decision: Distinguishing between free and tied assets: This thesis will show how
an individual investor’s risk aversion can be determined. The
resulting estimation applies only to that part of the investor’s
wealth that has not yet been budgeted for specific projects.
by structuring the decision in this way, as the two methods for allocating
the two parts of wealth differ significantly.32 For reserved wealth only the
shortfall-approach is adequate while for free wealth the only relevant criterion
is individual risk aversion.
The shortfall-approach has the single goal to minimize the probability of
not meeting a predefined minimum target return that depends on the investment goal. This target return usually represents the yearly return that
is necessary to meet a planned project. Roy (1952) initially phrased this
31 Cf.
Spremann (2000), 303ff .
wealth conceptually in this way contributes to solving the problem techni-
32 Separating
cally.
16
1.3. Research idea
concept “safety-first”. It later led to the development of shortfall-risk and
more generally to the implementation of lower partial moments into portfolio
management and financial planning33 .
The study at hand concentrates on the assessment of risk aversion and thus
on the question how to allocate free wealth. It is that part of wealth that has
not been reserved for any project at the time of survey - neither in terms
of time nor in terms of purpose. The investment horizon or the maturity
for this part of wealth is thus undetermined.34 In the analysis of the SCF
data this distinction between free and reserved wealth could unfortunately
not be considered. However, the data was ideal for portraying how a survey
assessing risk preferences must be designed and how the analysis must be
conducted. All questions in the SCF concern the respondent’s total financial
assets. Thus, questions of the investment horizon, possible budgeted projects
and similar issues interfered with the object of analysis, distorting both the
choice of the stock ratio and the independent factors used for explaining it.
Therefore, the independent factors will not have as much explanative power
as if the investors had been asked to answer only in the light of their free
wealth.
1.3.4
Outline of thesis
In a first empirical analysis, discrete predefined intervals of the stock ratio which serves as a proxy for the risk level - are regressed onto different factor
sets. Out of these the one with the highest overall significance was then
chosen. The resulting estimates and different models are compared by their
predictive power for in-sample assignment of the observations. As a result a
single factor set was created from those sets. The final group of factors consists
almost entirely of individual variables that are based on different hypotheses
of financial theory. For the proceeding of the analysis it is convenient to
have the same factors available in both the SCF 1995 and 1998. The SCF
guarantees this fact, as all factors relate to financial behavior, one could argue
that, in fact, we test for consistency of the overall financial decisions of an
33 Other
authors that dealt with this subject in detail are: Telser (1955), Kataoka (1963),
Leibowitz and Henriksson (1989) and Van Harlow (1991).
34 For a detailed discussion of this topic, see Spremann (2000), Ch. 8
Chapter 1. Introduction
17
investor.
It has been stated before that single investors might not be conversant
with fundamentals of financial markets, with its statistics and the quantitative measures of risk, return and skewness. The proceeding of the method,
however, implicitly assumes that at least on average the individual decides
correctly. Otherwise the regressions wouldn’t yield significant results for factors that are projected to be relevant. As such, the investors are assumed to
be boundedly rational. Their search for their optimal investment risk level
leads to asset allocations approaching the latent optimal result.
Surprisingly, the majority of individual investors have been shown to reconsider the risklevel of their portfolios seldomly even though they check their
values frequently.35
It must be noted that the emphasis of this study is not on factor analysis
or the derivation and selection of regressors that have the greatest explanative
power for the two specific samples that were employed. The goal is to describe
the procedure, methodology and testing possibilities for developing a tool to
determine investment risk preference.
We will proceed as follows: After structuring the asset allocation decision,
the theoretical part of the paper will outline the basic two-moment model, its
assumptions and the hypotheses motivating the choice of the factor sets. The
two-moment model lays the theoretical justification for using the stock ratio
as a proxy of risk aversion. The second part will present the empirical analysis
and determine the classification power of different regression models for the
final set of individual factors for in-sample and out-of-sample prediction of
the SCF 1995 and 1998.
The last part of the study offers an outlook on the extension of the model
and research idea to three moments. Figure 1.2 visualizes and sums up the
outline of this study.
35 Cf.
Lee (2001), Bertaut (1998), De Bondt (1998) and Roth and Mueller (1998).
18
1.3. Research idea
Figure 1.2: Structure and outline of thesis:
The first part comprises the determination of risk preferences in
two moments - mean and variance. An empirical analysis of the
Survey of Consumer Finances (SCF) helps to identify factors
that are significant in explaining different degrees of risk aversion. The determination of one’s certainty equivalent for specific
gambles is an alternative way of assessing one’s risk aversion. At
the end of part one a joint estimation procedure is presented. It
combines both methods to increase accuracy.
The second part extends the first by incorporating the third moment as an essential attribute and criterion of investment preference. A new risk aversion measure is derived to replace the
one by Pratt-Arrow for three-moment setting. The assessment
through gambles is extended by the third moment to yield the
investor’s skewness preference which in practice can be implemented by option strategies. The second part concludes with a
description of how to jointly estimate the regression model and
the gamble to achieve higher predictive precision in the investment choice.
Part I
Two-Moment Risk
Preference
19
Chapter 2
The CAPM and two
moment risk preference
“The one input to the optimal portfolio advice is risk
tolerance, and many providers of investment services
have started to think about how to measure risk
tolerance using a series of questionnaires. This is the
trickiest part of the conventional advice...”
– Cochrane (2000).
The CAPM (one factor model) is still one of the most widely used asset pricing
model - maybe less in academic research than in practice. Its most relevant
statements are:
1. there are only two criteria for the pricing of an asset j: its expected
return µj and its return volatility σj .
2. the Investment Opportunity Set (IOS) integrates all risky assets in its
boundaries. Different correlations between the assets give rise to the
Efficient Frontier that comprises those combinations of risky assets that
are highest in its return/volatility ratio.
3. adding a riskless asset to this setting and combining it with the efficient
frontier results in the Capital Market Line (CML) that is tangential to
the Efficient Frontier in the Market Portfolio (M).
21
22
4. as the Market Portfolio is the single optimal risky asset to hold, Tobin’s
Two-Fund-Separation Theorem tells us that the asset allocation decision
can be divided into the delegated management of M and the individual
choice for a point on the CML.
For the investor the last statement raises the single most important question about the degree of his risk aversion. The risk aversion determines the
percentage of risky assets, that is, the proportion of M in the investor’s portfolio. To simplify the analysis this study assumes that the market portfolio
consists solely of stocks. Bonds are assumed to be held unto maturity and
thus considered riskless. The asset allocation decision thus reduces to merely
choosing the individually desired stock ratio.
Figure 2.1: Dividing the Capital Market Line (CML) into 6
funds representing six different risk categories. The individual
investor will be assigned to one of these by the discrete choice
model.
The Capital Market Line is divided as illustrated above, yielding six funds
of distinctly different risk levels. This division is purely arbitrary and can be
modified easily to yield finer or cruder results of the risk aversion. The decision
Chapter 2. The CAPM and two moment risk preference
23
for five different stock ratios was motivated by the different risk profiles of
equity funds. For a derivation of the risk aversion within the CAPM setting
please refer to Appendix A.
2.1
Assumptions of mean-variance portfolio selection
The relevance of the investor’s risk taking capacity for the asset allocation
decision is ruled out by the before-mentioned conceptual division of assets
into free and reserved wealth. All foreseeable future short-time to mid-term
expenses must be provided for by the shortfall approach.
As frequent rebalancing of the portfolio is costly and yields below-average
returns1 , the study focuses on determining a risk level that suits the investor
in the long run. The study maintains the view that for an individual investor
a buy-and-hold strategy seems most profitable for longer investment horizons.
This view is supported by studies of Barber and Odean (1999), Siegel (1998)2
Schlarbaum, Lewellen, and Lease (1978) as well as Schlarbaum, Lewellen, and
Lease (1979). It is thereby refuted that markets are fully efficient. Systematic
out-performance of major market indices by price- and earnings-momentum
strategies has been proven by Jegadeesh and Titman (1993), La Porta (1996)
and Rouwenhorst (1998) among others. However, such strategies are researchintensive and extremely time-consuming and thus infeasible to pursue for an
individual investor. An investor could of course invest in funds exploiting
market inefficiencies, such as overreaction and other behavioral phenomena.
Such a strategy is well advised and supported, but it does not influence the
1 Cf.
the following in-text references as well as Malkiel (1999).
sources cite stocks as the superior investment vehicle when judged from a longer
time horizon. Siegel (1998) states that over the last 200 years stocks have beaten T-bonds
as measured by the average yearly return. Prerequisite is a well-diversified portfolio of at
least 12 stock titles (preferably blue chips) that is to be held for at least 15 years. He warns
against attempts to time the market and empirically finds that most timing and stockpicking strategies of individual investors end in a disaster. Even the selection of proven
growth stocks turns out to be difficult, as the development of the “Nifty Fifty” from 1972
- 1997 clearly illustrates. The technology shares within the nifty-fifty had high P/E-ratios
and underperformed the S&P500, while stocks of consumer goods companies contributed
the largest part of the growth.
2 Many
24
2.1. Assumptions of mean-variance portfolio selection
decision over what stock ratio the investor should choose. This decision must
be made solely on the basis of one’s risk aversion.
The rationale behind the importance of RAV lies in the fact that both
continuous rebalancing and active trading strategies cannot be implemented
by an individual investor. They are simply too costly in terms of time and
money for individuals have earn a living. The same holds true for any attempt
to time the market. Price decreases and increases happen much too quickly
for an individual investor to anticipate. Any delay in rebalancing and timing
is punished severely: The price increases that accounted for 80% magnitude
of the index increases in the last 25 years happened in about 20-30 days.
Everyone not invested in those days was never able to catch up with those
who had followed a buy-and-hold strategy. Therefore, stock picking, markettiming and switching strategies are considered futile attempts to beat the
market and just involve brokerage commissions, higher taxes and losses in
money and time.3
A critic could argue that by investing in actively managed mutual funds
one could delegate or evade the asset allocation decision. Such a view implies the belief that fund-managers have the ability to time the market and
could rule out the relevance of deciding for a specific static stock ratio. However, the skill of fund managers as compared to a benchmark of passive style
indices seems very questionable. Studies by Fisher and Statman (1997) and
Gruber (1996) have proven that in past crashes more than 80% of the actively
managed mutual funds have declined at least as much in value as the market
index.
The advice to the investors is thus to allocate their assets according to
their risk aversion in a fund that tracks one or several broad global indices
- depending on what kinds of risks the investor is willing to bear. - In this
context it is needless to repeat that they should not adjust their risk exposure
every time the market moves.
Volatility for such a long-term, buy-and-hold strategy is still important, as
the average investor checks his portfolio value frequently and worries if it falls
below a certain boundary. Studies about individual trading4 have revealed
that in times of market downturns investors frequently react in a panic and
3 For
a more detailed exposition and reasoning, see Malkiel (1999), 350ff.
and Odean (1999) among others.
4 Barber
Chapter 2. The CAPM and two moment risk preference
25
sell their assets, thereby materializing the former virtual losses. This kind of
behavior is in fact the greatest investment risk of all to the individual investor.
An initial adequate risk level might help to prevent it.
As we will see later, the adjustment of the skewness of expected returns
might also prove effective in preventing the investor from realizing losses or
from changing a chosen risk exposure. Option strategies can enable an investor
to take on more risk when investing in stocks. They are therefore integrated
into the analysis in the second part of the study.
As a summary and a set of assumptions for the first part, it can be stated
that the investor will hold the market as suggested by the two fund-separation.
The riskless asset is represented by money market accounts, while the market
portfolio will be a major market index such as the S&P500 total return index. The Pratt-Arrow measure of risk aversion is the quantity relevant to this
study’s first part’s problem set. As Appendix B proves, in the two-moment
model this measure is closely tied to the ratio of risky assets within the investor’s portfolio. For simplification, bonds possessed by the investor were
counted to the riskfree assets. It is understood that bonds generally constitute a part of the risky market portfolio, as they bear significant risks when
not held to maturity.
The character of the analysis is static. Though risk aversion is generally
assumed to be a quantity relatively stable over short time-horizons, it is bound
to change with age, wealth and varying circumstances in the investor’s life.
The assessment thus has to be repeated as those changes occur.
2.2
The concept of two-moment risk aversion
In the economics of uncertainty, the term risk aversion in its more general
meaning stands for a cautious attitude in the context of reasonable decision
making. A risk averse individual shows a diminishing willingness to accept
more and more risk, even if taking more risk is related to more positive mean
return. Thus “risk aversion refers to preferences with (i) decreasing rates of
substitution between risk and mean return, and (ii) signifies a behavior, where
diversification [...] pays.”5
5 Spremann and Kotz (1981). In contrast to risk-averse behavior a risk-neutral investor
attaches no importance to diversification. He aims at maximizing his return and accordingly
26
2.2. The concept of two-moment risk aversion
Figure 2.2: Typical concave utility function of a risk averse investor. At the current wealth level W = 50 the depicted gamble
pays off 60 with 40% probability and causes a loss of 40 with
60% probability. The investor’s certainty equivalent of the gamble is CE = U −1 [E(U (W ))]. The gamble’s risk premium equals
π = E(W ) − CE. A person who prefers the gamble to the certain payoff is a risk lover; one who is indifferent between taking
the gamble and accepting the actuarial amount is risk-neutral;
and one who avoids the gamble choosing the actuarial value is
risk-averse.
The higher the risk aversion the greater the utility gain implied by diversification. Deviations from the market portfolio should be the exception.
Under risk aversion the condition that an uncertain wealth gain is not
preferred to the expected value of the gain (w̃ ≺ E w̃) holds. This condition
is equivalent to the concavity of U .
Risk-aversion means that an investor when given the choice, will avoid to
take fair gambles. A gamble is called ‘fair’ when it has a zero risk premium
chooses the asset with the highest return disregarding its risk level. Risk neutral investors
judge risky prospects solely by their expected rates of return. The level of risk is irrelevant.
This type of investor will even consider taking out a credit in order to increase the expected
value of his investment. Such behavior patterns are known as ‘Plunging’. A behavior that
will be mentioned again in Section A.2 after Equation A.18.
Chapter 2. The CAPM and two moment risk preference
27
or in other words when its expected payoff is zero. Risk-averse investors
are willing to consider only risk-free or speculative prospects. The reason is
that the potential loss represents an amount of “displeasure” that is greater
than the amount of “pleasure” associated with the potential gain. Formally
speaking, a person is risk averse, if the utility of expected wealth is greater
than the expected utility of wealth: U [E(W )] > E[U (W )].
The additional assumption of insatiation, common in economics, together
with risk-aversion, causes indifference curves to be positively sloped and convex. Even though all investors are said to be risk-averse, they do not have
identical degrees of risk aversion. Different investors will have different maps
of indifference curves. The more risk-averse an investor, the steeper his indifference curves in the mean-variance-diagram.
In the classical sense, risk aversion measures attitudes towards pure wealth
bets. It is therefore conventionally captured by the second partial derivative
of the value function with respect to individual wealth. The value function
depends on individual wealth W .
The investor’s personal trade-off between portfolio risk and expected return is expressed by his utility function. The latter assumes that investors
can assign a welfare or a score to any investment portfolio depending on its
risk and return. Because we can compare utility values to the rate offered on
risk-free investments when choosing between a risky portfolio and a safe one,
we may interpret a portfolio’s utility value as its “certainty equivalent” rate of
return to an investor. The certainty equivalent rate of a portfolio is the rate
that risk-free investments would need to offer with certainty to be considered
as equally attractive to the risky portfolio. A portfolio is desirable only if its
certainty equivalent return exceeds that of the risk-free alternative.
In general, the risk premium must be added to the gamble in order to make
the investor indifferent between the gamble and its actuarial value. She’d be
willing to pay the premium in order to receive the expected value right away
instead of playing the gamble. However, the risk premium in itself does not
express whether the gamble is favorable or not. For that purpose the cost of
the gamble has to be calculated: C = CurrentW ealth − CE. If it is positive
then one would pay to avoid the gamble; if it is negative, one would pay to
take the gamble.
The overwhelming majority of financial models assumes individuals to be
28
2.2. The concept of two-moment risk aversion
risk averse. A reasonable utility function belonging to the class of HARA
utility functions is the following isoelastic utility which exhibits decreasing
absolute risk aversion and constant relative risk aversion6 :
U (W ) =
1 γ
W
γ
(2.1)
where 0 = γ < 1
2.2.1
The Markowitz Premium
The degree of the utility function’s curvature is measured by the difference
between the expected value of a gamble and the investor’s certainty equivalent.
The measure can also be interpreted as the amount that an investor would
pay to insure against a particular risk. Given the numbers of the gamble in
Figure 2.2, assuming initial wealth of W = 50 and the isoelastic utility of
Equation 2.1 with γ = 0.3, the Markowitz Risk Premium is
E(W ) − CE(W )
=
50 − 0.3 ·
=
17.72
1
0.3
0.6 0.3
0.4
0.3
100 +
10 ·
0.3
0.3
as the Certainty Equivalent, CE = U −1 [E[U (W )]] and U −1 = γW 1/γ .
In other words, the investor is willing to pay up to $17.72 to insure himself
against the gamble and the risk of losing $40.
2.2.2
The Pratt-Arrow measure of Risk Aversion
The most widely used measure of risk aversion is based on the insight that
the degree of curvature in a function is related to its second derivative. Pratt
(1964) and Arrow (1965) derived the risk premium by constructing an actuarially neutral gamble of Z dollars, equating the utility of expected wealth
and the expected utility of wealth.
By using a two-term Taylor approximation of the utility function they
disregard third and higher order terms or, in other words, skewness and higher
moments of the distribution. As a result, the methodology is numerically
6 Thus allowing for a much more realistic setting of the analysis than both the quadratic
and the exponential utility function.
Chapter 2. The CAPM and two moment risk preference
29
exact only for infinitesimal risks.7 The postponement of the truncation of
higher order terms as depicted in Chapter 7 permits the extension of the
analysis to larger risks.
For deriving his two-moment risk premium, Pratt (1964) uses the Taylor
series and equates the expected utility of a (random variable) gamble X̃ and
the utility of its expected value:
E[U (W + X̃)]
= U [W + E(X̃) − π(W, X̃)]
(2.2)
As it is a fair gamble, E(X̃) = 0:
E[U (W + X̃)]
= U [W − π(W, X̃)]
Expanding the left-hand side using Taylor series and disregarding third-order
and higher terms:
X̃ 2 U (W ) + ...
2!
E[X̃ 2 ] = E[U (W )] + E[X̃]U (W ) +
U (W )
2!
σ 2 = U (W ) +
U (W )
2
Expanding the right-hand side using Taylor series and disregarding secondorder and higher terms in { }-brackets:
2
π U (W ) − ...
U [W − π(W, X̃)] = U (W ) − πU (W ) +
2!
E[U (W + X̃)]
= E[U (W ) + X̃U (W ) +
Putting both sides together
σ 2 U (W ) = U (W ) − πU (W )
2
Solving for the Pratt-Arrow measure of a local risk premium π we get
1 2
U (W )
π = σZ − (2.3)
2
U (W )
U (W ) +
7 For large risks, Pratt/Arrow state that if, at every wealth level, a utility function U
1
displays greater local risk aversion than U2 , the risk premium associated with any risk
irrespective of its size, will be greater for U1 than for U2 . Thus, the cardinal measure of
risk aversion in the small corresponds to an ordinal measure in the large, provided that the
risk aversion functions do not intersect. However the measure does not permit inferences
about risk aversion in the large for all utility functions.
30
2.2. The concept of two-moment risk aversion
where σ 2 represents the variance of the random variable Z̃ which indicates the
outcome of the gamble in dollar terms. Whether the variance is an adequate
measure of portfolio risk will be discussed later, here it is just noted that
the extent to which variance lowers utility depends on α. Where α is the
investor’s degree of absolute risk aversion ARA, expressed as
U (W )
(2.4)
αA = ARA = − U (W )
the Pratt-Arrow measure risk aversion.
Multiplying the measure of absolute risk aversion by the level of wealth
W we obtain a measure of relative risk aversion (RRA):
U (W )
αR = RRA = W − (2.5)
U (W )
The RRA can be interpreted as an elasticity measuring the percentage change
in marginal utility given a percentage change in the individual’s wealth. ARA
on the opposite, measures the percentage change in marginal utility, given an
absolute change in wealth.
Each of these two measures indicates to what extent the odds have to be
better than fair in order to induce an individual to accept a bet of a certain
size, measured either as an absolute amount or as a percentage of income.
For the gamble in Figure 2.2 above which is actuarially neutral, the variance of the asymmetric risk is: 0.4(110 − 50)2 + 0.6(10 − 50)2 = 2400. The
Pratt-Arrow risk premium thus comes to
−0.7(50)−1.7
1
2400 −
= 16.8
π =
2
50−0.7
Thus the risk averse individual will be indifferent between winning $60 and
losing $40 when the actuarial value is $16.8.
The difference for the example between the Markowitz and Pratt/Arrow
premium is only small. It results from the fact that relative to the assumed
reference wealth level of $50 the risk is not small. The gamble is also skewed.
Consider a gamble that is even more skewed: On the right side in Figure 2.3
the gamble pays off $150 with a chance of 20%, while the gambler loses $37.5
with 80% probability. Initial wealth is again $50. Calculating the Pratt-Arrow
measure for this gamble yields a premium of $39.4.
Chapter 2. The CAPM and two moment risk preference
31
Figure 2.3: Two differently skewed gambles and their PrattArrow risk premia
The investor’s isoelastic utility function with γ = 0.3 and its derivatives
are
0.3−1 W 0.3
U (W )
=
U (W )
= W −0.7
U (W )
= −0.7W −1.7
U (W )
=
1.19W −2.7
From these derivatives the moments of the two gambles in Figure 2.3 and the
corresponding premia can be calculated:
σ2
Skewness
Pratt-Arrow πA
Markowitz
Gamble1
2’400
48’000
17
18
Gamble2
5’625
632’812
39
23
Due to a higher variance, the premium of Gamble2 is higher than that of the
less skewed Gamble1. In both cases the expected value of the gamble was 0,
both were actuarially neutral. With the mean being constant, an increase in
variance (from 2’400 to 5’625) and skewness (Fisher skewness grew from 0.41
to 1.50) results in a deterioration of the two-moment expected utility and the
local rejection of positive skewness-variance trade-offs.
Judging from the characteristics of the Pratt-Arrow risk measure, investors
will never accept a higher variance for an increase in skewness regardless of
32
2.3. Relevance of assessing the correct asset allocation
the size of the trade-off. Intuitively, this result is not satisfactory. Various
studies have proven that investors value positive skewness. The Markowitz
and Pratt/Arrow risk premium measures only consider the variance as a factor
of risk. Excluding negative skewness as a risk source seems only acceptable
when the source itself is symmetrically distributed. Explicit consideration
of the gamble’s third moment in an alternative risk premium measure might
prove valuable for weighing larger risks that have skewed distributions. Such
proceeding will be portrayed in Part II.
In general, it can be stated that the Pratt-Arrow definition of risk aversion
provides useful insight into the properties of ARA and RRA, but it assumes
that risks are small and actuarially neutral. The Markowitz concept, which
simply compares E[U (W )] with U [E(W )] is not limited by these assumptions.
The difference between these two measures becomes most accentuated when
the risk in question is large and very asymmetric (skewed). In such a case
the Pratt-Arrow measure tends to underestimate the risk. It tells us nothing
about the preferences of investors when return distributions are skewed. The
Markowitz measure of a risk premium is thus superior for large or asymmetric
risks.8
2.3
2.3.1
Relevance of assessing the correct asset allocation
Calculating Expected Utility loss
The expected utility approach assigns utiles to monetary values and thus
allows to measure the utility loss caused by a wrong asset allocation. The
magnitude of the calculated loss indicates whether the accurate assessment of
risk preferences is of any significant relevance for the investor.9
With regard to the return differences between different stock ratios - as depicted in Formula A.14 in Appendix A.1 - one might argue that the monetary
loss in expected utility for an assignment to a wrong risk class (stock ratio)
might be neglectable10 . As shown below, the precise assessment of individual
8 Cf.
Copeland and Weston (1992).
(2000), 328.
10 Even though it is believed that the expected utility approach is not a realistic reflection
9 Spremann
Chapter 2. The CAPM and two moment risk preference
33
preferences is indeed trivial when considering a short investment horizon of
one to three years, as the monetary disutility of a wrong assignment is very
small. However, when looking at a longer investment horizon, the loss in utility and the potential absolute monetary loss can be significant.
Time horizon of 1 year
We assume the following values for the parameters of the formula in Appendix A.14:
• financial wealth, W = 100 (in ’000 $)
• optimal asset allocation or stock ratio, b = 80%, where b =
µM −rf
2
αW σM
• return of riskfree asset, rf = 5% p.a.
• expected return of risky asset, µM = 10.5% p.a.
• annual volatility of risky asset, σM = 22% p.a.
With the above values, we obtain the investor’s risk aversion:
α=
.105 − .05
= 0.0142
0.8 · 100 · 0.222
The certainty equivalent, CE, of the investor’s end-of-period wealth, WT , for
his optimal asset allocation of 80% stocks amounts to:
α
Var[WT ]
2
0.0142
309.76 = 107.20
109.4 −
2
CE[WT ] = E[WT ] −
CE[WT ] =
where WT = W0 (1 + µw ) and µw = rf + b · (µM − rf )
Let’s assume the investor whose optimal asset allocation is 80% stocks is
assigned to a mutual fund with a stock ratio of only 20%. The certainty
of the average investor’s decision making criterion, we will utilize it here to analyze the
relevance of developing an accurate risk assessment method.
34
2.3. Relevance of assessing the correct asset allocation
equivalent of the investor’s end-of-period wealth, WT , for a stockratio of 20%
is:
0.0142
19.36 = 105.96
2
The relative utility loss thus equals 1 − 107.20
105.96 = 0.012 or approximately 1%.
A utility loss of such magnitude is in fact neglectable.
CE[WT ] = 106.1 −
Time horizon of 10 years
By prolonging the investment horizon, the utility loss increases as expected:
• return of riskfree asset, rf = 10 · 5% = 50%
• expected return of risky asset, µM = 10 · 10.5% = 105%
√
• annual volatility of risky asset, σM = 22% 10
• the investor’s risk aversion remains the same as for one-year horizon:
α=
1.05 − 0.5
= 0.0142
0.8 · 100 · 0.69572
The certainty equivalent, CE, of the investor’s end-of-period wealth, WT , for
his optimal asset allocation of 80% stocks amounts to:
α
Var[WT ]
2
0.0142
3097.6 = 172.01
194 −
2
CE[WT ] = E[WT ] −
CE[WT ] =
where WT = W0 · (1 + µw ) and µw = rf + b · (µM − rf ).
Let’s assume the investor whose optimal asset allocation is 80% stocks
is assigned to a mutual fund with a stock ratio of only 20%. The certainty
equivalent of the investor’s end-of-period wealth, WT , for a stockratio of 20%
is:
CE[WT ] = 161 −
0.0142
193.6 = 159.6
2
Chapter 2. The CAPM and two moment risk preference
35
The relative utility loss thus equals 1 − 172.01
159.6 = 0.078 or approximately 8%.
It increases almost proportionally with the investment horizon. In general,
the loss is more pronounced the larger the equity premium and the greater
the difference between the optimal and the assigned stock ratio.
The most considerable disutility, however, comes in the form of unplanned
portfolio rebalancing in unfavorable market situations when these rebalancings are triggered by an inadequate risk level e.g. a misspecified measuring of
risk aversion.
2.3.2
All stocks half the time or half stocks all the time?
When discussing the optimal asset allocation strategy for a given investor, it
is often argued that the answer depends on the time horizon, the dynamics of
the market or the timing abilities of the investor. Fortunately, this claim is a
misconception provided that there is a premium for risky assets. It is solely
the investor’s utility function, his risk preferences that determine the optimal
investment strategy. This section will show why.
The preceding analysis in Sections 1.3.1-1.3.3 has already raised a more
general question essential to asset allocation: Is it better to follow a constant
balanced strategy or to shift between extreme allocations by trying to time
the market?11
For the following brief exposition it will be assumed that the average return
on stocks is higher than the riskless rate and that stock returns follow a
random walk. Furthermore, transaction costs and taxes will be neglected.
The balanced strategy allocates 50% of W in stocks and the rest in the
riskless asset. The switching strategy invests 100% of W in stocks for half
of the investment horizon, the rest of the time it invests in the riskless asset.
Intuitively, one would presume that both strategies yield the same results.
And even though both strategies will yield the same expected return and
the same terminal wealth in the limit12 , the balanced strategy has a higher
average per period exposure (appe) than the switching strategy. As a higher
11 Among
others, Clarke and de Silva (1998) and Kritzman (2000) analyzed this problem.
fact, the expected cumulative wealth is not to be expected, as due to continuous
compounding the pdf of terminal wealth is positively skewed. The mean is thus greater
than both the median and the modus. The geometric average return would be a better
measure for what is to be expected even if it does not mirror the figure that will result on
average over many repetitions.
12 In
36
2.3. Relevance of assessing the correct asset allocation
exposure implies a better performance, the balanced strategy dominates the
switching strategy:
appe balanced strategy
=
appe switching strategy
=
N −1
1 1
Wn − W0
1+
N n=0 2
W0
K−1 Wk − W0
1 1+
N
W0
(2.6)
(2.7)
k=0
r˜S stands for the random return on the stock investment, rf depicts the
riskless rate, W0 symbolizes the total wealth in the period of the first stock
exposure of the corresponding strategy, n = 0, 1, 2, ..., N represents the investment period and k = 0, 1, 2, ...K is the index of the periods in which the
switching strategy is invested purely in stocks. By definition K = N/2.
As an example, the two strategies are compared for a time horizon of
N = 20 periods (therefore K = 10). Stocks are assumed to yield either
r˜S = +20% or r˜S = −6% alternately, while the riskfree rate returns a steady
rf = 6.2073%. The expected return for both strategies in the long run is:
rbs = 0.5 · 7% + 0.5 · 6.21% = 6.60%. For alternating risky returns the growth
paths of wealth for the balanced (B) and for the switching (S) strategy are:
t
0
1
2
3
4
5
6
B
1
1.13
1.13
1.28
1.28
1.45
1.45
S
1
1.2
1.13
1.35
1.27
1.53
1.44
t
7
8
9
10
11
12
13
B
1.64
1.64
1.86
1.86
2.1
2.11
2.38
S
1.72
1.62
1.94
1.83
1.94
2.06
2.19
t
14
15
16
17
18
19
20
B
2.38
2.7
2.7
3.05
3.06
3.46
3.46
S
2.32
2.47
2.62
2.78
2.96
3.14
3.33
An initial investment of 1 in the balanced strategy grows to 3.46 at the end
of period 20. The switching strategy yields a lower total capital of 3.33 which
stems from the lower average per period exposure (appe):
appe balanced strategy
=
19
1 1
Wn = 0.9919
20 n=0 2
appe switching strategy
=
9
1 Wk = 0.9013
20
k=0
The expected cumulative wealth for both strategies is 1 · (1.066)20 = 3.59 and
thus higher than the two results produced by the series above. The sequence
Chapter 2. The CAPM and two moment risk preference
37
of switching is thereby essentially irrelevant if long time horizons and thus
infinite repetitions of the investment strategies are considered.
Apart from higher performance, the balanced strategy has another advantage compared with the switching strategy: it bears less unsystematic risk as
measured by the return’s volatility13 and is thus more efficient. While the
balanced strategy’s volatility is 0.5 · σ 2 , the switching strategy’s volatility14
equals 0.5 · σ 2 + 0.52 · (r̃s − rf )2 ). Thus, even if an investor is a skillful
market-timer, he needs to achieve a substantial excess return over the balanced strategy in order to outperform it on a risk-adjusted basis and even
more so when considering transaction costs over longer investment periods.
At this point, one might argue that deciding whether switching pays solely
by the criterion of terminal wealth might be too one-dimensional. What if
an investor derives utility not only from terminal wealth, but also from the
interim realizations of wealth along its growth path?
Samuelson (1994) proves that even in such a case the balanced strategy
is still to be preferred provided the investor exhibits constant relative risk
aversion. Then, the sum of expected utilities over all periods is greater for
the balanced strategy. This is not surprising, as the switching strategy bears
higher unsystematic risk. For every point in time it thus has higher potential
deviations than the balanced strategy. The assumption of constant relative
risk aversion ascertains that the valuation of the volatility does not change
over the growth path of wealth. The specific form of the underlying utility
function does not influence this finding provided rf is the certainty equivalent
of r̃s .15
The above subsection tried to indicate why a steady, balanced strategy
beats a switching or timing strategy on average. This line of reasoning was
necessary as the balanced strategy serves as the default assumption in the
following chapters.
13 As
opposed to the before-mentioned average per period exposure
Clarke and de Silva (1998), 63.
15 U [W (1 + r )] = 0.5 · U [W (1 + r
sup )] + 0.5 · U [W (1 + rsdown )] must hold
f
14 See
38
2.4
2.4. Switching and the time horizon controversy
Switching and the time horizon controversy
The preceding chapter showed why a strategy of continuous dynamic adjustments16 to the asset allocation is on average inferior to a balanced buy-andhold strategy. This section briefly reviews the relevance of the investment
horizon for any asset allocation advice.17
In his 1963 paper Samuelson analyzes the behavior of a colleague who
turns down a single play of a bet that has a positive expected value though
exhibiting the possibility of a loss. Puzzlingly, the same colleague is willing to
accept multiple plays of the same bet.18 Samuelson’s famous theorem about
the ‘fallacy of large numbers’ states that this kind of behavior is inconsistent.
He accuses his colleague of erroneously believing that the variance of outcomes
decreases as the number of trials increases. In fact, the variance of terminal
wealth increases proportionally with an increasing time horizon. It is solely
the variance of the average annualized simple return that decreases with time.
The above line of reasoning indicates that the behavioral inconsistency is
of more than just academic interest.19 Repeated plays of independent gambles
(addition problems) essentially portray investments over time where returns
are compounded (multiplication problems).
Refusing a single gamble while accepting a series of them hints at the
hope of realizing the positive expected value rather than one extreme. It
also relates to the idea that there is greater probability of meeting a given
16 This
refers to adjustments that are not motivated by variations in underlying market
fundamentals that change the composition of the market portfolio.
17 In the study at hand the question of the investment horizon has conceptually been uncoupled from the link between asset allocation and risk aversion by the separation between
the tied and the free part of investors’ wealth. See Subsection 1.3.3
18 A finding also confirmed with experiments by Keren and Wagenaar (1987). Subjects
who were shown the explicit multi-year distribution were willing to accept more risks than
when faced with the characteristics of the one-year return distribution. Related studies
about the change of behavior induced by the increase of gambles played come from Lopes
(1984), Montgomery and Adelbratt (1982), Keren (1991) and Redelmeier and Tversky
(1992).
19 The role played by the investment horizon in optimal portfolio selection has been of
recurring interest in the financial literature. Under the term “Time diversification” this
topic received great attention by: Merton (1969), Samuelson (1989, 1990, 1994), Benartzi
and Thaler (1995), Bodie (1995), Thorley (1995), Merrill and Thorley (1996) and Bierman
(1998) among others.
Chapter 2. The CAPM and two moment risk preference
39
return objective by investing in high-risk/high-return assets than there is by
investing in low-risk/low-return assets and that the tendency for the high-risk
portfolio to dominate the low-risk portfolio increases with the length of the
investment horizon. This tendency has been called the “time diversification
effect”.
Merton (1969) and Samuelson (1969) showed that when returns follow a
random walk and utility functions exhibit constant relative risk aversion there
is no such effect. Asset allocation is then independent of the time horizon of
the investor.20
Samuelson’s contributions to this question can be separated into two paths:
first he assumed iid returns meaning that successive returns are independent of
each other and not autocorrelated. He thus assumed a random walk and concluded that an investor’s optimal choice of single-period portfolios will remain
constant over the investor’s life cycle. Later he examined the same question
allowing for autocorrelation of returns and assuming a mean-reversion process (rebound effect). Again he concluded that there is no argument for the
long-horizon investor to hold a riskier portfolio. Still later, he developed the
conventional wisdom to hold under both regimes (random walk and mean
reversion) - all under the assumption that the investor is anxious not to fall
under a subsistence level of terminal wealth.21
Empirically the “colleague’s” behavior could not be confirmed unambiguously. Evidence about time induced financial behavior is mixed. While Kahneman and Tversky (1979) justified the ‘colleague’s’ inconsistent decision by
myopic loss aversion22 , several studies proved that most subjects seemed to
20 These results have always been controversial and in reality investors were told to do
differently. Investment advisors explicitly recommend changing the asset mix as retirement
comes closer. Eventually the necessity to reduce portfolio risk due to retirement has to
be answered in a detailed financial plan and is not an issue of this analysis. However,
even if an investor’s risk taking capacity is considerable, it is conceivable that he feels very
uncomfortable with every day value changes of his portfolio. Though knowing that these
daily changes balance each other out in the long run if prices mean revert, such an investor
could be advised to buy portfolio insurance occasionally and in times of obvious ‘irrational
exuberance’ in order to hedge the portfolio’s value.
21 A similar argument was made by Roy (1952) and was later shown by Zelney to characterize the way investors perceive risk. Roy coined the well-known “safety-first” approach
that later led to the development of the concept of shortfall-risk and more general to the
implementation of lower partial moments in portfolio management and financial planning.
22 Loss aversion is an important property of the prospect theory value function introduced
40
2.4. Switching and the time horizon controversy
suffer from a fallacy of small rather than large numbers.23 Less well experienced subjects overestimate the variance of a multiple-play gamble rather
than underestimate it. As a consequence less risky investments are made.
This finding might explain why such a low proportion of the population holds
stocks and why the average relative stockholdings are so small.
If Samuelson’s colleague really mis-estimated the variance of the manybet-portfolio then he would presumably change his mind when shown the
correct distribution of final wealth resulting from multiple gambles. However,
experiments proved that this is not the case. It became clear that subjects
in fact favor repeated plays of a positive expected value gamble rather than
refrain from accepting them when shown the payoff distribution.24
Table 2.1: The relevance of the time horizon for the asset
allocation depicted in a matrix subdivided into return process
and form of utility function. The Bernoulli investor with U =
ln(W ) exhibiting constant relative risk aversion (crra) serves as
a benchmark for all conceivable utility functions.
Return
More Risk Averse
Bernoulli (Log-W)
Less Risk Averse
∗
†
process
than Log-W
Investor (crra)
than Log-W
Mean
reversion
Random
Walk
Relatively more
wealth is allocated
to the risky asset
Time horizon has no
impact on the asset
allocation decision
Longer time horizon
means that less
wealth is allocated
to the risky asset
Asset allocation is independent of time. It depends solely on
individual risk aversion. The optimal strategy is fully myopic.
∗ An
investor with a utility function logarithmic in wealth
generally, any investor exhibiting constant relative risk aversion. Any isoelastic
investor can be assumed: U (W ) = 1/γW γ .
† More
When assuming iid returns and consequently a constant and known investment opportunity set (IOS), then the time horizon is irrelevant for an
by them in 1979. It states that reductions in wealth, relative to the current reference point,
are weighted about twice as much gains.
23 See study by Benartzi and Thaler (1995).
24 Benartzi and Thaler (1995) show that by holding the distribution of final outcomes
constant they can predict which repeated gambles people will find attractive.
Chapter 2. The CAPM and two moment risk preference
41
isoelastic investor, as an isoelastic utility implies constant relative and decreasing absolute risk aversion. According to his degree of risk aversion the
investor allocates a constant proportion of his wealth in risky assets regardless of the length of the period considered. This equals Samuelson’s original
proposition in the ‘fallacy of large numbers’-debate in Samuelson (1963a).
The above result remains almost unchanged for the investor exhibiting crra
when returns are assumed to either mean-revert or to be partially predictable.
However, the investors who are more risk averse than log-wealth experience an
increase in expected utility under mean reversion as their horizon increases,
as Winhart (1999) and Kritzman (2000) point out. The reason is that mean
reversion25 causes that part of wealth allocated to the risky asset to disperse
at a “sufficiently slow rate so that conservative investors can tolerate greater
exposure over longer horizons.”
It can thus be concluded that there is no theoretical or empirical evidence
for the so-called “time-diversification”. If risky returns follow a random walk
then variance increases proportionally with time and the investor’s expected
utility is invariable to the investment horizon provided the investor has constant relative risk aversion. Thus, under iid returns investors will wish to hold
their asset allocation constant as their time horizon increases.
Apart from the specific form of the utility function, for the asset allocation
decision it is also crucial to define the correct objective function. Kränzlein
(2000) points out that maximizing expected terminal wealth might not be the
right goal given the fact that positive skewness of the terminal wealth’s pdf
increases with the time horizon.26 For a positively or right skewed pdf the
expected value overestimates the actual return that results in the end. Modus
or median would provide better target measures for the objective function.
A closely related and equally important factor concerning intertemporal
asset allocation is the retirement-induced reduction in portfolio risk.27 This
25 The data history is too short to enable an unambiguous statement about the specific
form of the return process. Evidence for mean reverting stock returns find Poterba and
Summers (1988) as well as Fama and French (1988).
26 Kahneman and Tversky (1979) have shown that investors care about the (interim)
change in wealth and not about the final asset value. That is after all why they are myopic
loss averters.
27 Other factors that play a role in life-time utility maximization and influence intertemporal asset allocation are: retirement postponement, the inflation rate, taxes, changing
42
2.5. Risk aversion and changes in wealth
effect is often extended to mean that investors become generally more risk
averse as they age. However, while there is considerable evidence of different
degrees of risk aversion within each age cohort, there is little evidence on
tolerances for risk bearing among different age cohorts. There are also no
reported long-term studies which show conclusively how attitudes toward risk
change as individuals age. Marshall (1994) presents the so-called draw-down
criterion as a plausible reason.28 With this criterion one aims at maximizing
terminal portfolio return subject to not falling below a prespecified threshold.
Even if an investor has the same risk aversion all his life, he will choose
progressively less risky portfolios as his investment horizon shortens due to the
rising relevance of the return variance of a high-risk portfolio. An investor’s
perception of the riskiness of any given single-period portfolio changes as his
horizon grows shorter. It is thus not necessary for an investor to become more
risk averse to observe a gradual shift from more to less risky portfolios with
the passage of time.
2.5
Risk aversion and changes in wealth
The previous section has pointed out that the relevance of the time horizon
depends largely on the course and the change of risk aversion with wealth.
In other words it depends on whether relative risk aversion is constant or
not. The following paragraphs will briefly examine what form of relative risk
aversion is best compatible with reality.
Intuitively, one would expect the coefficient of absolute risk aversion for a
given individual to decrease with wealth, while the coefficient of relative risk
aversion might be roughly constant across different wealth levels.29 That is,
consumption preferences, consumption opportunity sets and changing investment opportunity sets. These have been considered by Bodie and Crane (1997), Chen and Moore (1985),
Khaksari, Kamath, and Grieves (1989) and Bierman (1997, 1998) among others.
28 A drawdown is defined as a percentage of initial investment capital (similar to shortfall
risk), and as the lower bound of a confidence interval when measured over a continuum of
investment horizons.
29 Cf. Taggart (1996). Within the Consumption based asset pricing model with power
utility (CCAPM) the coefficient of relative risk aversion γ is typically estimated by
1 = Et [(1 + Ri,t+1 )δ
Ct+1 −γ
]
Ct
Chapter 2. The CAPM and two moment risk preference
43
we would expect an individual starting with little wealth to be more cautious
about entering a bet in which he could win or lose the amount of initial
endowment than the same individual who starts with a fortune. On the other
hand, there are reasons which support the assumption of at least constant
(very high) absolute risk aversion. Factors such as social status might play an
important role. Some people might dread the thought of losing 10% of their
wealth through plunges in stock market prices, especially if the decreases
necessitate restrictions in the standard of living.
The characteristics of an individual’s absolute risk aversion (ARA) allow
us to determine whether he treats a risky asset as a normal good when choosing between a single risky and a riskless asset.30 Arrow (1971) showed that
decreasing ARA over the entire domain of −U /U implies that the risky asset is a normal good - that is, the dollar demand for the risky asset increases
as the individual’s wealth increases.
Increasing absolute risk aversion as in the quadratic utility function suggested by the CAPM implies that the risky asset is an inferior good, and
constant absolute risk aversion implies that the individual’s demand for the
risky asset is invariant with respect to his initial wealth.
Under increasing relative risk aversion the wealth elasticity of the individual’s demand for the risky asset is strictly less than unity. In other words the
proportion of the individual’s initial wealth invested in the risky asset will
decline as his wealth increases.
For short time periods the question of whether the investor’s utility function exhibits constant absolute or constant relative risk aversion or even decreasing relative risk aversion has little relevance for the optimal asset allocation.31 It is longer investment horizons of ten years or more that call for
a more precise analysis of the higher moments of the utility-function. The
longer the investment horizon the more distinct the distribution’s skewness.32
that can be derived from U (Ct ) = Ct−γ . Ri,t+1 is the return on asset i at time t + 1, δ is
the time discount factor and Ct is aggregate consumption in period t. Asset returns and
aggregate consumption are assumed to be jointly homoskedastic and lognormal. Also see
Hansen and Singleton (1983).
30 Huang and Litzenberger (1988)
31 Cf. Spremann (2000), 331.
32 It must also be noted that the longer the horizon considered the more probable that
the investor’s preferences have changed. Time-varying preferences necessitate a periodic
44
2.6. Bounded rationality and goal of methodology
The assessment of moment preferences thus must aim at determining what
form of return distribution is desired by the investor. Some will want to hold
risky assets and buy portfolio insurance due to their high risk aversion. Others might be able to afford to sell insurance or cap their upside potential by
employing covered call strategies. In part two this study will show how to
determine such different preference patterns.
2.6
Bounded rationality and goal of methodology
The following subsections will briefly present some basic concepts that the
formal analysis rests on. One of them - the assumption that individuals are
boundedly rational - is a crucial prerequisite for the usefulness of advising
investors with an empirically calibrated questionnaire. The following section
will show why and discuss what criteria are appropriate for rating the performance of the econometric models.
Bounded rational decision makers are not only restricted by the availability
of information but also in their ability to learn. In order to save information
processing costs and time, boundedly rational decision makers apply simplified
thinking and calculation procedures. The irrational decision maker, too, is
restricted in his learning abilities but in contrast to the bounded rational
decision maker he does not take constraints suhc as processing information
cost or time into account. He just randomizes without reasoning and has thus
infinitely high information costs.33
2.6.1
Fundamentals of bounded rationality
In his Clarendon lectures on inefficient markets Shleifer (1999) names several
examples for behavioral mistakes of individual investors34 : Investors’ reasonre-assessment of the investor’s risk aversion.
33 For further details see Brunnermeier (1996)
34 “They trade on noise rather than information, follow the advice of financial gurus, fail
to diversify, actively trade stocks and churn their portfolios, sell winning stocks and hold
on to losing stocks increasing their tax liabilities, buy and sell actively and expensively
Chapter 2. The CAPM and two moment risk preference
45
ing errors are systematic, as they make use of ‘heuristics’ or rules of thumb
which fail to accommodate the full logic of a decision.35 The three main
deviations from standard decision making theory36 are:
First, investors do not assess risky gambles according to the von NeumannMorgenstern principle. They weight gains and losses by a reference point
rather than by their prospective final wealth37 . Secondly, individuals systematically violate Bayes’ rule38 , as they predict future uncertain events by
extrapolating a short history of data. Lastly, subjects make different investment choices depending on the framing of problems39 .
Perhaps the most relevant point of behavioral finance for this study is
that boundedly rational investors do not trade randomly, but rather tend to
make the same mistakes. Therefore, their deviations from traditional decision
making theory are expected to be systematic, synchronous and significant for
the pricing of securities even if experiments seem to prove that experienced
subjects move toward rational expectations.
As it is difficult to determine whether the investment level of a whole
sample is biased due to ‘herding’ or other situational circumstances, the data
sample in 3.3 will at least be examined and tested for systematic biases in the
independent factors.
2.6.2
Limits of Expected Utility Maximization
The Von Neumann and Morgenstern (1947) utility function for money income
assigns utilities to sums of money in such a way that the individual chooses
the action with the highest expected value of utilities.40 Unfortunately, the
axiomatic system cannot be directly subjected to empirical tests. In fact,
extensive experimental research indicates that the principle is widely violated
and that people often have motives inconsistent with the maximization of
expected utility. Certain observations created specific difficulties with the
system. One of them is the popularity of gambling. As the expected value of
managed mutual funds, follow stock price patterns.”
35 See Conlisk (1996)
36 See Kahneman and Riepe (1998)
37 Kahneman and Tversky (1979)
38 Cf. Kahneman and Tversky (1973).
39 See Benartzi and Thaler (1995).
40 Cf. Niehans (1990)
46
2.6. Bounded rationality and goal of methodology
most gambles is negative, they will only be accepted if the utility gain from
gaining amount x is regarded as much larger than the utility loss from losing
amount x.41 This implies that the function must be convex (rising marginal
utility) in that region. Such a risk-prone behavior - implying a convex utility
function - is hardly compatible with the widely observable buying of insurance, a risk-averse behavior that implies a concave utility function (decreasing
marginal utility).42 Friedman and Savage (1948) pursued this line of research
attempting to reconcile these two obviously contradicting behavioral patterns.
Another central problem concerns the nature of the postulated probabilities. In Von Neumann and Morgenstern’s theory they are portrayed as
objective frequencies. Experiments, however, suggest that human decisions
are rather guided by subjective judgments which are modified perceptions of
objective probabilities. These were later axiomatized together with utility in
an approach developed by Savage (1954).
The above mentioned violations of Expected Utility Maximization made
way for psychological models of preferences.43 One of the best-known is the
prospect theory by Kahneman and Tversky (1979) that defines preferences
over gains and losses relative to some benchmark outcome rather than over
consumption as the traditional models. Losses cause greater disutility than
gains utility and each possible outcome can be weighted in a nonlinear fashion
by its probability through the expectations operator.44
Some preferences models that have been applied to asset pricing include
Hogarth and Reder (1987) as well as Kreps (1988).
41 According
to the Bernoulli hypothesis.
style of representation is one of the great disadvantages of expected utility theory,
as it leaves no room for the pleasure or pain of risk itself. What appears as risk aversion is
rationalized as an implication of the diminishing marginal utility of income.
43 Campbell, Lo, and MacKinlay (1997) identify three main components that are altered
by psychological preference models: the period utility function, the geometric discounting
with discount factor delta and the mathemetical expectations operator Et :
42 This

Et 
+∞

δ U (Ct+j )
j
j=1
44 Some examples for general models of subjective expectations are Barberis, Shleifer, and
Vishny (1998), DeLong, Shleifer, Summers, and Waldmann (1990) and Froot (1989).
Chapter 2. The CAPM and two moment risk preference
2.6.3
47
Presentation determines investment choice
Not only probabilities are perceived subjectively, the choices themselves generally seem to be judged on the basis of their presentation. Benartzi and
Thaler (1999) studied how decision makers choose when faced with multiple
plays of a gamble or investment. When faced with simple mixed gambles
they are concerned about the amount they can lose on a single trial (holding
the distribution of returns for the portfolio constant). They display “myopic
loss aversion”. Many subjects who decline multiple plays of such a gamble
will accept it when shown the resulting distribution as opposed to the single
gamble. Benartzi and Thaler apply this analysis to the problem of retirement investing and show that workers invest more of their retirement savings
if they are shown long-term (rather than one-year) rates of return. This in
turn indicates that people cannot infer the probability distribution underlying
multiple plays of gambles.
2.6.4
Goal of questionnaire
At the outset of the study, the goal of analysis needs to be defined. One
important objective is the identification of the most significant factors explaining the dependent variable’s variance. Equally important is to identify
that model and method which - with the given factors - achieves the highest
rate of correctly predicted observations. However, it cannot be the goal to
design an econometric model that classifies all respondents correctly to the
choice they made. Apart from its impracticability, it would amount to asking directly each new out-of-sample individual what stock ratio he wishes.
The value added of the questionnaire method (regression models) must be
founded in its ability to derive and filter out underlying normative patterns
of investment choice. These patterns that are portrayed in the independent
factors must represent universally applicable correlations to the true latent
factors that determine investment choice. If they do not represent these correlations, new investors are mislead by the questionnaire’s result. Reasons for
such inconsistencies between actual and latent factors are multifarious.
48
2.6.5
2.6. Bounded rationality and goal of methodology
General problem of the approach
The main threat of the questionnaire-method arises from the fact that the
respondents’ answers are implicitly viewed as normatively correct, as they are
weighted to calibrate the final questionnaire.
The method uses a sample set of ‘ordinary’ respondents representative of
the whole population. However, among these respondents there are naturally
some whose asset allocation due to bounded rationality does not match their
own latent optimal allocation. In other words, some respondents’ stock ratio
is way too low or way too high for them. In the process of trial and error they
haven’t yet arrived at their own optimal investment strategy.
The question arises whether it would be wrong to assume unbounded rationality. - If people were fully rational, they would not need advice on how to
invest, as their self-assessment yielded the asset allocation optimal for them.
It thus seems straightforward for a study that aims at advising individual
investors to exclude the assumption of unbounded rationality and rational
expectations.
The fact that individuals are boundedly rational has been examined and
confirmed in many studies. The number of experiments reporting biases as
well as the number of books just reviewing the evidence is too enormous. The
article by Conlisk (1996) gives a good overview of the literature. Here it will
just be stated that even in situations where one is faced with decisions that
have objectively correct answers, individuals do not make the normatively
correct decisions.
One obvious sign for bounded rationality immanent in the data is the lack
of stock-ownership for investing individuals. This fact challenges the assessment method in general. In both surveys (SCF 1995 and 1998) more than half
of those people who had financial assets did not own stocks (61% and 56% in
the SCF95 and SCF98 respectively). Such asset allocation habits cannot be
explained by a high need for liquidity or caution due to close retirement.45
Judged from a normative point of view such allocation is suboptimal, especially in terms of diversification or the elimination of unsystematic risk. Thus,
if too few investors hold stocks, a calibration will probably lead to biased re45 It has been noted before that the low propensity to hold stocks, possibly a consequence
of high loss aversion or insufficient knowledge, leads to an above-average risk-premium - a
phenomenon described as the ‘equity-premium puzzle’.
Chapter 2. The CAPM and two moment risk preference
49
sults, to a questionnaire with a tendency to recommend a stock ratio that is
too low.
The above considerations question whether the sample-method might be
too susceptible to individual or cumulative errors in decision-making. They
also query whether there are different degrees of bounded rationality and
whether it is possible to test single observations for bounded rationality or
whole samples for bias. These questions need to be discussed in the following
chapter.
50
2.6. Bounded rationality and goal of methodology
Chapter 3
Empirical Analysis
This chapter introduces two data sets, presents the empirical econometric
analysis of a data set of individual investors and discusses what econometric
models are best suited for a specific data set and problem.
3.1
The Data set
Every two years the Federal Reserve Board has a comprehensive survey carried
out that collects data about financial habits and attitudes.1 Under the name
“Survey of Consumer Finances” (SCF) it is sponsored and published by the
Federal Reserve and the Department of Treasury. It is designed to provide
detailed information on U.S. families’ balance sheets and their use of financial
services, as well as on their pension rights, labor force participation, total
family income, and demographic characteristics at the time of the interview.
“The need to measure financial characteristics imposes special requirements on the survey design. The survey must provide reliable information
both on items that are broadly distributed in the population and on items
that are highly concentrated in a relatively small part of the population. To
address this problem, the SCF employs a dual-frame sample design that includes a standard geographically based random sample and a special oversample of relatively wealthy families. Weights are used to combine information
1 http://www.federalreserve.gov/pubs/OSS/oss2/scfindex.html
51
52
3.1. The Data set
from the two samples for estimates of statistics for the full population.”2
The survey thus not only draws a detailed picture of the financial situation
and financial attitudes of US households, its design also ensures that it is representative of the whole population. Different types of financial assets - such
as cash in transaction accounts, certificates of deposit, savings bonds, bonds,
stocks, mutual funds, retirement accounts, cash value of life insurance and
other assets - are recorded separately. The approximate value of real estate,
life insurance and expected inheritance and wage increases are asked as well.
The data set further contains information about household incomes, respondent’s attitudes toward financial risk, liquidity, use of credit, and reasons for
saving. This allows an easy calculation of each household’s stock ratio which
will serve as the dependent variable.
In fact, the SCF represents the ideal data source for a study of individual
choice behavior that intends to be descriptive rather than normative, abstract
rather than situational and operational rather than purely theoretical.
In general the models require that individuals act rationally which essentially means that they follow a consistent and calculated decision process in
which they pursue their own preferences and objectives. It excludes impulsiveness and implies consistent and transitive preferences. Under identical
circumstances individuals are assumed to make the same choices.
In the data sample the dependent variable’s character is continuous. In
order to apply Ordered, BNL, MNL, Conditional Logit or even Nested Logit
Models it had to be discretized. The advantage of models such as the MNL
is that they can account for nonlinearities in the relation of the dependent
variable and the independent factors. The reason for employing discrete data
at all roots in practical considerations: It is generally easier for investors to
determine their optimal stock ratio as an approximate interval instead of a
precise, single number. In the study these intervals, later called ‘risk classes’
are represented by the discrete dependent variable Y . For a thorough analysis it was examined whether the assignment precision could be increased by
creating different nests, different constellations of these classes. The different
settings are represented in Appendix D.
Even without discretizing the data, two distinct groups arise due to the
2 Verbatim from Kennickell, Starr-McCluer, and Sunden (1997). More details about the
survey procedures and statistical measures can be found in Appendix J
Chapter 3. Empirical Analysis
53
fact that for some people a stock ratio cannot be calculated, as they do not own
any financial assets. These individuals constitute the group for which aq cat
takes on the value 0. The ones who do have financial assets are assigned to
6 different groups. The first group comprises all individuals without stocks
while those with stocks are subdivided into 5 groups of equal intervals with a
span of 20%.
Figure 3.1: The data sets: SCF 1995 and 1998
Obs.* = Number of observations.
Dependent variable: Y = AQ CAT; Significance of its values:
0
1
2
3
=
=
=
=
do not own assets
0% stocks
1% - 19% stocks
20% - 39% stocks
Panel A: SCF1995
Y
0
1
2
3
Obs.∗
360
2399
472
334
4 = 40% - 59% stocks
5 = 60% - 79% stocks
6 = 80% - 100% stocks
Panel B: SCF1998
Y
Obs.∗
4
5
6
299
216
229
Y
0
1
2
3
Obs.∗
302
2245
349
340
Y
Obs.∗
4
5
6
364
329
375
Transforming the dependent, continuous variable into a discrete one does not
come without problems and, of course, not without a loss of information.
The intervals chosen to divide the IOS are admittedly arbitrary as mentioned
already. However, with the transformation an error term is inevitably added
54
3.1. The Data set
to the dependent variable. This increases its noise and impedes a reliable
analysis of its influencing factors.
For preservation of data accuracy an analysis of the continuous form of
the variable is thus clearly preferable whenever possible. Feasible models for
the continuous stock ratio would be OLS or the Tobit model. For using OLS
in setting 13 , the estimation of all classes in one step, the individuals without
financial assets had to be coded as (−1) so that they could be included in
the analysis. Of course, such arbitrary coding distorts the results and it
was carried out solely to compare the OLS with the Tobit model. For such
naturally censored data the Tobit model is obviously a better fit and can be
expected to yield superior results to OLS.
Data accuracy is not the only goal that needs to be considered. Continuous
data might not always be available, as in most samples the stock ratio will
be given as an approximate interval. It is conceivable that at a later time,
the results of the regression analysis will be evaluated to assign clients to 5-6
different investment funds rather than to a specific stock ratio. Also, by using
discretized data a vast number of additional models present themselves for
analysis, such as the Ordered Logit, MNL, Discrete Choice Model and the
Nested Logit. These models allow the consideration of individuals without
financial assets as well as all kinds of other groups and distinctions.
The models for analyzing discrete data mentioned above differ greatly
in their assumptions and specifications. A comprehensive comparison will
be difficult and we will therefore focus on the qualities such as the model’s
proximity to the underlying financial theory, the ability of predicting the single
choices correctly (out-of-sample) as well as several statistical measures like the
Akaike Information Criterion (AIC).4
3 See
Appendix D.
evaluating the different models the out-of-sample results are preferable, as insample estimates usually tend to overfit the data. Out-of-sample estimation was easy thanks
to the high standardization of the Survey of Consumer Finances which keeps the variables
virtually invariable over time.
4 When
Chapter 3. Empirical Analysis
3.2
55
Selection of factors and hypotheses
Below, a selection of those factors is given that according to theory and empirical studies have influence on the investment risk level.
The selection is arbitrary and non-scientific. The factors are tested by
surveys and grouped by factor analysis. Insignificant factors are omitted in
later analyses. Though anonymous, the surveys include demographic data in
order to control for biased sampling. Demographic factors proved insignificant
and played no further role within the analysis. This was convenient, as it is
not intelligable why for example a woman should invest differently from a
man.
The following factors are conjectured to have an impact on the choice of
risk level both normatively and positively:
• The number of people in the household, x101, and marital status, x7372 1,
are two factors that relate to financial obligations and unpredictable
events, as, on average, people who are married and have a family are
financially less flexible.
• Age, x8022.
According to the classic life cycle hypothesis by Brumberg and Modigliani5
and according to the behavioral life cycle hypothesis by Shefrin and
Thaler (1988) age impact the asset allocation over time.6 Goodfellow
and Schieber (1997) confirm that empirically the percentage in fixed income increases with age while the percentage in stocks declines. They
further find that higher income individuals are more inclined to invest
in stocks. For the SCF 95 and 98 the influence of income was however
not unambiguous.
The idea of the life-cycle hypothesis is to adjust an investor’s intertemporal asset allocation to changes in his human capital. Thus, when
ignoring other factors, the younger an investor the riskier he should invest. Two arguments support this hypothesis: the younger the greater
the possibility to use wages to cover losses caused by higher investment
5 Due
to Brumberg’s early death not published until 1980 in Modigliani (1980)
age is viewed to be negatively related with investment risk. Thus, the older the
investor the lower his stock ratio should be.
6 Where
56
3.2. Selection of factors and hypotheses
risk. Human capital can be regarded as a hedge against the risk of losses
from a high allocation to stocks. On the other hand, the longer an investor’s investment horizon, the more likely it is that stocks outperform
any other type of asset class.
Above all, the asset allocation demands most careful consideration, as
the largest portion of the portfolio’s total return is “determined by its
asset categories rather than by selection or timing. Whatever one’s
aversion to risk, one’s age, income from employment, and specific responsibilities in life go a long way to helping one to determine the mix
of assets in one’s portfolio.”7
• The level of education. Highest degree earned: x5905.
A common concern about this factor is the high correlation between
education, income and wealth. Evidence on the effects of education on
risk taking is mixed. Riley and Chow (1992) find that risk aversion decreases with education. Surprisingly, Jianakoplos and Bernasek (1998)
as well as Hersch (1996) come to the opposite result. A better and
related factor seems to be financial knowledge: Bayer, Bernheim, and
Scholz (1996) find that measures of savings are higher when employers
offer retirement seminars.
To the extent that education implies information collection, skill acquisition and general development of cognitive capacity, one’s investment
in human capital lowers bounded rationality and improves investment
decision. It can thus be expected that the better the education the lower
the decision error.
• Saving horizon and consumption patterns, x3008 1 and x3008 45.
Long investment horizons indicate the ability to ‘sit out’ market downturns. Under the term “time diversification” the relevance of the investment horizon received great attention from a normative perspective,
see Section 2.4. Samuelson (1963b), (1989), (1990), (1994) contributed
those articles most relevant to resolve this finance puzzle.
7 Malkiel
(1999), 351ff.
Chapter 3. Empirical Analysis
57
• Tolerance towards fluctuations, patience in sitting out market downturns and willingness to take risk, x3014 1 and x3014 4.
Frequent rebalancing is considered suboptimal, as transaction costs arise
from speculation tax, brokerage fees and time consumption.
• Low predictability of income and high dependency on savings and financial assets as a source of income: future pension: x5608, future
inheritance: x5821, foreseeable expenses: x7186 and profession: x7401.
These terms refer to the likelihood of being forced to liquidate investments due to insufficient funds, loss of job and probability of wage cuts.
Projects and other specific needs must be funded with specific assets
dedicated to that need in order to ensure their realization. If funds for
specific assets are invested as risky as the free, untied part of wealth,
the investor runs the risk of having to cancel the project. The old saying
“never gamble with the money that you need to pay your rent” is representative for this line of argument. The best stress test of a strategy
is probably: Even if the worst case occurs, the corresponding decrease
in portfolio value should not affect an individual’s standard of living.
• Credit behavior, x432 1 and x7131.
The impact of these two factor is theoretically ambiguous. People who
always pay off their credit card balance are rare in the US as these accounts are viewed as a permanent consumption credit. On the opposite,
it seems as if a constant level of credit over long periods is viewed as an
indicator for financial control.
• Expectations on economic development, x301 1 and x301 3. Optimistic
or positive expectations on the economy are an important prerequisite
for investing in stocks.
• Investment goal, x3006 1, x3006 5 and x7187. Aggressive goals can only
be achieved by accepting a corresponding amount of risk.
• Wish to leave one’s estate to others, x5832 5. The explicit desire for
leaving an estate can be a reason for maintaining a high stock ratio at
an old age.
58
3.3. Testing for bounded rationality
Some empirical studies on risk aversion examine independent factors such
as wealth and income. Bajtelsmit (1999) used a Tobit model to analyze
evidence of risk aversion in the Health and Retirement Study (Wave 2). The
results expectedly indicated that individuals with greater wealth allocate a
larger portion of their wealth to risky assets than those with less wealth. The
percentage allocation to risky assets was also shown to be lower for those with
lower education levels and for those with greater pension balances. Younger
investor groups allocate a greater proportion to risky assets with the most
significant difference for those in the 50-55 age group.
In this study where two Surveys of Consumer Finances (SCF) were employed, categorized wealth (wealth classes) proved insignificant in explaining
investment choice among the different groups of stock investors (those considered in setting 3c). Interestingly, when looking at the estimation result
where wealth is the single independent factor, the coefficient for the highest
risk class (80-100%) has a negative sign. This indicates that the higher the
wealth, the less likely the individual is to invest very risky. Even though the
factor is not significant, it shows that the assumption of decreasing relative
risk aversion does not hold for the SCF95 sample of stockowners.
Distribution of stockholders in the SCF95:
Wealth
in US$
0
110’00050’000100’000500’0001’000’000>5’000’000
3.3
Number of
people
360
1642
734
311
558
201
297
196
4299
Average
age
41
43
49
54
58
58
61
64
49
Average
risk class
–
1.2
1.9
2.2
2.7
3.3
3.4
3.3
1.9
Average
stock ratio %
–
3.88%
15.04%
19.05%
26.74%
36.60%
39.23%
36.29%
25.26%
Testing for bounded rationality
It was stated that one realistic threat to the applicability of the econometric
analysis of influential factors is a calibration-sample that is heavily biased.
The source of the bias can be ‘irrational exuberance in the financial markets’
implying entirely overvalued stocks, it can be the opposite - a deeply depressed
Chapter 3. Empirical Analysis
59
economy of overcautious subjects8 , it can be a non-representative sample of
the population or it can be the lack of investment knowledge in the majority
of the population.
Two basic prerequisites of the applicability and usefulness of the method
discussed are a) the sample of the population must be representative and b)
that financial markets must be in a non-exuberant and non-depressed state a condition which is difficult to measure.
One indication for an exuberant tendency on the markets are factor coefficients that fully contradict the hypotheses in sign and magnitude.
However, even if the sample is not completely biased, single hypotheses
can be violated by the respondents if they are boundedly rational. The latter
will be defined as the joint violation of those hypotheses specified in the tables
on pp. 63ff. Such joint violations will be tested further down.9
Over the years 1992-1998 the median family income as well as median family net worth rose in constant dollars. Reasons were the excellent economic
situation and the fact that both the ownership and the amount of holdings in
publicly traded stocks by families expanded greatly over that period. Even
though, the share of households who owned stocks increased from 34% in 92
to over 41% in 98, the vast majority of people who had assets did not invest
in stocks.10 45% of households with holdings of liquid riskless assets between
$60’000 and $100’000 did not directly hold stocks and 28% of households with
over $100’000 in liquid riskless assets did not hold stocks directly.11 In the
theory of the Consumption Capital Asset Pricing Model (CCAPM) this phenomenon is explained with monitoring costs which are assumed to be higher
for stocks than for any other asset class. One indication that costly infor8 In
times of market frenzy a significant proportion of investors succumb to the temptation of ‘herding’, thereby giving up their balanced asset allocation strategies. Even a
representative sample of the whole population will thus be biased towards extreme allocations.
9 The tests were carried out with boolean queries in Excel.
10 Politicians and economists alike have repeatedly argued that the average stock ratio
of individual households is in general too low to secure the accustomed standard of living
after retirement.
11 See Bertaut (1998) who states that “in addition to variables that capture access to
investment opportunities and information about the market, lower risk aversion, higher
expected future income, absence of income risk, and presence of a bequest motive will also
contribute positively to the probability of holding stocks.”
60
3.3. Testing for bounded rationality
mation about stock investments deter households from participation is that
household portfolios display persistent behavior: Individuals should remain
stock- or non-stockholders over some period of time. Such a persistence was
in fact confirmed by Bertaut (1998) in a panel study for the SCF of the years
1983 and 1992. Bertaut and Haliassos (1997) further show that the higher
the education level and the stronger the bequest motive, the lower is this
information or monitoring cost.12
Such costs partly justify the low ratio of stock-holding households indicating that either high education or specific motives have to be extant for
coping with the higher risk of stock investment. However, the argument given
does not suffice to refute the hypothesis of a biased sample and a generally
too low stock-ownership. The hypothesis could be invalidated, if households
without stocks were predominantly poor or lacked education. This however
is not the case: Wealth and the level of education vary unsystematically with
the investment risk both in the SCF 1995 and 1998.
The story that the data tells us is that, in addition to the strong restraint
against risky assets, there is obviously a lack of information among private
households about the general characteristics of asset classes, their return and
risk profile. This lack of information is not fully eliminated by a high level of
education. A BA or even a PhD or MD is no guarantor for possessing financial
knowledge. Thus, if the low ratio of stock-ownership is not a consequence of
a conscious decision process, the level of risk chosen can be expected to be
generally suboptimal.13
3.3.1
Putting things in perspective
In the preceding paragraphs, the wrongly assigned investors were interpreted
as deviations from the optimal choice, as single, non-fitting observations of
the model. By taking the opposite perspective, it is possible to view these
differently assigned observations as a correction of the model for boundedly
12 Bertaut
(1998) thus concludes that “education and advertising campaigns can be instrumental in helping households overcome reluctance to hold stocks caused by insufficient
information about the benefits, risk, and costs of market participation.
13 Another sign for this suboptimality and the poor average financial knowledge of households is the degree of diversification of the share of the risky assets: 52% of all stockholders
in 1999 held less than 8 different titles.
Chapter 3. Empirical Analysis
61
rational respondents. The following paragraph analyzes what kind of profile the high-risk investors have according to the Multinomial Logit model in
Setting 3c of SCF1995, a setting consisting of five risk classes:
Of 1550 stockowners, 229 held more than 80% in risky assets. Only 23 of
these 229 (roughly 10%), were correctly assigned to the high-risk class. The
typical high-risk investor as predicted by the MNL had the following profile:
• All respondents negated the question of whether their most important
reason for saving was liquidity and consumption: x3006 1 = 0.
• Roughly 80% opposed to having a saving and spending horizon of a few
months: x3008 1 = 0.
• 96% of the predicted high risk investors did not expect the US economy
to perform worse in the coming 5 years: x301 3 = 0.
• 85% confirmed they were willing to take financial risks: x3014 4 = 0.
• 90% had to admit that they did not always pay off the total balance
owed on their credit card account each month: x432 1 = 0.
• 70% did not expect to inherit any amount of money: x5821 = 0.
• Only 30% had earned a degree (Nursing, College, BA, MA, MBA, PhD,
Law): x5905 > 0.
• 95% had applied for a credit or a loan within the preceding 5 years:
x7131 = 0.
• 65% had wealth of less than US$50’000.
• The evidence for x101, x3006 5, x3008 45, x301 1, 5825 3, x7187, x7372 3
was mixed, as they showed no unambiguous pattern.
Contraintuitive results were produced for the expected amount of future
pension and the goal of saving. Almost no one estimated to receive even a
single dollar: x5608 = 0 and everyone saved for foreseeable expenses.
62
3.3. Testing for bounded rationality
Table 3.1: 50% of those who were characterized as
high-risk investors by the MNL were between 20 and 40
years old, the average age comes to 45 years.
Age
interval
0-20
21-30
31-40
41-50
Number of
individuals
0
11
23
15
Age
interval
51-60
61-70
71-80
81-100
Number of
individuals
11
8
4
2
As a brief conclusion it can be stated that the majority of factors confirmed
the hypotheses and predictions of the theory. Thus, the findings reinforce the
methodology and suggest advice should best be based on the principle of
exclusion of asset allocation strategies. The analysis of each group of respondents classified as a category by the model informs about common underlying
characteristics. If these characteristics are not met by a new respondent,
he shouldn’t be assigned to that class. For instance, a person whose saving
horizon spans only a few months should never invest 80-100% in stocks. Nevertheless 12% (27 of 229) of the high-risk investors admitted that they had
a saving horizon of only a few months - clearly a sign that there is a great
potential for investment optimization.
3.3.2
‘Correctly’ assigned observations
In setting 3c of the SCF1998 the MNL assigned about 25% of all investors
‘correctly’ to the classes that corresponded to their actual revealed choice.
‘Correctly’ means that the regression’s assignment of these people confirmed
their actual asset allocation. In order to put the results to the test, each
investor’s response set is analyzed for violating the hypotheses depicted in
Section 3.2.
Chapter 3. Empirical Analysis
63
Highest risk level: Class 4
In setting 3c for the SCF1998 the MNL assigned 123 people ’correctly’ to class
4. The following table lists those combinations of values for the independent
variables that are inconsistent with the highest risk class. Violations considered to be grave and irreconcilable with the assignment are labeled with
‘**’ two stars. The last column indicates the number of violations for the
corresponding variable.
Variable
X101
X3006 1
X3008 1
X301 3
X3014 4
X432 1
X5825 3
Type of Violation
> 2. a high number of dependents in connection with low wealth (< 100 000) and income
leaves the household too exposed to volatility
= 1. Liquidity and consumption are rather
short term goals. Stock investment is less
suited for such objectives.
= 1. People with a very short investment horizon should not invest heavily in stock.
= 1. A pessimistic view about the future economic development is possibly not the right
prerequisite for excessive stock investment.
= 1. Clear conflict of goals. Someone who is
not willing to take any financial risks should
not invest in stock.
= 0. Paying off credit card debt is an indicator
for having good control about financial issues.
= 1. In connection with the proximity of retirement and relatively modest wealth a high
stock ratio is not recommendable.
Relevance
Violations
3
**
2
**
5
22
**
3
29
**
0
Of 123 ‘correctly’ assigned observations, only 10 cases seemed to conflict
with the profile of the highest risk class after accounting for overlapping violations of the above hypotheses. This signifies that the recorded violations
came from only 10 different observations. In fact, the normative findings were
confirmed with high validity, as the group showed great homogeneity in terms
of stated risk preference (x3014), high wealth, largely managerial jobs and
long investment horizons.
64
3.3. Testing for bounded rationality
Lowest risk level: Class 1
Normatively the following values can be identified as violations for the people
in the lowest risk class: (inconsistency of goals and binding violations are
labeled with ‘**’ two stars):
Variable
X101
X3006 5
X3008 45
X301 1
X3014 1
X432 1
X5608
X5821
X7187
X7372 1
X8022
Type of Violation
< 3. A low number of dependents in connection with high wealth and income leaves freedom to invest riskily
=1. Retirement,Education and Family are
long-term goals that allow risky investment
= 1. People with a long investment horizon
should invest more in stock
= 1. An optimistic view about the future economic development suggests a higher stock ratio
= 1. Clear conflict of goals. Someone who is
not willing to take any financial risks should
not invest in stock.
= 1. Paying off credit card debt is an indicator
for having good control about financial issues.
> 3. High pension pay leaves the investor free
of worries about the retirement and increases
his risk taking capacity
> 3. High expected inheritance pay leaves the
investor free of worries about future funding
and increases his risk taking capacity
> 3. In connection with little wealth the necessity of a sizeable nest egg demands a higher
stock ratio
= 5. Singles who were never married have less
financial obligations and can afford a higher
stock ratio.
A young person can afford a high stock ratio, especially when he does not have a family
(x7372 1 = 0 and x101 = 1)
Relevance
Violations
44
**
5
**
35
12
**
0
40
0
8
**
3
10
10
The MNL model, setting 3c, SCF 1995, assigned 97 observations ‘correctly’
to the highest risk class. Each response was tested for violating the above
hypotheses. Again, the normative findings were confirmed with high validity
Chapter 3. Empirical Analysis
65
with one exception: about 30% of the respondents had a long investment horizon (x3006), but held less than 20% stocks. Apart from this, the group showed
great homogeneity, especially in terms of stated risk preference (x3014). No
respondent showed willingness to take substantial financial risk when investing. Surprisingly, most investors in this low risk class had managerial jobs.
In terms of wealth - a factor not included in the regression - the results
were contraintuitive, as 91% of the investors in the low risk class had total
assets of more than US$100’000.
3.3.3
‘Wrongly’ assigned observations
Highest risk level: Class 4
In setting 3c of the SCF1998 the MNL assigned 92 observations to the highest
risk class (80-100% stocks). According to revealed preference (actual stock
ratio), however, these should have been in the lowest risk class (0-20% stocks).
The question arises as to whether these cases were ‘wrongly’ assigned.
When analyzing the characteristics of those 92 respondents in detail it
becomes obvious that they do not fit into the risk class they have chosen
themselves. Most of them have a long investment horizon and long-term
saving goals, some are even willing to take substantial risks, half are optimistic
about the economic outlook and others expect a substantial windfall.
Variable
X101
X3006 56
X3008 45
X301 1
X3014 1
Type of Violation
< 3. A low number of dependents in connection with high wealth and income leaves freedom to invest riskily
Retirement, education and family are longterm goals that allow risky investment
= 1. People with a long investment horizon
should invest more in stocks
= 1. An optimistic view about the future economic development suggests a higher stock ratio
= 1. Clear conflict of goals. Someone who is
not willing to take any financial risks should
not invest in stocks.
Relevance
# of cases
44
**
51
**
57
18
**
19
66
X432 1
X5608
X5821
X7187
X7372 1
X8022
3.3. Testing for bounded rationality
=1. Paying off credit card debt is an indicator
for having good control over financial issues.
> 3. High pension pay leaves the investor free
of worries about the retirement and increases
his risk taking capacity
> 3. High sums of expected inheritance leaves
the investor free of worries about future funding and increases his risk taking capacity
> 3. In connection with little wealth the necessity of a sizeable nest egg demands a higher
stock ratio
= 5. Singles who were never married have less
financial obligations and can afford a higher
stock ratio.
< 40. A young person can afford a high stock
ratio, especially when he does not have a family (x7372=0 and x101=1)
15
1
9
**
2
6
2
Naturally, for some observations there are overlapping violations. However,
even when accounting for these overlappings and even when only the highly
relevant violations (marked with ‘**’) are considered, 88 responses seem incompatible with the lowest risk class. They thus seem to be ‘correctly’ assigned by the model when judging by the underlying hypotheses.
Lowest risk level: Class 0
In setting 3c for the SCF1998 the MNL assigned 59 observations to the lowest
risk class (0-20% stocks). According to revealed preference (actual stock ratio), these should have been in the highest risk class (80-100% stocks). Again
one has to examine whether these cases were really ‘wrongly’ assigned or
whether the model corrected these individual choices.
Even though the picture is less unambiguous than for the cases that were
’wrongly’ assigned to the highest risk class, there are many violations among
these cases, too. Most of the respondents have a short investment horizon,
short-term saving goals, half of them are not willing to take any financial risk
and expect an economic deterioration, while others are too close to retirement
to invest heavily in stock.
Chapter 3. Empirical Analysis
Variable
X101
X3006 1
X3008 1
X301 3
X3014 4
X432 1
X5825 3
Type of Violation
> 2. A high number of dependents in connection with low wealth and income leaves
the household too exposed to volatility risk of
stock investment
= 1. Liquidity and consumption are rather
short term goals. Stock investment is less well
suited for such objectives.
= 1. People with a very short investment horizon should not invest heavily in stocks.
= 1. A pessimistic view about the future economic development is possibly not the right
prerequisite for excessive stock investment.
= 1. Clear conflict of goals. Someone who is
not willing to take any financial risks should
not invest in stock.
= 0. Paying off credit card debt is an indicator
for having good control about financial issues.
= 1. In connection with the proximity of retirement and relatively modest wealth a high
stock ratio is not recommendable.
67
Relevance
# of cases
20
**
0
**
8
22
**
26
11
**
10
After accounting for overlapping violations, there remain 33 observations
that do not seem to belong into the highest risk class.
The analysis of contradictions can be concluded by stating that the model
and the empirical data strongly support the discussed hypotheses. The deviation from the hypotheses in the form of a cumulation of grave decision errors
amounted to less than 3% of each risk class. This indicates that although
investors commit single mistakes that concern some hypotheses and factors,
they do not err cumulatively.
3.3.4
Stated and observed preferences
Various empirical studies about risk preferences of individual investors as
the one by Bajtelsmit (1999) found that hypothetical questions are not a
reliable source of information. Respondents of the SCF 1989, for example,
were asked how much risk they would be willing to take for a certain return on
a hypothetical investment. What people chose in response was not consistent
68
3.4. Structure of Analysis and Nests
with their asset allocation. Thus, surveys about stated preferences have to be
judged with caution. It seems as if what people say is not always compatible
with what they do. If this holds true, then approaches involving gambles
won’t yield reliable results even if subjects were to act according to expected
utility theory.
Another reason for the inconsistency between stated and observed preference might base on a mis-understanding of the concept of risk. Possibly
not all people primarily think of return volatility when asked to define risk.
Communicating risk accurately and completely is thus of foremost relevance
if investors’ decision making is to be improved.
The incompatibility between observed and stated preferences concerning
risk was less pronounced in the two surveys examined. In the SCF 95 survey,
221 of 4300 people stated that they were not prepared to take any risk at all.
Half of these 221 respondents also actually held no stocks at all. Only 10%
of the people showed inconsistency between stated and revealed preferences.
Actual
Stock Ratio
0%
1% - 20%
21% - 40%
41% - 60%
61% - 80%
81% - 100%
3.4
Number of
Respondents
109
30
20
20
22
20
221
Structure of Analysis and Nests
For the analysis each sample was divided into different subsets. Each subset
was then estimated separately. For an overview of the structure and nests
please refer to Appendix D.1
In the first setting - labeled ‘1’ - all individuals were included in the model.
Those with no assets at all and those with no stocks were examined together
with stock-owners. While the first two groups were assigned quite well in
this setting, the prediction of the different risk classes was quite poor. This
happens often when we have groups of unequal sizes as is the case in the 1995
SCF. Cases are more likely classified to the larger groups, regardless of how
Chapter 3. Empirical Analysis
69
well the model fits. A further reason for the imbalance of predicted outcomes
xik we see that the
are the weights. If they are plotted as a function of βjk
xik = 0, implying
maximum weight will be given to observations for which βjk
that Pt = 0.5, while relatively little weight will be given to observations for
which Pt is close to 0 or 1.
In the second setting, a two-step procedure was adopted: in a first step labeled ‘2a’ in Appendix D.1 - we only distinguished whether respondents had
financial assets or not. In a second step - labeled ‘2b’ - we excluded respondents without financial assets from the samples. However, as the assignment
to classes with high risk was still not satisfactory, a third structure was set
up.
In the third setting the assignment problem was divided into three steps:
first we adopted the binary model from structure ‘2a’ distinguishing only
whether people owned financial assets or not. In a second binary model labeled ‘3b’ in Appendix D.1, we distinguished whether people owned stocks
or not. Finally in the last step - ‘3c’ -, we distinguished between the different
risk levels among the stockowners.
It is conceivable to combine the earlier two- and three step regressions in
a single nested logit model estimating the coefficients for all nests simultaneously. This was labeled as setting ‘4’ in Appendix D.1. However, as explained
in Subsection 3.5.7 the NLM is primarily used for the analysis of choice attributes rather than investors’ characteristics. It was not considered here,
as the SCF did not contain any data referring to a subjective judgement of
different risk classes and investment choices.
A benefit of simultaneous estimation with the NLM is that it enables
the consideration of different utility functions for each ‘knot’ in the structure
and may obtain the effects on probabilities of all choices in the model. It is
possible to calculate marginal effects of a change in any factor in the utility
function for any alternative. With separate models for each step this is not
feasible. A different sample, however, containing preferences on risk classes
would possibly shed some light on the question why many investors refrain
from stocks.
When evaluating the performance of the different models characterized
hereafter, it is crucial to keep in mind that not all of them can be compared
with each other. For a continuous dependent variable the OLS and Tobit
70
3.5. Characterization of econometric models
model were considered. For a discrete binomial dependent variable, the WLS
and the Multinomial logit model were put side by side and for a discrete multinomial dependent variable, the OLS, the Ordered Logit and the Multinomial
Logit model were compared with each other.
3.5
Characterization of econometric models
This section will briefly characterize the objective function and the different
models of estimation used in the empirical analysis. All models were estimated using the full version of LIMDEP7.014 . Although this program proved
extremely helpful and flexible for all models considered in this study, the classification tables, AIC, out-of-sample estimation and several other calculations
had to be programmed with Excel macros. In all regressions a constant term
was used to account for systematic deviations not explained by the independent factors.15
3.5.1
The Objective Function
Investors are assumed to maximize expected utility. The objective function
thus plays a crucial role for the estimation procedure and for the applicability
of different regression models.
For portfolio choice the traditional approach has been to define the objective function in terms of the moments of the return distribution. Sharpe
(1964), Lintner (1965), Mossin (1966) portrayed individuals as single-period
maximizers of expected utility of their future wealth. They limited their
valuation model to the first two moments. This well-known mean-variance
valuation model was extended by Rubinstein (1973) to a general parameterpreference model which will be applied for the second part of this study.
In this first part, the objective function of the mean-variance model will
suffice. The utility of investor i can thus be defined in terms of expected
return µ and volatility σ of investment opportunity j.
αi
(3.1)
Ui = µj − σj2
2
14 Developed
by Econometric Software, Inc., whose founder is Prof. W.H. Greene.
Greene (1998), p. 514 suggests normalizing β0 = 0 for the MNL model in
order to identify the parameters of the model.
15 Although
Chapter 3. Empirical Analysis
71
αi depicts the investor’s Pratt-Arrow measure of relative risk aversion. At
the same time it is the dependent variable of the regression models to be
explained by k exogenous factors xik that were presented in Section 3.2. The
regression model16 therefore is
Ui = µj −
xik 2
βjk
σj
2
(3.2)
In the case where discrete choice models are considered17 , a finite number
of investment opportunities must be given. In setting 1 there are 7 discrete
investment choices. One consisted entirely of the risk-free asset and one consisted entirely of stocks. The remaining four choices are linear combinations
of these two extremes, as shown in Figure 2.1 on page 22.
In combination with time series data that allows the calculation of the
investment opportunities’ means and variances, threshold values of the relative
risk aversion αi can be calculated. These threshold values mark those degrees
of relative risk aversion for which the utilities of two adjoining investment
opportunities are the same. By calculating these threshold values it is possible
to regress directly onto relative risk aversion instead of making the detour via
the regression (αi = β xi ) embedded in the utility function.
The threshold values for αi can be calculated by
αk 2
αk 2
µk −
σ = µk+1 −
σ
(3.3)
2 k
2 k+1
for k=1,2,...6; αk marks the relative risk aversion for which the utilities of the
investment opportunities’ (k and k + 1) are equal.
As an example the following yearly Swiss Market returns from 1925-1997
(source: Bank Pictet & Cie.) are depicted for six different risk classes in the
model.18
k
1
2
3
16 Subscripts
µk
4.43%
5.16%
5.90%
σk
3.52%
5.47%
8.54%
k
4
5
6
µk
6.64%
7.38%
8.11%
σk
11.89%
15.34%
18.83%
j and k will be dropped hereafter.
B shows how the Multinomial Logit Model can be derived from Utility Maximization and an Extreme Value Distributed Error term.
18 The time series data on market portfolio returns is analyzed in greater depth in Appendix I.
17 Appendix
72
3.5. Characterization of econometric models
For these moments of returns the following threshold values can be calculated
with Equation 3.3:
k+1=k
6=5
5=4
4=3
αk
1.22
1.51
2.16
regr.value
1.00
1.37
1.84
k+1=k
3=2
2=1
1=0
αk
3.11
8.33
71.51
regr.value
2.64
5.72
39.92
The first column of the above table indicates for which two risk classes the
threshold αk in the second column was calculated. Thus, for a relative risk
aversion of αk = 1.22 the two investment opportunities k = 6 and k = 5
with their mean and volatility given above yield the same utility according to
Equation 3.1.
The goal of this proceeding is to rephrase the objective function 3.1 and to
solve for relative risk aversion in order to regress directly onto αk -classes. As
it is impossible in discrete choice models to regress onto non-integer values,
the regression itself will be carried out using the k’s which represent different threshold values of risk aversion. The k’s are thus used as the values
of a multinomial dependent variable Y in a discrete choice model and stand
for classes of increasing risk. The third column labeled ‘regr.value’ gives the
‘real’ values for the relative risk aversion that are represented by k. The higher
k, the higher the investment opportunity’s risk and the lower the investor’s
risk aversion who chooses the corresponding asset allocation. The ’regr.value’
refers to the mean of two adjoining thresholds and can be interpreted as the
risk aversion representative for the risk class k: For example, the representative relative risk aversion for the risk class 5 is [1.22 + (1.22 + 1.51)/2] = 1.37.
Generally, it does not matter whether the integer values of the dependent
variable in the discrete choice models are converted into a measure for relative
risk aversion or a stock ratio, as these two quantities are connected anyway
through Formula A.14 in Section A.1 of the Appendix.
The purpose of this subsection was to show how the threshold values of
relative risk aversion. These values correspond to the finite number of risk
classes in the settings depicted in Figure D.1 of Appendix D.
Chapter 3. Empirical Analysis
3.5.2
73
Ordinary and Weighted Least Squares Model
Ordinary Least Squares (OLS) estimation was carried out for settings where
the dependent variable Y is continuous or polytomous discrete, while Weighted
Least Squares (WLS) was applied for settings with a dichotomous dependent
variable19 , as the OLS does not yield efficient estimates in that case.20
Goldberger (1964) proposed the following two-step, weighted estimator to
obtain unbiased and efficient estimates of the LPM:
The standard linear regression model for OLS21
yi = β0 + β xi + εi
(3.4)
where i = 1, 2, ...n and xi is a K-vector of known factors, is used to yield
unbiased estimates βˆj . From these estimates, weights wi are calculated22 for
each observation i
wi =
− 12
β̂ xi
β̂ xi 1 −
(3.5)
All terms on both sides of the linear regression model in Eq. 3.4 are then
multiplied by wi
(wi Yi ) =
β̂wi xi + wi εi
(3.6)
and a second regression for these terms yields unbiased and efficient estimates.
However, coefficients can no longer be interpreted.
In setting 1, observations are considered for which no stock ratio could be
calculated, as the corresponding individuals did not own any financial assets.
For these observations the dependent variable was arbitrarily set to -1.
19 Linear regression models with a dependent variable that is either zero or one are often
called linear probability models.
20 To obtain BLU estimators, serial independence and homoscedasticity of the error term
as well as non-collinearity of the independent variables must be given. Clearly for a dichotomous Y the error term cannot have constant variance, as the variance of εi varies
systematically with the regressors xi .
21 Subscript j on β and k on x are suppressed.
22 The weights are just the reciprocals of the estimated standard error of ε . Problems
i
with the formula 3.5 may arise if the predicted value β xi is greater than 1. In that case it
can be truncated to 0.999, see Aldrich and Nelson (1984), 84.
74
3.5. Characterization of econometric models
3.5.3
The Tobit Model
When the dependent variable, the percentage of wealth allocated to risky
assets, is given in a continuous form in setting 1, it represents a naturally
censored variable. The reason is that one cannot calculate a stock ratio for
individuals who do not own any financial assets. This particularity destroys
the linearity assumption so that the OLS seems clearly inappropriate. The
model is censored as one can at least observe the exogenous variables. Thus, a
standard Tobit model (or Type I Tobit) can be used for estimating regression
coefficients:
yi∗
= β0 + β xi + εi ,
yi
=
yi∗
yi
=
0
if
if
yi∗
yi∗
εi ∼ N [0, σ 2 ]
>0
(3.7)
(3.8)
≤0
εi are assumed to be iid drawings from N (0, σ 2 ) and can be interpreted as
the collection of all the unobservable variables that affect the utility function.
We can write the likelihood function L for n independent observations of
the model as
x β −1 (yi − xi β)
]
(3.9)
[1 − Φ( i )]
σ φ[
L=
σ
σ
0
1
where Φ and φ are the distribution and density function, respectively, of the
standard normal variable.
∗
0 means the product over those i for which yi ≤ 0, and
1 means the
∗
product over those i for which yi > 0.
The method of estimation is maximum likelihood. The prediction for the
Tobit model in LIMDEP - which is also used for setting up the classification
tables - is calculated as the conditional mean of yi given xi :
E[yi |xi ] = Li ΦL + Ui (1 − ΦU ) + (ΦU − ΦL )βN xi + σi (φL − φU )
where Li = lower bound (0 in our case),
Ui = upper bound (+∞ in our case)
φL = φ[ 0−βσ x) ]
φU = 0
(3.10)
Chapter 3. Empirical Analysis
3.5.4
75
The Ordered Logit Model
The Ordered Logit model is based on the following specification:
yi∗
= β xi + εi ,
Yi
Yi
Yi
yi∗
εi ∼ N [0,
π2
]
3
= 0 iff
≤ µ0 ,
= j iff µj−1 < yi∗ ≤ µj ,
= J iff yi∗ > µJ−1 ,
(3.11)
(3.12)
where j = 0, 1, 2, ..., J and J = 6 for structure one23
The probabilities for each observation are calculated as follows24 :
P [Yi = j] = f [µj − β xi ] − f [µj−1 − β xi ]
where f
with µ−1
µ0
µJ
(3.13)
= density function of logistic distribution
= −∞
=0
= +∞
The particular algorithm for estimating the threshold parameters of the
Ordered Logit in LIMDEP did not produce predicted choices that are evenly
spread over risk classes, as in the sample. To correct for this unsatisfactory
result, the threshold parameters were manually fixed to equal the appropriate
coding of the risk class. For example µ1 =2, ... , µJ−1 = J. Imposing fixed
values yielded significantly better classification tables for the Ordered in all
settings.
The Ordered Logit model can be applied since the respondents express a
preference in the form of an ordinal ranking with regard to risk and return.
23 J varies among the structures depicted in Figure D.1. J = 6 for structure 1, J=5 for
structure 2b, J=4 for structure 3c.
24 The algorithm used to obtain the maximum likelihood estimates is DFP (by Davidson,
Fletcher and Powell, see Fletcher (1980)). Starting values are obtained by OLS. This
initial regression is based on the dichotomy formed by using the binary indicator 1, as if
a univariate probit model applied. For grouped data, p+ and p0 = 1 − p+ provide the
dichotomy, and minimum chi-squared estimates are obtained. The constant term and the
values of the thresholds are estimated by using the cell frequencies under the assumption
that all of the slopes are zero. The real line is segmented in such a way that the logistic
probabilities corresponding to this partition match the sample cell frequencies, cf. Greene
(1998).
76
3.5. Characterization of econometric models
Therefore, “[...] the values that y takes correspond to a partition of the real
line, whereas in the unordered model they correspond either to a nonsuccessive
partition of the real line or to a partition of a higher-dimensional Euclidean
space.”25
Amemiya (1985) urges caution in using an ordered model “because if the
true model is unordered, an ordered model can lead to serious biases in the
estimation of the probabilities. The cost of using an unordered model when
the true model is ordered is a loss of efficiency rather than consistency.”
According to the underlying financial theory, however, the employment
of an Ordered model seems more obvious than the use of any discrete nonordered model (MNL, CLM), as it preserves the linearity of the risk-returnrelationship. We will discuss this point in more detail when evaluating the
various models and their performance.
The use of a discrete regression model, of course, has its disadvantages.
The continuous dependent variable - the stock ratio - is transformed into a
discrete variable representing a fund that covers a certain interval of the CML:
continuous: yi
= β0 + β1 x1i + ... + ε
discrete: Yi
= β0 + β1 x1i + ... + ν
Yi
=
Yi
= k iff k + 1 >
(3.14)
0 if yi = 0
yi
≥ k for k = 1,2,3,4,5
0.2
By discretizing the dependent variable in this way, an additional error
(ν − ε) is added to the error term of the continuous model:
ν = ε + [yi − (0.1 + (Yi − 1) · 0.2)]
(3.15)
Thus, for the precision of estimation it is obviously more advantageous
and more efficient to use a regression model that conserves the information
contained in the continuous variable if possible. The Tobit model that was
presented earlier certainly has that feature. It conserves both the linearity of
risk as well as the significance of the unit distance.
Even though the Ordered Logit model does not retain the information of
the unit distance, it has one important advantage over the Multinomial, the
25 Amemiya
(1985), p.292
Chapter 3. Empirical Analysis
77
Conditional and the Nested Logit model: It conserves the ordinal ranking of
the dependent variable. The linearity of risk is not lost in the Ordered Logit,
while the other mentioned models assume diverse alternatives that cannot
easily be ordered in one dimension.
3.5.5
The Binomial- and Multinomial Logit Model
The multinomial logit (MNL) model’s application26 to economic consumer
theory is based on the random utility approach formalized by Manski (1977).
It states that the individual always selects the alternative with the highest
utility. These utilities, however, are not known with certainty and thus treated
as random variables. Assuming iid random utilities yields a simple scalable
model of choice probabilities. Appendix B illustrates how the MNL model
can be derived from utility maximization with an error term that exhibits an
extreme value distribution.
In the MNL model the independent variables are individual specific characteristics, such as credit behavior, education, savings horizon etc. that explain
the variability of tastes across the portion of the population to which our
model of investment behavior applies. The vector of socio-economic characteristics is the same for all choices.
In contrast to the Conditional Logit or Discrete Choice Model27 where attributes of the choices j also enter the analysis, only characteristics xi of the
individual i are considered in the Multinomial Logit model. The MNL is typically employed for individual data in which the x variables are characteristics
of the observed individuals, not the choices.
While the independent factors are the same for all alternatives, the characteristics xi of the individual are assumed to be the same for all choices j
both in the MNL and the CLM. The set of alternatives and choice characteristics can differ for each observation only in the CLM. That is why the choice
subscript on x is dropped in the MNL formula below.
Both the MNL and the CLM were originally derived as special cases of
a general model of utility maximization where individuals i are assumed to
26 Its
original formulation is due to Luce (1959).
originally referred to this model as Conditional Logit (CLM), see McFadden
(1973). However, both terms - Conditional Logit (CLM) and Discrete Choice Model (DCM)
- are used synonymously here.
27 McFadden
78
3.5. Characterization of econometric models
have preferences defined over a set of alternatives j:
U (ij)
Observed Y
xik + εj
= β0 + βjk
=
choice j if U (j) > U (k) ∀ j...k
(3.16)
(3.17)
where i = 1, 2, ...n ; j = 1, 2, ..., J and k = 1, 2, ..., K.
The disturbances or individual heterogeneity terms εj are assumed to be
iid with extreme value distribution exp[−exp(−ε)].28
The choice probabilities of the MNL can thus be defined as:
P (Yi = j)
=
exp(β0 + βj xi )
J
exp(β0 + βj xi )
(3.18)
j=0
where i = 1, 2, ...n and j = 1, 2, ..., J
Both the MNL and the CLM, characterized hereafter, share one restrictive property that has come to be known as the ‘independence from irrelevant
alternatives’ (iia). Amemiya (1985) notes that the MNL “implies that the
alternatives are dissimilar”, as it calculates the relative probabilities between
a pair of alternatives (the category j and the base category 0) ignoring the
other alternatives. If two alternatives are very similar, their error terms can
no longer be assumed to be independent.29 Consequently, the relative probabilities are not independent of each either and can no longer be calculated
pairwise without yielding biased results.30
3.5.6
The Conditional Logit Model (CLM)
In contrast to the MNL model the CLM analyzes primarily the attributes of
the choices rather than the characteristics of the individual. These choice attributes typically have a factual individual specific nature, but they can also
28 McFadden proved that the MNL is derived from Utility Maximization iff (ε ) are inj
dependent and follow an extreme value or log Weibull distribution (EVD). For a proof, see
Appendix B or Amemiya (1985), p.297, Equation (9.3.42)
29 See also Greene (1993).
30 The Hausman test is similar to a Likelihood Ratio test. The unrestricted model’s
loglikelihood is compared with that of a restricted model that comprises a smaller set of
choices. If restricting the choice set leads to a singularity, Hausman and McFadden (1984)
suggest to reduce the number of regressors as well.
Chapter 3. Empirical Analysis
79
have a subjective ‘coloring’. For example, in the models analyzing transport
choice variables such as the individual travel time and travel costs of alternative travel modes are factual figures, objectively assessable. The comfort,
prestige and safety of a transport mode, on the other hand, are variables that
possess a much more subjective element.
Nevertheless it is possible to include socio-economic characteristics in the
model. Apart from this, the two models (MNL, CLM) are very much alike31
Unfortunately, the SCF does not include attributes of the investment
choices. Also, the number of choice attributes is limited from the perspective
of financial theory. There are basically only the return moments that characterize different investment choices. These return moments could be estimated
empirically and were then added into the data set. The main consideration
in this context concerned the selection of the correct set of factors. It had to
be a set that equally applied to all investors.
The most obvious choice would be the incorporation of the moments of
the return distribution of each investment class, as they are the constituent
parts of all asset pricing models. Estimates for the first three moments - mean
return, variance and skewness - can be obtained empirically from the S&P500
total return index for the last 25 years (1975-2000).32 These moment estimates
were then included as attributes and independent variables interacting with
alternative specific constants as well as with socio-economic characteristics.
Thus every individual in the sample is assumed to choose from the same choice
set and all alternatives are assumed to be available to all investors.
Like the MNL model, the CLM can be expressed by utility U of choice j
for individual i:
U (ij)
= β0 + βj xi + εj
(3.19)
where i = 1, 2, ...n and j = 1, 2, ..., J.
The probability that the investor chooses alternative j is
Observed Y
31 The
=
choice j if U (j) > U (k) ∀ j = k
(3.20)
random, individual specific terms are assumed to be iid, each with an extreme
value distribution (Gumbel). Under this condition the CLM can be derived from utility
maximization just as the MNL model before, cf. Domencich and McFadden (1975), ch. 4
and 5.
32 Estimates for the Swiss Market were obtained from the MSCI Switzerland Total Return
Index in Appendix I.
80
3.5. Characterization of econometric models
P (Yi = j) =
exp(βj xi )
J
exp(βj xi )
j=0
Apart from the independent variables that provide the choices attributes
(return moments), socio-economic variables can be included in the model.
They cannot simply be incorporated in the same way as the choices’ attributes, as the discrete choice probabilities are homogeneous of degree zero
in the parameters. Attributes which are the same for all outcomes for every
single individual thus drop out of the probability model. The socio-economic
characteristics would clearly qualify as such attributes as they stay the same
for all risk categories. They can only be integrated by using the equivalent of
dummy variable interaction terms and are thus expanded by interacting with
choice specific binary variables.
The result of this kind of model is a large set of fixed parameters with
few variables that account for variations in taste. Thus, the model was not
considered in the empirical analysis.
3.5.7
The Nested Logit Model
Even though the Nested Logit model (NLM) is not used for estimation in the
empirical analysis of this study, it will be briefly characterized here for the sake
of completeness. Its application is well conceivable if one wishes to examine
the characteristics of the different investment choices rather than those of
the investor. The availability of data on the investments’ characteristics is a
necessary prerequisite for the applicability of the Nested Logit model. While
it is possible to include factors that portray investor characteristics, these take
the role of alternative specific constants, as for a single observation they do
not vary across choices.
The study refrained from applying the NLM, as according to portfolio theory, investment choices are completely characterized by the moments of their
return distribution - mean, variance, skewness. However, these attributes are
objective in character and lack the subjective judgement of the investor. This
subjective judgement, however, is exactly what the Conditional Logit and
the Nested Logit mean to measure. Utilizing return moments as independent
variables for explaining risky choice in these models would come down to a
Chapter 3. Empirical Analysis
81
tautology.
Nevertheless, there are other independent factors imaginable representing
choice attributes that could be employed: the ethical purpose of a fund for
example, its brand name or the name of the fundmanager or the interest in a
specific industry.
The choice among different investment alternatives modeled above in the
MNL model and CLM may be viewed as taking place at more than one level.
This so-called ‘hierarchical’ choice is comfortably handled by a nested logit
model which is essentially an extension of the CLM. In contrast to the latter, the NLM, however, is not subject to the ‘independence from irrelevant
alternatives’ (iia).
The NLM’s Likelihood-function is given in Appendix C. Provided the data
is appropriately formatted, the NLM can be estimated using NLOGIT2.033
3.6
Goodness of fit and hypotheses testing
3.6.1
Classification Tables and Error Distance
One measure for the quality and fit of the model is the number of correct
predictions it makes. These can be read from classification tables that list
the predicted choices of the model34 together with the actual choices of the
investors (observations). An example is the following classification table from
the TOBIT model in setting 1 for the SCF 1998:
33 By
Econometric Software, Inc.
predicted choices had to be calculated separately from the estimation of the factor
coefficients in LIMDEP. The estimates were imported into MSExcel where the predicted
choices were calculated. A Macro finally assigned each observation into the appropriate cell
of the classification table.
34 These
82
3.6. Goodness of fit and hypotheses testing
Setting 1 - Tobit, SCF1998
Classification table
Predicted Choice
0
1
2
3
4
183
71
48
0
0
455
439
1051
234
60
4
11
134
136
56
4
5
116
110
90
2
5
115
143
93
2
12
90
122
94
5
11
112
124
101
655
554
1666
869
494
Percent correct: 22.49%
Error Distance: 8’967’647, ln(ED): 16.01
Actual
0
1
2
3
4
5
6
5
0
7
8
15
6
9
22
67
6
0
0
0
0
0
0
0
0
302
2246
349
340
364
329
375
4305
The last column lists the number of actual observations per category, while the
last row depicts the number of predicted observations. In the diagonal there
are the number of observations for which actual and predicted values of Y
coincide. The sum of these values divided by the total number of observations
gives the ratio of correct assignments (labeled ‘Percent correct’). That
ratio in itself, though, is not a sufficient measure for comparison of different
regressions, as it lacks to account for the degree of misassignment. To compare
regression results, both the distance and the number of wrong assignments
need to be considered. Both deviations are weighted accordingly with the
Error Distance:
J J
[(1 + (r − c)3 )(Prc − Trc )2 ]
(3.21)
c=0 r=0
where J = number of categories - 1,
r = row in classification table, r = 0, 1 ..., J.
c = column in classification table, c = 0, 1 ..., J.
Trc = total number of actual observations for cell in row r in column c,
Prc = total number of predicted observations for cell in row r in column c.
As the resulting number for the Error Distance is rather large, its log was
taken in order to ease the comparison of the models (see the previous table
for example).
Chapter 3. Empirical Analysis
3.6.2
83
Akaike Information Criterion (AIC)
When choosing one regression model out of many competing models, we
can use the Likelihood ratio test LRT > d where d is determined so that
P [LRT > d|L(α)] = c with c a certain prescribed constant such as 5%. Such
proceeding is unfortunately time consuming and often even inconclusive35 .
Akaike36 formulated a loss function on the basis of the sum of squares corrected for by the number of regressors. As the loss function is to be minimized
among competing models, the model with the lowest AIC is considered to yield
the best results37 .
AIC = −
2
2K
log L(α̂) +
T
T
(3.22)
where T = number of observations,
α̂ = maximum likelihood estimators
K = number of independent variables
3.6.3
Likelihood ratio tests
The likelihood ratio (LR) test in discrete choice models is used in the same
way that the F-test is used in linear regression models for joint tests of several
parameters.38
LIMDEP’s standard output for each model comprises a LR test of the
hypothesis that all coefficients are zero. The results for these tests were not
included in this study, as the null hypothesis could always be rejected at a
35 Cf.
Amemiya (1985), p.146
(1973)
37 For full details cf. Amemiya (1980)
38 Of the three classical test statistics (LR, Lagrange multiplier and Wald test) that are
asymptotically equivalent, the LR test is conceptually the simplest. In general it denotes
twice the difference between the restricted θ̂ and unrestricted θ̃ values of the loglikelihood
function
36 Akaike
2(logL(θ̂) − logL(θ̃))
θ̂ refers to the unrestricted ML estimate and θ̃ denotes the ML estimate subject to r
restrictions. See Davidson and Mackinnon (1993), 275.
84
3.6. Goodness of fit and hypotheses testing
very low level of significance39 (5% and 1% respectively).
It is more informative to test the null hypothesis that only some coefficients
are zero or that all the coefficients except for the alternative-specific constants
are zero.40 The test statistic for the latter is
−2(logL(c) − logL(β))
(3.23)
with K − J + 1 degrees of freedom, where J is the number of alternatives in
the choice set (7 in setting 1) and logL(c) is the log-Likelihood of a model
with only constants. LogL(c) can be obtained by estimating a model with
J − 1 alternative-specific constants41 or from
logL(c) =
J
i=1
Ni ln
Ni
N
(3.24)
where Ni is the number of observations selecting alternative i and N is the
total sample size.
For the SCF samples LR tests were carried out both for single and for
multiple factor coefficients. LR tests simply compare the Loglikelihoods of
the restricted and the unrestricted model: The test statistic (Eq. 3.23) is
distributed as χ2 (df ) under the null. For a single restriction the critical value
CV (df = 1) at the 5% level is 3.841 because the probability of obtaining
a random drawing from a χ2 -distribution that is greater than 3.841 is 5%.
Thus, if the test statistic LR > CV , the null (that the single factor coefficient
equals zero) can be rejected at the 5% level.
The tables in Appendix E.3 summarize several LR tests for the MNL and
Ordered Logit in different settings of the SCF 1998. For the MNL the LR
tests gave the following results:
• In setting 1, SCF 1998, the null that the four least significant factor
coefficients jointly equal zero can not be refuted at the 5% level in the
MNL.
• In setting 2a and 2b, SCF 1998, the null that the nine least significant
factor coefficients jointly equal zero can not be refuted at the 5% level.
39 Also
called the ‘size of test’.
test was refuted for the MNL in all settings at the 5% level.
41 Cf. Ben-Akiva and Lerman (1985).
40 This
Chapter 3. Empirical Analysis
85
• In setting 3b, SCF 1998, there are 12 factor coefficients for which the
null cannot be refuted at the 5% level.
• Finally, in setting 3c, SCF 1998, there are altogether 18 factor coefficients for which the null can not be refuted at the 5% level. The model
can be estimated with only 6 variables and one alternative specific constant.
3.7
3.7.1
Results of regressions for Setting 1 - “All
Categories”
Coefficients
SCF1998 In-Sample-Estimation
For the OLS (see Table E.1 on page 180) 17 out of 25 factors proved significant
at the 5% level. 8 coefficients did not have the expected sign.42
In the Tobit model (see Table E.2 on page 182) 16 out of 25 factors were
significant at the 5% level. Only 5 coefficients did not have the expected sign.
Even though the AIC of the OLS is lower than the one of the Tobit, the
predictive power of the Tobit as given by the classification table proves to be
better than that of the OLS. The percentage of correctly classified choices is
higher in the Tobit and the Error Distance is slightly lower than in the OLS.
While the OLS fails to assign any choices to the two highest risk classes, the
Tobit at least classifies a few observations to category 5.
In the Ordered Logit model (see Table E.3 on page 184) 23 out of 25
factors are significant at the 5% level, while 6 coefficients do not have the
expected sign. Even though 42% of all observations are correctly classified by
the Ordered, especially the majority of cases in the low risk classes, it does
not spread the predicted choices as evenly over the higher risk classes as the
MNL.
In the MNL model (see Table E.4 on page 186) the logits of the different risk classes had on average at least 14 significant factors. The number
42 The expected signs are discussed in Appendix F, Table F.3 on Page 258. In the tables
wrong coefficient signs are encircled.
86
3.7. Results of regressions for Setting 1 - “All Categories”
of coefficients with a wrong sign varied between 7 and 9. More important
than the sign, however, is the change in magnitude of each coefficient among
the different risk classes. Non-monotone increasing coefficients reveal irregularities among different risk levels and can thus unveil non-linear relations
between a factor and the risk level. The course of the coefficients over classes
of increasing risk can be comprehended best through the graph in Figure 3.2.
For setting 1 of the in-sample estimation, the MNL clearly produces the
best result in terms of correctly assigned choices and lowest Error Distance.
Chapter 3. Empirical Analysis
Figure 3.2: The course of the coefficients over classes of increasing risk for setting 1, in-sample estimation of the SCF 1998 with
a MNL model. The dependent variable Y represents the level
of risk measured as return volatility; people for whom Y =0 do
not have any financial assets; Y =1 depicts pure fixed income investors; Y =2 stands for a stock ratio of 1-20%, Y =3 ⇒ 21-40%,
... Y =6 ⇒ 81-100%.
87
88
3.7. Results of regressions for Setting 1 - “All Categories”
SCF1995 in 1998 data Out-of-Sample-Estimation
For OLS see Table E.21 on page 216, for the Tobit model see Table E.22
on page 218, for Ordered Logit see Table E.23 on page 220, for MNL see
Table E.24 on page 222.
Not all the coefficient estimates are significantly different from zero at
the 5% or 1% levels. When LR tests were carried out for the non-significant
factors in the discrete choice models, the null could not always be refuted (see
Subsection 3.6.3).
3.7.2
Results of Classification Tables
Empirical Analysis of Structure1: Comparison of Models
Dataset: SCF 1998, In-sample Estimation
Observations = 4305, Parameters = 25, Deg.Fr.= 4274
Model
OLS
Tobit
Ordered
MNL
Log L
-1679.83
-2363.74
-7833.00
-5350.63
AIC
0.792
1.110
3.650
2.500
Percent
20.74%
22.49%
41.79%
54.94%
Error Dist
9’562’127
8’967’647
4’265’498
11’219’224
Ln ED
16.07
16.01
15.27
16.23
Empirical Analysis of Structure1: Comparison of Models
Out-of-sample-Estimation SCF1995 in 1998 data
Observations = 4305, Parameters = 25, Deg.Fr.= 4274
Model
OLS
Tobit
Ordered
MNL
Log L
-1558.57
-2187.82
-7024.75
-5024.36
AIC
0.737
1.029
3.280
2.350
Percent
17.19%
16.68%
34.05%
43.46%
Error Dist
11’864’863
16’002’731
7’455’722
19’807’854
Ln ED
16.29
16.59
15.82
16.80
Chapter 3. Empirical Analysis
3.8
3.8.1
89
Results of regressions for Setting 2 - Twostep estimation
Setting 2a - “Assetholder or Non-asset holder”
Coefficients
SCF1998, in-sample estimation: For the WLS (see Table E.5 on page 189)
21 out of 25 factors proved significant at the 5% level. 11 coefficients did not
have the expected sign.43
In the Tobit model (see Table E.6 on page 191) 16 out of 25 factors were
significant at the 5% level. Only 8 coefficients did not have the expected sign.
As in setting 1, the AIC of the OLS is lower than the one of the Tobit,
however, the predictive power of the Tobit as given by the classification table
proves to be better than that of the OLS. The percentage of correctly classified
choices is higher in the Tobit and the Error Distance is slightly lower than in
the OLS.
In the Binomial Logit model (see Table E.7 on page 193) 18 out of
25 factors are significant at the 5% level and 9 coefficients do not have the
expected sign. Even though the percentage for the number of correctly predicted choices are higher than in the Tobit, the Binomial Logit assigns too
few observations to the category 0.
For setting 2a of the in-sample estimation, the Tobit thus produces the
best results in terms of correctly assigned choices and lowest Error Distance.
SCF1995 in 1998 data Out-of-Sample-Estimation: for OLS see Table E.25 on
page 225, for MNL see Table E.26 on page 227.
43 The expected signs are discussed in Appendix F, Table F.3 on Page 258. In the tables
wrong coefficient signs are encircled.
90
3.8. Results of regressions for Setting 2 - Two-step estimation
Figure 3.3: The course of the coefficients over classes of increasing
risk for setting 2b, in-sample estimation of the SCF 1998 with a MNL
model. The dependent variable Y represents the level of risk measured
as return volatility; people for whom Y =0 are pure fixed income investors; Y =1 stands for a stock ratio of 1-20%, Y =2 ⇒ 21-40%, ...
Y =5 ⇒ 81-100%.
Chapter 3. Empirical Analysis
91
Figure 3.4: The course of the coefficients over classes of increasing
risk for setting 2b, in-sample estimation of the SCF 1998 with a MNL
model. The dependent variable Y represents the level of risk measured
as return volatility; people for whom Y =0 are pure fixed income investors; Y =1 stands for a stock ratio of 1-20%, Y =2 ⇒ 21-40%, ...
Y =5 ⇒ 81-100%.
Results of Classification Tables
Empirical Analysis of Structure2a: Comparison of Models
Dataset: SCF 1998, In-sample Estimation
Observations = 4305, Parameters = 25, Deg.Fr.= 4274
Model
WLS
Tobit
BNL
Log L
183.61
-618.63
-678.87
AIC
-0.074
0.299
0.327
Percent
90.94%
92.89%
93.59%
Error Dist
264’654
165’030
168’414
Ln ED
12.49
12.01
12.03
92
3.8. Results of regressions for Setting 2 - Two-step estimation
Empirical Analysis of Structure2a: Comparison of Models
Out-of-sample-Estimation SCF1995 in 1998 data
Observations = 4305, Parameters = 25, Deg.Fr.= 4274
Model
WLS
BNL
3.8.2
Log L
-93.26
-737.67
AIC
0.055
0.355
Percent
92.31%
92.54%
Error Dist
231’123
231’855
Ln ED
12.35
12.35
Setting 2b - “Assetholders only”
Coefficients
SCF1998, In-Sample Estimation: For the OLS (see Table E.9 on page 195)
only 10 out of 25 factors proved significant at the 5% level. 5 coefficients did
not have the expected sign.44
In the Ordered Logit model (see Table E.11 on page 197) 22 out of 25
factors are significant at the 5% level and only 6 coefficients do not have the
expected sign. Even though the bulk of cases for category 0 are correctly classified, the Ordered Logit fails to assign a satisfactory number of observations
to the highest risk classes. The MNL model (see Table E.12 on page 199)
however, not only classifies the bulk of cases in category 0 correctly, it also
evenly spreads predicted choices over the high risk classes 4 and 5.
For setting 2b of the in-sample estimation, the MNL yields the best result
in terms of correctly assigned choices and lowest Error Distance. The MNL’s
advantage becomes evident from the Figures in Table 3.3: No independent
factor in the MNL is strictly monotone increasing or decreasing in the risk
classes depicted on the x-axis. In contrast to all other models, the MNL can
imitate these non-linearities between independent and dependent variables to
give superior results.
SCF1995 in 1998 data Out-of-Sample-Estimation: for OLS see Table E.28
on page 229, for Ordered Logit see Table E.30 on page 231, for MNL see
Table E.31 on page 233.
44 The expected signs are discussed in Appendix F, Table F.3 on Page 258. In the tables
wrong coefficient signs are encircled.
Chapter 3. Empirical Analysis
93
Results of Classification Tables
Empirical Analysis of Structure2b: Comparison of Models
Dataset: SCF 1998, In-sample Estimation
Observations = 4003, Parameters = 25, Deg.Fr.= 3978
Model
OLS
Ordered
MNL
Log L
-547.28
-8846.00
-4672.59
AIC
0.286
4.43
2.347
Percent
23.06%
38.67%
58.41%
Error Dist
10’113’838
6’395’874
10’996’615
Ln ED
16.13
15.67
16.21
Empirical Analysis of Structure2b: Comparison of Models
Out-of-sample-Estimation SCF1995 in 1998 data
Observations = 4003, Parameters = 25, Deg.Fr.= 3978
Model
OLS
Ordered
MNL
3.9
Log L
-139.87
-7672.27
-4287
AIC
0.084
3.908
2.189
Percent
22.16%
40.47%
51.04%
Error Dist
15’259’122
10’630’777
26’479’635
Ln ED
16.54
16.18
17.09
Results of regressions for setting 3 - Threestep estimation
The following analysis follows a three-step estimation of risk-taking. It was
depicted in Figure D.1 as the ‘third setting’. The assignment of respondents
for our data sample SCF 1995 was divided into three steps.
In the first step - labeled 3a in Figure D.1 - we distinguished only between
people with and without assets (coded 0 and 1 respectively). This step was
already evaluated in setting 2a.
In the second step - called 3b in Figure D.1 - we excluded the respondents
without assets and distinguished only between non-stockholders (coded as 0)
and stockholders (coded as 1).
In the third step - marked 3c in Figure D.1 - we excluded both people
without assets and people without stocks. The remaining respondents were
analyzed for their different stock ratios.
Following are the tables that summarize the regression results.
94
3.9.1
3.9. Results of regressions for setting 3 - Three-step estimation
Setting 3b - “Stock- or Non-stock holder”
Coefficients
SCF1998, in-sample estimation: For the WLS (see Table E.13 on page 202)
16 out of 25 factors proved significant at the 5% level. 6 coefficients did not
have the expected sign.45
In the Tobit model (see Table E.14 on page 204) 16 out of 25 factors were
significant at the 5% level. Only 5 coefficients did not have the expected sign.
In contrast to setting 1, the AIC of the OLS is lower than the one of the
Tobit and its predictive power is better. The percentage of correctly classified
choices is higher in the OLS and the Error Distance is significantly lower than
in the Tobit.
The Binomial Logit model (see Table E.7 on page 193) proves to be
slightly better than the OLS. Its Error Distance is lower and the percentage
of correctly assigned choices is higher, although only 9 out of 25 factors are
significant at the 5% level. 7 coefficients do not have the expected sign.
For setting 3b of the in-sample estimation, the MNL thus produces the
best result in terms of correctly assigned choices and lowest Error Distance.
SCF1995 in 1998 data Out-of-Sample-Estimation: for OLS see Table E.32
on page 236, for MNL see Table E.33 on page 238.
Results of Classification Tables
Empirical Analysis of Structure3b: Comparison of Models
Dataset: SCF 1998, In-sample Estimation
Observations = 4003, Parameters = 25, Deg.Fr.= 3978
Model
WLS
Tobit
BNL
Log L
-1987.91
-3173.2
-1906.34
AIC
1.006
1.598
0.965
Percent
74.62%
71.15%
77.44%
Error Dist
1’575’318
3’245’919
1’231’107
Ln ED
14.27
14.99
14.02
45 The expected signs are discussed in Appendix F, Table F.3 on Page 258. In the tables
wrong coefficient signs are encircled.
Chapter 3. Empirical Analysis
95
Empirical Analysis of Structure3b: Comparison of Models
Out-of-sample-Estimation SCF1995 in 1998 data
Observations = 4003, Parameters = 25, Deg.Fr.= 3978
Model
WLS
BNL
3.9.2
Log L
-2029.91
-1941.21
AIC
1.043
0.998
Percent
64.20%
65.88%
Error Dist
4’687’071
2’876’910
Ln ED
15.36
14.87
Setting 3c - “Stockholders only”
Coefficients
For the OLS (see Table E.17 on page 208) as few as 3 factors out of 25 proved
significant at the 5% level. 6 coefficients did not have the expected sign.46
Even though the AIC of the OLS was low, its predictive power as given by
the classification table is so poor that it shouldn’t be considered at all for
distinguishing between risky choices in this setting.
In the Ordered Logit model (see Table E.19 on page 210) 7 out of 25
factors are significant at the 5% level, while 7 coefficients do not have the
expected sign. Similar to the result of the OLS, the Ordered Logit produces a
very poor classification table in this setting. Both, the OLS and the Ordered
Logit spread the observations only over two classes.
The MNL model (see Table E.20 on page 212) does a much better job in
this respect. Though there are very few significant factors, the MNL spreads
all observations evenly over all risk categories. It also produces the highest
percentage of correctly classified cases and the lowest Error Distance.
For setting 3c of the in-sample estimation, the MNL clearly produces the
best result in terms of correctly assigned choices and lowest Error Distance.
SCF1995 in 1998 data Out-of-Sample-Estimation: for OLS see Table E.35
on page 240, for Ordered Logit see Table E.37 on page 242, for MNL see
Table E.38 on page 244.
46 The expected signs are discussed in Appendix F, Table F.3 on Page 258. In the tables
wrong coefficient signs are encircled.
96
3.10. Conclusion of the two-moment setting
Results of Classification Tables
Empirical Analysis of Structure3c: Comparison of Models
Dataset: SCF 1998, In-sample Estimation
Observations = 1758, Parameters = 25, Deg.Fr.= 1733
Model
OLS
Ordered
MNL
Log L
-330.96
-3899.90
-2766.67
AIC
0.405
3.219
3.178
Percent
21.23%
21.46%
27.95%
Error Dist
2’629’736
2’147’852
1’803’590
Ln ED
14.78
14.58
14.41
Empirical Analysis of Structure3c: Comparison of Models
Out-of-sample-Estimation SCF1995 in 1998 data
Observations = 1758, Parameters = 25, Deg.Fr.= 1733
Model
OLS
Ordered
MNL
Log L
-284.49
-3516.39
-2345.46
AIC
0.399
4.570
3.059
Percent
20.20%
20.94%
21.74%
Error Dist
2’231’527
4’675’546
5’743’122
Ln ED
14.62
15.36
15.56
3.10
Conclusion of the two-moment setting
3.10.1
The factors and their explanative power
The predictive power of the OLS and the Ordered Logit significantly decreased
over the settings 1 - 3, as non-linearities in the independent variables became
more important to predict risky choices. The MNL performed consistently
well over all settings and should be the model of choice for determining risk
preferences in the presented framework.
In general when more than two risk classes are considered the OLS does
not perform as well as the MNL. This can be attributed to the fact that the
MNL considers the risk classes in a non-linear fashion. The logit of every
risk class has its own coefficient for each factor.47 In this way, the model can
account for non-monotone decreasing or increasing relations between Y and
xik . In fact, the estimation shows that the factor combinations do not directly
47 The MNL employs the factor set for every risk class except for the base category. It
also includes a constant term for every category’s logit. In this way, it uses j × k factors as
compared to only k factors used in the OLS.
Chapter 3. Empirical Analysis
97
and linearly translate into Yj , the risk classes. There are kinks in the ordinal
structure of explanative variables that are best visible in the MNL regression
tables.
3.10.2
Performance of the econometric models
Aim of this empirical analysis was the explanation of the choice for investment risk by investor characteristics and a comparison of models in terms of
predictive power.
The limited number of independent factors could expectedly not explain
all of the variation in investment choice. However, given the large sample
size and the diversity of tastes, intents and preferences, the MNL was able to
weight the factors in such a way that the choices were evenly spread over all
categories. The predictive power was also quite good. Most coefficients had
the expected signs.
The predictive power varied significantly between the SCF 1995 and 1998.
The 1998 data produced a much better in-sample fit of the actual choice.
Out-of-sample prediction with the 1995 estimates in the 1998 data gave classifications very similar to those by the 1998 in-sample estimation.
The predictive power also varied greatly among the different models. For
the in-sample estimation, the MNL tended to perform best, followed by the
Ordered Logit, the Tobit and the OLS. This order was maintained in the outof-sample prediction, with the sole exception of setting 2b where assignment
was very poor in all models.
For the empirical analysis the number of risk classes chosen was arbitrary.
There were seven risk classes in setting 1. Six classes were derived from the
division of the CML and one class consisted of people without financial assets.
Important information is lost when the continuous stock ratio is transformed
into an ordinal variable. This is a plain disadvantage of working with discrete
data.48 A higher number of classes would of course be also conceivable, though
that would inevitably change the predictive power and the assignment precision of the models. By increasing the number of classes, more logit equations
48 The
boundaries of the classes are determined arbitrarily. An observation with a stock
ratio of 21% is assigned to class 1 in setting 3c while a ratio of 19% is assigned to class 0. The
true latent risk segments may thus become blurred and similarities between observations
of different classes at each boundary cannot be considered in the estimation.
98
3.10. Conclusion of the two-moment setting
are created by the MNL. This leads to a decrease in the maximum probability
of the predicted class. A higher number of factors per logit can be expected
to be less significant implying a lower predictive power. The goodness of fit
of the model will decrease at the same time.
3.10.3
Transferability of results
The aging of the population is a development all industrialized countries share
- the U.S. as well as Switzerland and Germany. The proportion of families
headed by individuals between 45 and 54 years has risen by around 2 percentage points between 1995 to 1998.49 The financial decisions of families with
heads in this group are most likely dominated by the cost of college education
for their children and the need to save for their own retirement. The predominance of these two factors are specific to the U.S., as in Europe higher
education is much less costly and the public pension system still provides
more security for retired people.
On average, private pensions of US investors are to a higher degree invested
in stocks than private investors in Germany or Switzerland. Also, it is well
known that investors in the U.S. hold on average more stocks50 than European
investors. Furthermore, it can only be speculated whether the investment
mentality far West is more trading-orientated than in Europe. After all, speculation and short-term buying and selling involves serious tax-disadvantages
in the U.S. However, there are studies indicating that the average holding
period of risky assets is higher in Europe.51
49 See
Bertaut (1998) and Kennickell, Starr-McCluer, and Sunden (1997).
average proportion of stocks in US private investor’s portfolio (market value) is
higher than that of European investors. Also, when counting direct and indirect stockholdings, more than 45% of the US workforce holds stocks as compared to an average 30% in
Europe. Surprisingly, the spread of direct stock-ownership is greater in Switzerland than
in the U.S. or in Germany, see Coccart and Volkart (2001).
51 Recent behavior does not confirm the common view that investors are long term holders.
According to Sanford Bernstein, a money management firm, in the U.S. the average holding
period 10 years ago for a NASDAQ stock was 751 days. At the time of this study it was
181 days. ‘Equity buyers’ held a stock for an average of 8 months in 1999 as compared with
an average holding period of 2 years in 1989. Those who purchased the 50 most heavily
traded NASDAQ stocks kept their shares for an average of only three weeks. A decade ago,
the average mutual fund was held for 11 years. Today, the lack of patience and perspective
has reduced that to a 4 year average holding period, cf. Lee (2001).
50 The
Chapter 3. Empirical Analysis
99
One aspect that might lightly hamper the transferability of results is the
difference in credit-behavior and private debt, as credit-taking is generally
more socially and economically accepted in the U.S.
Overall, the results presented for U.S. data can be expected to have good
predictive power also for Switzerland and Germany, especially as financial
behavior of private investors worldwide becomes increasingly similar. Reforms
of the private pension systems in Europe52 , the shift of the responsibility for
retirement insurance from the public to the private sector, are indicators for
a growing commitment of private investors in the stock market.
The method shown can be easily transferred to fit investment behavior
across different countries. All that is necessary is a representative sample53
to calibrate the weights of each question in the questionnaire.
52 From 2002 onwards statutory payments in Germany are gradually replaced by private
voluntary retirement provision. The latter is financially supported by the state at 35-55%.
Keeping in mind that in 2000 alone, private investors allocated DM200bn. in equity assets,
this reform is bound to impact the German stockmarket.
53 ‘Small’ relates to a number of at least 500 observations in the sample.
100
3.10. Conclusion of the two-moment setting
Chapter 4
Joint estimation by
gambles and observed
stock ratio
This chapter will extend the previously presented econometric model by incorporating gambles in order to arrive at more robust estimation results.
The disadvantages of gambles have already been briefly discussed in Subsection 1.2.2. The value added of incorporating them lies in the fact that they
are complementing the factor model by bringing a more direct preference test
to the estimation procedure. The factor model focuses on data and characteristics that are a little more remote to the asset allocation decision, whereas
the gamble directly confronts the investor to choose among risk levels. The
choice, in combination with the current wealth level, yields an estimate for
the Pratt-Arrow measure of an individual investor’s risk-aversion. These estimates can be used together with the presented factor model to form a joint
estimation model for risk aversion.
As there is no SCF1995- and SCF1998-survey data available that relates to
the investors’ choice of gambles, the joint estimation method will be discussed
and portrayed theoretically.
101
102
4.1. Determining Two-Moment Risk Aversion by Gambles
4.1
Determining Two-Moment Risk Aversion
by Gambles
The derivation and the logic of the Pratt-Arrow measure of risk aversion
has been explained in Subsection 2.2.2. Building on Formula A.14 in Appendix A.1 the investor is asked to quantify a sure outcome, called certainty
equivalent CE, for a given symmetric gamble [x,p,y].1
As the main concern of the study is with return data rather than with
absolute amounts, gambles for ‘payoffs in return form’ are considered in the
following formulas.
In the picture below, the relative risk aversion for a two moment setting
is derived from a symmetric, tree-structured gamble:
Two-moment risk aversion
The investor I is given with his wealth W=200’000
Expected Annual Return ER=9%
Volatility σ2 =22%
+31%
262’000
✒
✒
0.5
0.5
200’000
200’000
❅ 0.5
❅ 0.5
❅
❅
❘ -13%
❅
❘ 174’000
❅
The investor declares his Certainty Equivalent CE = 5.00%
Apart from the probabilities for the outcomes, the expected return ER
and volatility σ2 of the 2-moment gamble are presented to the investor. He
then needs to name his Certainty Equivalent for that gamble as well as his
1 This amount, called certainty equivalent, makes the individual indifferent between playing the gamble or receiving the sure amount CE. In a more intuitive way, the certainty
equivalent CE can be understood as the value the investor assigns to the gamble presented
to him. If he does not approve of the gamble and its potential payoffs (x,y), he will name
a low certainty equivalent for it. The higher CE, the less risk averse the investor.
Chapter 4. Joint estimation by gambles and observed stock ratio
103
current wealth level.
The two-moment preference is determined through the Certainty Equivalent the investor i names. For reasons of transparency and comprehensibility
the gamble is portrayed by two illustrations which are complemented by the
moments of the underlying distribution. The first illustration gives the payoffs in return style, the second one in absolute amounts assuming the investor
considers allocating all of his wealth W .2
The Pratt/Arrow risk premium π is − 12 σ22 W UU . The strong assumption
that must be made is: risk premium π2 = ER−CE, where ER is the Expected
Return and CE stands for Certainty Equivalent. Solving for relative risk
aversion αr yields
αr = −W
U ER − CE
=2
U
σ22
(4.1)
Dividing the above result by wealth W gives the A/P measure of absolute
risk aversion αa .
Based on the gamble, Equation 4.1 can be rewritten to yield a regression
based relationship:
2
ER − CEi
σ22
= βjk
xik + εi
(4.2)
where β xi stands for the risk aversion measure for investor i. Different regression models can be used here to estimate the factor weights β. The model
format depends on the distribution assumption of ε. For a MNL model the
error term ε is assumed to be logistic distributed with mean 0 and a scale
parameter λ.3
The log-likelihood function for the above model can be expressed by
logL(β|xi ) =
ER − CEi
ln 2
−
β
x
i
σ22
i=1
I
(4.3)
2 Note: In order to avoid typical behavioral biases such as loss aversion, it would be
beneficial to portray a gamble with purely positive or negative payoffs in a survey.
3 For a normal logistic distribution λ = π 2 /3. For the standardized logistic distribution
√
λ = π/ 3.
104
4.2
4.2. Joint estimation of econometric choice and gamble
Joint estimation of econometric choice and
gamble
In Subsection 3.5.5 and Appendix B, the application of the discrete choice
model to the problem of portfolio selection was justified by the basic framework of utility maximization. The investor is assumed to choose that investment that maximizes his expected utility. A conceivable utility function for
his decision-making is:
Ui = µj −
αri 2
σ
2 j
(4.4)
µj and σj represent the expected return and volatility of investor i’s portfolio
choice j, αri stands for the investor i’s relative risk aversion. Replacing αri
by β xi yields the regression model:
Ui = µj −
β xi 2
σ
2 j
(4.5)
The weights β of the independent factors xi are to be estimated with the
models described in Section 3.5.4
When applying a MNL model to the problem of utility maximization5 the
choice probability for Equation 4.5 is
exp Ui
Pi (j) = I
i=1 exp Ui
(4.7)
where ji is the choice of investor i for one of the J portfolios.
The corresponding Log-Likelihood function is
logL(β|xi ) =
I
lnPi (ji )
(4.8)
i=1
4 An
alternative relationship solves for the relative risk aversion and can thus be regressed
directly:
µj − r f
β xi = αri =
(4.6)
yi · σj2
µj and σj are again expected return and volatility of investment choice j, rf is the riskfree
rate and yi is the investor’s current stock ratio.
5 For a derivation of the MNL from utility maximization see Appendix B.
Chapter 4. Joint estimation by gambles and observed stock ratio
105
The equations 4.8 and 4.3 can be estimated jointly. It is likely that the
estimates for risk aversion resulting from the two equations differ. A joint
estimation of the two models will increase the precision and efficiency in the
estimation of β. The joint Log-Likelihood function will be
logL(β, γ|xi ) =
I i=1
ER − CEi
−
γβ
x
(j
)
ln 2
+
lnP
i
i i
σg2
(4.9)
106
4.2. Joint estimation of econometric choice and gamble
Part II
Three-moment risk
preference
107
109
In this second part the main focus of the analysis lies on the investor’s preference structure when the third moment skewness is considered in addition
to the first two - expected return and variance. In the subsequent sections
skewness S will be defined as the non-normalized third central moment of the
gamble or the return distribution. This definition is to be distinguished from
the Fisher skewness which is normalized by the cubed volatility.
In the discrete case:
3
(xi − E(X)) · pi
(4.10)
S=
i
where xi stands for return i and its probability pi .
In the continuous case where simple lognormal returns are considered:
(4.11)
S = exp (σ 2 − 1) 2 + exp σ 2
where σ depicts the dispersion parameter of the normal density function corresponding with the lognormal.
Two lines of thought will be intertwined in this second part:
1. How to assess an investor’s three moment risk preference for the return
distribution’s moments and their trade-offs in particular. The certainty
equivalence method will be employed to determine these trade-offs which
indicate how much skewness is demanded by the investor per unit of
expected return or variance.
2. How to formally describe the portfolio’s distribution and its modification
by different option strategies. Adding options to the portfolio decreases
expected return, decreases shortfall risk and increases skewness. The
degree of these changes depends on the well-known parameters of the
Black-Scholes pricing formula. The strike price, however, is the main
driver for the change in the portfolio’s moment trade-offs. The higher
the strike price of puts, the higher the degree of protection and the more
costly the loss in expected return and the lower σ, µ and the higher is
the skewness.
These two lines of thought meet at the specific shape of the portfolio’s
return distribution demanded by the investor. The distribution can be sufficiently represented by its first three moments. Different moment combinations
110
that are of equal utility for the investor can be identified by so-called moment
trade-offs. The gambles define what trade-offs the investor demands and the
strike price of the option strategy determines how to implement these preferences.
The second part concludes with a brief theoretical instruction on how to
estimate three moment preference jointly with discrete choice models and
gambles.
Chapter 5
Shortcomings of
two-moment asset pricing
5.1
Critique of the mean-variance approach
Traditional mean-variance analysis has a number of shortcomings that will be
briefly pointed out in order to motivate a mean-variance-skewness approach.
While the approximation of a utility function by a quadratic in a certain
range is central to the Markowitz (1959) rational for mean and variance, normal distributions or other two-parameter families of probability distributions
were not part of his justification for mean and variance. It was Tobin (1958)
who offered two alternative ways of deriving mean and variance as the investor’s decision making criteria1 : (1) “the investor evaluates the future of
consols only in terms of some two-parameter family of probability distributions, such as the normal, or (2) the assumption that the utility function
1 The
third moment or skewness was not considered by Markowitz or Sharpe, Lintner
and Mossin for reasons of cost and inconvenience. Partly as a consequence, most work
on “nonstandard” portfolio optimization has remained purely theoretical in nature, confer
Mao (1970), as well as Hogan and Warren (1974). Models of portfolio optimization based
on the semivariance were developed by Chen and Park (1991) and Ouederni and Sullivan
(1991) to name only a few. Then as today is true that the use of variance produces the
same set of efficient portfolios as the use of semi-variance iff all distributions of returns are
symmetric or have the same degree of asymmetry.
111
112
5.1. Critique of the mean-variance approach
is quadratic2 , and returns do not exceed the point at which the quadratic
reaches its maximum”.
Thus the traditional mean-variance approach can be derived in two alternative ways:
Either it must be assumed that
• the investor has a quadratic utility function or that
• returns are normally distributed.
Unfortunately both assumptions are not only unrealistic, but prove insufficient and unsatisfactory in describing investment preferences of individuals.
5.1.1
Quadratic utility
The assumption of a quadratic utility function eliminates all moments of the
return distribution that are higher than the variance from the objective function of optimal asset allocation. It rules out the possibility that investors
could have any preference for skewness, e.g. an asymmetric return distribution. Any investment strategy aiming at modifying the skewness of the return
distribution - such as option strategies for example - cannot be examined with
the traditional mean-variance model.
The quadratic utility also implies increasing absolute risk aversion, a feature that is empirically most unrealistic. The quadratic utility has been called
fatuous by Pratt (1964) and Arrow (1971) for this reason. All polynomous
utility functions share this grave disadvantage, as U > 0 and U < 0 never
occur together.
Quadratic utility also displays satiation. This property implies that an
increase in wealth beyond the satiation point decreases utility.3 - A grave
disadvantage as individuals always prefer more wealth to less and treat risky
investments as normal goods. In order to circumvent this flaw the quadratic
2 Markowitz’
approach of assuming a quadratic utility did not mean to persuade the
investor that some prepackaged utility function has desirable features. It refers rather to
soliciting the investor’s preferences among various gambles and summarizing these in a
utility function.
3 Dybvig and Ingersoll (1982) among others observe that quadratic utility implies that
very high return states will have negative marginal utility and thus negative state prices
which contradicts the no-arbitrage condition of equilibrium prices.
Chapter 5. Shortcomings of two-moment asset pricing
113
utility function is usually limited to a narrow range of wealth. As a consequence it is appropriate only for relatively low returns which precludes its use
from investments that like lotteries have some very high values of the potential
payoff, albeit with low probabilities of occurrence.
5.1.2
Normal distribution
In answering how many moments are needed to describe the investor’s assessment of the probability distribution adequately, most asset pricing models
assume that the importance of all moments beyond the variance is much
smaller than that of the expected value and variance.
The normality of returns relies heavily on the extent of portfolio diversification4 and the length of the holding period. For the mean-variance-results
to hold, the portfolios must be very broadly diversified and the analysis must
be limited to a short time interval. The assumption of perfectly diversified
portfolios, however, is unsatisfactory - especially for a study on individual investors’ risk aversion. Empirical studies have clearly shown that the majority
of shareholders own less than 10 different stocks.
Alternatively, normality is given when it is assumed that portfolios are
revised continuously. Samuelson (1972) tries to prove that disregarding moments higher than the variance will not affect portfolio choice. The flaws
inherent in this paper were indicated by Loistl (1976). Brockett and Garven
(1998) add essential comments to this discussion.
Samuelson’s major assumption to arrive at his conclusion concerns the
“compactness” of the distribution of stock returns. The distribution of the
rate of return on a portfolio is said to be compact if the risk can be controlled
by the investor. In general, compactness can be interpreted as the continuity
of stock prices. If stock prices do not take sudden jumps, then the uncertainty
of stock returns over smaller and smaller time periods decreases. Under these
circumstances investors who can rebalance their portfolios frequently will do
so in order to render higher moments of the stock return distribution irrelevant.
4 The Central Limit Theorem (CLT) states that the sum (or average) of a large number of
- possibly dependent, possibly time-varying - random variables will converge to the Normal
distribution, no matter what distribution the variables have.
114
5.1. Critique of the mean-variance approach
It is not that skewness does not matter in principle. Rather, the actions
of investors who are frequently revising their portfolio limits higher moments
to negligible levels. Continuity or compactness, however, is not an innocuous assumption. Portfolio revisions entail transaction costs, meaning that
rebalancing must necessarily be limited and that skewness and other higher
moments cannot entirely be ignored. Continuous rebalancing implicitly leads
to infinite large continuous trading volumes - again an assumption that is not
unproblematic. Compactness rules out certain phenomena such as the major
stock price jumps that occur in response to takeover attempts. It also rules
out such dramatic events as the 25% one-day decline of the stock market in
October 1987.
The normality assumption of equity returns was also questioned by Mandelbrot (1963) and Fama (1965) upon observing the presence of leptokurtosis
(fat tails) in the empirical distribution of price changes.5
Mean-variance analysis is adequate if the portfolio may be revised frequently and if there are no sudden price jumps. Unfortunately, these conditions are not satisfied. The study at hand aims at advising individual investors
who consider allocating their free wealth for an undetermined period of time.
These individuals cannot and will not rebalance their portfolio continuously
- be it for transaction costs, time or other reasons. They are thus exposed to
sudden extreme price jumps that they might want to hedge against.
Simple returns in discrete time are lognormally distributed and thus positively skewed.6 The quantities decisive for the individual investor are
the change in and the current level of the portfolio value. As the portfolio
value can be written as the product of simple returns, the portfolio value is
lognormally distributed. The product of a lognormal distribution is again a
lognormal distribution. Its density function and its first four moments are
characterized in Section 8.1.
Given that a stock price cannot be negative, the lognormal as a representation of the return distribution is more realistic than the normal as it can be
5 The Pareto-Levy distribution originally proposed by Mandelbrot (1960) has been shown
to yield empirically much better results than the Gaussian (normal) distribution.
6 The lognormal model is not fully consistent with all the properties of historical stock
returns. At short horizons, historical returns show weak evidence of skewness and strong
evidence of excess kurtosis, cf. Campbell, Lo, and MacKinlay (1997).
Chapter 5. Shortcomings of two-moment asset pricing
115
modified to exclude outcomes lower than -100%. The effective annual rate is
re (t) = exp(rt) − 1. For short holding periods where t is small, the approximation of re (t) by rt is quite accurate and the normal distribution provides
a good approximation to the lognormal. For short holding periods, therefore,
the mean and standard deviation of the effective holding period returns are
proportional to the mean and standard deviation of the annual, continuously
compounded rate of return on the stock and to the time interval. For longer
holding periods, however, the normal and the lognormal distribution differ
distinctly.
5.1.3
Performance measurement of optioned portfolios
Mean-variance based pricing models (like the CAPM) will mismeasure the performance of portfolios containing fairly priced option positions.7 When skewness is positively valued, mean-variance performance measures will overrate
the rebalancing or ‘enhancement’-strategies8 which reduce skewness. Equally,
it will underrate the momentum strategies9 which buy skewness.10 The reason is that beta as the sole risk measure is incorrect and that the traditional
CAPM formula does not hold when the market is lognormally distributed.
The use of only two moments gives a distorted view of the relative performance of alternative investment strategies that result in return distributions
that predominantly change the return-skewness ratio. Skewness is not considered in the world of the Capital Market Line and the mean-variance-model.
As a risk-averse investor will prefer the positively skewed return distributions, it is worthwhile to quantify this characteristic. The asymmetry is
relevant although it is not as important as the magnitude of the standard
deviation.
In reality, however, the covered-call-strategy’s immanent severe downside
risk is not shown, as the third dimension, skewness is missing. The Put
Option Strategy on the other hand seems to perform very poorly. It reduces
the ex-ante expected return for reducing the downside-risk and thus increasing
7 Cf.
Leland (1999), 29.
hold the market portfolio and write one-year covered calls on it.
9 They hold the market portfolio and buy one-year protective puts on it.
10 The standard CAPM also empirically fails to explain the asset returns of the smallest
market-capitalized deciles, as they are the ones with the most skewed returns.
8 They
116
5.2. Summary of critique
Figure 5.1: Distorted performance of option strategies in the traditional CAPM,
source: Bookstaber and Clarke (1984).
Traditional performance and risk
measures do not apply to portfolios
that are supplemented with options.
In a mean-variance world as shown
to the left, writing call options seems
to have improved performance relative to the market portfolio.
skewness. It is obvious, that the complex return structure of the total optionstock portfolio, cannot be priced within the CAPM framework.
5.2
Summary of critique
The probability distribution of the rate of return can be characterized by
its moments. The reward for taking risks is measured by the first moment,
which is the mean of the return distribution. Higher moments characterize
the volatility or risk and the asymmetry in payoffs.
Investors’ risk preferences can be characterized by their preferences for
the various moments of the distribution. The fundamental approximation
theorem by Samuelson (1972) shows that when portfolios are revised often
enough, and prices are continuous, the desirability of a portfolio can be measured by its mean and variance alone. The rates of return on well-diversified
portfolios for holding periods that are not too long can be approximated by a
normal distribution. For short holding periods (up to one month) the normal
distribution is a good approximation for the lognormal.
Based on the above, the mean-variance model is not a general model of asset choice. Its central role in financial theory can be attributed to its analytical
tractability and mathematical simplicity. In situations in which variance alone
is not adequate to measure risk the mean-variance assumption is potentially
restrictive.
Chapter 6
Skewness
The literature that deals with the role of skewness is mostly concerned with
the equilibrium structure of returns. Less attention has been given to the
impact of skewness on risk taking. Considering skewness in portfolio choice
improves accuracy and captures investors’ preferences more comprehensively.1
Even though small losses occur more likely than with a normal distribution private investors prefer positively skewed distributions of their portfolio
value. They do so because big losses are less probable than in a normally or
a negatively skewed distribution. As most investors are risk averse, they view
the disutility of a loss as unevenly greater than the utility of a gain in the
same proportion.
Kraus and Litzenberger (1976)2 show that systematic skewness3 earns a
risk premium and thus matters in the composition of portfolios. As investors
desire positive skewness in portfolio returns, the risky asset will be held in
higher proportion than predicted by a mean-variance framework due to its
1 Even
Tobin (1958) admits that ‘mean-variance analysis is not satisfying as a framework
for representing portfolio choice’.
2 They extend the CAPM by the third moment to incorporate the effect of skewness on
valuation. Empirically their three-moment CAPM provides a better fit of the data than
the traditional two-moment model. For their model three rates of return must be known
to determine the structure of capital asset prices: market portfolio return, riskless rate of
return and the return of the zero beta and non-zero gamma portfolio.
3 Ingersoll revises this view and develops a model that prices only conditional skewness,
see Ingersoll (1975, 1987).
117
118
6.1. Higher moment preferences
positive skewness. Aversion to standard deviation and preference for positive
skewness are general characteristics of all investors having utility functions
displaying the desirable behavioral attributes of decreasing marginal utility
of wealth and non-increasing absolute risk aversion.
6.1
Higher moment preferences
Under the premise that individual investors cannot achieve compactness in
their portfolio distribution, their payoff structure will be asymmetric. In that
case volatility4 or variance become inadequate as risk measures, as the average risk-averse investor desires low downside risk and high upside potential.
Determining preferences in terms of three rather than two moments will therefore result in a more precise picture of the portfolio distribution desired by
the investor.5
All the even moments represent the likelihood of extreme values. The
odd moments represent measures of asymmetry. We can characterize the risk
aversion of any investor by the preference scheme that the investor assigns to
the various moments of the distribution.6
It must be emphasized again that both the lower and the upper end of the
distribution matter for the preference structure. Risk measures such as Valueat-Risk or Lower Partial Moments focus on the lower end of the distribution
and do not capture the degree of upside potential the investor desires to
achieve. Skewness, by contrast, in combination with the first two moments is
able to mirror the attitude towards both the upper and the lower part of the
distribution.7
4 The
characterization of good and bad volatility by Sortino and van der Meer (1991)
recognized the fundamental importance of downside risk.
5 Experimental tests have demonstrated with clear evidence that individuals attach significant importance to moments in final asset value higher than two (refer to Gordon,
Paradis, and Rorke (1972) and Coombs and Lehner (1981)). In particular, the third moment - skewness - has been shown to have a great deal of importance in measuring what is
conceived of as “risk”.
6 A number of articles has shown that moment preference does not match up with a sequence of utility derivatives: Menezes, Geiss, and Tressler (1980), Meyer (1987), Rothschild
and Stiglitz (1970) and Whitmore (1970).
7 Skewness preference becomes increasingly important as a decision making criterion
when the investor has to choose among different risk levels that correspond to distinctly
Chapter 6. Skewness
119
To know if a particular moment is desirable or not, one has to know the
coefficient of this moment - in other words the appropriate derivative of the
utility function at the point E(W ). An increase in the expected return (ceteris
paribus) of a rational individual who prefers more money to less will increase
his expected utility.8 For the higher moments of the return distribution theoretical analysis reveals a preference for positive skewness as investors are loss
averse. Empirically, it is possible to test for the direction of preferences with
a simple regression analysis:
Levy and Sarnat (1972) evaluated the impact of the distribution moments
on expected utility with a sample of US mutual funds and derived the relationship of the distribution moments to investors’ utility from the regression
analysis itself.9 For the period 1946-67 only the variance and the third moment are significant. For the period 43-67, the fourth moment is also significant, but the remaining higher moments are not significant in any of the time
periods examined. Rsquared is very high and ranges from .65 (’56-’67) to .94
(’43-’67). In all regressions the coefficient of the variance is positive. Consequently it was concluded that investors are typically averse to variance.10 The
regression coefficient of the third moment (skewness) is negative and highly
significant, suggesting that the average investor likes positive and avoids negative asymmetry. Thus, increasing the skewness of the distribution reduces
differently skewed return distributions. Skewness increases both with the risk level as well
as with the time horizon of the investment due to the compounding effect over long time
periods.
8 Brockett and Kahane (1992) and Brockett and Garven (1998) claim that expected
utility preferences never universally translate into moment preferences thereby refuting the
results presented by Scott and Horvath (1980).
9 Other studies examining the direction of moment preference are Arditti (1967), Arditti
and Levy (1972), Beedles (1979), Booth and Smith (1987), Jean (1971), Kraus and Litzenberger (1976), Scott and Horvath (1980) and Tsiang (1972). In the regression analysis a
negative coefficient for the second derivative is usually interpreted as implying an aversion
to variance. A positive third derivative is seen as indicative of skewness preference.
10 A logical flaw might be inherent in this conclusion according to Brockett and Garven
(1998): If the return distributions were normally distributed, actual statistical independence
of the sample mean and variance should be a consequence and a characterizing property of
the normal distribution. “Thus positive coefficients can arise only as a sampling artifact.
However, if the distribution were not normally distributed then the sample mean and variance should be necessarily dependent. Thus, correlation may be not so much a statement
indicative of a relationship between risk and return as much as a statement concerning the
lack of normality of the distributions.”
120
6.2. Prospect theory and adaptive aspiration
the required average return.11 The fourth moment (kurtosis) is significant in
only one of the four regressions and even there its coefficient is very small.
Thus, although expected utility depends on all the moments of the distribution function, it is often assumed that the first three distribution moments
reflect most of the required information. This does not imply that all investors
have cubic utility functions.
6.2
Prospect theory and adaptive aspiration
It was shown that the utility function and the asset pricing formulas are
the core of the investor’s asset allocation decision. Utility can be expressed
in terms of the moments of return or by any other function of purely arbitrary parameters. Whatever its mathematical form, its relation with wealth
W matters greatly to account for behavioral phenomena. One alternative
to the traditional objective function of Maximizing Expected Utility is the
value function in the prospect theory of Kahneman and Tversky (1979). The
function is defined with respect to a natural reference point which expresses
the transition from taking chances to playing the ‘safe side’.12 As this reference point is the ‘point of inflection’ it marks the change from convexity to
concavity.
For deterministic choice problems final asset positions are perceived as less
relevant than changes in wealth. Individuals try to minimize their losses. They
take chances to decrease them, while ‘collecting’ gains instead of ‘gambling’
for higher profits. To avoid a sure loss of a given quantity, people risk incurring
an even greater loss if there is a chance for a better outcome.
The value function incorporates the common view that the difference between $0 and $100 is seen as greater as the difference between $1000 and
$1100 - irrespective of the sign of the magnitude. In addition, the function is
steeper for losses than for gains expressing the empirical finding of loss aversion. The disutility of a loss weighs 2.5 times heavier than the utility of a gain
of the same magnitude. This finding is consistent with investors’ preference
11 Which partly explains why people participate in lotteries that typically have negative
expected values but are characterized by high positive skewness.
12 Here as in traditional portfolio selection models it is crucial to define how the reference
point (benchmark) defining gains and losses evolves over time.
Chapter 6. Skewness
121
for positive skewness. Risk-averse individuals seek protection from loss by
insurance.
Stated as above the concept is static and refers to a fixed risk preference.
Most studies however indicate that risk preference is not fixed, but depends
on the context of a choice.
The investor’s asset level and the degree of risk aversion are interdependent
and should be modeled accordingly. One possibility is to assume that the
reference point is changed every time the individual reaches a new level of
wealth, also called aspiration level by March (1987). Observations of human
decision making suggest that risk preferences do depend on the values of
possible outcomes relative to levels of aspiration. “A model of variable risk
preference suggests that some risk averse behavior may result from a human
tendency to focus on targets and from the adaptation of those targets to
experience rather than from a fixed trait of risk aversion.”13
The view that risk preferences change with respect to the level of wealth
or status is in line with Friedman and Savage (1948) as Robson (1992) proved.
Friedman and Savage (1948) examined the phenomenon that individuals simultaneously purchase insurance and participate in lotteries which again
refers to investors’ preference of positive skewness of returns and of final
wealth. Robson (1992) provides a natural explanation of the concave-convexconcave utility function by explicitly incorporating the relative standing that
wealth induces.
The behavioral implications of variable risk preferences help explain why
investors show a strong demand for low risk alternatives. Preferred risk varies
inversely with accumulated resources of the investor. Demand for riskier alternatives stems mainly from larger amounts of initial investments. The larger
these amounts, the slower the elimination of decision makers with losses. Another reason for riskier alternatives are situations in which the investor faces
only negative cumulative effects on expected returns. This view is consistent
with studies of gamblers involved in track betting. As the likelihood of a
horse winning will be monotone decreasing with the length of the odds, track
betting is, on average, a losing bet. Therefore, when betting over the course
of a day, the bettor will, on average, become increasingly risk prone as the
day goes on.
13 March
(1987).
122
6.2. Prospect theory and adaptive aspiration
“All of these findings show that the propositions about fixed risk preferences arising from exogenous forces need to be supplemented with contextdependent considerations.” Factors such as the success level at which choices
occur, overconfidence and overreaction have a strong influence on investment
behavior. “Aspirations and financial goals may change on the basis of experience or imitation of others. Prior histories of greater wealth produce higher
aspirations and therefore a preference for greater risk.”14
In the long run, the adaptation of aspiration levels to a decision maker’s
own experience tends to make risk taking independent of that person’s wealth.
The theory of adaptive aspiration level picks up this finding arguing that
the risk preference of a single individual depends on his current wealth level
and that the observed risk seeking when facing potential losses contributes to
the chances of ‘financial survival’ for the individual. Thus people who change
their risk preferences according to their current level of wealth will most likely
have the best investment performance. The assumption of constant relative
risk aversion might therefore be suboptimal from a normative viewpoint.15
The above properties are in conformance with empirical findings by Swalm
(1966). He finds that for any obtained level of wealth, a new utility schedule
comes into play - a conclusion with intuitive appeal as corresponding to one’s
own experience. It is a result that strengthens the plausibility of a cubic
utility: Actual wealth never reaches a point where increasing marginal utility
is expressed. The convex tail represents feelings about a possible richness
which has extra aura. As one becomes wealthier such a level is shifted away.
At the same time, this convex tail is instrumental in explaining observed
behavior patterns that are shared by many individuals. Furthermore, it is
a result that implies status as a critical influential factor in decision making
under uncertainty. Status as a function of wealth might account for behavior
that has so far been characterized as inconsistent with rationality.
To summarize the results it can be stated that investors will gamble:
• to avoid a sure medium-sized loss even if that implies that they run the
danger of suffering a larger loss.
14 March
(1987).
finance seems to suggest that risk preference is not fixed. It depends on
the context of a choice. This is what Pratt/Arrow modeled as the interdependence of asset
level and the degree of risk aversion.
15 Behavioral
Chapter 6. Skewness
123
• if the outcomes are below a certain target or an aspiration level.
To account for this behavioral peculiarity both the utility function of Friedman and Savage (1948) and the value function of Kahneman and Tversky
(1979) have a region that is convex and then concave again.
Goals are context-dependent and aspiration levels are responsive to the
degree of investment success. Thus, investors’ preferences for risk depend on
a target which is a function of their experience.
6.3
6.3.1
Gambling and insurance habits
Cubic utility and skewness
It needs to be emphasized that truncating the Taylor approximation of the (expected) utility function after n terms does not mean that the utility function
is replaced by an n-th degree polynomial. Rather U (W ) is locally approximated at every wealth level by such a polynomial. Thus the utility function
is an envelope of such polynomials and not a polynomial itself.
The Taylor approximation is used here to account for the impact of skewness, it is not the goal of this study to describe investors’ behavior by a cubic
utility function.
6.3.2
Implications of the cubic utility
A cubic utility function typically has the following form (W again equals
W0 + x):
U (W ) = aW + bW 2 + cW 3 + d
(6.1)
The only separable cubic utility concave over positive wealth levels near
zero that can be derived from the above equation is according to Rubinstein
(1973):
U (W ) = W̃ − γi W̃ 2 +
γ2 3
W̃
3
(6.2)
where γ −1 > 0
The cubic utility function exhibits several controversial properties that
need to be discussed briefly.
124
6.3. Gambling and insurance habits
• A cubic utility function is inconsistent with the log normality of the first
differences of market price changes.16
• It does not exhibit decreasing marginal utility for all wealth levels.17
When it has decreasing marginal utility for positive wealth levels near
zero, it has increasing absolute risk aversion within that same range.
• The cubic function is not bounded so that in order to avoid the revival
of the St.Petersburg Paradox, it must be assumed that beyond a certain point (W0 ) increments to wealth do not increase utility18 , that is,
U (W ) = U (W0 ) for all W ≥ W0 .
Restricting the range of permissible W in the function is also a necessary
condition for its separability and a closed equilibrium solution, as it
assures increasing marginal utility U > 0.
Unfortunately this implies a negative slope coefficient to the risk tolerance function which results in a negative wealth elasticity of demand
for risky assets. Another price for the theoretical convenience lies in the
fact that risk tolerance is non-linear which makes equilibrium
valuation theoretically intractable.
When properly restricted the cubic utility yields non-satiation and preference for positive skewness, it does however not exclude risk-loving for
all levels of wealth.
• The function has first a concave and then a convex segment. For low
values of W̃ the investor is a risk averter and dislikes variance, but
for relatively high values of W̃ the investor becomes a risk lover and
consequently desires variance.
Arditti (1975) criticized this characteristic, as it seems unreasonable that
at high wealth levels individuals should desire investments characterized
by both high variance and skewness. Wealthy individuals are per se not
risk lovers. As the cubic utility does not exclude risk-loving for all levels
of wealth it might seem unsuitable for purposes of portfolio analysis.
16 Arditti
(1967).
(1969).
18 As there is no distribution for which the expected value of the cubic U and its cash
equivalent will be infinite.
17 Levy
Chapter 6. Skewness
6.3.3
125
Insurance and Gambling
Friedman and Savage successfully argued that, while an individual in a low
income group might have the type of utility function described in the section
above, this same individual’s utility function becomes concave as he moves
into a high income group.
This finding would be consistent with an individual who buys insurance
against losses, takes small bets and seeks to diversify his portfolio. It is
similar in many important respects to the one presented by Friedman and
Savage (1948).19
With the above utility function the purchase of insurance contracts can be
reconciled with gambling, without recourse to subjective utility20 since in both
cases investors prefer positive asymmetrical distributions (like a lottery’s) but
dislike negative asymmetry and therefore buy insurance policies.21
19 They
attempted to reconcile the observed phenomenon that many individuals simultaneously purchase lottery tickets (or gamble) and take out insurance. The first activity
hints at risk-loving behavior while the other depicts risk-averse behavior. To remove the
contradiction, the utility function can be neither strictly concave nor strictly convex.
20 An approach chosen by Kahneman and Tversky (1979) whose value function accounts
for the investor’s loss aversion, as it is convex below the endowment point and at the same
time explains the preference for high unlikely payoffs observable in gambling.
21 For strictly concave utility functions an alternative approach to Friedman and Savage’s proposition suggests that simultaneous gambling and acquisition of insurance can be
explained by assuming that subjective and objective probabilities diverge. Subjective probability are viewed as exceeding their objective counterparts when the latter are low. This
assumption is sufficient to show that risk-averse individuals will also be willing to gamble
or to purchase lottery tickets.
126
6.3. Gambling and insurance habits
Chapter 7
Determining skewness
preference through
gambles
This chapter will present a novel approach of assessing an individual’s risk
aversion for two moments and his preference for the third moment of the
return distribution or of a gamble.
The determination of the investor’s three-moment risk preference will uncover the moments’ trade-offs (units of expected return per additional unit
variance, units of skewness per lost unit expected return and units of skewness
per additional unit variance) and thus help decide what option strategy best
suits that particular investor.
In the first section of this chapter the risk premia for a three moment
approximation will be derived. These are essential for the determination of
the investor’s skewness preference in the second section.
127
128
7.1. Risk premia for three moment approximation
7.1
Risk premia for three moment approximation
The insufficiency of the Pratt/Arrow premium to account for higher moment
preferences has already been treated in Section 2.2. The following sections will
derive new risk premia that can capture trade-offs in the moment preferences
of investors when payoffs are skewed.
We have
• W – current wealth
• µ̃ – random rate of return
• π – relative risk premium
(W )
• α = −W UU (W
) – Pratt/Arrow measure of relative risk aversion
We are calculating the risk premium π that leaves the decision maker
indifferent between risking the investment and the actuarial value of the investment:
U [W (1 + µ̄ − π)] = E{U [W (1 + µ̃)]}
(7.1)
where U (•) is a von Neumann-Morgenstern utility function and µ̄ ≡ E(µ̃).
The right-hand side represents the expected utility of the current wealth given
the investment, the left-hand side is the current wealth plus the utility of the
actuarial value of the investment. We can use a Taylor’s series approximation
to expand the utility function of wealth around both sides:
U [W (1 + µ̄)] − πW U (W + µ̄) +
(−1)n
π 2 W 2 U (W + µ̄)+ (7.2)
2
∞
π n W n (n)
U [W (1 + µ̄)] =
n!
n=3
1
U [W (1 + µ̄)] + µ̄W U (W + µ̄) + E[µ̃]2 W 2 U (W + µ̄)+
2
∞
1
E[µ̃]n W n U (n) (W + µ̄)
n!
n=3
From here we can choose two roads to three-moment approximation:
Chapter 7. Determining skewness preference through gambles
129
1. We neglect terms of order π 2 and higher on the left side. In addition we
neglect terms of order higher than the third moment on the right side
and solve directly for the riskpremium or
2. We neglect terms of order higher than the third moment on the right
side, but include π 2 on the left side. This leaves us with a quadratic
equation for which there are two solutions.
Following the first approach we get
1
µ̄W U (W + µ̄) + E[µ̃]2 W 2 U (W + µ̄)
(7.3)
2
1
+ E[µ̃]3 W 3 U (W + µ̄)
6
U (W + µ̄)
1
U (W + µ̄) 1
− E[µ̃]3 W 2 = − σ2 W 2
U (W + µ̄)
6
U (W + µ̄)
−πW U (W + µ̄) =
π
where E[µ̃]3 represents the non-normalized third moment of the return distribution.
Following the second approach we find
−πW U (W + µ̄)
π 2 W 2 U (W + µ̄)
2
1
= µ̄W U (W + µ̄) + E[µ̃]2 W 2 U (W + µ̄)
2
1
+ E[µ̃]3 W 3 U (W + µ̄)
6
+
Writing U n for U n (W + µ̄) and solving for π
2
1 ± 1 + σ 2 W UU + 13 m3 UU UU π =
W UU 3
1 ± 1 + αr2 σ 2 − m3 αr UU =
−αr
1
1
1 m3 U 2+
= −
±
+
σ
αr
αr2
αr 3 U (7.4)
(7.5)
This second approach would supposedly yield more precise results for higher
moment preferences than the first approach, however it is less tractable and
130
7.2. Creating gambles for skewness preference
will for a skewness of 0 not equal the Pratt-Arrow measure of risk aversion.
It is for this reason that it will not be dealt with hereafter. Instead the first
approach will be pursued and employed in the subsequent analysis.
7.2
Creating gambles for skewness preference
Building on the insights of Section 2.2 that illustrated the insufficiency of the
traditional measures of risk aversion, the investor’s porfolio selection model
is expanded by the third moment, based on the new three-moment risk premium that derived in Equation 7.3 in Section 7.1 and the investor’s moment
preferences will be derived in this section.
The assessment of any individual’s risk premium is closely connected to
that person’s utility function. For determining utility functions there are four
principal categories that can be distinguished1 : 1. the preference comparison
methods, 2. probability equivalence methods, 3. value equivalence methods,
and 4. certainty equivalence methods.
According to the literature2 , the approach best suited for risk premium
assessment seems to be the certainty equivalence method where an individual
is asked to specify a sure outcome CE for a gamble [x,p,y]. This amount,
called certainty equivalent, makes the individual indifferent between playing
the gamble or receiving the sure amount CE. In a more intuitive way, the
certainty equivalent CE can be understood as the value the investor assigns
to the gamble presented to him. If he does not approve of the gamble and its
potential payoffs, he will name a low certainty equivalent for it. The higher
CE, the less risk averse the investor.
As the main concern of the study is with return data rather than with
absolute amounts, gambles for ‘payoffs in return form’ are considered in the
following formulas.
1 Farquhar (1984) gives a good, non-technical overview of all utility assessment methods
and further subdivides each of the four categories.
2 See Becker, DeGroot, and Marschak (1964), Birnbaum (1992), Farquhar (1984), Hershey, Kunreuther, and Schoemaker (1982), Fishburn (1967, 1988) and Schoemaker and
Hershey (1992).
Chapter 7. Determining skewness preference through gambles
131
The derivation of skewness preference
The proceeding can be divided into two steps:3 In a first step, relative risk
aversion for a two moment setting, a gamble without skewness, has to be
derived. The investor is thus presented with a tree-structured gamble as in
the Figure on page 137.
Apart from the probabilities of the outcomes the investor is given the
expected return ER and volatility σ2 of the 2-moment gamble. He then needs
to name his Certainty Equivalent for that gamble as well as his current wealth
level that will be relevant for determining his skewness preference.
2 refers
Pratt/Arrow’s risk premium π2 - the subscript
to its applicability
to 2-moment gambles or distributions - is − 21 σ22 W UU . The strong assumption that must be made is: risk premium π2 = ER − CE, where ER stands
for Expected Return and CE stands for Certainty Equivalent. Solving for
relative risk aversion αr yields
αr = −W
U ER − CE
=2
U
σ22
(7.6)
Dividing the above result by wealth W gives the Pratt/Arrow-measure of
absolute risk aversion αa .
In the second step, the investor is presented with a second tree-structured
gamble for which he has to name his certainty equivalent CE. This time
however, the gamble is skewed and CE will contain additional information
about the third moment of the gamble. To distinguish the second gamble’s
parameters from those of the first one that considered only two moments,
the subscript 3 was introduced (π3 , σ32 ). As the payoff structure is skewed,
the Pratt/Arrow-risk premium is not applicable4 . Instead Formula 7.3 of
(W +µ̄)
section 7.1 is employed and solved for UU (W
+µ̄) which will be abbreviated as
U U
hereafter:
3 In
6 π3 − 12 αr σ32
U =−
U
E[µ̃]3 W 2
(7.7)
this section ‘skewness’ will denote E[µ̃]3 , the non-normalized third moment. The
traditional skewness measure, the Fisher skewness, is the third moment normalized by σ33 .
4 An exact preference ordering for risky portfolios using the first three moments of portfolio return can in general be determined only for an investor having a cubic utility function
for wealth (see Subsection 6.3.1 for further discussion). Unfortunately a third degree polynomial would be an unsuitable utility function for a risk averse investor.
132
7.2. Creating gambles for skewness preference
where π3 stands for the three-moment risk premium and again π3 = ER−CE;
αr depicts the relative risk aversion estimated in the first step above, σ32
represents the skewed gamble’s variance and E[µ̃]3 its non-normalized third
moment.5 For ease of presentation E[µ̃]3 will be abbreviated as m3 hereafter.
Having established U /U , it is simple to derive −U /U by dividing the
above result by αr :
U
U − = UU =
U
− U
6(π3 − 12 αr σ 2 )
E[µ̃]3 W 2
−2 Wπσ2 2
2
3 σ22
=−
W E[µ̃]3
π3
σ2
− 32
π2
σ2
(7.8)
where m3 = E[µ̃]3 .
With these formulas and after only two gambles a relatively comprehensive
picture of the investor’s preferences can be drawn. Of course, this necessitates
that the individual names two certainty equivalents that reliably reflect his
tastes.
The two newly derived ratios need to be explained: While U /U stands
for the preference for the third moment relative to the first moment of the
distribution, −U /U can be interpreted as the skewness ratio that reflects
the investor’s preference for the third moment relative to aversion to variance.6
The lower U /U the more skewness is needed to compensate for a loss in
expected return. Similarly, the lower −U /U the more skewness is needed
to compensate for a rise in the gamble’s variance.7
A positive influence of skewness might offset the negative influence of the
variance of the risk premium. The local three-term approximation has convex
regions even though the utility function is globally concave. This may lead
5 Pratt
(1964) shows that decreasing absolute risk aversion in two moments corresponds
to a decreasing risk premium. Arditti (1967) proves that a decreasing risk premium corresponds to U (W ) > 0. This was confirmed by Bawa (1975) who applies the property to
develop what came to be known as ‘stochastic dominance’.
6 −U /U is also known as (the degree of absolute) Prudence, a term established by
Kimball (1990) relating to the desire to avoid disappointment and linked to the precautionary savings motive. Prudence implies non-increasing absolute risk aversion. Based on
this concept Menezes, Geiss, and Tressler (1980) define ‘downside risk aversion’ as having
a utility function with a uniformly positive third derivative.
7 Variations of financial holdings often involve trade-offs between moments. For a cubic
utility the value of the trade-off between 3rd moment and the variance (−U /U ) decreases
monotonically with W. At high levels of wealth skewness preference (U /U ) decreases,
while third moment preference is constant.
Chapter 7. Determining skewness preference through gambles
133
to local risk seeking behavior in spite of global risk aversion. However, the
question whether a utility function exhibits increasing, constant or decreasing
risk aversion can only be answered with respect to a specific gamble and
investor.
The changes in the ratios and in the moments’ trade-offs are proportional:
An investor A with a ratio U /U twice as high as that of investor B will
demand half the skewness per unit loss of expected return. Similarly, an
investor C with a ratio −U /U twice as high as that of investor D will
demand half the skewness per unit rise in variance. In the next paragraphs
these trade-offs will be derived analytically.
First consider the trade-off between expected return and skewness: A
decrease in expected return of one percentage point lowers the risk premium
by that amount.8 To keep the risk premium constant, the compensating
skewness can be calculated by expressing the new risk premium (sub-subscript
2) in terms of the original one (sub-subscript 1). Solving for the skewness of
the modified gamble with lower expected return yields:
π2
U
1
1
αr σ32 − W 2 m32
2
6
U
m3 2
= π1 − 0.01
U 1
1
αr σ32 − W 2 m31 − 0.01
=
2
6
U
0.06U =
− m 31
W 2 U (7.9)
where the second subscript of m32 stands for the skewness of the gamble
with a single percentage-point decrease in expected return compared with the
original gamble’s skewness m31 .
The change in skewness necessary to compensate for a 1% loss in expected
return is thus:
0.06U ∆m3
= 2m31 − 2 ∆ER
W U
(7.10)
Next consider the trade-off between skewness and variance: An increase
in variance of one percentage point raises the risk premium. To keep the risk
premium constant, the compensating skewness can be calculated by equating
the new risk premium (sub-subscript 3) and the original one (sub-subscript 1).
8 Empirical
studies confirm a positive tradeoff between mean return and skewness.
134
7.2. Creating gambles for skewness preference
Solving for the skewness of the modified gamble with higher variance yields:
1
U 1
αr σ32 − W 2 m31
2
6
U
m33
U 1
1
αr σ32 − 0.01 − W 2 m33
2
6
U
αr U = m31 − 0.03 2 W U
U = m31 + 0.03
W U =
(7.11)
where the second subscript of m33 stands for the skewness of the gamble with
a one percentage-point increase in variance.
The change in skewness necessary to compensate for a 1% increase in
variance is thus:
∆m3
∆σ 2
=
=
0.03
αr
W 2 UU U 0.03
W U (7.12)
As the volatility is the more intuitive and traditional measure of risk,
it might be of greater interest to know the trade-off between skewness and
volatility. An increase of one percentage point in volatility necessitates the
following skewness to keep the risk premium constant:
U 1
1
αr σ32 − W 2 m31
2
6
U
m34
U 1
1
αr (σ3 − 0.01)2 − W 2 m34
2
6
U
1
1
σ
−
3
10 000
= m31 − 3αr 50
(7.13)
W 2 UU 1
3U 1
σ
−
= m31 +
3
W U 50
10 000
=
where the second subscript of m34 stands for the skewness of the gamble with
a one percentage-point decrease in volatility.
The change in skewness necessary to compensate for a 1% increase in
volatility is thus:
∆m3
= 3αr
∆σ
− 101000
W 2 UU 1
50 σ3
(7.14)
Unfortunately, even volatility is a poor risk measure for hedged (addition
of puts) or enhanced (addition of calls) portfolio distributions. Lower partial
Chapter 7. Determining skewness preference through gambles
135
moments capture the modifications of put- and call strategies more appropriately and thus reflect their benefit with more accuracy. However, they
cannot capture the form of the distribution’s upper end that is relevant for
the propensity to gamble.
7.2.1
Interpretation of three-moment preferences
Looking at Equations 7.3, 7.12 and 7.10 it can be learned that an increase
in variance requires a compensation in the form of positive skewness when
assuming a concave utility function and when holding the end-of-period (eop)
wealth constant.
For the negative exponential utility the relative risk premium increases
with the variance of returns when W increases and the distribution is symmetric (skewness=0)9 . The premium remains constant for logarithmic utility
when W increases.
For the negative exponential utility the variance’s compensation with
skewness in Equation 7.10 decreases proportionally with W , as the implicit
skewness premium for a higher variance decreases with increasing wealth. For
the logarithmic utility it is constant and equals 1/2.
When increasing the variance and keeping the mean constant in a gamble10 , the expected utility of the changed prospect will be lower than for the
original gamble for all concave utility functions. The EU deteriorates simply
by concavity as fair games are rejected by risk averters. This implies that
positive skewness-variance trade-offs are rejected locally. It explains why call
options with identical expected returns are priced lower the further out of the
money they are.
The effect of the change in variance splits investors into two groups: The
first group of investors will require an increase in the risk premium to accept
the second game while the second group of investors are willing to take a cut
in EU because they prefer the higher positive skewness. The groups can be
distinguished by −U /U .
9 As
the relative risk aversion increases for the negative exponential utility.
other words the probability of the good outcome decreases. Tilting a prospect in
such a way introduces “in the large” aspects to risk taking. For prospects of very small risks,
tilting is undesirable. Remaining “in the small”, as Samuelson (1972) shows for concave
utilities, leaves the first two moments as sole potential factors.
10 In
136
7.2. Creating gambles for skewness preference
The investors with more conservative utility are increasingly attracted to
positive skewness while the bolder investors are invariant to it. This seems
paradoxical considering Keynes’ famous quotation “The poor should not gamble while the millionaires should do nothing else”. The dollar premium for
the negative exponential utility (NEU) is constant in W , increasing with
the variance and decreasing with mean/variance. For the logarithmic utility it is decreasing with W , increasing with the variance and decreases with
mean/variance at a decreasing rate with W .
The reason is that the relative conservatism of the NEU-behavior decreases
as downside risk is reduced and the ratio mean/variance increases. Aggressive
behavior represented by the logarithmic utility is decreasingly concerned with
downside risk as wealth increases. The relevance of this observation is exposed
particularly with financial instruments such as options.
7.2.2
Example for the assessment of risk aversion
To illustrate the benefit and practical use of the above trade-off formulas, the
assessment of an imaginary investor i’s preferences is simulated hereafter.
In the first step, the two-moment preference is determined and the investor
i is presented with a gamble. For reasons of transparency and comprehensibility the gamble is portrayed by two illustrations which are complemented
by the moments of the underlying distribution. The first illustration gives
the payoffs in return style, the second one in absolute amounts assuming the
investor considers allocating all of his wealth W.11
11 Note: In order to avoid typical behavioral biases such as loss aversion, it would be
beneficial to portray a gamble with purely positive or negative payoffs in a survey.
Chapter 7. Determining skewness preference through gambles
137
Step 1: Two-moment risk aversion
The investor i is given with his wealth W =200’000
Expected Return ER2 =9%
Volatility σ2 =22%
262’000
+31%
✒
✒
0.5
0.5
200’000
200’000
❅ 0.5
❅ 0.5
❅
❅
❘ 174’000
❅
❘ -13%
❅
The investor declares his Certainty Equivalent CE2 = 5.00%
With these parameters it is possible to solve for investor i’s relative risk
aversion αri and his absolute risk aversion αai :
αri
αai
=
U ER − CE
9% − 5%
=2
=2
U
σ22
20%2
1.653
=
8.264 · 10−6
= −W
Step 2: Three-moment risk preference
Expected Return ER3 =7%
Volatility σ3 =21.31%
Non-normalized Skewness, 3rd moment m3 =0.0175
+65%
330’000
0.1 ✒
0.1 ✒
200’000
0.5✲
+9%
❅
0.4❅
❘ -10%
❅
200’000
❅
0.5✲
218’000
0.4❅
❘ 180’000
❅
The investor declares his Certainty Equivalent CE3 = 6.00%
138
7.2. Creating gambles for skewness preference
With the information of this second gamble, U /U can be derived:
6 π3 − 12 αr σ32
U = −
U
E[µ̃]3 W 2
6 (ER3 − CE3 ) − 12 αr σ32
= −
m3 W 2
6 (7% − 6%) − 12 · 1.653 · 21.31%2
= −
0.0175 · 200 0002
= 2.360 · 10−10
Dividing U /U from above by αr yields −U /U :
U − U
=
6(π3 − 12 αr σ32 )
E[µ̃]3 W 2
−2 Wπσ2 2
2
3 σ22
W E[µ̃]3
3
·
= −
200 000
= −
=
π3
σ2
− 32
π2
σ
2
2
22%
(7% − 6%) 21.31%2
−
0.0175 (9% − 5%)
22%2
2.855 · 10−5
The investor i is now completely described by his relative two-moment risk
aversion, his skewness-return preference and skewness-variance preference.12
The trade-offs can be calculated using the formulas 7.10, 7.12 and 7.14:
The trade-off (change in skewness necessary to compensate for a 1% loss
in expected return) between skewness m3 and expected return ER is:
∆m3
∆ER
12 Note
0.06
W 2 UU =
2m31 −
=
2 · 0.0175 −
=
0.0286
200 0002
0.06
· 2.360 · 10−10
that for practical purposes e.g. when carrying out a survey about investment
preferences, one will be especially interested in depicting the moments of empirical portfolio distributions. Portraying portfolio skewness with gambles is particularly difficult, as
in tree structured gambles high skewness values cannot be achieved unless one assumes
unrealistically high or low payoffs. As an alternative it might be easier to directly present
graphs of different return distributions to the investor and ask his certainty equivalent for
these.
Chapter 7. Determining skewness preference through gambles
139
The change in skewness necessary to compensate for a 1% increase in
variance is:
∆m3
∆σ 2
=
0.03 ·
=
0.03 ·
=
αr
W 2 UU 1.653
200 0002 · 2.360 · 10−10
0.00525
Similarly, the change in skewness necessary to compensate for a 1% increase in volatility is:
∆m
∆σ
− 101000
W 2 UU 1
50 σ3
=
3αr
=
3 · 1.653 ·
=
0.00229
1
1
50 21.31% − 10 000
200 0002 · 2.360 · 10−10
140
7.2. Creating gambles for skewness preference
Chapter 8
Creating portfolio
skewness with options
In the preceding chapter the moment preferences of investors were examined.
It was shown how the trade-offs among pairs of moments desired by the investor can be determined analytically. This chapter will examine how these
preferred trade-offs, or moment preferences can be implemented through option strategies. The first section deals with the question how options change
the distribution of portfolio returns. Different option strategies will achieve
different trade-offs among the moments of the return distribution.
Investors who appreciate skewness are willing to accept lower expected
portfolio return in exchange for protection from losses or in exchange for
higher potential payoffs. While portfolio protection - a so-called ‘hedge’ - can
be achieved via buying put options, higher potential payoffs - called ‘enhancement’ of the portfolio hereafter - are established via writing call options.
In general positive skewness can be achieved by less than perfect diversification, through call and put options as well as through lotteries. Many
common trading strategies also result in skewed returns. The most common
is “cut your losses and let profits run”. It results in lots of small losses and a
few big gains. Two other strategies that yield skewed returns are those that
use leverage dynamically and those that follow a trend. The former adds to
winning trades and reduces losers, the latter corresponds to posting many
141
142
8.1. Portfolio distribution
small losses in volatile markets and a few large gains in trending markets.
8.1
Portfolio distribution
The first step in modelling distributional effects of options is to choose the
appropriate assumption about the form of the portfolio’s return distribution.
As the study is foremost concerned with long investment horizons, single
period returns and the portfolio’s terminal value at each period, the lognormal
model will be employed. It assumes that continuously compounded singleperiod returns are iid normal implying that single-period gross simple returns
are distributed as iid lognormal variates.1 Furthermore, it is assumed that
call and put options can be bought on the particular portfolio held by the
investor. This assumption is not too restricting, as today options on all major
market indices with varying strike prices and maturities are available. Also,
the sole investment in the market-portfolio has been an explicit assumption
in the earlier empirical analysis of two-moment preference. Considering only
portfolios that are tracking major indices is thus an assumption consistent
with the preceding chapters.
The mean, variance and skewness of simple returns are thus given by
σi2
−1
(8.1)
E[Rit ] = exp µi +
2
(8.2)
V ar[Rit ] = exp 2µi + σi2 exp σi2 − 1
2 (8.3)
Skew[Rit ] =
exp (σ) − 1 2 + exp σ
The lognormal density is
1
√
xσi 2π
· exp
−(ln x − µi )2
2σi2
(8.4)
On financial markets skewness is produced by sudden jumps in asset price
processes. These jumps are partly perceived as momentum. In index-trackingstock-portfolios, different measures of skewness can be achieved by adding
or selling options on the underlying index. The skewness preferences were
1 This model has the further advantage of not violating limited liability, since limited
liability yields a lower bound of zero on gross return, see Campbell, Lo, and MacKinlay
(1997), 15.
Chapter 8. Creating portfolio skewness with options
143
assessed by the gambles in Section 7. If the investor is willing to sacrifice some
(Fisher) skewness to obtain a higher expected return (if both Formula 7.10
and Formula 7.14 are positive for investor i), he should write call options. If
the investor primarily wants to avoid high losses and is prepared to trade in
some expected return for higher positive (Fisher) skewness and less volatility
(if Formula 7.14 is negative and Formula 7.10 positive for investor i), then he
should buy put options.
8.2
Truncating the distribution’s lower end with
Puts
Adding put-options to the portfolio truncates the lower end of the portfolio
distribution. If the density’s x-axis depicts the portfolio’s value at the end of
period 1, the cumulated probabilities of the portfolio realizations to the left
of the strike price are shifted to the point: StrikePricePuts − TotalCostPuts .
In addition to the truncation and the translation, the rest of the distribution
to the right of the Strike price X is shifted left by the total cost of the put
options. Figure 8.1 illustrates these changes in the return distribution of the
portfolio.
The total cost of the put-strategy can be calculated applying the BlackScholes’ pricing formula for a European put on a stock index (accounting for
a continuous dividend yield) :
p = X exp(−r(T − t))N (−d2 ) − S exp(−q(T − t))N (−d1 )
(8.5)
where X=Strike price of put option, S=index value, r=risk-free rate, (T −
t)=time to maturity, N (x) is the cumulative pdf for a standardized normal
variable, q=average annualized dividend yield during the life of the option,
σ= volatility of the index and
S r−q+σ2
ln X
+
(T − t)
√ 2
(8.6)
d1 =
σ T −t
√
(8.7)
d2 = d1 − σ T − t
Unfortunately the moments of the shifted portfolio’s distribution cannot be
derived in closed-end form. However, with any spreadsheet program the effect
144
8.2. Truncating the distribution’s lower end with Puts
Figure 8.1: Lognormal portfolio returns hedged with puts
The protective put strategy cuts off the lower end of the distribution and shifts its
upper part to the left due to the incurred option premia. The put options’ strike
price used for hedging the density below was at 85% of the portfolio’s value. The
other parameters were determined according to empirical findings: rf = 3.5%, σ =
15.78%, (T − t) = 1 year, q = 3%.
The unhedged and
Mean:
Volatility:
FS Skewness:
hedged moments of the distribution are:
unhedged=8.98%
hedged=7.73%
unhedged=15.39%
hedged=14.82%
unhedged=0.424
hedged=0.766
of the option’s pricing variables on the portfolio’s moments can be simulated.
In general, the trade-off between the first and the third moment is more
proportional than the trade-off between the second and the third moment.
Obviously, relatively more (Fisher) skewness is needed to achieve a decrease
in volatility than an increase in expected return.2 One reason is rooted in the
fact that volatility alone is not an appropriate risk measure for hedging- or
2 Figures
5.1 and 8.3 clearly illustrate this fact.
Chapter 8. Creating portfolio skewness with options
145
Figure 8.2: Lognormal portfolio returns hedged with puts exhibiting different strike
prices
The protective put strategy cuts off the lower end of the distribution and shifts its
upper part to the left by the incurred option premia. The higher the put options’
strike price used for hedging the greater is the protection against losses and the larger
the cost of the options. The portfolio’s expected return µ is 8.31%, its volatility
σ = 21.84%, the non-normalized third moment m3 = 0.0062 and its Fisher skewness
F S = 0.5958. Calculations were carried out for the parameters rf = 3.5%, (T −
t) = 1 year, and dividend yield q = 3%. The first column gives the moments of
the unhedged portfolio (MuPf), the following columns show the moments of that
portfolio hedged with puts whose strike prices amount to 80%, 95% and 115%,
respectively, of the portfolio’s market value:
MuPf
8.31%
21.84%
0.0062
0.5958
µ
σ
m3
FS
X=80
7.93%
20.47%
0.0075
0.8745
X=95
6.98%
17.32%
0.0074
1.4312
X=115
4.32%
11.4%
0.0044
2.9513
enhancement strategies.
The tables assist in choosing the option strategy appropriate for investor i
146
8.2. Truncating the distribution’s lower end with Puts
Table 8.1: Impact of different strike prices on the moments of a lognormally distributed portfolio with high volatility.
All values are calculated for the following portfolio volatility σ, riskfree rate rf ,
maturity T and dividend yield q for the index put options:
σ = 22%, rf = 3.5%, T = 1y, q = 3%
Each block of four rows depicts a different portfolio that is hedged with put options
whose strike prices amount to X% of the portfolio’s current market value. The
four rows of each block show the portfolio’s mean return µ, its volatility σ, its
non-normalized third moment m3 and its Fisher skewness F S = m3 /σ 3 .
MuPf
7.23%
21.62%
0.006
0.5958
8.31%
21.84%
0.0062
0.5958
9.4%
22.06%
0.0064
0.5958
10.5%
22.28%
0.0066
0.5958
11.61%
22.5%
0.0068
0.5958
12.73%
22.73%
0.007
0.5958
µ
σ
m3
FS
µ
σ
m3
FS
µ
σ
m3
FS
µ
σ
m3
FS
µ
σ
m3
FS
µ
σ
m3
FS
X=80
6.95%
20.17%
0.0073
0.8907
7.93%
20.47%
0.0075
0.8745
8.9%
20.81%
0.0077
0.8502
9.91%
21.1%
0.0079
0.8354
10.94%
21.4%
0.008
0.8211
11.96%
21.73%
0.0082
0.7999
X=85
6.76%
19.27%
0.0075
1.044
7.64%
19.65%
0.0077
1.0133
8.59%
19.96%
0.0079
0.9951
9.51%
20.33%
0.0081
0.9665
10.49%
20.65%
0.0084
0.9495
11.45%
21.01%
0.0086
0.923
X=90
6.53%
18.16%
0.0074
1.2366
7.34%
18.56%
0.0077
1.2009
8.17%
18.96%
0.008
1.1664
9.07%
19.29%
0.0082
1.1469
9.94%
19.69%
0.0085
1.1144
10.83%
20.08%
0.0088
1.083
X=95
6.25%
16.9%
0.0071
1.4719
6.98%
17.32%
0.0074
1.4312
7.72%
17.73%
0.0078
1.3918
8.49%
18.15%
0.0081
1.3537
9.29%
18.56%
0.0084
1.3169
9.95%
18.77%
0.0087
1.3182
X=100
5.74%
15.43%
0.0065
1.7767
6.37%
15.85%
0.0069
1.7305
7.03%
16.27%
0.0073
1.6859
7.71%
16.69%
0.0076
1.6427
8.41%
17.11%
0.008
1.6009
9.13%
17.53%
0.0084
1.5602
X=105
5.07%
14.02%
0.0058
2.1058
5.72%
14.35%
0.0061
2.0828
6.29%
14.76%
0.0065
2.0316
6.88%
15.18%
0.0069
1.9823
7.48%
15.61%
0.0074
1.9346
8.11%
16.03%
0.0078
1.8885
X=110
4.46%
12.53%
0.005
2.5149
4.92%
12.94%
0.0053
2.4538
5.51%
13.26%
0.0057
2.4291
6%
13.68%
0.0061
2.3714
6.52%
14.09%
0.0065
2.316
7.05%
14.51%
0.0069
2.2626
X=115
3.85%
10.91%
0.004
3.0608
4.32%
11.4%
0.0044
2.9513
4.71%
11.8%
0.0047
2.8787
5.11%
12.2%
0.0051
2.8093
5.53%
12.61%
0.0055
2.743
6.11%
13.13%
0.006
2.6497
Chapter 8. Creating portfolio skewness with options
147
Table 8.2: Impact of different strike prices on the moments of a lognormally distributed portfolio with low volatility.
All values are calculated for the following portfolio volatility σ, riskfree rate rf ,
maturity T and dividend yield q for the index put options:
σ = 16%, rf = 3.5%, T = 1y, q = 3%
Each block of four rows depicts a different portfolio that is hedged with put options
whose strike prices amount to X% of the portfolio’s current market value. The
four rows of each block show the portfolio’s mean return µ, its volatility σ, its
non-normalized third moment m3 and its Fisher skewness F S = m3 /σ 3 .
MuPf
6.32%
16.03%
0.0019
0.4568
7.39%
16.2%
0.0019
0.4568
8.47%
16.36%
0.002
0.4568
9.56%
16.52%
0.0021
0.4568
10.66%
16.69%
0.0021
0.4568
11.77%
16.86%
0.0022
0.4568
µ
σ
m3
FS
µ
σ
m3
FS
µ
σ
m3
FS
µ
σ
m3
FS
µ
σ
m3
FS
µ
σ
m3
FS
X=80
6.29%
15.56%
0.0023
0.6093
7.31%
15.79%
0.0023
0.5896
8.36%
15.99%
0.0024
0.5776
9.41%
16.21%
0.0024
0.561
10.48%
16.42%
0.0024
0.546
11.57%
16.62%
0.0025
0.5371
X=85
6.1%
15.01%
0.0025
0.7442
7.07%
15.27%
0.0026
0.716
8.06%
15.54%
0.0026
0.6897
9.06%
15.79%
0.0026
0.6652
10.11%
16.01%
0.0027
0.6504
11.15%
16.26%
0.0027
0.6288
X=90
5.9%
14.17%
0.0027
0.9354
6.59%
14.38%
0.0027
0.9193
7.5%
14.68%
0.0028
0.885
8.43%
14.98%
0.0029
0.8523
9.38%
15.27%
0.0029
0.8211
10.36%
15.56%
0.003
0.7914
X=95
5.37%
12.88%
0.0026
1.2374
6.16%
13.21%
0.0028
1.1944
6.98%
13.55%
0.0029
1.1529
7.76%
13.94%
0.003
1.0951
8.63%
14.27%
0.0031
1.057
9.52%
14.59%
0.0032
1.0202
X=100
4.83%
11.5%
0.0024
1.6057
5.52%
11.85%
0.0026
1.5561
6.23%
12.2%
0.0027
1.508
6.96%
12.54%
0.0029
1.4614
7.66%
12.96%
0.003
1.3926
8.45%
13.31%
0.0032
1.3492
X=105
4.18%
9.97%
0.0021
2.1012
4.75%
10.32%
0.0022
2.0417
5.26%
10.75%
0.0024
1.9502
5.88%
11.1%
0.0026
1.8964
6.69%
11.56%
0.0028
1.8131
7.37%
11.92%
0.003
1.7624
X=110
3.39%
8.45%
0.0016
2.7056
3.95%
8.89%
0.0018
2.5923
4.43%
9.24%
0.002
2.5207
4.84%
9.67%
0.0022
2.4109
5.37%
10.02%
0.0024
2.3475
5.92%
10.37%
0.0026
2.2865
X=115
2.7%
7.03%
0.0012
3.4409
3.13%
7.45%
0.0014
3.2901
3.39%
7.85%
0.0015
3.1377
3.78%
8.19%
0.0017
3.0481
4.3%
8.63%
0.0019
2.9238
4.73%
8.98%
0.0021
2.8446
148
8.2. Truncating the distribution’s lower end with Puts
Table 8.3: Impact of different strike prices on the moments of a lognormally distributed portfolio with high volatility and higher riskfree rate.
All values are calculated for the following portfolio volatility σ, riskfree rate rf ,
maturity T and dividend yield q for the index put options:
σ = 16%, rf = 4.5%, T = 1y, q = 3%
Each block of four rows depicts a different portfolio that is hedged with put options
whose strike prices amount to X% of the portfolio’s current market value. The
four rows of each block show the portfolio’s mean return µ, its volatility σ, its
non-normalized third moment m3 and its Fisher skewness F S = m3 /σ 3 .
MuPf
7.23%
21.62%
0.006
0.5958
8.31%
21.84%
0.0062
0.5958
9.4%
22.06%
0.0064
0.5958
10.5%
22.28%
0.0066
0.5958
11.61%
22.5%
0.0068
0.5958
12.73%
22.73%
0.007
0.5958
µ
σ
m3
FS
µ
σ
m3
FS
µ
σ
m3
FS
µ
σ
m3
FS
µ
σ
m3
FS
µ
σ
m3
FS
X=80
7.18%
20.28%
0.0073
0.8764
8.17%
20.57%
0.0075
0.8599
9.18%
20.87%
0.0077
0.8441
10.17%
21.21%
0.0078
0.8209
10.94%
21.4%
0.008
0.8211
11.99%
21.69%
0.0082
0.8074
X=85
6.97%
19.39%
0.0075
1.0278
7.91%
19.71%
0.0077
1.0085
8.82%
20.08%
0.0079
0.9785
9.8%
20.4%
0.0082
0.9606
10.74%
20.76%
0.0084
0.9329
11.75%
21.08%
0.0086
0.9162
X=90
6.72%
18.29%
0.0075
1.2183
7.6%
18.62%
0.0077
1.1971
8.44%
19.02%
0.008
1.1619
9.29%
19.42%
0.0083
1.1281
10.23%
19.75%
0.0085
1.1087
11.12%
20.14%
0.0088
1.0769
X=95
6.51%
16.95%
0.0072
1.4705
7.24%
17.37%
0.0075
1.4289
7.99%
17.79%
0.0078
1.3887
8.77%
18.2%
0.0081
1.3499
9.56%
18.62%
0.0085
1.3124
10.45%
18.96%
0.0088
1.2921
X=100
5.99%
15.47%
0.0066
1.7777
6.63%
15.89%
0.0069
1.7306
7.29%
16.31%
0.0073
1.6851
7.97%
16.74%
0.0077
1.6412
8.68%
17.16%
0.0081
1.5986
9.48%
17.5%
0.0085
1.5775
X=105
5.56%
14.08%
0.0059
2.1137
6.12%
14.5%
0.0063
2.0598
6.69%
14.93%
0.0067
2.0078
7.29%
15.35%
0.0071
1.9577
7.91%
15.77%
0.0075
1.9094
8.64%
16.12%
0.0079
1.8873
X=110
5.05%
12.5%
0.005
2.5664
5.4%
13%
0.0054
2.4665
5.99%
13.33%
0.0058
2.4395
6.49%
13.75%
0.0062
2.3799
7.01%
14.17%
0.0066
2.3226
7.55%
14.59%
0.007
2.2675
X=115
4.4%
11.06%
0.0041
3.0484
4.77%
11.46%
0.0045
2.9702
5.16%
11.86%
0.0048
2.8955
5.69%
12.38%
0.0053
2.7923
6.13%
12.79%
0.0057
2.7246
6.58%
13.2%
0.0061
2.6598
Chapter 8. Creating portfolio skewness with options
149
Figure 8.3: Assuming a portfolio of stocks with lognormally distributed returns, the
following graphs show the mean, volatility and skewness of portfolios hedged with
put options of strike prices that amount to 80%, 85%, ..., 115% of the portfolio’s
value. The figures visualize the previous tables where the riskfree rate rf = 3.5%,
the maturity of the options T = 1y and the normally distributed return had a
volatility σ = 20% and a mean varying from µ = 5%, 6%, ..., 10%.
150
8.2. Truncating the distribution’s lower end with Puts
according to his skewness preference that was determined with the certainty
equivalence gambles. For example, an investor who stated his CE to equal
7% in gamble 1 of Section 7.2.2, Page 137, and CE = 5.25% in gamble 2
of Section 7.2.2, Page 137, exhibits the following trade-off ratios: m3 /µ =
0.00138 and m3 /σ = 0.00733. Thus, for a 1% decrease in expected return he
demands an increase in skewness of at least 0.000138. For a 1% increase in
volatility he expects an increase in skewness of at least 0.00733. Assuming the
riskfree rate is rf = 3.5% and the time horizon of the option strategy should
be T = 1y, it becomes obvious that the above ratios restrict the possible
strategies depicted in Table 8.1 on Page 146. A m3 /µ-trade-off of 0.00138
excludes all put options with a strike price higher than 90% of the current
portfolio price. The trade-off ratio of skewness and volatility is no restriction
in this case, as the volatility of the protective put strategy decreases more
than proportionally with an increasing exercise price. However, the investor
is only willing to purchase out-of-the-money puts. All other options cause the
mean to decrease by too much compared to the compensation with skewness.
Blake (1996) finds that the cost of portfolio insurance expressed by put
premiums is less than the maximum that investors would be willing to pay to
avoid risk. He concludes that the demand for portfolio insurance is high in
all wealth ranges. This holds even though the proportionate cost of the put
premium increases with wealth, as the richer investors hold more risky assets.
The expected returns increase at a faster rate than the put costs, so that the
portfolio insured returns also increase monotonically with wealth.
Thus, if investors were offered portfolio insurance whose costs as a proportion of their wealth (putcosts/wealthlevel) is less than π (Pratt-Arrow risk
premium), we would expect them to take it.3
Judging from the above exposition and the analysis of preferences, the
market’s demand for portfolio insurance should be strong. Unfortunately, it
is not widely available to private investors.
In considering what portfolio insurance strategy to adopt, put-options
prove to be the only workable possibility. Dynamic portfolio insurance strate3 In
this context Beighley (1994) states that even though a continuous, dynamic strategy
with out-of-the-money options would not be expected to pay off in every quarter, it would
be expected to pay off in those quarters when protection is really needed; that is, when
significant downside price movements occur.
Chapter 8. Creating portfolio skewness with options
151
gies replicate put options. In contrast to the latter, however, these strategies
prove to fail when they are needed most. That is, in times of market panics
and crashes when market liquidity falls heavily and no one is prepared to
take the other side of the sales deal. The October 1987 crash with its automated program sales of dynamic insurance strategies is a warning reminder
illustrating how dynamic strategies fail to fulfill their very purpose.
The question whether one wishes to simply follow a protective put strategy
or whether financing the puts by writing calls additionally can be answered
by the examining the investor’s skewness preference and the current market
conditions for options, e.g. their prices.
When considering a put option strategy over longer-term horizons (over
one year) one is forced to apply a roll-over strategy. Variables that need to
be decided on include: maturity T of the options used, strike price X (floor
of the hedge strategy), and general strategy of the insurance program. The
last variable refers to the variety of programs imaginable. Naturally, the
program’s cost - the endogenous variable of the decision problem - also has
to be considered.
Figlewski, Chidambaran, and Kaplan (1993) consider three basic programs: the ‘fixed strike’ program which is similar to the ‘time invariant portfolio protection (TIPP)’, the ‘fixed percentage’ program which is similar to
the ‘constant proportion portfolio insurance (CPPI)’ and the ‘ratchet strategy’ which employs one of the first two depending on the direction of the
market movement.
With the fixed strike strategy an investor selects a desired strike level
which is fixed for the whole period. If the stock price rises sharply in the
beginning of the strategy, rolling over the put becomes very cheap, though
the strategy then proves not very protective. If the market drops heavily, the
puts at roll-over will become very expensive, rendering the strategy essentially
into a riskless asset. With the fixed percentage strategy the strike price is
set at a given percentage of the stock price at the time of rollover. This
strategy has the advantage of continued capital protection in rising markets:
When the market increases, the absolute floor of the hedge program does
as well. However, in times of continued downward movement, this strategy
provides no adequate protection from losses, as the floor continues to drop
with the market price at every rollover. The ratchet strategy sets the strike
152
8.2. Truncating the distribution’s lower end with Puts
price of the initial put to a given percentage of the stock price, as in the fixed
percentage strategy. If the stock price rises and the put matures, the fixed
percentage strategy is employed. If the stock price has dropped at the time
of maturity, the fixed strike strategy is applied. Thus, in case of a market
rise, the strike price is raised to lock in some of the gain and the rollover is
carried into a new put with a higher strike, equal to the chosen percentage of
current stock price. If the stock has dropped over the option’s life, the strike
is not lowered. With this program, the investor locks in early gains without
accepting the downside risk of the fixed percentage strategy.
For this study, put options with a maturity of three months will be used,
the strike price will be arbitrarily set to 95% of the stock price at rollover and
moreover, the fixed percentage strategy will be employed. The decision for
three-month-options is a compromise: longer term options are more expensive
and, assuming that they will be held to maturity, they will not allow continuing adaptation of the floor to the current market price. Shorter-term options
will yield more losses over continued market declines. As the most severe price
drops in the form of crashes usually happen within few days time and harsh
market declines can prevail for longer time than one or two months, the three
month-options seem to offer the best features in terms of cost, protection and
potential.
Furthermore, Leland (1999) found that using in-the-money options with
this strategy provides the only degree of protection acceptable for a portfolio
insurance strategy with options. Rolling over out-of-the-money options (5%
or 10% out of the money) does not yield sufficient protection from disastrous
movements at low volatility rates (sigma=10%). While a 105-strike fixed
percentage strategy offers low risk, it also offers a mean return well above the
riskless rate (> 2%) and good possibilities on the upside.
The fixed percentage strategy in combination with three-month-options
offers reasonable protection in the case of a sudden crash that happens in
the course of a few days while it allows the floor to increase with the market
over time. On the other hand, it does not become increasingly expensive in
the case of long-term market declines like the fixed strike strategy.4 It seems
4 The diffusion process and the positive drift contribute to the price moving further and
further away from its starting value, the longer the time period. This affects the performance
of the fixed strike strategy significantly, as the optional character of the position disappears.
Chapter 8. Creating portfolio skewness with options
153
more reasonable anyway to protect oneself against sudden market crashes
than against slow steady downward movements that happen over the course
of more than three months. In such a case an individual reaction like selling
parts of one’s risky assets probably proves more sensible than protection via
puts.
8.3
Enhancing the distribution’s upper end with
Calls
Leland (1999), (1980) and Booth, Tehranian, and Trennepohl (1985) showed
that an investor whose risk tolerance grows with wealth more quickly than
the average investor’s risk tolerance will want portfolio insurance (exhibiting
convexity); if his risk tolerance grows less quickly than the market’s, a covered
call strategy (exhibiting concavity) is optimal. With growing risk tolerance
the investor’s preference for greater skewness increases. Optimal strategies,
therefore, are preference dependent, and no preference or performance measure that depends on the first two moments of the return distribution alone
will correctly rank all alternatives for all investors.
Kraus and Litzenberger (1976) - assuming investors have a cubic utility
function - demonstrate that when the market portfolio has iid returns, the
average investor must have a preference for skewness. This implies a positive
third derivative of the investor’s utility function which in turn means that his
risky investments increase as his wealth increases.5
Empirical evidence for the preference of positive skewness is omnipresent.
Almost every house owner has a fire- and water- insurance; life insurances are
no rarity. While the buying of insurance and simultaneous purchase of lottery
tickets is incompatible with traditional utility functions, it makes perfect sense
when considering positively skewed return distributions. Investors desire to
The strategy is similar either to simply holding the stock (in the case of a market rise) or
to holding the riskless asset (if the price falls far below the starting value), with the former
dominating over the long run because of the drift. Since fixing the strike tends to lead to
rolling into puts that are increasingly deep in or out-of-the-money, either the protection
or the market exposure of the strategy disappears as time passes. It is therefore not a
procedure that many investors would follow over the long run.
5 Pratt (1964), Arrow (1965).
154
8.4. Implementation and Limits of Options strategies
sell potential losses for a certain low cost (they pay the put option premium)
while at the same time they are willing to purchase high upside potential
for a small price (the call option premium). The success of insurances and
state lotteries are an obvious proof that most people exhibit this kind of
behavior. They do not esteem low, certain incomes and are averse to losses.
As a consequence they like to gamble for large amounts that can be gained
only with very low probability. They appreciate high potential gains as an
additional investment, even if receiving them is highly improbable.
8.4
Implementation and Limits of Options strategies
The previous sections primarily discussed the impact of option strategies on
the moments of the underlying portfolio. Nothing was said about how to best
design and implement an option strategy. Static protective put strategies
where the level of protection is not adjusted as the underlying’s price increases,
generally tend to underperform. Dynamic protective put strategies where the
options are replaced with puts of higher strikes as the price of the underlying
increases, effectively protect from losses, but involve significant transaction
costs. A cheaper way of protection offer barrier options. However, their
disadvantage comes in the form of lower liquidity, non-standardized trading
and less choice and supply.
In a two-moment setting a covered call strategy seems to improve the
risk-adjusted performance of a portfolio, as the expected return is increased
by the option premium and the volatility decreased by truncating the upper
end of the distribution. This is a major flaw of the CAPM and two-moment
asset pricing frameworks, as the asymmetry of risk is not considered in these
models.
It can easily be shown that the systematic writing of covered call options
or the systematic purchase of protective puts cannot stochastically dominate
the simple long stock (buy-and-hold) strategy. These ‘additions’ should only
be considered as ad-hoc measures to increase performance in specific personal
or particular market situations.
At low exercise prices the isolated covered call strategy approximates the
Chapter 8. Creating portfolio skewness with options
155
risk-free asset, while it almost resembles a stock for high strikes.
The picture is reversed for the isolated protective put strategy. For low
exercise prices it is similar to a pure stock, while imitating a risk-free bond
for high strike prices. Increasing the strike price of a protective put position
signifies moving from a pure stock to a pure bond investment.6
Finally, a general remark of caution about the efficiency of option hedges
needs to be made at this point. While, compared with dynamic trading
strategies such as TIPP or CPPI, option strategies are very reliable in crash
situations and market crises, their protection is limited by the issuing counterparty. Put options cannot cushion the impact of an overall economic crash
or worldwide depression. If a freefall similar to the one in 1929 was to happen
again, the consequences on the banking system and guarantors of financial
instruments in general is uncertain.
6 and
vice versa for a covered call strategy.
156
8.4. Implementation and Limits of Options strategies
Chapter 9
Joint estimation of three
moment preference
This last section will show theoretically how three-moment risk aversion can be
estimated jointly from the checklist questions and the three moment gamble.
As mentioned in subsection 3.5.1 the utility function of the general parameter
preference model by Rubinstein (1973) will be employed. Limited to the first
three moments, the utility of investor i is defined in terms of expected return
µ, volatility σ and skewness m3 (unnormalized third moment) of investment
opportunity j.
Ui
1 U 1 U 2 3
W σj2 +
W mj
2U
6 U
1
1
= µj + αi σj2 + ψi m3j
2
6
= µj −
(9.1)
where −W · U /U equals αi , the investor’s Pratt-Arrow measure of relative
risk aversion. This aversion to variance, αi , is determined by the first gamble
on page 137 and equation 7.6.
The second determinant of the investor’s utility function is skewness preference, depicted by ψi or W 2 · U /U . It is assessed by the second gamble on
page 7.2.2 and equation 7.7.
Both quantities, αi and ψi can also be determined through two separate
regressions αi = β xi and ψi = γ zi . Factors explaining the aversion to vari157
158
ance αi that can be used as independent factors have already been discussed.
Those accounting for skewness preference will need to be determined in a
separate examination and are not a subject matter of this study.
The regression model is thus
1
1
= µj − β xi σj2 + γ zi m3j + νi j
2
6
Ui
(9.2)
where νi j is an iid Gumbel random variable capturing measurement errors
not accounted for by the model.
The joint estimation model for the regression and the gambles has the
log-Likelihood function
logL =
I
{ln [αi − ϑβ xi ] + ln [ψi − θγ zi ] + lnPi (ji )}
(9.3)
i=1
with αi
ψi
= 2
=
ER − CEi
σg2
6((ER3 − CE3i ) − 0.5αi σ32 )
E[µ̃]W 2
σg2 stands for the gamble’s variance and is not to be confounded with the
variance of the investment σj2 ; ϑ and θ are coefficients of the joint estimation
and Pi (ji ) = exp(Ui )/( exp(Ui )).
Chapter 10
Summary and Conclusion
This study has presented three novel modules that were combined in an integrated approach to structuring the allocation of an investor’s free wealth in a
two- and three-moment asset pricing framework.
Firstly, econometric models of discrete choice were applied to portfolio
selection in a two-moment, traditional mean-variance, setting. It was shown
that the choice of the best model depends on the format of the data - or more
specific on the dependent variable, the stock ratio. The best performance was
defined a) as predictive power of the model to classify observations correctly
and b) the ability to cover and seize the whole spectrum of risk classes by
assigning observations evenly and consistently well over all risk segments.
Both criteria a) and b) could be improved considerably by dividing the
estimation in two and three steps, respectively. Especially the prediction of
choices between different levels of stock ratio was significantly better when
analyzed separately (setting 3c).
When considering samples that contain people without any financial assets
and when the dependent variable is in continuous format, then the Tobit
Model performs better than the OLS. When the dependent variable is in
discrete format and multinomial, the MNL Model performs better than the
Ordered Logit Model.
In the case where the sample contains only people that dispose of financial assets, the MNL Model performs best, regardless whether the dependent
159
160
variable is in continuous or discrete format.
In the case where there are very few risk classes as in the binomial settings1 ,
the WLS and MNL performed almost equally well with the MNL slightly
superior.
Apart from a higher number of independent factors, its obvious advantage is that it can account for nonlinearities in the relation of the dependent
variable and the independent factors. This advantage became evident from
the Figures in Table 3.3. No independent factor in the data sample is strictly
monotone increasing or decreasing in the risk classes.
In addition to applying econometric models in a two-moment setting, gambles were employed to measuring risk aversion. Also, a simultaneous equation
model for the two estimates of risk aversion was derived and explained theoretically. This joint estimation by stated and observed preferences given in
the form of gambles and observed stock ratios leads to more robust results,
as gambles can correct for biases and mistakes in the choice of the investor’s
actual, current stock ratio.
In the second part of the study the portfolio selection model was extended
to three moments. It was shown that two moments do not suffice for portraying the investor’s preferences appropriately, as portfolios are usually not
rebalanced continuously. The consideration of the third moment, skewness,
matters as all investors prefer positively skewed return distributions. Some
individuals will want to buy additional skewness, while others might want
to trade in some skewness for obtaining a riskfree premium. These preference patterns are determined by moment preference trade-offs which can be
determined and calculated for individual investors through gambles. It was
further demonstrated how these preference trade-offs can be implemented in
an investment strategy through option strategies that are based on different
strike prices.
Thirdly, the simultaneous equation model for three moments of econometric choice and the gamble was derived theoretically.
The study’s goal was to survey different methods for determining individual risk aversion as well as examine and develop new ways of assessing it with
1 If the dependent variable is in continuous format, a binomial setting implies discretizing
it and thus essentially increasing the error term, as pointed out in Subsection 3.5.4 on
Page 76
Chapter 10. Summary and Conclusion
161
econometric models. Its goal did not consist in determining the factors most
relevant in choosing the portfolio’s risk level.
The limited number of only 25 factors in the empirical analysis of the large
SCF samples was bound to result in a correspondingly modest predictive
power. The likelihood ratio tests proved that in all models those variables
not significant at the 5% level can be jointly left out without changing the
classification results. Nevertheless, the assignment performance of the three
step estimation by the Multinomial Logit model can be rated as quite good.
Overall, the thesis can be characterized as a practical study on how to
determine risk aversion. This topic is relevant for a number of reasons, but
particularly because private clients often lack financial knowledge while investment advisors lack the standardized tools to assess their clients’ preferences.
As a consequence, these preferences are never determined nor documented
and investors keep their misconceptions about what their investment strategies can achieve. The evolving field of behavioral finance plays a prominent
role in detecting these misconceptions and finding ways to eliminate them.
The study at hand has tried to contribute its part to this mission.
The possible extensions of this line of research are manifold. A more
profound insight into the individual improvement of investment decisions and
the change of risk taking over time is given by longitudinal studies (panel
data). It is evident that there is great potential for optimizing individual
portfolios given the financial objectives of investors. The internet provides an
ideal platform for offering risk assessment and allocation tools to the public
at no cost. These tools can also be a signal of quality for the corresponding
institute and offer a good possibility to make a first contact with a new client.
After all, both the individual investor and the advising financial institution
have the same goal - to profoundly comprehend the individual’s preferences.
“Know thyself” were the two words inscribed in the Apollo temple in Delphi.
- An almost ironical welcome for visitors of an oracle, but possibly still valid,
even when considering issues of asset allocation ...
162
Appendix A
Asset Allocation and
Higher Moment Models
A.1
A.1.1
Optimal Asset Allocation in the 2-Moment
Model
Objective Function
We analyze the utility Uij for investor i, i = 1, ..., N who can choose k investment alternatives j = 1, ..., k. Essentially these are all funds lying on the
Capital Market Line. They are correctly priced and clearly distinguishable by
their risk - return profile. As all three moments of the return (expected value,
standard deviation (risk) and skewness) are known, the individual’s choice
depends only on his risk aversion.
We have
• W0 – Wealth at the outset of the period
(W )
• α = − UU (W
) – the Pratt-Arrow Measure of absolute risk aversion
• µPf = rf + βPf (µM − rf ),
describes the portfolio return µPf where M stands for Marketportfolio,
163
164
A.1. Optimal Asset Allocation in the 2-Moment Model
rf for riskfree rate and βPf for the beta of the portfolio (according to
the CAPM)
• WT = (1 + µPf )W0
is a random variable defining terminal value of wealth at time T (end
of period)
Assumptions
1. U (w) = − exp(−αW ), where W > 0.
2. The investor maximizes his expected utility: Max E[U (W )]
3. Stock returns are normally distributed: µi ∼ N [µ, σ 2 ]
4. Utilities of different individuals are independently distributed
Assuming that a rational investor maximizes his expected utility, we have
M ax E[U (WT )]
(A.1)
For clarifying the connection between expected utility and mean-variance
decision criterion we approximate the investor’s utility function at E(WT ]
with a Taylor Polynom
U (WT ) = U (E[WT ]) + U (E[WT ])(WT − E[WT ])
1
+ U (E[WT ])(WT − E[WT ])2
2
∞
1 (i)
U (E[WT ])(WT − E[WT ])i
+
i!
i=3
(A.2)
where U (i) stands for the i-th derivative of the utility function. Assuming
that the Taylor polynom converges and that E and
are interchangeable,
we can write
E[U (WT )]
1
2
= U (E[WT ]) + U (E[WT ])σw
2
∞
1 (i)
+
U (E[WT ])m(i)
i!
i=3
(A.3)
Chapter A. Asset Allocation and Higher Moment Models
165
2
where σw
describes the variance of terminal wealth and m(i) refers to the ith central moment of the probability distribution of WT : E[(WT − E(WT ))i ].
Expected utility thus depends not only on mean and variance, but in addition
on higher central moments such as skewness and curtosis. For now, in order
to have a mean-variance portfolio selection lead to utility maximization, we
can choose between two possible roads of simplification:
A
we assume that discrete returns of risky assets are distributed multivariate normally N (µ, σ 2 ). As the normal distribution is symmetric in its
mean, all higher odd central moments cancel out and all portfolios can
be fully characterized by their mean-variance profile. Regardless of their
specific utility function all risk averse investors will then for a given mean
minimize their portfolio’s variance.
B
we can assume the investor has a quadratic utility function. In that case
all moments higher n = 2 have no impact on the expected utility, as those
derivatives become zero. This alternative however is not very realistic as
has been pointed out.
Proceeding with option A, our investor n has a utility function of
Un = − exp(−αn WT )
(A.4)
where WT = W0 [ω (µi − rf ) + rf ], with ω being the vector of weights of
the single assets i within the portfolio. The investor maximizes his expected
utility, risky terminal wealth being normally distributed:
Max E[− exp(−αn WT )]
(A.5)
Since we assumed WT = W0 (1+ω (µi −rf )+rf ] to be normally distributed
with mean µ and variance σ 2 , we can write the expectation as -φ(ia) where φ
is the characteristic function of a normal distribution with mean and variance
equal to that of the portfolio.
The problem known as the “hybrid model”1 can thus be restated as
Max [− exp(−αn (ω (µi − rf ) + rf ) +
1 Spremann
(2000).
αn2 2
σ )]
2 Pf
(A.6)
166
A.1. Optimal Asset Allocation in the 2-Moment Model
The utility function to be maximized is an ordinal function of mean and
variance. As it is ordinal, we can operate on it with the increasing function
1
θ(x) = −
log(−x)
(A.7)
α
As maximizing expected utility equals minimizing the argument of the
exponential function, we arrive at
Max [ω (µi − rf ) + rf ] −
αn 2
σ
2 Pf
(A.8)
The above expression equals the certainty equivalent return corresponding
to a risky investment that is made up of the riskfree and one efficient risky
asset - namely the market portfolio. The term behind the square bracket is
the Pratt-Arrow measure of the local risk premium.
By substituting terminal wealth with the mean-variance features of the
portfolio that the investor chooses, the objective function of the maximization
problem can be rewritten as:
Max E[Un ] = µPf −
αn 2
σ + εn
2 Pf
(A.9)
where εn is a random disturbance capturing measurement errors and factors not accounted for by our model.
A.1.2
Solution of the Objective Function
The following closed-end solution for 2-moment risk aversion (Pratt-Arrow
measure), given the stock ratio, the portfolio’s mean expected return, its
variance and the risk-free rate, is thought to be a comparison for the estimates
arrived at in the regression models.
It lays the foundation for using the stock ratio as a proxy for risk aversion
in the two-moment model framework. As can be seen, the Pratt/Arrow 2moment risk aversion is indeed closely tied to the stock ratio. The formula
has been used in various studies that estimated relative risk aversion. Among
them are: Blume and Friend (1975), Morin and Suarez (1983), Siegel and
Hoban (1982) as well as Riley and Chow (1992). An investor values his
asset allocation strategy according to all possible, random events ZW likely
Chapter A. Asset Allocation and Higher Moment Models
167
to happen in the course of the year
Max [E[WT ] −
αi 2
σ ]
2 Pf
(A.10)
The investor is given by:
• his wealth w,
• his absolute risk aversion as measured by Pratt/Arrow’s coefficient αA
• and his asset allocation strategy x - that is, the ratio of the market
portfolio (stocks) in his portfolio.
According to the CAPM two types of assets are given: the risk-free described by rf , and the risky Marketportfolio given by its expected return
E(rM ) and its volatility σM .
With a stock ratio of x the investor yields a portfolio return of µw =
rf + x(µM − rf ), the risk being σw = xσM . At the end of one year his
portfolio has thus an expected value of w(1 + µw ) with a volatility of w ∗ σw .
If the investor values his decision according to the classic criterion his asset
allocation strategy is given by:
Max E[Un (x)]
αA 2 2
[w σw (x)]
(A.11)
2
αA 2 2 2
[w x σM ]
= w(1 + rf + x(µM − rf )) −
2
= w(1 + µw (x)) −
The objective function is quadratic in x. Its first-order condition
2
=0
w(µM − rf ) − αA xw2 σM
(A.12)
yields a maximum, as it is concave in x. The optimal 2-moment asset allocation is thus:
µM − rf
(A.13)
x2m =
2
αA wσM
Solving for Pratt/Arrow’s 2-moment measure of absolute risk aversion
yields
µM − rf
αA =
(A.14)
2
x2m wσM
168
A.2. Optimal Asset Allocation in the 3-Moment Model
A.2
Optimal Asset Allocation in the 3-Moment
Model
We have
• W0 – Wealth at the outset of the period
• x – Amount of wealth to be invested in the well-diversified risky asset
(Marketportfolio)
• r̃Pf – Random excess portfolio return over the risk-free rate rf
• W – End-of-period wealth W ≡ W0 + W0 [x(r̃Pf + rf ) + (1 − x)rf )]
Assumptions
1. The investor maximizes his expected utility: Max E[U (W )]
2. Utilities of different individuals are independently distributed
We expand expected utility to include the third moment:
E{U [W (1 + xr̃Pf )]}
= U (W ) + xW U (W )E(r̃Pf )
1
2
)
+ x2 W 2 U (W )E(r̃Pf
2
1
3
+ x3 W 3 U (W )E(r̃Pf
) + ...
6
(A.15)
The first-order condition (FOC) for a maximum of the above objective
function using three moment approximation is
W U (W )E(r̃Pf )
2
+ xW 2 U (W )E(r̃Pf
)
1 2 3 3
x W U (W )E(r̃Pf
)=0
+
2
(A.16)
We are faced with an objective function quadratic in x. Applying the
standard solution yields two results. Only one of these two has a positive
Chapter A. Asset Allocation and Higher Moment Models
169
second derivative and thus satisfies the sufficient condition for a maximum.
The optimal 3-moment asset allocation is
−1 − 1 − 2x2m ξA(W )
(A.17)
x3m =
ξA(W )
with x2m =
r̃Pf
2
αAP wσPf
,ξ=
3
E(r̃Pf
)
2 )
E(r̃Pf
=
m3P f
2
σPf
(W )
and A(W ) = −W UU (W
)
Comparing the optimal solutions for 2- and 3-moments (equations A.17
and A.12) we find that the two are equal when the skewness measure, the third
moment of the return distribution, approaches zero: limm→0 x2m = x3m .
As one would expect, the allocation to the risky asset with the three
moment approximation exceeds the one with the mean-variance solution when
the skewness of returns is positive. The opposite holds when the skewness is
negative.
The existence of a three-moment solution requires that x2m ξA(W ) ≤ 0.5.
This condition has to be examined more closely:
When analyzing each variable separately, we see that x2m > 0 except
for the unlikely case where µPf < rf . A(W ) > 0 ∀ W > 0, as U < 0
and U > 0.2 The sign of ξ depends solely on m3 . Thus, the 3-moment
approximation always has a solution if m < 0 and µPf > rf .
If m > 0, the existence of an optimal solution heavily depends on the ratio
m3
σ2 .
m3 <
σ 2 σ 2 U U 2r̃U U (A.18)
When skewness exceeds the threshold characterized above, there is no interior
solution to the allocation decision, meaning that a risk-averse individual would
2 Pratt (1964) and Arrow (1971) suggest that desirable properties for an investor’s utility
function are:
1. U (x) > 0, non-satiation, positive marginal utility for wealth which means nonsatiety
with respect to wealth.
2. U (x) < 0, decreasing marginal utility for wealth, i.e. risk aversion and
3. non-increasing or decreasing absolute risk aversion i.e. risky assets are not inferior
goods.
170
A.2. Optimal Asset Allocation in the 3-Moment Model
invest as much as possible in the risky asset, even if that meant taking infinite
liabilities. This kind of behavior was coined ‘plunging’ by Tobin (1958). It
depicts the trade-off between mean-variance and higher-moment models, as
the advantages won by incorporating skewness are partly lost by the limited
range of its applicability: For portfolios which are sufficiently skewed, the
three-moment approximation will be inferior to the mean-variance solution.
Appendix B
Deriving the MNL model
from Utility maximization
The following lines are taken from Amemiya (1985), 297: Consider an individual i whose utilities associated with three alternatives j are given by
Uij = xi + εij j=0,1 and 2
(B.1)
where xi is a nonstochastic function of explanatory variables and unknown
parameters and εij is an unobservable random variable (we will write εj from
here on for simplification). It is assumed that the individual chooses the alternative for which the associated utility is the highest. McFadden (1973) proved
that the multinomial logit model is derived from utility maximization iff {εj }
are independent1 and the distribution function of εj is given by exp[exp(−εj )],
the Type I extreme-value distribution (Gumbel), or log Weibull distribution2 .
Its density is given by f (.) = exp(εj ) exp[− exp(εj )]
Proof of the iff part: The probability that the ith person chooses alterna1 IID
disturbances constrain all the disturbances to have the same scale parameter implying that the variances of the random components of the utilities are equal (homoscedastic).
The independence of the random components of the utilities give rise to the IIA problem
that both the MNL and the CLM share.
2 See Johnson and Kotz (1970), 272. It states that the maximum of independent Gumbel
variates with a common scale parameter is itself Gumbel distributed.
171
172
tive j can be expressed as (suppressing the subscripts i from µ):
P (yi = 2)
=
P (Ui2 > Ui1 , Ui2 > Ui0 )
=
P (ε2 + µ2 − µ1 > ε1 , ε2 + µ2 + µ0 > ε0 )
 ε +µ −µ

ε2 +µ
2 2
1
∞
2 −µ0
f (ε2 ) 
f (ε1 )dε1 ·
f (ε0 )dε0  dε2
=
−∞
−∞
(B.2)
−∞
∞
exp(ε2 ) exp[− exp(−ε2 )] × exp[− exp(−ε2 − µ2 − µ2 + µ1 )]
=
−∞
=
× exp[− exp(−ε2 − µ2 − µ2 + µ0 )]dε2
exp(µi2 )
exp(µi0 ) + exp(µi1 ) + exp(µi2 )
This result is equal to Pij in the definition 3.18 of a multinomial logit model,
if we put µi2 − µi0 = xi2 β and µi1 − µi0 = xi1 β. The expressions for Pi0 and
Pi1 can be similarly derived.
It has been shown that Maximization of a utility function (UM) with iid
error terms that exhibit an extreme value distribution (EVD) give rise to a
multinomial logit model (MLM). Or expressed with symbols:
U M, IID, EV D ❀ M LM
A proof utilizing the approach above shows that the same constellation conditions the indepence from irrelevant alternatives (IIA):
U M, IID, EV D ❀ IIA
Proof:
P (Ui1 > Ui2
=
=
P (µ1 + ε1 > µ2 + > ε2 )
∞ ε1 +µ
1 −µ2
f (ε2 )dε2 f (ε1 )dε1
−∞
(B.3)
−∞
∞
exp[− exp(−ε1 + µ1 − µ2 )] exp(−ε1 ) exp[− exp(−ε1 )]dε1
=
−∞
=
+∞
1
exp[−(1 + exp(µ1 + µ2 ))]
1 + exp(−µ1 + µ2 )
−∞
Chapter B. Deriving the MNL model from Utility maximization
=
exp(µ1 )
exp(µ1 ) + exp(µ2 )
=
P (U1 > U2 |U1 > U3 or U2 > U3 )
173
Thus:
P (y1 |y1 or y2 )
When carrying out the same calculations for a utility with normally distributed error terms (normal), it can be shown that:
U M, IID, normal ❀ IIA
Thus, the Probit model does not share the sometimes disadvantageous IIA
feature with the MNL model. It poses a useful alternative for problems where
the IIA is clearly not met.
174
Appendix C
Likelihood function of the
Nested Logit Model
Following the likelihood function for simplest nested logit model - one with
three responses depicted below - is derived. The distribution of the error
terms is as follows:
F (ε3 )
=
F (ε1 , ε2 )
=
exp [− exp (−ε3 )]
ρ ε1
ε2
exp − exp −
+ exp −
ρ
ρ
Figure C.1: Simplest Nested Logit Model with three responses. Utilities are
assumed to be positively correlated.
175
176
The latter represents a ‘Type b’ Gumbel bivariate distribution with the correlation between ε1 and ε2 ∼
= 1 − ρ2 and 0 < ρ ≤ 1.
As F (ε1 , ∞) = exp[− exp(−ε1 )] is a natural generalization of the MNL
model
δ2F
δε1 δε2
f (ε1 , ε2 ) =
f can be negative if ρ > 1.
Define
yji
=
1 if yj = j with j = 1, 2, 3
=
0 otherwise
Then (subscript will be omitted after the second line)
L =
=
Πi P (yi = 1)y1i P (yi = 2)y2i P (yi = 3)y3i
y1
Π [P (y = 1|y = 1 or y = 2)P (y = 1 or 2)]
y2
· [P (y = 2|y = 1 or y = 2)P (y = 1 or 2)]
=
· P (yi = 3)y3
ΠP (y = 1|y = 1 or y = 2)y1 P (y = 2|y = 1 or 2)y2
·ΠP (y = 1 or y = 2)y1 +y2 · P (y = 3)y3
≡ L1 · L2
It follows that
P (y = 1|y = 1 or 2)
=
=
As µji = xi βj :
P (y = 1|y = 1 or y = 2)
exp
=
exp µρ1 + exp
(µ1 − µ2 )
Λ
ρ
=
µ1
ρ
µ1
ρ
ρ
exp µρ1 + exp µρ2
exp( µρ1 ) + exp( µρ2 ) + exp(µ3 )
ρ
2
+1
exp(µ2 − µ3 ) exp µ1 −µ
ρ
ρ
2
+
1
exp(µ2 − µ3 ) exp µ1 −µ
+1
ρ
Appendix D
Structure of Estimations
177
178
Figure D.1: Structure and nests of empirical analyses
Appendix E
Empirical Results of
Regressions
E.1
In-sample estimation SCF1998
179
180
E.1. In-sample estimation SCF1998
Table E.1: Setting 1 - Results for OLS in-sample-regression: The dependent
variable is continuous: the stock ratio is given as a percentage. Dataset: SCF
1998.
Observations = 4305, Parameters = 25, Deg.Fr.= 4280
Adjusted R-squared = 0.33066, Log-L = -1679.8332
Akaike Info. Crt. = 0.792
Variable
Coeff.
Constant
X101
X1706
X3006 1
X3006 5
X3008 1
X3008 45
X301 1
X301 3
X3014 1
X3014 4
X432 1
X5608
X5821
X5825 3
X5905
X7131
X7186
X7187
X7372 1
X7401 1
X7401 2
X7401 5
X8021 1
X8022
-0.1212
-0.0083
0.0144
-0.1158
0.0117
-0.0575
0.0506
0.0139
-0.0224
0.0073
-0.1661
0.1211
0.0150
0.0112
-0.0864
0.0260
0.1015
0.0307
0.0185
⊕ 0.0517
0.0664
0.0781
⊕ 0.0066
0.0108
⊕ 0.0042
Standard
Error
0.0325
0.0047
0.0035
0.0387
0.0157
0.0164
0.0125
0.0138
0.0131
0.0233
0.0138
0.0133
0.0102
0.0041
0.0124
0.0049
0.0127
0.0067
0.0055
0.0165
0.0153
0.0176
0.021
0.0176
0.0004
b/St.Er.
-3.735
-1.751
4.169
-2.989
-0.747
-3.495
4.035
-1.012
-1.705
-0.312
-12.043
9.099
1.467
2.754
-6.957
5.293
-8.003
-4.603
3.361
3.141
4.333
4.449
0.315
0.614
9.731
P of
|Z| > z
0.0002
0.0799
0
0.0028
0.4551
0.0005
0.0001
0.3117
0.0882
0.755
0
0
0.1424
0.0059
0
0
0
0
0.0008
0.0017
0
0
0.7528
0.5392
0
Mean
of X
2.6492
1.2393
0.0207
0.1554
0.1619
0.4476
0.2434
0.2767
0.0609
0.3034
0.5094
0.1131
0.5422
0.3933
1.121
0.3654
1.1779
1.9331
0.5988
0.3443
0.1466
0.0904
0.7807
49.8404
Chapter E. Empirical Results of Regressions
181
Setting 1 - OLS, SCF 1998 (cont.)
Classification table
Predicted
2
3
10
0
667
292
100
152
96
130
90
158
73
137
81
146
1117 1015
0
1
271
21
919
297
14
15
10
13
9
12
13
12
17
18
1253
388
Percent correct: 20.74%
Error Distance: 9’562’127
Actual
0
1
2
3
4
5
6
4
0
70
65
85
93
92
106
511
5
0
1
3
6
2
2
7
21
6
0
0
0
0
0
0
0
0
302
2246
349
340
364
329
375
4305
182
E.1. In-sample estimation SCF1998
Table E.2: Setting 1 - Results for Tobit in-sample-regression: The dependent variable is continuous: the stock ratio is given as a percentage. Dataset:
SCF 1998.
Observations = 4305, Parameters = 25, Deg.Fr.= 4280
Log-L = -2363.740, Akaike Info. Crt. = 1.110
Variable
Coeff.
Standard
Error
Primary Index Equation for Model
Constant
-0.5437
0.0612
X101
-0.0084
0.0087
X1706
0.0248
0.0053
X3006 1
-0.0101
0.0641
0.017
0.0274
X3006 5
-0.0832
0.0329
X3008 1
0.1285
0.0212
X3008 45
0.0112
0.0241
X301 1
-0.0145
0.0223
X301 3
0.0353
0.0365
X3014 1
-0.3793
0.0278
X3014 4
X432 1
0.2207
0.023
X5608
0.0175
0.0157
X5821
0.0163
0.0063
-0.1561
0.0223
X5825 3
X5905
0.046
0.0078
X7131
0.0777
0.0223
X7186
0.0306
0.0111
X7187
0.0332
0.0085
⊕0.073
0.0295
X7372 1
0.0716
0.0252
X7401 1
X7401 2
0.069
0.0305
-0.0663
0.0428
X7401 5
0.0158
0.0335
X8021 1
X8022
⊕0.0051
0.0008
Disturbance Standard Deviation
Sigma
0.5006
0.0094
b/St.Er.
P of
|Z| > z
Mean
of X
-8.883
-0.963
4.698
-0.158
-0.62
-2.524
6.073
0.465
-0.65
0.966
-13.638
9.613
1.113
2.614
-6.988
5.93
-3.487
-2.743
3.908
2.478
2.838
2.266
-1.549
0.471
6.214
0
0.3357
0
0.8746
0.5356
0.0116
0
0.642
0.5156
0.3339
0
0
0.2658
0.009
0
0
0.0005
0.0061
0.0001
0.0132
0.0045
0.0234
0.1214
0.6376
0
2.6492
1.2393
0.0207
0.1554
0.1619
0.4476
0.2434
0.2767
0.0609
0.3034
0.5094
0.1131
0.5422
0.3933
1.121
0.3654
1.1779
1.9331
0.5988
0.3443
0.1466
0.0904
0.7807
49.8404
53.456
0
Chapter E. Empirical Results of Regressions
183
Setting 1 - Tobit, SCF1998 (cont.)
Classification table
Predicted
2
3
48
0
1051
234
134
136
116
110
115
143
90
122
112
124
1666
869
0
1
183
71
455
439
4
11
4
5
2
5
2
12
5
11
655
554
Percent correct: 22.49%
Error Distance: 8’967’647
Actual
0
1
2
3
4
5
6
4
0
60
56
90
93
94
101
494
5
0
7
8
15
6
9
22
67
6
0
0
0
0
0
0
0
0
302
2246
349
340
364
329
375
4305
184
E.1. In-sample estimation SCF1998
Table E.3: Setting 1 - Results for Ordered model in-sample-regression:
The dependent variable (stock ratio) is discrete, 7 risk classes are considered.
Dataset: SCF 1998.
Observations = 4305, Parameters = 25, Deg.Fr.= 4280
Log-L = -7833.028, Akaike Info. Crt. = 3.651
Variable
Coeff.
Standard
Error
Index function for probability
Constant
1.112
0.0667
X101
-0.0301
0.0101
X1706
0.0896
0.0061
-0.2245
0.0773
X3006 1
0.078
0.0317
X3006 5
-0.2242
0.0371
X3008 1
0.3742
0.0236
X3008 45
X301 1
0.0039
0.0267
-0.0748
0.0255
X301 3
0.0694
0.0413
X3014 1
-0.9338
0.0314
X3014 4
X432 1
0.6213
0.0251
X5608
0.0853
0.02
X5821
0.0594
0.0069
-0.4518
0.0247
X5825 3
X5905
0.1491
0.0088
X7131
0.2805
0.0254
X7186
0.1075
0.0128
X7187
0.1188
0.0098
⊕0.2599
0.0328
X7372 1
0.2513
0.0283
X7401 1
0.2609
0.0336
X7401 2
-0.1379
0.0495
X7401 5
0.0124
0.0365
X8021 1
X8022
⊕0.0162
0.0009
b/St.Er.
P of
|Z| > z
Mean
of X
16.677
-2.972
14.738
-2.904
-2.46
-6.041
15.875
-0.145
-2.934
1.68
-29.704
24.783
4.273
8.635
-18.322
16.877
-11.021
-8.389
12.156
7.914
8.891
7.766
-2.788
0.34
17.886
0
0.003
0
0.0037
0.0139
0
0
0.8847
0.0033
0.093
0
0
0
0
0
0
0
0
0
0
0
0
0.0053
0.7337
0
2.6492
1.2393
0.0207
0.1554
0.1619
0.4476
0.2434
0.2767
0.0609
0.3034
0.5094
0.1131
0.5422
0.3933
1.121
0.3654
1.1779
1.9331
0.5988
0.3443
0.1466
0.0904
0.7807
49.8404
Chapter E. Empirical Results of Regressions
185
Setting 1 - Ordered model, SCF1998 (cont.)
Variable
Coeff.
Standard b/St.Er.
P of
Mean
Error
|Z| > z
of X
Threshold parameters for index
Mu(1)
2.0000
.... (Fixed Parameter) ....
Mu(2)
3.0000
.... (Fixed Parameter) ....
Mu(3)
4.0000
.... (Fixed Parameter) ....
Mu(4)
5.0000
.... (Fixed Parameter) ....
Mu(5)
6.0000
.... (Fixed Parameter) ....
Classification table
Actual
0
1
2
3
4
5
6
0
1
2
55
241
6
56
1443
417
0
59
76
0
49
69
0
52
64
0
39
60
1
57
62
112
1940
754
Percent correct: 41.79%
Error Distance: 4’265’498
Predicted
3
0
260
146
121
146
128
131
932
4
0
68
65
95
101
99
112
540
5
0
2
3
6
1
3
12
27
6
0
0
0
0
0
0
0
0
302
2246
349
340
364
329
375
4305
186
E.1. In-sample estimation SCF1998
Table E.4: Setting 1 - Results for Multinomial Logit model in-sampleregression. The standard errors of the regression coefficients are given in
brackets. The significance levels are abbreviated with asterisks: ‘*’ and ‘**’
are significant at the 5, and 1 percent level.
Observations = 4305, Parameters = 25, Deg.Fr.= 4280
Log-L = -5350.629, Akaike Info. Crt. = 2.497
Constant
X101
X1706
X3006 1
X3006 5
X3008 1
X3008 45
X301 1
X301 3
X3014 1
X3014 4
X432 1
X5608
Prob[Y=1]
1.0538 **
(0.3994)
-0.0339
(0.0523)
0.4146 **
(0.159)
-1.7349 **
(0.4149)
-0.1063
(0.1939)
-0.4319 **
(0.1658)
-0.1119
(0.1802)
-0.3736 *
(0.1686)
-0.1474
(0.1777)
-0.9411 **
(0.3438)
-0.6546 **
(0.1651)
1.1629 **
(0.2551)
0.1729
(0.2175)
Prob[Y=2]
-3.5672 **
(0.5752)
-0.0562
(0.077)
0.5051 **
(0.1619)
-1.6767 **
(0.5651)
-0.284
(0.2658)
-0.5417 *
(0.2688)
0.2242
(0.2224)
-0.4932 *
(0.2331)
0.0247
(0.2239)
-0.9286 *
(0.416)
-1.4066 **
(0.2347)
2.1681 **
(0.2936)
0.2228
(0.2423)
Prob[Y=3]
-2.9594 **
(0.575)
-0.1245
(0.0791)
0.5589 **
(0.162)
-1.3543 *
(0.5598)
-0.3208
(0.2685)
-0.4074
(0.2764)
0.4887 *
(0.2266)
-0.4579
(0.2364)
0.0681
(0.226)
-0.798
(0.4084)
-2.2076 **
(0.2748)
1.894 **
(0.2942)
0.3629
(0.2365)
Prob[Y=4]
-2.4082 **
(0.5588)
-0.1306
(0.0786)
0.4944 **
(0.1619)
-1.4221 *
(0.5595)
0.1105
(0.2538)
-0.7436 *
(0.2901)
0.4772 *
(0.2235)
-0.3995
(0.2287)
-0.1483
(0.2272)
-0.9396 *
(0.4111)
-2.2787 **
(0.2719)
1.9302 **
(0.2934)
0.1589
(0.2452)
Prob[Y=5]
-3.1688 **
(0.5823)
-0.008
(0.0775)
0.5416 **
(0.162)
-1.7018 **
(0.5995)
-0.229
(0.267)
-0.925 **
(0.3138)
0.4886 *
(0.2273)
-0.2277
(0.2306)
-0.1243
(0.2321)
-0.7429
(0.4064)
-2.2043 **
(0.2841)
1.9268 **
(0.2968)
0.3093
(0.2395)
Prob[Y=6]
-2.6335 **
(0.5614)
-0.0568
(0.0759)
0.5904 **
(0.1617)
-1.6559 **
(0.5822)
-0.2186
(0.2596)
-0.7137 *
(0.2873)
0.4872 *
(0.2231)
-0.2361
(0.2233)
-0.3184
(0.2293)
-0.6175
(0.3971)
-2.0304 **
(0.2636)
1.9268 **
(0.2919)
0.3389
(0.2369)
Chapter E. Empirical Results of Regressions
187
Setting 1 - Results for Multinomial Logit model, SCF1998, in-sample-regression.
(cont.)
X5821
X5825 3
X5905
X7131
X7186
X7187
X7372 1
X7401 1
X7401 2
X7401 5
X8021 1
X8022
Prob[Y=1]
0.2132 *
(0.1026)
-0.4328 **
(0.1563)
0.3968 **
(0.1418)
-1.0497 **
(0.1569)
-0.4994 **
(0.1042)
0.3971 **
(0.143)
0.4332 *
(0.2007)
1.1768 **
(0.3245)
1.1236 **
(0.2563)
0.2441
(0.2078)
0.1976
(0.1845)
0.0387 **
(0.005)
Prob[Y=2]
0.272 *
(0.11)
-0.7954 **
(0.2098)
0.5928 **
(0.1489)
-1.1817 **
(0.2107)
-0.5603 **
(0.1256)
0.6279 **
(0.1518)
0.6744 *
(0.2731)
1.6201 **
(0.3608)
1.5056 **
(0.3238)
0.2593
(0.3602)
0.5389
(0.2953)
0.0593 **
(0.0073)
Prob[Y=3]
0.292 **
(0.1096)
-0.7071 **
(0.2131)
0.6075 **
(0.1493)
-1.2205 **
(0.2146)
-0.7976 **
(0.1263)
0.6133 **
(0.1526)
0.9834 **
(0.2836)
1.4964 **
(0.3626)
1.5388 **
(0.3243)
0.5048
(0.3545)
0.101
(0.2993)
0.0585 **
(0.0075)
Prob[Y=4]
0.2927 **
(0.1092)
-1.0674 **
(0.2157)
0.6534 **
(0.1488)
-1.0723 **
(0.2101)
-0.7377 **
(0.1253)
0.6996 **
(0.1519)
0.811 **
(0.2785)
1.2386 **
(0.3588)
1.214 **
(0.3231)
-0.1002
(0.3944)
0.0536
(0.2875)
0.0539 **
(0.0073)
Prob[Y=5]
0.2656 *
(0.1099)
-1.2747 **
(0.2263)
0.6775 **
(0.1495)
-1.3139 **
(0.2165)
-0.5239 **
(0.1277)
0.5893 **
(0.1527)
0.5703 *
(0.2806)
1.3041 **
(0.3629)
1.5492 **
(0.3223)
-0.1422
(0.4226)
0.234
(0.2987)
0.0594 **
(0.0076)
Prob[Y=6]
0.3092 **
(0.1087)
-0.9593 **
(0.2145)
0.5593 **
(0.1487)
-1.3443 **
(0.2116)
-0.6094 **
(0.1251)
0.6307 **
(0.1518)
0.7814 **
(0.2759)
1.4739 **
(0.3577)
1.1924 **
(0.3242)
-0.0783
(0.3834)
0.1102
(0.2902)
0.0529 **
(0.0073)
188
E.1. In-sample estimation SCF1998
Setting 1 - Multinomial model, SCF1998 (cont.)
Classification table
Predicted
2
3
0
0
10
10
9
17
4
30
6
17
7
13
8
25
44
112
0
1
73
229
45
2097
0
250
1
209
0
217
0
190
0
218
119
3410
Percent correct: 54.94%
Error Distance: 11’219’224
Actual
0
1
2
3
4
5
6
4
0
30
24
36
65
43
38
236
5
0
22
17
24
17
30
25
135
6
0
32
32
36
42
46
61
249
302
2246
349
340
364
329
375
4305
Chapter E. Empirical Results of Regressions
189
Table E.5: Setting 2a - Results for WLS in-sample-regression: The
dependent variable is binomial, investors without assets are to be separated
from investors. Dataset: SCF 1998.
Observations = 4305, Parameters = 25, Deg.Fr.= 4280
Adjusted R-squared = 0.17118 , Log-L = 183.6142
Akaike Info. Crt. = -0.074
Variable
WConstant
WX101
WX1706
WX3006 1
WX3006 5
WX3008 1
WX3008 45
WX301 1
WX301 3
WX3014 1
WX3014 4
WX432 1
WX5608
WX5821
WX5825 3
WX5905
WX7131
WX7186
WX7187
WX7372 1
WX7401 1
WX7401 2
WX7401 5
WX8021 1
WX8022
Coeff.
1.0375
-0.0062
0.0023
-0.0764
0.0085
-0.0353
0.0032
0.0188
-0.009
0.0241
-0.0516
0.0407
0.0025
0.0028
-0.0244
0.0028
0.0656
0.0199
0.0007
⊕0.0246
0.0398
0.0515
⊕0.0271
0.0101
⊕0.0024
Standard
Error
0.0112
0.0012
0.0007
0.0161
0.0044
0.0051
0.0028
0.0035
0.0029
0.0052
0.004
0.0031
0.0024
0.001
0.0034
0.0009
0.0038
0.0016
0.0012
0.0043
0.0038
0.0045
0.0042
0.0051
0.0001
b/St.Er.
92.616
-4.994
-3.029
-4.739
-1.933
-6.963
-1.162
-5.454
-3.121
-4.659
-12.818
13.229
1.062
2.78
-7.267
2.973
-17.247
-12.717
-0.624
5.689
10.408
11.542
6.511
1.997
18.356
P of
|Z| > z
0
0
0.0025
0
0.0532
0
0.2451
0
0.0018
0
0
0
0.2884
0.0054
0
0.0029
0
0
0.5324
0
0
0
0
0.0458
0
Mean
of X
4.4055
13.9801
7.9597
0.075
0.7525
0.6012
2.7309
1.1722
1.5339
0.3655
1.0782
3.2446
0.662
2.9799
1.6916
6.9568
1.7373
6.0276
10.9689
3.6183
2.0572
0.8631
0.4756
4.4393
274.983
190
E.1. In-sample estimation SCF1998
Setting 2a - WLS,SCF1998 (cont.)
Classification table
Predicted
0
1
185
117
273
3730
458
3847
Percent correct: 90.94%
Error Distance: 264’654
Actual
0
1
302
4003
4305
Chapter E. Empirical Results of Regressions
191
Table E.6: Setting 2a - Results for Tobit in-sample-regression: The dependent variable is binomial, investors without assets are to be separated from
investors. Dataset: SCF 1998.
Observations = 4305, Parameters = 25, Deg.Fr.= 4280
Log-L = -618.6330, Akaike Info. Crt. = 0.299
Variable
Coeff.
Standard
Error
Primary Index Equation for Model
Constant
0.8193
0.0226
X101
-0.0089
0.0033
X1706
0.0033
0.0024
X3006 1
-0.1106
0.027
0.0113
0.0109
X3006 5
-0.0485
0.0115
X3008 1
0.006
0.0087
X3008 45
0.0256
0.0096
X301 1
-0.014
0.0091
X301 3
0.0327
0.0162
X3014 1
-0.0701
0.0096
X3014 4
X432 1
0.0549
0.0092
X5608
0.0037
0.0071
X5821
0.004
0.0028
-0.035
0.0086
X5825 3
X5905
0.0037
0.0034
X7131
0.0906
0.0088
X7186
0.0281
0.0046
X7187
0.0011
0.0038
⊕0.034
0.0115
X7372 1
0.0521
0.0107
X7401 1
X7401 2
0.0725
0.0122
0.0349
0.0146
X7401 5
0.0173
0.0123
X8021 1
X8022
⊕0.0033
0.0003
Disturbance Standard Deviation
Sigma
0.2487
0.0029
b/St.Er.
P of
|Z| > z
Mean
of X
36.244
-2.697
-1.363
-4.09
-1.037
-4.229
-0.685
-2.67
-1.535
-2.014
-7.299
5.941
0.521
1.402
-4.047
1.074
-10.261
-6.057
-0.277
2.971
4.891
5.944
2.387
1.408
10.743
0
0.007
0.1728
0
0.2997
0
0.4933
0.0076
0.1247
0.044
0
0
0.6027
0.1608
0.0001
0.2828
0
0
0.7818
0.003
0
0
0.017
0.1592
0
2.6492
1.2393
0.0207
0.1554
0.1619
0.4476
0.2434
0.2767
0.0609
0.3034
0.5094
0.1131
0.5422
0.3933
1.121
0.3654
1.1779
1.9331
0.5988
0.3443
0.1466
0.0904
0.7807
49.8404
86.836
0
192
E.1. In-sample estimation SCF1998
Setting 2a - Tobit, SCF1998 (cont.)
Classification table
Predicted
0
1
85
217
89
3914
174
4131
Percent correct: 92.89%
Error Distance: 165’030
Actual
0
1
302
4003
4305
Chapter E. Empirical Results of Regressions
193
Table E.7: Setting 2a - Results for Binomial Logit model in-sampleregression: The dependent variable (stock ratio) is discrete, 6 risk classes are
considered, Dataset: SCF 1998.
Observations = 4305, Parameters = 25, Deg.Fr.= 4280
Log-L = -678.8651, Akaike Info. Crt. = 0.327
Variable
Coeff.
Standard b/St.Er.
Error
Characteristics in numerator of Prob[Y = 1]
Constant
1.148
0.3993
2.875
X101
-0.0361
0.0524
-0.688
X1706
0.4498
0.1573
2.859
X3006 1
-1.7036
0.4087
-4.169
0.1143
0.1942
-0.589
X3006 5
-0.4435
0.1662
-2.668
X3008 1
0.1799
-0.224
X3008 45 0.0403
0.3784
0.1687
-2.243
X301 1
-0.1418
0.1777
-0.798
X301 3
0.9157
0.3399
-2.694
X3014 1
-0.7956
0.1649
-4.826
X3014 4
X432 1
1.3092
0.2555
5.125
X5608
0.1844
0.2168
0.851
X5821
0.2236
0.1021
2.19
-0.4924
0.1562
-3.151
X5825 3
X5905
0.4385
0.1413
3.103
X7131
1.0615
0.1569
-6.765
X7186
0.5137
0.1042
-4.93
X7187
0.4369
0.1422
3.073
⊕0.4644
0.2008
2.312
X7372 1
1.2243
0.3244
3.774
X7401 1
X7401 2
1.1555
0.2562
4.51
⊕0.2235
0.2079
1.075
X7401 5
0.204
0.1846
1.105
X8021 1
X8022
⊕0.0405
0.005
8.188
P of
|Z| > z
Mean
of X
0.004
0.4915
0.0042
0
0.5562
0.0076
0.8228
0.0249
0.4251
0.0071
0
0
0.3949
0.0285
0.0016
0.0019
0
0
0.0021
0.0208
0.0002
0
0.2823
0.2691
0
2.6492
1.2393
0.0207
0.1554
0.1619
0.4476
0.2434
0.2767
0.0609
0.3034
0.5094
0.1131
0.5422
0.3933
1.121
0.3654
1.1779
1.9331
0.5988
0.3443
0.1466
0.0904
0.7807
49.8404
194
E.1. In-sample estimation SCF1998
Table E.8: Setting 2a Binomial Logit model, SCF1998 (cont.)
Classification table
Predicted
0
1
69
233
302
43
3960 4003
112
4193 4305
Percent correct: 93.59%
Error Distance: 168’414
Actual
0
1
Chapter E. Empirical Results of Regressions
195
Table E.9: Setting 2b - Results for OLS in-sample-regression: The dependent variable is continuous: the stock ratio is given as a percentage. Dataset:
SCF 1998.
Observations = 4003, Parameters = 25, Deg.Fr.= 3978
Adjusted R-squared = 0.23837 , Log-L = -547.2792
Akaike Info. Crt. = 0.286
Variable
Coeff.
Constant
X101
X1706
X3006 1
X3006 5
X3008 1
X3008 45
X301 1
X301 3
X3014 1
X3014 4
X432 1
X5608
X5821
X5825 3
X5905
X7131
X7186
X7187
X7372 1
X7401 1
X7401 2
X7401 5
X8021 1
X8022
0.056
0
0.0168
-0.0064
0.0011
-0.0162
0.0584
0.0125
-0.009
0.0302
-0.1085
0.069
0.0117
0.0073
-0.0569
0.022
0.0185
0.0046
0.0195
0.0188
0.0148
0.0082
-0.0283
-0.0063
⊕0.0012
Standard
Error
0.0266
0.0039
0.0027
0.0328
0.0127
0.0137
0.01
0.0112
0.0105
0.0187
0.0112
0.0105
0.008
0.0032
0.01
0.0039
0.0104
0.0053
0.0043
0.0134
0.0121
0.014
0.0174
0.0146
0.0004
b/St.Er.
2.106
-0.012
6.197
-0.194
-0.083
-1.182
5.824
1.122
-0.852
1.617
-9.669
6.555
1.454
2.291
-5.676
5.694
-1.787
-0.87
4.518
1.398
1.221
0.587
-1.629
-0.427
3.317
P of
|Z| > z
0.0352
0.9907
0
0.8458
0.9337
0.2371
0
0.2618
0.394
0.1059
0
0
0.1461
0.022
0
0
0.0739
0.3841
0
0.1622
0.2221
0.557
0.1034
0.6691
0.0009
Mean
of X
2.6453
1.3283
0.0187
0.1531
0.1461
0.4634
0.2343
0.277
0.0615
0.273
0.5428
0.1182
0.5723
0.3717
1.1974
0.3417
1.1511
1.9928
0.6233
0.367
0.1516
0.0859
0.7974
50.4217
196
E.1. In-sample estimation SCF1998
Table E.10: Setting 2b OLS, SCF1998 (cont.)
Classification table
Predicted
0
1
2
3
581
1233
396
36
10
95
200
44
7
94
183
56
6
90
204
64
6
77
179
67
8
92
195
80
618
1681 1357
347
Percent correct: 23.06%
Error Distance: 10’113’838
Actual
0
1
2
3
4
5
4
0
0
0
0
0
0
0
5
0
0
0
0
0
0
0
2246
349
340
364
329
375
4003
Chapter E. Empirical Results of Regressions
197
Table E.11: Setting 2b - Results for the Ordered model in-sampleregression: The dependent variable (stock ratio) is discrete, 6 risk classes
are considered. Dataset: SCF 1998.
Observations = 4003, Parameters = 25, Deg.Fr.= 3978
Log-L = -8845.996, Akaike Info. Crt. = 4.432
Variable
Coeff.
Standard
Error
Index function for probability
Constant
-0.6315
0.0616
X101
-0.0352
0.0093
X1706
0.1158
0.0053
⊕0.1426
0.0714
X3006 1
0.0946
0.0284
X3006 5
-0.3239
0.0364
X3008 1
0.5339
0.0211
X3008 45
X301 1
0.1289
0.0241
-0.0284
0.0229
X301 3
0.2289
0.0367
X3014 1
-1.6068
0.0342
X3014 4
X432 1
0.7888
0.0222
X5608
0.105
0.0179
X5821
0.0628
0.0061
-0.5703
0.0227
X5825 3
X5905
0.2036
0.0076
X7131
0.1596
0.0234
X7186
0.1187
0.0114
X7187
0.1644
0.0085
⊕0.2639
0.0301
X7372 1
0.1844
0.0252
X7401 1
0.1923
0.0305
X7401 2
-0.4341
0.0492
X7401 5
-0.0272
0.034
X8021 1
X8022
⊕0.0149
0.0009
b/St.Er.
P of
|Z| > z
Mean
of X
-10.25
-3.781
22.032
1.998
-3.331
-8.89
25.33
5.342
-1.236
6.229
-46.95
35.577
5.878
10.276
-25.173
26.836
-6.831
-10.394
19.236
8.777
7.321
6.303
-8.816
-0.801
17.282
0
0.0002
0
0.0457
0.0009
0
0
0
0.2163
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0.423
0
2.6453
1.3283
0.0187
0.1531
0.1461
0.4634
0.2343
0.277
0.0615
0.273
0.5428
0.1182
0.5723
0.3717
1.1974
0.3417
1.1511
1.9928
0.6233
0.367
0.1516
0.0859
0.7974
50.4217
198
E.1. In-sample estimation SCF1998
Setting 2b - Ordered model, SCF1998 (cont.)
Variable
Coeff.
Standard b/St.Er.
P of
Mean
Error
|Z| > z
of X
Threshold parameters for index
Mu(1)
2.0000
.... (Fixed Parameter) ....
Mu(2)
3.0000
.... (Fixed Parameter) ....
Mu(3)
4.0000
.... (Fixed Parameter) ....
Mu(4)
5.0000
.... (Fixed Parameter) ....
Classification table
Predicted
0
1
2
3
1203
907
94
41
35
192
70
47
31
159
70
73
21
175
90
76
25
141
80
77
31
158
92
79
1346 1732
496
393
Percent correct: 38.67%
Error Distance: 6’395’874
Actual
0
1
2
3
4
5
4
1
5
7
2
6
14
35
5
0
0
0
0
0
1
1
2246
349
340
364
329
375
4003
Chapter E. Empirical Results of Regressions
199
Table E.12: Setting 2b - Results for Multinomial Logit model, SCF1998,
in-sample-regression. The standard errors of the regression coefficients are
given in brackets. The significance levels are abbreviated with asterisks: ‘*’
and ‘**’ are significant at the 5, and 1 percent level
Observations = 4003, Parameters = 25, Deg.Fr.= 3978
Log-L = -4672.588, Akaike Info. Crt. = 2.347
Constant
X101
X1706
X3006 1
X3006 5
X3008 1
X3008 45
X301 1
X301 3
X3014 1
X3014 4
X432 1
X5608
Prob[Y=1]
-4.6093 **
(0.4286)
-0.0214
(0.0586)
0.091 **
(0.0342)
0.0582
(0.4215)
-0.1806
(0.1886)
-0.1103
(0.218)
0.3361 *
(0.1374)
-0.1164
(0.1673)
0.1787
(0.1428)
0.0221
(0.2557)
-0.7457 **
(0.1727)
1.0095 **
(0.1547)
0.0498
(0.114)
Prob[Y=2]
-3.9979 **
(0.4276)
-0.09
(0.0612)
0.1446 **
(0.035)
0.3789
(0.4147)
-0.218
(0.192)
0.0279
(0.227)
0.6017 **
(0.1438)
-0.0795
(0.1717)
0.2211
(0.1457)
0.1506
(0.2435)
-1.5474 **
(0.2244)
0.7332 **
(0.1547)
0.191
(0.1009)
Prob[Y=3]
-3.4461 **
(0.4057)
-0.0951
(0.0605)
0.0802 *
(0.0342)
0.3096
(0.415)
0.2124
(0.1712)
-0.3059
(0.2436)
0.5907 **
(0.1392)
-0.0235
(0.161)
0.0062
(0.1475)
0.0127
(0.249)
-1.6193 **
(0.221)
0.7698 **
(0.1533)
-0.0134
(0.1201)
Prob[Y=4]
-4.199 **
(0.4374)
0.0266
(0.059)
0.1276 **
(0.0349)
0.0295
(0.4671)
-0.1258
(0.19)
-0.4935
(0.2713)
0.599 **
(0.145)
0.1481
(0.1634)
0.0304
(0.1549)
0.2103
(0.2403)
-1.5441 **
(0.2357)
0.7671 **
(0.1595)
0.137
(0.1078)
Prob[Y=5]
-3.6678 **
(0.4099)
-0.0225
(0.057)
0.1762 **
(0.0337)
0.0746
(0.4449)
-0.116
(0.1796)
-0.2814
(0.2404)
0.5992 **
(0.1384)
0.1408
(0.1532)
-0.1639
(0.1508)
0.3342
(0.2248)
-1.3724 **
(0.2106)
0.7669 **
(0.1504)
0.1672
(0.102)
200
E.1. In-sample estimation SCF1998
Setting 2b - Results for Multinomial Logit model, SCF1998,
in-sample-regression. (cont.)
X5821
X5825 3
X5905
X7131
X7186
X7187
X7372 1
X7401 1
X7401 2
X7401 5
X8021 1
X8022
Prob[Y=1]
0.0589
(0.0427)
-0.3604 *
(0.1458)
0.1965 **
(0.0497)
-0.132
(0.147)
-0.0613
(0.0736)
0.2334 **
(0.0555)
0.2264
(0.1935)
0.4369 **
(0.1678)
0.379
(0.2061)
0.0155
(0.3034)
0.3537
(0.2382)
0.0201 **
(0.0055)
Prob[Y=2]
0.0789
(0.0418)
-0.2729
(0.1501)
0.2108 **
(0.0509)
-0.1712
(0.1525)
-0.2988 **
(0.0747)
0.2195 **
(0.0575)
0.5349 *
(0.2076)
0.3134
(0.1717)
0.4122 *
(0.2067)
0.2523
(0.2959)
-0.0851
(0.2428)
0.0193 **
(0.0058)
Prob[Y=3]
0.0791
(0.0408)
-0.6335 **
(0.1541)
0.2571 **
(0.0495)
-0.0235
(0.146)
-0.2385 **
(0.0732)
0.3058 **
(0.0556)
0.3618
(0.2007)
0.0548
(0.1638)
0.0868
(0.205)
-0.3524
(0.3428)
-0.1355
(0.2282)
0.0147 **
(0.0055)
Prob[Y=4]
0.0522
(0.0424)
-0.8423 **
(0.1686)
0.2817 **
(0.0515)
-0.2657
(0.1549)
-0.0244
(0.0771)
0.1953 **
(0.0578)
0.1191
(0.2033)
0.1195
(0.1725)
0.423 *
(0.2036)
-0.3928
(0.3749)
0.047
(0.242)
0.02 **
(0.0059)
Prob[Y=5]
0.096 *
(0.0395)
-0.5252 **
(0.1525)
0.1626 **
(0.0492)
-0.2947 *
(0.1483)
-0.1103
(0.0728)
0.2369 **
(0.0553)
0.3319
(0.1972)
0.2916
(0.1616)
0.068
(0.2068)
-0.3254
(0.3302)
-0.078
(0.2318)
0.0136 *
(0.0056)
Chapter E. Empirical Results of Regressions
201
Setting 2b - Multinomial model, SCF1998 (cont.)
Classification table
Predicted
0
1
2
3
2142
10
10
30
250
10
17
24
210
4
30
36
217
6
17
65
188
7
13
43
218
8
25
38
3225
45
112
236
Percent correct: 58.41%
Error Distance: 10’996’615
Actual
0
1
2
3
4
5
4
22
17
24
17
31
26
137
5
32
31
36
42
47
60
248
2246
349
340
364
329
375
4003
202
E.1. In-sample estimation SCF1998
Table E.13: Setting 3b - Results for WLS in-sample-regression: The
dependent variable is binomial, investors without stocks are to be separated
from stock-owning investors. Dataset: SCF 1998.
Observations = 4003, Parameters = 25, Deg.Fr.= 3978
Adjusted R-squared = 0.35424 , Log-L = -1987.9069
Akaike Info. Crt. = 1.006
Variable
WConstant
WX101
WX1706
WX3006 1
WX3006 5
WX3008 1
WX3008 45
WX301 1
WX301 3
WX3014 1
WX3014 4
WX432 1
WX5608
WX5821
WX5825 3
WX5905
WX7131
WX7186
WX7187
WX7372 1
WX7401 1
WX7401 2
WX7401 5
WX8021 1
WX8022
Coeff.
3.1433
-0.0021
0.0086
⊕0.1686
0.0073
-0.0499
0.048
0.0063
-0.0041
0.1861
-0.1774
0.1908
0.0275
0.02
-0.059
0.0745
0.0056
0.0155
0.0032
⊕0.1386
0.0636
0.0153
-0.034
-0.0781
⊕0.0006
Standard
Error
0.4611
0.0043
0.0042
0.0373
0.0155
0.0136
0.014
0.013
0.0125
0.0246
0.0154
0.018
0.0082
0.0034
0.0124
0.0057
0.0122
0.0069
0.0056
0.0152
0.0155
0.0166
0.0187
0.0162
0.0004
b/St.Er.
6.817
-0.477
2.05
4.519
0.473
-3.662
3.431
0.489
-0.325
7.551
-11.543
10.613
3.356
5.816
-4.756
13.077
-0.463
-2.256
0.576
9.134
-4.099
0.921
-1.819
-4.821
1.378
P of
|Z| > z
0
0.6331
0.0404
0
0.6364
0.0002
0.0006
0.6247
0.7449
0
0
0
0.0008
0
0
0
0.6433
0.0241
0.5647
0
0
0.357
0.0688
0
0.1683
Mean
of X
0.2056
7.929
4.2415
0.0577
0.4499
0.5235
1.3145
0.7149
0.8616
0.1654
1.105
1.4616
0.379
1.74
1.189
3.4631
1.0873
3.5573
6.1949
1.7951
1.0423
0.4486
0.2903
2.3232
151.6105
Chapter E. Empirical Results of Regressions
203
Setting 3b - WLS, SCF1998 (cont.)
Classification table
Predicted
0
1
1805
441
575
1182
2380 1623
Percent correct: 74.62%
Error Distance: 1,575,318
Actual
0
1
2246
1757
4003
204
E.1. In-sample estimation SCF1998
Table E.14: Setting 3b - Results for Tobit in-sample-regression: The dependent variable is binomial, investors without stocks are to be separated from
stock-owning investors. Dataset: SCF 1998.
Observations = 4003, Parameters = 25, Deg.Fr.= 3978
Log-L = -3173.196, Akaike Info. Crt. = 1.598
Variable
Coeff.
Standard
Error
Primary Index Equation for Model
Constant
-0.745
0.0925
X101
-0.0166
0.0131
X1706
0.0337
0.0079
X3006 1
⊕0.0595
0.0985
0.0315
0.0413
X3006 5
-0.1004
0.0495
X3008 1
0.1978
0.0318
X3008 45
0.0006
0.0363
X301 1
⊕0.022
0.0334
X301 3
0.0426
0.0555
X3014 1
-0.5708
0.0415
X3014 4
X432 1
0.365
0.0343
X5608
0.0236
0.0236
X5821
0.0241
0.0094
-0.2178
0.0335
X5825 3
X5905
0.075
0.0116
X7131
0.0822
0.0337
X7186
0.0548
0.0167
X7187
0.0606
0.0128
0.1194
0.0444
X7372 1
0.1193
0.0379
X7401 1
X7401 2
0.1229
0.0457
-0.0802
0.0641
X7401 5
0.0442
0.0507
X8021 1
X8022
⊕0.0076
0.0012
Disturbance Standard Deviation
Sigma
0.7558
0.0146
b/St.Er.
P of
|Z| > z
Mean
of X
-8.057
-1.266
4.268
0.604
-0.763
-2.029
6.224
0.017
0.658
0.768
-13.75
10.645
1
2.562
-6.502
6.452
-2.443
-3.278
4.743
2.691
3.145
2.688
-1.251
0.873
6.154
0
0.2055
0
0.5458
0.4456
0.0425
0
0.9861
0.5105
0.4427
0
0
0.3174
0.0104
0
0
0.0146
0.001
0
0.0071
0.0017
0.0072
0.211
0.3828
0
2.6453
1.3283
0.0187
0.1531
0.1461
0.4634
0.2343
0.277
0.0615
0.273
0.5428
0.1182
0.5723
0.3717
1.1974
0.3417
1.1511
1.9928
0.6233
0.367
0.1516
0.0859
0.7974
50.4217
51.614
0
Chapter E. Empirical Results of Regressions
205
Setting 3b - TOBIT, SCF1998 (cont.)
Classification table
Predicted
0
1
2124
122
1033
724
3157
846
Percent correct: 71.15%
Error Distance: 3’245’919
Actual
0
1
2246
1757
4003
206
E.1. In-sample estimation SCF1998
Table E.15: Setting 3b - Results for Multinomial Logit model in-sampleregression: The dependent variable is binomial, investors without stocks are
to be separated from stock-owning investors. Dataset: SCF 1998.
Observations = 4003, Parameters = 25, Deg.Fr.= 3978
Log-L = -1906.335, Akaike Info. Crt. = 0.965
Variable
Coeff.
Standard b/St.Er.
Error
Characteristics in numerator of Prob[Y = 1]
Constant
-2.3607
0.2484
-9.505
X101
-0.0399
0.0367
-1.088
X1706
0.1237
0.0237
5.213
X3006 1
⊕0.1732
0.2963
0.585
0.08
0.1173
-0.682
X3006 5
-0.2148
0.1352
-1.588
X3008 1
0.5408
0.0877
6.165
X3008 45
0.02
0.102
0.196
X301 1
⊕0.0619
0.0963
0.643
X301 3
0.1546
0.1637
0.944
X3014 1
-1.3125
0.1115
-11.766
X3014 4
X432 1
0.8141
0.0893
9.12
X5608
0.1137
0.0803
1.416
X5821
0.0742
0.029
2.556
-0.5144
0.0896
-5.744
X5825 3
X5905
0.219
0.0334
6.566
X7131
0.1738
0.0977
-1.778
X7186
0.147
0.0478
-3.077
X7187
0.2395
0.0399
5.999
⊕0.313
0.1221
2.564
X7372 1
0.2468
0.1073
2.301
X7401 1
X7401 2
0.2726
0.1269
2.148
-0.1386
0.1714
-0.808
X7401 5
0.0233
0.1372
0.17
X8021 1
X8022
⊕0.0175
0.0034
5.118
P of
|Z| > z
Mean
of X
0
0.2768
0
0.5588
0.4953
0.1122
0
0.8443
0.52
0.3451
0
0
0.1568
0.0106
0
0
0.0754
0.0021
0
0.0104
0.0214
0.0317
0.4188
0.8653
0
2.6453
1.3283
0.0187
0.1531
0.1461
0.4634
0.2343
0.277
0.0615
0.273
0.5428
0.1182
0.5723
0.3717
1.1974
0.3417
1.1511
1.9928
0.6233
0.367
0.1516
0.0859
0.7974
50.4217
Chapter E. Empirical Results of Regressions
207
Table E.16: Setting 3b Multinomial Logit model, SCF1998 (cont.)
Classification table
Predicted
0
1
1831
415
2246
488
1269 1757
2319 1684 4003
Percent correct: 77.44%
Error Distance: 1’231’107
Actual
0
1
208
E.1. In-sample estimation SCF1998
Table E.17: Setting 3c - Results for OLS in-sample-regression: The
dependent variable is continuous: the stock ratio is given as a percentage.
Dataset: SCF 1995.
Observations = 1757, Parameters = 25, Deg.Fr.= 1732
Adjusted R-squared = 0.01430 , Log-L = -330.9550
Akaike Info. Crt. = 0.405
Variable
Coeff.
Constant
X101
X1706
X3006 1
X3006 5
X3008 1
X3008 45
X301 1
X301 3
X3014 1
X3014 4
X432 1
X5608
X5821
X5825 3
X5905
X7131
X7186
X7187
X7372 1
X7401 1
X7401 2
X7401 5
X8021 1
X8022
0.5703
⊕0.0059
0.0069
-0.0211
0.008
-0.0306
0.0178
0.0293
-0.0367
0.0284
-0.0657
0.0186
0.0052
0.0012
-0.0398
0.0001
0.0194
0.0102
0.0012
-0.0132
0.0231
0.0356
-0.0602
-0.0273
-0.0004
Standard
Error
0.0508
0.007
0.0036
0.0457
0.0209
0.0286
0.0161
0.0185
0.0165
0.026
0.0257
0.0186
0.0104
0.0043
0.0182
0.0056
0.0168
0.0082
0.0058
0.0234
0.0189
0.0238
0.0405
0.0282
0.0007
b/St.Er.
11.234
0.841
1.913
-0.461
0.383
-1.072
1.103
1.578
-2.228
1.092
-2.561
-0.999
0.5
0.27
-2.191
0.01
-1.153
1.248
-0.211
-0.564
-1.224
-1.495
-1.488
-0.969
-0.633
P of
|Z| > z
0
0.4003
0.0557
0.6447
0.7015
0.2839
0.2701
0.1145
0.0259
0.2747
0.0104
0.318
0.6171
0.7874
0.0284
0.9924
0.2491
0.212
0.8328
0.5729
0.2208
0.135
0.1367
0.3327
0.527
Mean
of X
2.6693
2.2106
0.025
0.1485
0.0768
0.6181
0.2089
0.2977
0.0825
0.0916
0.7718
0.1628
0.7803
0.218
1.8036
0.3204
1.1013
2.5726
0.7513
0.5179
0.1491
0.0364
0.8839
53.2692
Chapter E. Empirical Results of Regressions
209
Table E.18: Setting 3c OLS, SCF1998 (cont.)
Classification table
Predicted
0
1
2
0
67
282
0
53
287
0
44
320
0
24
305
0
30
344
0
218
1538
Percent correct: 21.23%
Error Distance: 2’629’739
Actual
0
1
2
3
4
3
0
0
0
0
1
1
4
0
0
0
0
0
0
349
340
364
329
375
1757
210
E.1. In-sample estimation SCF1998
Table E.19: Setting 3c - Results for Ordered model in-sample-regression:
The dependent variable (stock ratio) is discrete, 5 risk classes are considered.
Dataset: SCF 1998.
Observations = 1757, Parameters = 25, Deg.Fr.= 1732
Log-L = -3899.910, Akaike Info. Crt. = 4.468
Variable
Coeff.
Standard
Error
Index function for probability
Constant
2.7241
0.1057
X101
⊕0.0237
0.0146
X1706
0.0403
0.0077
-0.0477
0.1028
X3006 1
0.0655
0.0452
X3006 5
-0.2076
0.0574
X3008 1
0.1497
0.0334
X3008 45
X301 1
0.1761
0.039
-0.2276
0.0342
X301 3
0.1651
0.0541
X3014 1
-0.4895
0.0493
X3014 4
X432 1
0.1091
0.0388
X5608
0.042
0.022
X5821
0.015
0.009
-0.2383
0.0372
X5825 3
X5905
⊕0.002
0.0119
X7131
0.1105
0.0356
X7186
0.0266
0.0173
X7187
0.0029
0.0125
-0.0406
0.0483
X7372 1
0.1355
0.039
X7401 1
0.1559
0.0497
X7401 2
-0.2943
0.0824
X7401 5
-0.2107
0.0599
X8021 1
X8022
-0.003
0.0014
b/St.Er.
P of
|Z| > z
Mean
of X
25.778
1.628
5.261
-0.464
1.449
-3.616
4.477
4.517
-6.648
3.052
-9.928
-2.81
1.904
1.663
-6.398
-0.164
-3.104
1.537
-0.229
-0.841
-3.475
-3.14
-3.573
-3.515
-2.129
0
0.1035
0
0.6424
0.1474
0.0003
0
0
0
0.0023
0
0.0049
0.057
0.0963
0
0.8693
0.0019
0.1242
0.819
0.4005
0.0005
0.0017
0.0004
0.0004
0.0333
2.6693
2.2106
0.025
0.1485
0.0768
0.6181
0.2089
0.2977
0.0825
0.0916
0.7718
0.1628
0.7803
0.218
1.8036
0.3204
1.1013
2.5726
0.7513
0.5179
0.1491
0.0364
0.8839
53.2692
Chapter E. Empirical Results of Regressions
211
Setting 3c - Ordered model, SCF1998 (cont.)
Variable
Coeff.
Standard b/St.Er.
P of
Mean
Error
|Z| > z
of X
Threshold parameters for index
Mu(1)
2.0000
.... (Fixed Parameter) ....
Mu(2)
3.0000
.... (Fixed Parameter) ....
Mu(3)
4.0000
.... (Fixed Parameter) ....
Classification table
Predicted
0
1
2
0
200
149
0
164
174
0
152
212
0
142
186
0
152
221
0
810
942
Percent correct: 21.46%
Error Distance: 2’147’852
Actual
0
1
2
3
4
3
0
2
0
1
2
5
4
0
0
0
0
0
0
349
340
364
329
375
1757
212
E.1. In-sample estimation SCF1998
Table E.20: Setting 3c - Results for Multinomial Logit model in-sampleregression: The dependent variable (stock ratio) is discrete, 5 risk classes
are considered. Dataset: SCF 1998. The standard errors of the regression
coefficients are given in brackets. The significance levels are abbreviated with
asterisks: ‘*’ and ‘**’ are significant at the 5, and 1 percent level
Observations = 1757, Parameters = 25, Deg.Fr.= 1732
Log-L = -2766.666, Akaike Info. Crt. = 3.178
Constant
X101
X1706
X3006 1
X3006 5
X3008 1
X3008 45
X301 1
X301 3
X3014 1
X3014 4
X432 1
X5608
Prob[Y=1]
0.5443
(0.5644)
-0.0661
(0.0778)
0.0533
(0.0397)
0.2976
(0.4793)
-0.0398
(0.2399)
0.1434
(0.2881)
0.27
(0.1759)
0.0252
(0.2144)
0.0433
(0.1746)
0.1362
(0.2957)
-0.7874 **
(0.2693)
-0.2777
(0.2047)
0.1354
(0.1153)
Prob[Y=2]
1.2078 *
(0.5505)
-0.0757
(0.0773)
-0.0095
(0.0388)
0.2311
(0.4807)
0.4154
(0.2236)
-0.1947
(0.3019)
0.2487
(0.1724)
0.0705
(0.2057)
-0.1725
(0.1756)
-0.0216
(0.3)
-0.8552 **
(0.2657)
-0.2365
(0.204)
-0.074
(0.1334)
Prob[Y=3]
0.3806
(0.5734)
0.0513
(0.0761)
0.0366
(0.0395)
-0.0454
(0.5252)
0.0404
(0.2378)
-0.381
(0.3249)
0.2437
(0.1769)
0.2506
(0.207)
-0.1572
(0.1814)
0.1539
(0.2921)
-0.8216 **
(0.2779)
-0.2182
(0.2087)
0.0864
(0.1212)
Prob[Y=4]
0.9173
(0.5536)
0.0004
(0.0742)
0.0853 *
(0.0386)
0.014
(0.5065)
0.0716
(0.2296)
-0.1634
(0.3001)
0.2603
(0.1719)
0.2371
(0.1991)
-0.3403
(0.178)
0.3037
(0.2806)
-0.6423 *
(0.2575)
-0.2084
(0.2017)
0.1138
(0.1163)
Chapter E. Empirical Results of Regressions
213
Setting 3c - Results for Multinomial Logit model, SCF1998,
in-sample-regression. (cont.)
X5821
X5825 3
X5905
X7131
X7186
X7187
X7372 1
X7401 1
X7401 2
X7401 5
X8021 1
X8022
Prob[Y=1]
0.0243
(0.0485)
0.0729
(0.1895)
0.0114
(0.062)
-0.0444
(0.1852)
-0.2312 *
(0.09)
-0.014
(0.0642)
0.2906
(0.2589)
-0.1378
(0.2115)
0.0234
(0.2621)
0.2715
(0.4052)
-0.4068
(0.3165)
0.0002
(0.0074)
Prob[Y=2]
0.0246
(0.0477)
-0.2624
(0.1929)
0.057
(0.0606)
0.0982
(0.1796)
-0.1709
(0.0887)
0.0702
(0.0626)
0.1277
(0.2525)
-0.4232 *
(0.205)
-0.339
(0.2614)
-0.3555
(0.4421)
-0.4663
(0.3048)
-0.0059
(0.0072)
Prob[Y=3]
-0.0045
(0.0491)
-0.4859 *
(0.2049)
0.0836
(0.0623)
-0.1593
(0.1866)
0.042
(0.092)
-0.0421
(0.0644)
-0.1301
(0.2542)
-0.3301
(0.2116)
0.0466
(0.2595)
-0.3637
(0.4681)
-0.2715
(0.3154)
0.0005
(0.0075)
Prob[Y=4]
0.0408
(0.0465)
-0.1802
(0.1921)
-0.0428
(0.0603)
-0.1743
(0.1813)
-0.0468
(0.0885)
0.0011
(0.0621)
0.0892
(0.2498)
-0.1573
(0.2035)
-0.3199
(0.2628)
-0.2819
(0.4336)
-0.4213
(0.3088)
-0.0058
(0.0073)
214
E.1. In-sample estimation SCF1998
Setting 3c - Multinomial model, SCF1998 (cont.)
Classification table
Predicted
0
1
2
97
48
74
56
82
76
61
46
128
47
38
88
59
59
88
320
273
454
Percent correct: 27.95%
Error Distance: 1’803’590
Actual
0
1
2
3
4
3
38
45
40
61
46
230
4
92
81
89
95
123
480
349
340
364
329
375
1757
Chapter E. Empirical Results of Regressions
E.2
215
Out-of-sample estimation SCF 1995 in SCF1998
216
E.2. Out-of-sample estimation SCF 1995 in SCF1998
Table E.21: Setting 1 - Results for OLS in-sample-regression: The dependent variable is continuous: the stock ratio is given as a percentage. Dataset:
SCF 1995.
Observations = 4299, Parameters = 25, Deg.Fr.= 4274
Adjusted R-squared = 0.29850, Log-L = -1558.5730
Akaike Info. Crt. = 0.737
Variable
Coeff.
Constant
X101
X1706
X3006 1
X3006 5
X3008 1
X3008 45
X301 1
X301 3
X3014 1
X3014 4
X432 1
X5608
X5821
X5825 3
X5905
X7131
X7186
X7187
X7372 1
X7401 1
X7401 2
X7401 5
X8021 1
X8022
-0.1925
-0.0222
0.0093
-0.0341
0.0293
-0.0412
0.012
0.0436
-0.0616
0.0014
-0.1332
0.1218
0.0007
0.0088
⊕0.0335
0.021
0.111
0.0799
0.015
⊕0.0964
0.0485
0.093
⊕0.0408
0.0191
⊕0.0044
Standard
Error
0.0313
0.0048
0.0034
0.0351
0.0161
0.0166
0.0125
0.0125
0.0139
0.0247
0.0129
0.0129
0.0054
0.0041
0.0124
0.0037
0.0125
0.012
0.0053
0.0164
0.0153
0.0164
0.0196
0.0178
0.0004
b/St.Er.
-6.157
-4.661
2.747
-0.972
1.823
-2.489
0.959
-3.489
-4.417
-0.055
-10.291
9.428
-0.136
2.159
2.711
5.733
-8.908
-6.642
2.81
5.884
3.177
5.685
2.079
1.073
10.204
P of
|Z| > z
0
0
0.006
0.3312
0.0683
0.0128
0.3375
0.0005
0
0.9563
0
0
0.8921
0.0309
0.0067
0
0
0
0.005
0
0.0015
0
0.0376
0.2833
0
Mean
of X
2.6218
1.268
0.024
0.141
0.164
0.5301
0.2901
0.2175
0.0514
0.3438
0.5194
0.2645
0.5459
0.3957
1.4836
0.3559
0.6855
1.9035
0.6215
0.3112
0.1661
0.0989
0.789
49.7704
Chapter E. Empirical Results of Regressions
217
Setting 1 - OLS Out-of-Sample results
SCF1995 estimates in 1998 data
Classification table
0
1
2
144
42
98
568
261
784
21
21
152
12
26
124
21
16
136
24
19
122
23
26
137
813
411
1553
Percent correct: 17.19%
Error Distance: 11’864’863
Actual
0
1
2
3
4
5
6
Predicted
3
15
476
115
127
134
120
131
1118
4
3
140
36
46
47
34
51
357
5
0
16
4
5
10
9
7
51
6
0
1
0
0
0
1
0
2
302
2246
349
340
364
329
375
4305
218
E.2. Out-of-sample estimation SCF 1995 in SCF1998
Table E.22: Setting 1 - Results for Tobit in-sample-regression: The dependent variable is continuous: the stock ratio is given as a percentage. Dataset:
SCF 1995.
Observations = 4299, Parameters = 25, Deg.Fr.= 4274
Log-L = -2187.820, Akaike Info. Crt. = 1.029
Variable
Coeff.
Standard
Error
Primary Index Equation for Model
Constant
-0.5995
0.0611
X101
-0.0375
0.0094
X1706
0.0224
0.0053
X3006 1
-0.0576
0.0609
0.0836
0.0284
X3006 5
-0.0245
0.0333
X3008 1
0.051
0.0223
X3008 45
0.0533
0.0217
X301 1
-0.0659
0.026
X301 3
0.0434
0.0397
X3014 1
-0.3094
0.0258
X3014 4
X432 1
0.1829
0.0233
X5608
0.0011
0.0089
X5821
0.0128
0.0065
⊕0.0779
0.0213
X5825 3
X5905
0.0393
0.006
X7131
0.0416
0.0224
X7186
0.0882
0.0211
X7187
0.0407
0.0084
⊕0.1515
0.0309
X7372 1
0.0252
0.0257
X7401 1
X7401 2
0.0888
0.0292
-0.0165
0.0402
X7401 5
-0.021
0.0352
X8021 1
X8022
⊕0.0048
0.0008
Disturbance Standard Deviation
Sigma
0.4879
0.0098
b/St.Er.
P of
|Z| > z
Mean
of X
-9.805
-3.989
4.201
-0.946
2.945
-0.736
2.285
-2.453
-2.532
1.094
-11.982
7.856
0.124
1.975
3.664
6.594
-1.855
-4.188
4.82
4.894
0.978
3.044
-0.41
-0.595
5.899
0
0.0001
0
0.3442
0.0032
0.4616
0.0223
0.0142
0.0113
0.274
0
0
0.9011
0.0483
0.0002
0
0.0636
0
0
0
0.3279
0.0023
0.6821
0.5515
0
2.6218
1.268
0.024
0.141
0.164
0.5301
0.2901
0.2175
0.0514
0.3438
0.5194
0.2645
0.5459
0.3957
1.4836
0.3559
0.6855
1.9035
0.6215
0.3112
0.1661
0.0989
0.789
49.7704
49.866
0
Chapter E. Empirical Results of Regressions
219
Setting 1 - Tobit Out-of-Sample results
SCF1995 estimates in 1998 data
Classification table
0
1
2
245
28
25
1353
306
369
136
59
96
107
57
107
96
68
126
100
51
116
113
71
108
2150
640
947
Percent correct: 16.68%
Error Distance: 16’002’731
Actual
0
1
2
3
4
5
6
Predicted
3
4
150
37
43
43
34
50
361
4
0
56
14
18
19
17
19
143
5
0
10
7
7
10
9
14
57
6
0
2
0
1
2
2
0
7
302
2246
349
340
364
329
375
4305
220
E.2. Out-of-sample estimation SCF 1995 in SCF1998
Table E.23: Setting 1 - Results for Ordered model in-sample-regression:
The dependent variable (stock ratio) is discrete, 7 risk classes are considered.
Dataset: SCF 1995.
Observations = 4299, Parameters = 25, Deg.Fr.= 4274
Log-L = -7024.7, Akaike Info. Crt. = 3.280
Variable
Coeff.
Standard
Error
Index function for probability
Constant
0.8007
0.0733
X101
-0.0899
0.0112
X1706
0.0804
0.0066
-0.2374
0.0826
X3006 1
0.2206
0.033
X3006 5
-0.1053
0.0398
X3008 1
0.1326
0.027
X3008 45
X301 1
0.1794
0.0263
-0.2609
0.0323
X301 3
0.11
0.0453
X3014 1
-0.748
0.0314
X3014 4
X432 1
0.5372
0.0279
X5608
0.0125
0.0121
X5821
0.0382
0.0082
⊕0.2252
0.0261
X5825 3
X5905
0.1231
0.0074
X7131
0.2727
0.0278
X7186
0.3144
0.0261
X7187
0.1228
0.0107
⊕0.4212
0.038
X7372 1
0.0963
0.0327
X7401 1
0.3082
0.0351
X7401 2
⊕0.0728
0.045
X7401 5
0.0047
0.0425
X8021 1
X8022
⊕0.0163
0.001
b/St.Er.
P of
|Z| > z
Mean
of X
10.93
-8.057
12.114
-2.873
6.688
-2.644
4.915
-6.822
-8.069
2.429
-23.796
19.242
-1.036
4.666
8.641
16.603
-9.819
-12.028
11.506
11.089
2.95
8.78
1.617
0.111
16.18
0
0
0
0.0041
0
0.0082
0
0
0
0.0152
0
0
0.3001
0
0
0
0
0
0
0
0.0032
0
0.1059
0.9113
0
2.6218
1.268
0.024
0.141
0.164
0.5301
0.2901
0.2175
0.0514
0.3438
0.5194
0.2645
0.5459
0.3957
1.4836
0.3559
0.6855
1.9035
0.6215
0.3112
0.1661
0.0989
0.789
49.7704
Chapter E. Empirical Results of Regressions
221
Setting 1 - Ordered model, SCF1995 (cont.)
Variable
Coeff.
Standard b/St.Er.
P of
Mean
Error
|Z| > z
of X
Threshold parameters for index
Mu(1)
2.0000
.... (Fixed Parameter) ....
Mu(2)
3.0000
.... (Fixed Parameter) ....
Mu(3)
4.0000
.... (Fixed Parameter) ....
Mu(4)
5.0000
.... (Fixed Parameter) ....
Mu(5)
6.0000
.... (Fixed Parameter) ....
Setting 1 - Ordered Logit Out-of-Sample results
SCF1995 estimates in 1998 data
Classification table
0
1
2
11
218
56
12
1158
578
0
82
123
0
74
95
0
62
109
0
66
93
0
83
110
23
1743 1164
Percent correct: 34.05%
Error Distance: 7’455’722
Actual
0
1
2
3
4
5
6
Predicted
3
16
412
115
136
154
132
131
1096
4
1
83
28
33
34
33
49
261
5
0
2
1
2
5
4
2
16
6
0
1
0
0
0
1
0
2
302
2246
349
340
364
329
375
4305
222
E.2. Out-of-sample estimation SCF 1995 in SCF1998
Table E.24: Setting 1 - Results for Multinomial Logit model, SCF1995,
in-sample-regression. The dependent variable (stock ratio) is discrete, 7 risk
classes are considered. The standard errors of the regression coefficients are
given in brackets. The significance levels are abbreviated with asterisks: ‘*’
and ‘**’ are significant at the 5, and 1 percent level
Observations = 4299, Parameters = 25, Deg.Fr.= 4274
Log-L = -5024.357, Akaike Info. Crt. = 2.349
Constant
X101
X1706
X3006 1
X3006 5
X3008 1
X3008 45
X301 1
X301 3
X3014 1
X3014 4
X432 1
X5608
Prob[Y=1]
1.0963 **
(0.3734)
-0.0937
(0.0505)
0.5384 **
(0.1553)
-0.3447
(0.4925)
-0.0668
(0.1951)
-0.2209
(0.1647)
0.065
(0.1656)
-0.3757 *
(0.1701)
-0.4897 **
(0.1571)
-0.5208
(0.3491)
-0.5975 **
(0.1574)
1.7983 **
(0.2667)
0.0153
(0.1118)
Prob[Y=2]
-3.0716 **
(0.5162)
-0.2034 **
(0.0743)
0.6883 **
(0.1577)
-0.416
(0.5863)
-0.0914
(0.2598)
-0.1019
(0.2602)
0.5442 **
(0.2094)
-0.4185 *
(0.2088)
-0.2767
(0.2115)
-0.5474
(0.413)
-1.4758 **
(0.2138)
2.5944 **
(0.2967)
0.0968
(0.1198)
Prob[Y=3]
-3.0243 **
(0.5484)
-0.1861 *
(0.0808)
0.6164 **
(0.1585)
-0.4335
(0.623)
0.1636
(0.2682)
-0.1342
(0.2808)
0.4095
(0.2199)
-0.8029 **
(0.222)
-0.6405 **
(0.2296)
-0.5143
(0.4342)
-1.885 **
(0.2373)
2.6838 **
(0.3084)
-0.0116
(0.1272)
Prob[Y=4]
-3.0706 **
(0.5744)
-0.1856 *
(0.0834)
0.6269 **
(0.1589)
-0.2968
(0.6331)
0.2864
(0.2708)
-0.6221
(0.3294)
0.1509
(0.2235)
-0.7054 **
(0.2241)
-0.9807 **
(0.2563)
-0.5496
(0.4383)
-2.1931 **
(0.2786)
2.5184 **
(0.3158)
0.1464
(0.1219)
Prob[Y=5]
-3.4945 **
(0.6414)
-0.2683 **
(0.0917)
0.6295 **
(0.16)
-0.3621
(0.676)
0.2315
(0.2892)
-0.285
(0.3514)
0.3362
(0.2432)
-0.7643 **
(0.2388)
-0.918 **
(0.2766)
-0.0659
(0.4354)
-1.9919 **
(0.3019)
2.6108 **
(0.33)
-0.1374
(0.1435)
Prob[Y=6]
-2.6013 **
(0.588)
-0.3347 **
(0.0911)
0.7268 **
(0.1597)
-1.2552
(0.7823)
0.4626
(0.2792)
-0.0229
(0.2947)
0.2963
(0.2352)
-0.4791 *
(0.2312)
-0.6895 **
(0.2558)
-0.2015
(0.433)
-1.6345 **
(0.2556)
2.0423 **
(0.3141)
-0.0165
(0.1329)
Chapter E. Empirical Results of Regressions
223
Setting 1 - Results for Multinomial Logit model, SCF1995, in-sample-regression.
(cont.)
X5821
X5825 3
X5905
X7131
X7186
X7187
X7372 1
X7401 1
X7401 2
X7401 5
X8021 1
X8022
Prob[Y=1]
0.2739 **
(0.0936)
0.1201
(0.1815)
0.4504 **
(0.1058)
-1.1994 **
(0.1464)
-0.9935 **
(0.1924)
0.0047
(0.1113)
0.78 **
(0.1907)
1.1321 **
(0.3119)
0.8898 **
(0.2264)
0.4136 *
(0.1989)
0.1192
(0.1774)
0.0363 **
(0.0049)
Prob[Y=2]
0.3874 **
(0.0996)
0.4924 *
(0.2147)
0.6346 **
(0.1103)
-1.0034 **
(0.1916)
-1.2032 **
(0.2252)
0.1674
(0.1199)
1.1657 **
(0.2562)
1.3688 **
(0.342)
1.1607 **
(0.283)
0.596
(0.3102)
0.0752
(0.2689)
0.0525 **
(0.0068)
Prob[Y=3]
0.3654 **
(0.102)
0.3695
(0.2248)
0.6055 **
(0.1117)
-0.9756 **
(0.2032)
-1.2742 **
(0.2345)
0.236
(0.1222)
1.5339 **
(0.2908)
1.149 **
(0.3511)
1.2596 **
(0.2917)
0.1466
(0.3702)
-0.4285
(0.301)
0.0574 **
(0.0073)
Prob[Y=4]
0.3479 **
(0.1024)
0.5886 *
(0.2318)
0.6948 **
(0.1128)
-1.133 **
(0.2127)
-1.4634 **
(0.239)
0.3203 **
(0.1233)
1.285 **
(0.2913)
1.0063 **
(0.3561)
1.1629 **
(0.3018)
-0.296
(0.4566)
0.0545
(0.3253)
0.0522 **
(0.0077)
Prob[Y=5]
0.3168 **
(0.1051)
0.5704 *
(0.2453)
0.6084 **
(0.1146)
-1.5036 **
(0.2356)
-1.559 **
(0.2508)
0.2381
(0.1274)
1.8232 **
(0.336)
1.5237 **
(0.3738)
1.4387 **
(0.3284)
0.4773
(0.4193)
0.0692
(0.3976)
0.0522 **
(0.0085)
Prob[Y=6]
0.3375 **
(0.1052)
0.3715
(0.2382)
0.5897 **
(0.1138)
-1.4245 **
(0.2254)
-1.1777 **
(0.2483)
0.2351
(0.126)
1.2313 **
(0.2938)
0.9469 **
(0.3638)
0.9906 **
(0.311)
0.4812
(0.353)
0.1508
(0.3187)
0.0517 **
(0.0079)
224
E.2. Out-of-sample estimation SCF 1995 in SCF1998
Setting 1 - Multinomial Logit Out-of-Sample results
SCF1995 estimates in 1998 data
Classification table
0
1
2
32
244
0
52
1722
24
1
243
32
0
221
40
0
224
41
0
214
36
1
247
39
86
3115
212
Percent correct: 43.46%
Error Distance: 19’807’854
Actual
0
1
2
3
4
5
6
Predicted
3
9
151
30
29
39
26
30
314
4
0
1
1
3
9
4
6
24
5
17
295
40
46
49
46
51
544
6
0
1
2
1
2
3
1
10
302
2246
349
340
364
329
375
4305
Chapter E. Empirical Results of Regressions
225
Table E.25: Setting 2a - Results for Weighted Least Squares (WLS)
in-sample-regression: The dependent variable is binomial, investors without
assets are to be separated investors. The portrayed mean has no interpretation, as each variable was weighted by W , the reciprocal of the observations’
standard errors. Dataset: SCF 1995.
Observations = 4299, Parameters = 25, Deg.Fr.= 4274
Adjusted R-squared = 0.19860, Log-L = -93.2645
Akaike Info. Crt. = 0.055
Variable
WConstant
WX101
WX1706
WX3006 1
WX3006 5
WX3008 1
WX3008 45
WX301 1
WX301 3
WX3014 1
WX3014 4
WX432 1
WX5608
WX5821
WX5825 3
WX5905
WX7131
WX7186
WX7187
WX7372 1
WX7401 1
WX7401 2
WX7401 5
WX8021 1
WX8022
Coeff.
1.0627
-0.0108
0.0021
-0.0064
0.0031
-0.0298
0.0012
0.0198
-0.0271
0.0224
-0.0504
0.0632
0.0018
0.0053
⊕0.0078
0.0062
0.0717
0.0485
0.0066
⊕0.048
0.0416
0.0599
⊕0.0364
0.017
⊕0.0026
Standard
Error
0.0144
0.0016
0.0008
0.01
0.005
0.0064
0.0035
0.0036
0.0041
0.0075
0.0042
0.0038
0.0011
0.001
0.0033
0.001
0.0042
0.0036
0.0013
0.0052
0.0042
0.0052
0.0055
0.0067
0.0002
b/St.Er.
73.848
-6.856
-2.659
-0.641
-0.63
-4.69
0.331
-5.534
-6.655
-3.008
-11.86
16.677
1.731
5.239
2.378
6.387
-16.954
-13.577
-5.177
9.213
9.89
11.561
6.663
2.556
16.302
P of
|Z| > z
0
0
0.0078
0.5217
0.5288
0
0.7406
0
0
0.0026
0
0
0.0835
0
0.0174
0
0
0
0
0
0
0
0
0.0106
0
Mean
of X
3.7764
12.6871
7.6259
0.1169
0.6437
0.5509
2.8431
1.5168
0.9162
0.246
1.2473
3.0855
1.6937
2.9259
2.2083
8.779
1.6508
3.254
10.3022
3.514
1.8178
0.8948
0.4661
4.1698
253.8758
226
E.2. Out-of-sample estimation SCF 1995 in SCF1998
Setting 2a - WLS
Out-of-Sample results
SCF1995 estimates in 1998 data
Classification table
Predicted
0
1
Actual
0
31
271
1
60
3943
91
4214
Percent correct: 92.31%
Error Distance: 231’123
302
4003
4305
Chapter E. Empirical Results of Regressions
227
Table E.26: Setting 2a - Results for Multinomial Logit model in-sampleregression: The dependent variable is binomial, Assetholders are separated
from people who do not own any financial assets. Dataset: SCF 1995.
Observations = 4299, Parameters = 25, Deg.Fr.= 4274
Log-L = -737.6719, Akaike Info. Crt. = 0.355
Variable
Coeff.
Standard b/St.Er.
Error
Characteristics in numerator of Prob[Y = 1]
Constant
1.1679
0.3728
3.133
X101
-0.1039
0.0506
-2.053
X1706
0.565
0.1541
3.666
X3006 1
-0.3656
0.4937
-0.741
0.0447
0.1949
-0.229
X3006 5
-0.2147
0.1648
-1.303
X3008 1
0.1024
0.1656
0.618
X3008 45
0.3944
0.1702
-2.318
X301 1
-0.5039
0.157
-3.21
X301 3
0.5118
0.3464
-1.477
X3014 1
X3014 4
-0.7152
0.1568
-4.562
1.9187
0.2668
7.191
X432 1
X5608
0.0187
0.1115
0.167
X5821
0.2814
0.0934
3.013
⊕0.1646
0.1811
0.909
X5825 3
X5905
0.4807
0.1054
4.562
X7131
1.1953
0.1464
-8.166
X7186
1.0232
0.1923
-5.32
X7187
0.0428
0.1102
0.388
⊕0.8307
0.1906
4.358
X7372 1
1.1408
0.3116
3.661
X7401 1
X7401 2
0.9173
0.226
4.059
⊕0.409
0.1989
2.056
X7401 5
0.1045
0.1773
0.589
X8021 1
X8022
⊕0.038
0.0049
7.806
P of
|Z| > z
Mean
of X
0.0017
0.0401
0.0002
0.4589
0.8185
0.1926
0.5363
0.0205
0.0013
0.1396
0
0
0.8672
0.0026
0.3634
0
0
0
0.698
0
0.0003
0
0.0398
0.5558
0
2.6218
1.268
0.024
0.141
0.164
0.5301
0.2901
0.2175
0.0514
0.3438
0.5194
0.2645
0.5459
0.3957
1.4836
0.3559
0.6855
1.9035
0.6215
0.3112
0.1661
0.0989
0.789
49.7704
228
E.2. Out-of-sample estimation SCF 1995 in SCF1998
Table E.27: Setting 2a - Multinomial Logit model
Out-of-sample estimation
SCF1995 estimates in 1998 data
Classification table
Predicted
0
1
Actual
0
28
274
1
47
3956
75
4230
Percent correct: 92.54%
Error Distance: 231’855
302
4003
4305
Chapter E. Empirical Results of Regressions
229
Table E.28: Setting 2b - Results for OLS in-sample-regression: The
dependent variable is continuous: the stock ratio is given as a percentage.
Dataset: SCF 1995.
Observations = 3939, Parameters = 25, Deg.Fr.= 3914
Adjusted R-squared = 0.16465 , Log-L = -139.8736
Akaike Info. Crt. = 0.084
Variable
Coeff.
Constant
X101
X1706
X3006 1
X3006 5
X3008 1
X3008 45
X301 1
X301 3
X3014 1
X3014 4
X432 1
X5608
X5821
X5825 3
X5905
X7131
X7186
X7187
X7372 1
X7401 1
X7401 2
X7401 5
X8021 1
X8022
0.0376
-0.0099
0.0111
-0.0331
0.0364
-0.0022
0.0104
0.0183
-0.0263
0.0318
-0.0828
0.0412
0.0024
0.0022
⊕0.0262
0.013
0.015
0.0218
0.0233
⊕0.0408
0.0048
0.0184
-0.0058
-0.0053
⊕0.0011
Standard
Error
0.0241
0.0037
0.0025
0.0263
0.0122
0.0129
0.0094
0.0094
0.0107
0.0185
0.0098
0.0095
0.004
0.003
0.0092
0.0027
0.0097
0.0089
0.0039
0.0125
0.0113
0.0123
0.0151
0.014
0.0003
b/St.Er.
1.563
-2.682
4.497
-1.26
2.98
-0.166
1.1
-1.949
-2.466
1.719
-8.413
4.327
-0.611
0.746
2.853
4.858
-1.555
-2.432
5.928
3.257
-0.43
1.499
-0.383
-0.378
3.363
P of
|Z| > z
0.118
0.0073
0
0.2075
0.0029
0.868
0.2712
0.0513
0.0137
0.0856
0
0
0.5411
0.4559
0.0043
0
0.1199
0.015
0
0.0011
0.6675
0.1338
0.7019
0.7053
0.0008
Mean
of X
2.6111
1.379
0.0244
0.1391
0.146
0.5519
0.2917
0.2031
0.0518
0.3112
0.5626
0.2777
0.5826
0.4174
1.6078
0.3343
0.6677
1.9667
0.654
0.3359
0.1731
0.0955
0.8116
50.5463
230
E.2. Out-of-sample estimation SCF 1995 in SCF1998
Table E.29: Setting 2b OLS, Out-of-sample estimation
SCF1995 estimates in 1998 data
Classification table
Predicted
2
3
198
1
66
0
81
0
96
0
80
1
94
0
615
2
0
1
545
1502
22
261
17
242
18
250
17
231
19
262
638
2748
Percent correct: 22.16%
Error Distance: 15’259’122
Actual
0
1
2
3
4
5
4
0
0
0
0
0
0
0
5
0
0
0
0
0
0
0
2246
349
340
364
329
375
4003
Chapter E. Empirical Results of Regressions
231
Table E.30: Setting 2b - Results for Ordered model in-sample-regression:
The dependent variable (stock ratio) is discrete, 6 risk classes are considered.
Dataset: SCF 1995.
Observations = 3939, Parameters = 25, Deg.Fr.= 3914
Log-L = -7672.327, Akaike Info. Crt. = 3.908
Variable
Coeff.
Standard
Error
Index function for probability
Constant
-1.1537
0.0705
X101
-0.1321
0.0112
X1706
0.1025
0.0056
-0.1893
0.0699
X3006 1
0.3785
0.0306
X3006 5
-0.1808
0.0422
X3008 1
0.2403
0.0244
X3008 45
X301 1
0.2711
0.0239
-0.337
0.0312
X301 3
0.2123
0.0413
X3014 1
-1.2624
0.0323
X3014 4
X432 1
0.716
0.0257
X5608
0.0076
0.0104
X5821
0.0453
0.0071
0.3611
0.0233
X5825 3
X5905
0.1459
0.0065
X7131
0.1071
0.0264
X7186
0.362
0.0238
X7187
0.1843
0.0094
⊕0.5389
0.0367
X7372 1
0.0491
0.0284
X7401 1
0.2868
0.0325
X7401 2
-0.2179
0.0493
X7401 5
-0.0525
0.0418
X8021 1
X8022
⊕0.0171
0.001
b/St.Er.
P of
|Z| > z
Mean
of X
-16.361
-11.84
18.153
-2.708
12.383
-4.286
9.853
-11.354
-10.795
5.14
-39.06
27.895
-0.731
6.407
15.53
22.602
-4.055
-15.238
19.578
14.674
1.726
8.835
-4.42
-1.255
17.583
0
0
0
0.0068
0
0
0
0
0
0
0
0
0.465
0
0
0
0.0001
0
0
0
0.0843
0
0
0.2095
0
2.6111
1.379
0.0244
0.1391
0.146
0.5519
0.2917
0.2031
0.0518
0.3112
0.5626
0.2777
0.5826
0.4174
1.6078
0.3343
0.6677
1.9667
0.654
0.3359
0.1731
0.0955
0.8116
50.5463
232
E.2. Out-of-sample estimation SCF 1995 in SCF1998
Setting 2b - Ordered model, SCF1995 (cont.)
Variable
Coeff.
Standard b/St.Er.
P of
Mean
Error
|Z| > z
of X
Threshold parameters for index
Mu(1)
2.0000
.... (Fixed Parameter) ....
Mu(2)
3.0000
.... (Fixed Parameter) ....
Mu(3)
4.0000
.... (Fixed Parameter) ....
Mu(4)
5.0000
.... (Fixed Parameter) ....
Out-of-sample estimation
SCF1995 estimates in 1998 data
Classification table
Predicted
0
1
2
3
1389
803
44
9
125
206
14
4
93
226
16
5
90
250
16
8
91
216
16
5
113
236
17
9
1901 1937
123
40
Percent correct: 40.47%
Error Distance: 10’630’777
Actual
0
1
2
3
4
5
4
1
0
0
0
1
0
2
5
0
0
0
0
0
0
0
2246
349
340
364
329
375
4003
Chapter E. Empirical Results of Regressions
233
Table E.31: Setting 2b - Results for Multinomial Logit model, SCF1995,
in-sample-regression. The dependent variable (stock ratio) is discrete, 6 risk
classes are considered. The standard errors of the regression coefficients are
given in brackets. The significance levels are abbreviated with asterisks: ‘*’
and ‘**’ are significant at the 5, and 1 percent level
Observations = 3939, Parameters = 25, Deg.Fr.= 3914
Log-L = -4287.000, Akaike Info. Crt. = 2.189
Constant
X101
X1706
X3006 1
X3006 5
X3008 1
X3008 45
X301 1
X301 3
X3014 1
X3014 4
X432 1
X5608
Prob[Y=1]
-4.1796 **
(0.3719)
-0.1088
(0.0564)
0.1492 **
(0.0302)
-0.0396
(0.3404)
-0.0252
(0.1797)
0.1144
(0.2091)
0.4733 **
(0.1349)
-0.0465
(0.1275)
0.2182
(0.1492)
-0.0379
(0.2419)
-0.8767 **
(0.1514)
0.7962 **
(0.1396)
0.0806
(0.0476)
Prob[Y=2]
-4.1282 **
(0.416)
-0.0914
(0.0648)
0.0775 *
(0.0341)
-0.0513
(0.4009)
0.2315
(0.1916)
0.0802
(0.234)
0.3379 *
(0.1507)
-0.4292 **
(0.1481)
-0.1448
(0.174)
-0.0035
(0.2773)
-1.2879 **
(0.1832)
0.8879 **
(0.1632)
-0.0282
(0.064)
Prob[Y=3]
-4.175 **
(0.4487)
-0.0906
(0.0677)
0.0877 *
(0.0355)
0.0885
(0.4152)
0.3545
(0.1944)
-0.4109
(0.2899)
0.0786
(0.1552)
-0.3324 *
(0.1508)
-0.4853 *
(0.2075)
-0.0407
(0.2816)
-1.5963 **
(0.234)
0.7236 **
(0.1754)
0.1304 *
(0.0521)
Prob[Y=4]
-4.6043 **
(0.5314)
-0.1731 *
(0.0777)
0.0904 *
(0.0404)
0.0151
(0.4773)
0.2983
(0.2191)
-0.0713
(0.3145)
0.2654
(0.1822)
-0.3914 *
(0.1716)
-0.4234
(0.2318)
0.4424
(0.2765)
-1.3914 **
(0.2609)
0.814 **
(0.1995)
-0.1536
(0.092)
Prob[Y=5]
-3.7115 **
(0.4689)
-0.2386 **
(0.0774)
0.1878 **
(0.0399)
-0.885
(0.6208)
0.5274 *
(0.208)
0.1924
(0.2516)
0.2249
(0.1729)
-0.1072
(0.1626)
-0.194
(0.2083)
0.304
(0.2773)
-1.0345 **
(0.2072)
0.2446
(0.1733)
-0.0331
(0.0751)
234
E.2. Out-of-sample estimation SCF 1995 in SCF1998
Setting 2b - Results for Multinomial Logit model, SCF1995,
in-sample-regression. (cont.)
X5821
X5825 3
X5905
X7131
X7186
X7187
X7372 1
X7401 1
X7401 2
X7401 5
X8021 1
X8022
Prob[Y=1]
0.1131 **
(0.0368)
0.3781 **
(0.1224)
0.1857 **
(0.0339)
0.1967
(0.1305)
-0.207
(0.1237)
0.1635 **
(0.0496)
0.3758 *
(0.1792)
0.2417
(0.1496)
0.2773
(0.1789)
0.1905
(0.2481)
-0.0324
(0.2111)
0.0162 **
(0.0049)
Prob[Y=2]
0.0912 *
(0.0427)
0.2551
(0.1393)
0.1565 **
(0.0383)
0.2231
(0.1469)
-0.2769 *
(0.1399)
0.2318 **
(0.0553)
0.7445 **
(0.2259)
0.02
(0.1688)
0.3723
(0.1922)
-0.2616
(0.3195)
-0.5392 *
(0.2506)
0.021 **
(0.0056)
Prob[Y=3]
0.0734
(0.0435)
0.4752 **
(0.1499)
0.2459 **
(0.0411)
0.0681
(0.1594)
-0.4664 **
(0.1469)
0.3166 **
(0.0574)
0.4935 *
(0.2259)
-0.1212
(0.1786)
0.2797
(0.2067)
-0.6962
(0.4162)
-0.0561
(0.2787)
0.0157 *
(0.0061)
Prob[Y=4]
0.0428
(0.0495)
0.4559 **
(0.1698)
0.1594 **
(0.0458)
-0.304
(0.1888)
-0.5629 **
(0.1654)
0.2348 **
(0.0654)
1.0315 **
(0.2812)
0.3953
(0.2116)
0.5539 *
(0.2437)
0.071
(0.3747)
-0.0417
(0.3604)
0.0159 *
(0.0071)
Prob[Y=5]
0.0636
(0.0502)
0.2563
(0.1611)
0.1402 **
(0.0444)
-0.2269
(0.1774)
-0.1855
(0.163)
0.2323 **
(0.0638)
0.4405
(0.2308)
-0.1811
(0.1946)
0.1042
(0.2214)
0.0792
(0.3008)
0.0376
(0.2728)
0.0155 *
(0.0064)
Chapter E. Empirical Results of Regressions
235
Setting 2b - Multinomial model
Out-of-sample estimation, SCF1995 estimates in 1998 data
Classification table
Predicted
0
1
2
3
1999
0
2
0
312
3
0
0
292
7
1
0
305
3
0
2
286
4
0
1
321
5
0
2
3515
22
3
5
Percent correct: 51.04%
Error Distance: 26’479’635
Actual
0
1
2
3
4
5
4
245
34
40
54
38
47
458
5
0
0
0
0
0
0
0
2246
349
340
364
329
375
4003
236
E.2. Out-of-sample estimation SCF 1995 in SCF1998
Table E.32: Setting 3b - Results for WLS in-sample-regression: The
dependent variable is binomial, investors without stocks are to be separated
from stock-owning investors. The portrayed mean has no interpretation, as
each variable was weighted by W , the reciprocal of the observations’ standard
errors
Observations = 3939, Parameters = 25, Deg.Fr.= 3914
Adjusted R-squared = 0.30814 , Log-L = -2029.9107
Akaike Info. Crt. = 1.043
Variable
WConstant
WX101
WX1706
WX3006 1
WX3006 5
WX3008 1
WX3008 45
WX301 1
WX301 3
WX3014 1
WX3014 4
WX432 1
WX5608
WX5821
WX5825 3
WX5905
WX7131
WX7186
WX7187
WX7372 1
WX7401 1
WX7401 2
WX7401 5
WX8021 1
WX8022
Coeff.
5.1786
-0.0169
0.0247
-0.0186
0.0239
⊕0.0273
0.0346
0.0417
-0.0125
0.024
-0.1559
0.1238
0.0094
0.0101
⊕0.075
0.0415
0.0145
0.0261
0.0428
⊕0.1137
0.0093
0.0566
-0.0243
-0.0511
⊕0.0013
Standard
Error
1.8554
0.0031
0.004
0.0264
0.0145
0.0122
0.0103
0.0109
0.0097
0.0263
0.0126
0.015
0.0051
0.0039
0.0131
0.0039
0.0095
0.0117
0.0052
0.0136
0.0153
0.0144
0.0123
0.0143
0.0004
b/St.Er.
2.791
-5.393
6.161
-0.704
1.645
2.231
3.357
-3.815
-1.283
0.914
-12.379
8.235
1.849
2.576
5.707
10.681
1.528
-2.223
8.159
8.342
0.609
3.915
-1.983
-3.568
3.586
P of
|Z| > z
0.0053
0
0
0.4814
0.1001
0.0257
0.0008
0.0001
0.1993
0.3607
0
0
0.0645
0.01
0
0
0.1265
0.0262
0
0
0.5423
0.0001
0.0473
0.0004
0.0003
Mean
of X
0.0478
7.9872
3.779
0.0773
0.4007
0.5181
1.5073
0.8188
0.6828
0.1365
1.3259
1.3829
0.768
1.6019
1.1176
4.1887
1.0113
2.1663
5.7473
1.8505
0.8597
0.4959
0.3654
2.3439
152.4247
Chapter E. Empirical Results of Regressions
Setting 3b - WLS
Out-of-sample estimation
SCF1995 estimates in 1998 data
Classification table
Predicted
0
1
Actual
0
2047
199
2246
1
1234
523
1757
3281
722
4003
Percent correct: 64.21%
Error Distance: 4’687’071
237
238
E.2. Out-of-sample estimation SCF 1995 in SCF1998
Table E.33: Setting 3b - Results for Multinomial Logit model in-sampleregression: The dependent variable is binomial, investors without stocks are
to be separated from stock-owning investors. Dataset: SCF 1995.
Observations = 3939, Parameters = 25, Deg.Fr.= 3914
Log-L = -1941.207, Akaike Info. Crt. = 0.998
Variable
Coeff.
Standard b/St.Er.
Error
Characteristics in numerator of Prob[Y = 1]
Constant
-2.5324
0.2452
-10.328
X101
-0.1303
0.0391
-3.328
X1706
0.1209
0.0226
5.342
X3006 1
-0.1209
0.265
-0.456
0.2434
0.1203
2.024
X3006 5
⊕0.0198
0.1388
0.143
X3008 1
0.303
0.092
3.293
X3008 45
0.2366
0.0917
-2.579
X301 1
-0.1238
0.109
-1.136
X301 3
0.1061
0.177
0.6
X3014 1
X3014 4
-1.1626
0.1043
-11.146
0.708
0.0904
7.834
X432 1
X5608
0.0326
0.0375
0.87
X5821
0.0843
0.0288
2.922
⊕0.3587
0.086
4.173
X5825 3
X5905
0.18
0.0247
7.291
X7131
0.0535
0.0963
0.555
X7186
0.3121
0.0873
-3.574
X7187
0.2273
0.0382
5.954
⊕0.5568
0.1276
4.364
X7372 1
0.0788
0.1078
0.731
X7401 1
X7401 2
0.3022
0.1209
2.498
-0.0593
0.1652
-0.359
X7401 5
-0.1413
0.1442
-0.98
X8021 1
X8022
⊕0.0171
0.0034
5.018
P of
|Z| > z
Mean
of X
0
0.0009
0
0.6483
0.043
0.8865
0.001
0.0099
0.2559
0.5487
0
0
0.3845
0.0035
0
0
0.5787
0.0004
0
0
0.4646
0.0125
0.7196
0.327
0
2.6111
1.379
0.0244
0.1391
0.146
0.5519
0.2917
0.2031
0.0518
0.3112
0.5626
0.2777
0.5826
0.4174
1.6078
0.3343
0.6677
1.9667
0.654
0.3359
0.1731
0.0955
0.8116
50.5463
Chapter E. Empirical Results of Regressions
Table E.34: Setting 3b Multinomial Logit model
Out-of-sample estimation
SCF1995 estimates in 1998 data
Classification table
Predicted
0
1
Actual
0
1449
797
2246
1
569
1188
1757
2018 1985
4003
Percent correct: 65.88%
Error Distance: 2’578’910
239
240
E.2. Out-of-sample estimation SCF 1995 in SCF1998
Table E.35: Setting 3c - Results for OLS in-sample-regression: The
dependent variable is continuous: the stock ratio is given as a percentage.
Dataset: SCF 1995.
Observations = 1550, Parameters = 25, Deg.Fr.= 1525
Adjusted R-squared = 0.01510 , Log-L = -284.4867
Akaike Info. Crt. = 0.399
Variable
Coeff.
Constant
X101
X1706
X3006 1
X3006 5
X3008 1
X3008 45
X301 1
X301 3
X3014 1
X3014 4
X432 1
X5608
X5821
X5825 3
X5905
X7131
X7186
X7187
X7372 1
X7401 1
X7401 2
X7401 5
X8021 1
X8022
0.5217
-0.0114
0.0004
-0.0563
0.0498
-0.0191
0.0318
0.0065
-0.0517
0.0472
-0.0349
0.0427
0.0105
0.0073
-0.0038
0.003
0.0459
0.0078
0.0111
⊕0.0143
0.0287
0.0116
⊕0.0015
0.0185
-0.0002
Standard
Error
0.0532
0.0081
0.0039
0.0467
0.0226
0.0302
0.0184
0.017
0.0216
0.0295
0.0241
0.0204
0.0066
0.0046
0.017
0.0045
0.0177
0.0172
0.006
0.0262
0.0199
0.0239
0.0389
0.0317
0.0007
b/St.Er.
9.812
-1.4
-0.091
-1.206
2.2
-0.633
-1.727
-0.385
-2.394
1.602
-1.448
-2.095
-1.592
-1.574
-0.221
-0.678
-2.589
-0.452
1.843
0.545
-1.444
-0.486
0.038
0.583
-0.341
P of
|Z| > z
0
0.1615
0.9277
0.2279
0.0278
0.527
0.0842
0.7006
0.0166
0.1091
0.1475
0.0362
0.1113
0.1155
0.8247
0.4975
0.0096
0.6514
0.0653
0.5859
0.1489
0.6271
0.9694
0.5597
0.733
Mean
of X
2.5555
2.3277
0.0271
0.1477
0.0845
0.6794
0.3077
0.1574
0.0723
0.1284
0.7871
0.3368
0.8
0.5916
2.4135
0.3626
0.6381
2.5852
0.7723
0.4729
0.1761
0.0458
0.8852
54.6277
Chapter E. Empirical Results of Regressions
241
Table E.36: Setting 3c OLS Out-of-sample estimation
SCF1995 estimates in 1998 data
Classification table
Predicted
0
1
2
1
126
221
0
109
230
0
119
244
0
109
219
0
123
250
1
586
1164
Percent correct: 20.20%
Error Distance: 2’231’527
Actual
0
1
2
3
4
3
1
1
1
1
2
6
4
0
0
0
0
0
0
349
340
364
329
375
1757
242
E.2. Out-of-sample estimation SCF 1995 in SCF1998
Table E.37: Setting 3c - Results for Ordered model in-sample-regression:
The dependent variable (stock ratio) is discrete, 5 risk classes are considered.
Dataset: SCF 1995.
Observations = 1550, Parameters = 25, Deg.Fr.= 1525
Log-L = -3516.390, Akaike Info. Crt. = 4.570
Variable
Coeff.
Standard
Error
Index function for probability
Constant
2.4061
0.1076
X101
-0.0748
0.0164
X1706
0.0064
0.0081
-0.2652
0.1013
X3006 1
0.337
0.047
X3006 5
-0.0581
0.059
X3008 1
0.0382
X3008 45 0.2067
X301 1
0.1048
0.0345
-0.3872
0.0435
X301 3
0.2985
0.0587
X3014 1
-0.2645
0.0477
X3014 4
X432 1
0.299
0.0404
X5608
0.0743
0.0133
X5821
0.0411
0.0092
⊕0.0064
0.035
X5825 3
X5905
0.0107
0.0093
X7131
0.3311
0.0362
X7186
0.1189
0.0355
X7187
0.0728
0.0125
⊕0.1639
0.0509
X7372 1
0.1918
0.0416
X7401 1
0.0605
0.05
X7401 2
-0.1248
0.0738
X7401 5
0.0548
0.0624
X8021 1
X8022
-0.0005
0.0014
b/St.Er.
P of
|Z| > z
Mean
of X
22.354
-4.561
-0.782
-2.619
7.17
-0.984
-5.41
-3.042
-8.907
5.089
-5.55
-7.396
-5.603
-4.474
0.181
-1.157
-9.141
-3.345
5.828
3.217
-4.61
-1.21
-1.692
0.879
-0.33
0
0
0.4344
0.0088
0
0.3252
0
0.0023
0
0
0
0
0
0
0.856
0.2471
0
0.0008
0
0.0013
0
0.2261
0.0906
0.3795
0.7413
2.5555
2.3277
0.0271
0.1477
0.0845
0.6794
0.3077
0.1574
0.0723
0.1284
0.7871
0.3368
0.8
0.5916
2.4135
0.3626
0.6381
2.5852
0.7723
0.4729
0.1761
0.0458
0.8852
54.6277
Chapter E. Empirical Results of Regressions
243
Setting 3c - Ordered model, SCF1995 (cont.)
Variable
Coeff.
Standard b/St.Er.
P of
Mean
Error
|Z| > z
of X
Threshold parameters for index
Mu(1)
2.0000
.... (Fixed Parameter) ....
Mu(2)
3.0000
.... (Fixed Parameter) ....
Mu(3)
4.0000
.... (Fixed Parameter) ....
Out-of-sample estimation
SCF1995 estimates in 1998 data
Classification table
Predicted
0
1
2
0
323
24
0
313
26
0
310
53
0
292
35
0
328
46
0
1566
184
Percent correct: 20.94%
Error Distance: 4’675’546
Actual
0
1
2
3
4
3
2
1
1
2
1
7
4
0
0
0
0
0
0
349
340
364
329
375
1757
244
E.2. Out-of-sample estimation SCF 1995 in SCF1998
Table E.38: Setting 3c - Results for Multinomial Logit model in-sampleregression: The dependent variable (stock ratio) is discrete, 5 risk classes
are considered. Dataset: SCF 1995. The standard errors of the regression
coefficients are given in brackets. The significance levels are abbreviated with
asterisks: ‘*’ and ‘**’ are significant at the 5, and 1 percent level
Observations = 1550, Parameters = 25, Deg.Fr.= 1525
Log-L = -2345.463, Akaike Info. Crt. = 3.059
Constant
X101
X1706
X3006 1
X3006 5
X3008 1
X3008 45
X301 1
X301 3
X3014 1
X3014 4
X432 1
X5608
Prob[Y=1]
0.1129
(0.5181)
0.009
(0.0792)
-0.0775 *
(0.0379)
0.0291
(0.4399)
0.2607
(0.2304)
-0.0058
(0.2883)
-0.1432
(0.1817)
-0.3845 *
(0.1693)
-0.3547
(0.2009)
0.0849
(0.3074)
-0.454 *
(0.2236)
0.0796
(0.2024)
-0.1078
(0.0686)
Prob[Y=2]
0.0138
(0.5435)
0.0203
(0.0818)
-0.0679
(0.0388)
0.2046
(0.4456)
0.3768
(0.2311)
-0.5741
(0.3335)
-0.4317 *
(0.1842)
-0.272
(0.17)
-0.7043 **
(0.2289)
0.0417
(0.3092)
-0.743 **
(0.2678)
-0.0686
(0.2113)
0.0522
(0.0569)
Prob[Y=3]
-0.4407
(0.6183)
-0.0772
(0.0907)
-0.0684
(0.0438)
0.0805
(0.5081)
0.3226
(0.2528)
-0.205
(0.3564)
-0.2222
(0.2093)
-0.3285
(0.1896)
-0.6356 *
(0.2525)
0.5145
(0.306)
-0.5089
(0.2922)
0.0251
(0.2335)
-0.2318 *
(0.0958)
Prob[Y=4]
0.4576
(0.5773)
-0.1354
(0.0919)
0.0334
(0.044)
-0.8612
(0.6473)
0.5412 *
(0.2483)
0.1135
(0.3062)
-0.24
(0.2034)
-0.069
(0.1853)
-0.3611
(0.2352)
0.3458
(0.314)
-0.1988
(0.2484)
-0.5659 **
(0.2117)
-0.1235
(0.0801)
Chapter E. Empirical Results of Regressions
245
Setting 3c - Results for Multinomial Logit model, SCF1995,
in-sample-regression. (cont.)
X5821
X5825 3
X5905
X7131
X7186
X7187
X7372 1
X7401 1
X7401 2
X7401 5
X8021 1
X8022
Prob[Y=1]
-0.0213
(0.045)
-0.1321
(0.1641)
-0.0353
(0.0434)
0.0588
(0.1692)
-0.0835
(0.1684)
0.0787
(0.0589)
0.3586
(0.2631)
-0.2267
(0.1924)
0.0897
(0.2337)
-0.5399
(0.3838)
-0.5145
(0.304)
0.0048
(0.0067)
Prob[Y=2]
-0.0373
(0.0455)
0.1095
(0.1735)
0.0591
(0.046)
-0.1225
(0.1777)
-0.2242
(0.1726)
0.1547 **
(0.0598)
0.1602
(0.2605)
-0.35
(0.2002)
-0.0048
(0.2437)
-0.9636 *
(0.4676)
-0.1026
(0.3272)
-0.0001
(0.0071)
Prob[Y=3]
-0.0671
(0.0513)
0.1147
(0.1926)
-0.0178
(0.0505)
-0.5052 *
(0.2053)
-0.3583
(0.1906)
0.0801
(0.0677)
0.6719 *
(0.3126)
0.1345
(0.2321)
0.2497
(0.2779)
-0.1089
(0.433)
-0.0763
(0.4014)
0.0009
(0.008)
Prob[Y=4]
-0.048
(0.053)
-0.1157
(0.1865)
-0.0482
(0.0493)
-0.4059 *
(0.1993)
-0.0026
(0.1927)
0.0739
(0.0677)
0.0548
(0.2724)
-0.4321 *
(0.2168)
-0.2006
(0.2605)
-0.1223
(0.3743)
0.0643
(0.3285)
0.0004
(0.0076)
246
E.2. Out-of-sample estimation SCF 1995 in SCF1998
Setting 3c - Multinomial Logit model
Out-of-Sample estimation
SCF1995 estimates in 1998 data
Classification table
Predicted
0
1
2
231
16
18
216
21
25
213
17
39
209
14
21
226
17
28
1095
85
131
Percent correct: 21.74%
Error Distance: 5’743’122
Actual
0
1
2
3
4
3
79
75
89
78
91
412
4
5
3
6
7
13
34
349
340
364
329
375
1757
Chapter E. Empirical Results of Regressions
E.3
247
Likelihood Ratio Tests
All LR tests are set up in such way that the null could not be refuted for the
maximal number of least significant factor coefficients.
LR tests, setting 1
In setting 1 the dependent variable has 7 categories. The number of observations are 4305, there are 25 parameters and thus 4280 degrees of freedom.
MNL model. H0 : CoefficientVarX =0, Var X: X101, X3006 5, X301 3,
X5608. 24 Restrictions: 4 variables in 6 logits. P-value=0.345
Var X
see above
Likelihood Ratio test
LR Test
θ̃
θ̂
-5350.63 -5363.71
26.16
Tabl-V.
36.42
Result
n.r.
θ̃ = LogL of unrestricted model, θ̂ = LogL of restricted model
Tabl-V. = Critical Value of chisquare distribution
ref. = refute the null, n.r. = the null cannot be refuted
Ordered Logit model. H0 : CoefficientVarX =0. 6 Restrictions:
X3006 1, X3006 5, X301 1, X3014 1, X7401 5, X8021 1. P-Value=0.14
Var X
see above
Likelihood Ratio test
LR Test
θ̃
θ̂
-7833.03 -7837.90
9.74
Tabl-V.
12.59
Result
n.r.
LR tests, setting 2a
In setting 2a the dependent variable has 2 categories. The number of observations are 4305, there are 25 parameters and thus 4280 degrees of freedom.
MNL model.
H0 : CoefficientVarX =0, VarX: X101, X3006 5,
X3008 45, X301 3, X5608, X7401 5, X8021 1, X5821, X301 1. Nine
Restrictions. P-value=0.05
Var X
see above
Likelihood Ratio test
LR Test Tabl-V.
θ̃
θ̂
-678.87 -687.32
16.918
16.92
Result
n.r.
248
E.3. Likelihood Ratio Tests
LR tests, setting 2b
In setting 2b the dependent variable has 6 categories. The number of observations are 4003, there are 25 parameters and thus 3978 degrees of freedom.
MNL model. H0 : CoefficientVarX =0, VarX: X101, X3006 1, X3008 1,
X301 1, X3014 1, X5608, X7131, X7186, X7401 5. Nine Variables, 6
Risk Classes = 45 Restrictions. P-value=0.14
Var X
see above
Likelihood Ratio test
LR Test
θ̃
θ̂
-4672.59 -4700.15
55.13
Tabl-V.
61.66
Result
n.r.
Ordered Logit model. H0 : CoefficientVarX =0. 5 Restrictions:
X101, X3006 1, X3006 5, X301 3, X8021 1. P-Value=0.13
Var X
see above
Likelihood Ratio test
LR Test
θ̃
θ̂
-8846.00 -8850.23
8.47
Tabl-V.
11.07
Result
n.r.
LR tests, setting 3b
In setting 3b the dependent variable has 2 categories. The number of observations are 4003, there are 25 parameters and thus 3978 degrees of freedom.
MNL model. H0 : CoefficientVarX =0, VarX: X101, X3006 1, X3006 5,
X3008 1, X301 1, X301 3, X3014 1, X5608, X7131, X7401 2, X7401 5,
X8021 1. 12 Restrictions. P-value=0.08
Var X
see above
Likelihood Ratio test
LR Test
θ̃
θ̂
-1906.34 -1915.93
19.18
Tabl-V.
21.03
Result
n.r.
LR tests, setting 3c
In setting 3c the dependent variable has 2 categories. The number of observations are 1757, there are 25 parameters and thus 1732 degrees of freedom.
MNL model. H0 : CoefficientVarX =0, VarX: X101, X3006 1, X3006 5,
X3008 1, X301 1, X301 3, X3014 1, X432 1, X5608, X5821, X5825 3,
X5905, X7131, X7187, X7401 1, X7401 2, X7401 5, X8022. 18
Variables, 5 Risk classes = 72 Restrictions. P-value=0.47
Chapter E. Empirical Results of Regressions
Var X
see above
Likelihood Ratio test
LR Test
θ̃
θ̂
-2766.66 -2802.76
72.18
249
Tabl-V.
92.81
Result
n.r.
Ordered Logit model.
H0 : CoefficientVarX =0, VarX: X101,
X3006 1, X3006 5, X5608, X5821, X5905, X7131, X7186, X7187,
X7372 1, X7401 2, X8022. 12 Restrictions. P-value=0.19
Var X
see above
Likelihood Ratio test
LR Test
θ̃
θ̂
-3899.91 -3907.92
16.02
Tabl-V.
21.03
Result
n.r.
250
E.3. Likelihood Ratio Tests
Appendix F
Independent Factors
From a total of over 3’000 SCF-variables, 150 were selected for the sample.
30 of these were chosen according to the study’s hypotheses and significance
tests. The best 17 out of these 30 were chosen as the Independent Variable
Set.
For most of the variables the original coding as given by the SCF had to be
modified: Codes were combined, values adjusted to yield an ordinal ranking
and continuous variables had to be discretized to end up with a limited number
of separate classes with an approximately equal number of observations per
class. The variables are:
x101, x1706, x3006, x3008, x301, x3014, x432, x5608, x5821, x5825,
x5905, x7131, x7186, x7187, x7401, x8021, x8022
Some of the above are multidimensional variables, their dimensions not
obeying any ordinal ranking. They had to be split into several dummy variables to identify the effect of each dimension. The number of factors defined
in the following thus increased from 17 to 21.
251
252
Table F.1: Calculation of financial assets
Composition of total financial assets and the stock ratio.
tmv = total market value
tdv = total dollar value
Variable
Element
Stock
ratio
Amount
invested
in stocks
Financial
Assets
= Amount in stocks / Total financial assets
=
+
+
=
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
x3822
x3915
x6704
x3506
x3529
x3610
x3706
x3718
x3721
x3804
x3818
x3822
x3824
x3826
x3828
x3830
x3902
x3915
x3930
x6704
x6706
x7635
x7636
x7637
x7638
x7639
Significance
tmv of all Stock Funds
tmv of publicly traded stock
tmv of all of the mutual funds
amount in checking account
amount in all other checking accounts
how much in IRA or KEOGH?
amount in tax-free money market account
amount in remaining money market accounts
tdv of all these CDs
amount in savings account
amount in all your remaining savings accounts
total market value of all of the Stock Funds
tmv of all of the Tax-Free Bond Funds
tmv of all of the Government Bond Funds
tmv of all of the Other Bond Funds
tmv of any other mutual funds
total face value of all the savings bonds
tmv of publicly traded stock
tdv of all the cash or call money accounts
tmv of all of the mutual funds
tmv of all of your bonds
tmv of all of the Mortgage-backed bonds
tmv of US Gov bonds
tmv of these state/municipal bonds
tmv of your foreign bonds
tmv of all of the other type of bonds
Chapter F. Independent Factors
253
Table F.2: Overview of variables
Overview of variables used in the empirical analysis as independent
factors.
Variable
Significance
Coding
x101
Number of people in the
household
Do you have any lines
of credit, not counting
credit cards?
How much is your real
estate property worth
if sold today
Code amount
x1101
x1706
x1715
How much is still owed
of the mortgage on the
property?
x3006
What are your most
important reasons for
saving?
x3006 1
Is your most important
reason for saving liquidity
and consumption?
0 - Yes
1 - No
0
1
2
3
4
5
6
0
1
2
3
4
5
6
0
1
2
3
4
5
6
0
1
-
$0
$1 to $10’000
$10’000 to $50’000
$50’000 to $100’000
$100’000 to $500’000
$500’000 to $1’000’000
over $1’000’000
$0
$1 to $10’000
$10’000 to $50’000
$50’000 to $100’000
$100’000 to $500’000
$500’000 to $1’000’000
over $1’000’000
do not save
Liquidity
Consumption
Events
House, Car
Education, Family
Retirement, Reserves
No
Yes
continued on next page
254
Variable
Significance
Coding
x3006 5
Is your most important
reason for saving education and family?
In planning your saving
and spending, which of
the time periods is
most important for
you?
In planning your saving
and spending, are the next
few months most important for you?
In planning your saving
and spending, do you have
a longer than 5 years?
Do you expect the
US economy to perform
better in the next 5 years
than it has over the last 5?
Do you expect the
US economy to perform
better in the next 5 years
than it has over the last 5?
What kind of financial
obligations will you have
to meet in the next five
years?
What amount of financial
risk would you be willing
to take when you save
or make investments?
0 - No
1 - Yes
x3008
x3008 1
x3008 45
x301 1
x301 3
x3011 4
x3014
1
2
3
4
5
0
1
-
Next few months
Next year
Next few years
Next 5-10 years
Longer than 10 years
No
Yes
0 - No
1 - Yes
0 - else
1 - better
0 - else
1 - worse
0 - else
1 - education
1 - substantial risks to earn
substantial returns
2 - above average risks to
earn above average returns
3 - average risks to earn
average returns
continued on next page
Chapter F. Independent Factors
Variable
x3014 1
x3014 4
x401 0
x413
x432
x432 1
x432 3
x5318
Significance
Would you be willing to
take substantial risks to
earn substantial returns
when you save or make investments?
You are not willing to
take any financial risks?
How do you judge buying
things on the installment
plan?
After the last payments
were made on these
accounts roughly what
was the balance still owed
on these accounts?
Do you pay off the total
balance owed on your
credit card account each
month?
Do you always pay off
the total balance owed on
your credit card account
each month?
Do you hardly ever pay
off the total balance owed
on your credit card account each month?
How much social benefits
are received each month or
255
Coding
4 - Not willing to take any
financial risks
0 - No
1 - Yes
0
1
0
1
-
Wrong
Right
else
good idea
0
1
2
3
4
5
6
1
2
3
-
$0
$1 to $499
$500 to $999
$1’000 to $1’999
$2’000 to $4’999
$5’000 to $9’999
over $10’000
Always or almost always
Sometimes
Hardly ever
0 - No
1 - Yes
0 - No
1 - Yes
0 - $0
1 - $1 to $499
continued on next page
256
Variable
Significance
Coding
year?
2 - $500 to $999
3 - $1’000 to $1’999
4 - $2’000 to $4’999
5 - $5’000 to $9’999
6 - over $10’000
0 - $0
1 - $1 to $499
2 - $500 to $999
3 - $1’000 to $1’999
4 - $2’000 to $4’999
5 - $5’000 to $9’999
6 - over $10’000
0 - $0
1 - $1 to $10’000
2 - $10’000 to $50’000
3 - $50’000 to $100’000
4 - $100’000 to $500’000
5 - $500’000 to $1’000’000
6 - over $1’000’000
0 - Wrong
1 - Right
1 - Nursing, Chiropratic, other
2 - Associate’s, junior college
3 - Bachelor’s degree
4 - MA, MS, MBA
5 - PhD, MD, Law, JD, other
doct.
0 - Yes
1 - No
x5608
About how much do you
expect your future
pension to be?
x5821
About how much in future
inheritance (or transfer
of assets) do you
expect?
x5825 3
You do not expect to leave
a sizable estate to others?
What is the highest
degree you earned?
x5905
x7131
x7141
Have you applied for any
type of credit or loan in
the last 5 years?
How much did you
borrow the last time
you used your largest
0 - $0
1 - $1 to $999
2 - $1’000 to $4’999
continued on next page
Chapter F. Independent Factors
Variable
x7186
x7187
257
Significance
Coding
line of credit secured
by the equity in your
home?
3 - $5’000 to $9’999
4 - $10’000 to $49’999
5 - $50’000 to $100’000
6 - over $100’000
0 - Yes
1 - No
0 - $0
1 - $1 to $10’000
2 - $10’000 to $50’000
3 - $50’000 to $100’000
4 - $100’000 to $500’000
5 - $500’000 to $1’000’000
6 - over $1’000’000
1 - married
2,3,4 - separated, divorced,
widowed
5 - never married
0 - No
1 - Yes
1 - Managerial, Executive
2 - Technical, Sales, Administrative
5 - Operator, Fabricator, Laborer
1 - male
2 - female
0 - No
1 - Yes
Are you saving for
foreseeable expenses now?
About how much do
you think you need to
have in savings for
emergencies and other
unexpected things
that may come up?
x7372
Marital status
x7372 1
Are you married?
x7401
Is the official title
of your job: (y/n) ?
x8021
Gender
x8021 1
Are you male?
x8022
Age in years
258
Table F.3: Expected signs of factor coefficients
Based on the hypotheses in Section 3.2 the estimated coefficients of
the independent variables are expected to have the following signs:
Variable
Sign
Justification
X101
−
X1706
+
X3006 1
−
X3006 5
+
X3008 1
−
X3008 45
+
X301 1
+
X301 3
−
X3014 1
X3014 4
X432 1
+
−
+
X5608
+
X5821
+
The higher the number of people in the household
the higher the financial obligations and the lower
will be the ability to take risks
More wealth in the form of real estate property implies higher risk taking capacity
Liquidity and consumption are short-term saving
goals and call for less risky investments
Long-term saving goals such as education and family allow for higher risk in the portfolio
A short investment horizon requires a less risky asset allocation
A long investment horizon makes higher stock ratio
in the portfolio possible
Optimistic economic expectations are one prerequisite for taking a higher investment exposure
Pessimistic economic expectations impede high
stakes in risky assets
The investor is willing to take substantial risks
The investor is not willing to take any financial risks
Investors who always pay off their credit signal financial discipline and thus the ability to take financial risks
The higher the expected future pension the higher
the risk to be taken
Future windfalls in the form of inheritance allow an
investor to take higher risks
continued on next page
Chapter F. Independent Factors
259
Variable
Sign
Justification
X5825 3
−
X5905
+
X7131
+
X7186
+
X7187
+
X7372 1
−
X7401 1
+
X7401 2
+
X7401 5
−
X8021 1
−
X8022
−
Those who do not plan to leave an estate will consume their wealth and should take increasingly less
risk as they grow older
The higher the educational background the higher
the probability that the investor has an understanding about the financial markets
Investors who haven’t applied for credit or loan are
more likely to be in control of their finances and
thus can take higher risks
Investors not saving for foreseeable expenses are
considering to invest their free wealth and thus focus on a longer time perspective
The higher the savings needed the more risky one
needs to invest
Married investors are less independent and have a
lower potential for risky investments
Managers and Executives are willing to take more
risks in order to earn higher returns
Just as Managers and Executives, Technical, Sales
and Administrative employees due to relatively
high wages also have a higher ability to compensate for losses in their investments
Operators and Laborers have less leeway to compensate for losses with their income
Even though there is no unambiguous interpretation of the gender coefficient, some studies found
that men in their futile attempts to time the market trade more often than female investors. Male
investor should thus hold less risky portfolios
The older the investor the lower his capacity to offset investment losses by human capital (future income)
260
Appendix G
Various Riskrulers
G.1
Risk Quota by Fidelity Investments
G.2
Allianz Anleger Analyse
G.3
Union Investment
261
262
G.3. Union Investment
Figure G.1: ‘Risk Quota by Fidelity Investments’: Example of traditional risk
rulers
Chapter G. Various Riskrulers
263
Figure G.2: ‘Allianz Anleger-Analyse’: Example for a psychologically motivated risk ruler. These are typically not empirically tested with quantitative
methods.
264
G.3. Union Investment
Figure G.3: ‘Union Investment - Test zur persönlichen Risikobereitschaft’:
Example for a psychological risk ruler. Answering Sheet.
Chapter G. Various Riskrulers
265
Figure G.4: Example for a psychological risk ruler: ‘Union Investment - Test
zur persönlichen Risikobereitschaft’. Evaluation Sheet.
266
G.3. Union Investment
Appendix H
Practical implementation
of a Risk Ruler
H.1
Proceeding when developing a Risk Ruler
The first step when developing a questionnaire for the assessment of risk
aversion or risk preferences in general is to consult the underlying theory and
empirical studies on the topic in order to derive workable hypotheses about
financial behavior. On the basis of these hypotheses the factors and indicators
most relevant for financial decision making can be determined.
In a first number of estimations these selected factors will be tested for
their significance and collinearity. Insignificant factors can be dropped and
similar, collinear factors can be condensed to create new ones. In order to
minimize cost and effort these pretests can be carried out using existing sample
data.
With the final set of factors a new questionnaire is set up that can be
used in a survey. The survey design must ensure that the resulting sample is
representative. The collected answers are then used to estimate the weights
of each factor e.g. question in the questionnaire. In this sense, the data serves
the purpose of a calibration sample.
267
268
H.2
H.2. Example of an interactive Risk Ruler
Example of an interactive Risk Ruler
For experimental reasons I set up a Sample-Risk Ruler under the Web-address
http://riskaversion.hypermart.net. Any investor can visit this site and determine his optimal stock ratio and skewness preference by answering the questions on that page.1 The questions differ slightly from the ones in the SCF
used for the empirical analysis in the sense that they are more investmentspecific and thus have hopefully more predictive power.
Alternatively, the Risk Ruler test can be carried out by oneself using the
same questions displayed below. After answering all questions, the responses
must be weighted by using the scheme in Figure H.2. The questionnaire leaves
the investor with three stock ratios: The econometric model yields the optimal
stock ratio according to the questions answered. The second estimate for the
optimal stock ratio comes from the gamble’s certainty equivalent and the third
stock ratio is the one the investor currently holds. Typically this last figure
is not a result of conscious decision-making. Portfolios evolve over time as
an investor buys new shares due to capital increases, inherits financial assets
or disinvests because of changing circumstances. The result of the gamble
captures usually a rough estimate of one’s risk aversion. However, for an
average investor gambles are difficult to understand. An accurate answer for
the certainty equivalent will thus result seldomly. The most reliable estimate
for the optimal stock ratio is therefore the result of the questionnaire. The
other two numbers provide useful comparative values.
The online Risk Ruler was programmed in PERL and set up on a Host
that supports cgi-scripts thus enabling the processing and saving of the input
data. In order to ensure that the sample is at least partly representative,
demographic factors were included as questions. Also, the input file automatically stored the visitor’s IP address. In that way, experimenting visitors
who are sending multiple rounds of answers can be eliminated, as they would
otherwise distort the sample.
1 The corresponding risk aversion coefficients are not displayed, but can easily be calculated using the equations in Chapter 7.
Chapter H. Practical implementation of a Risk Ruler
Your Financial Risk Preference
By filling in the following questionnaire you can determine your personal risk
preference (risk aversion) measured as the stock ratio optimal for you. The
ratio of stocks is the percentage of stocks to your total financial assets. Apart
from your optimal exposure you will learn about your degree of skewness
preference regarding the distribution of investment returns. In other words
you will find out whether an option strategy would be suitable for you either
to prevent you from downside risk or to enhance your upside potential. Your
results have been calculated utilizing a multinomial logit model and factor
analysis based on a data set collected on the internet.
Bold text in [brackets] symbolizes the key for each question. The sample
data and regression results later refer to these keys.
1. Please state the country of your main residence: [nat]
......................................................
2. Please name your age in years: [age]
(a) < 20
(b) 21-34
(c) 35-44
(d) 45-54
(e) 55-64
(f) 65-74
(g) > 75
3. Are you ... [sex]
(a) female
(b) male
4. What is your marital status? [mar]
(a) single
(b) married
(c) divorced
(d) widowed
269
270
H.2. Example of an interactive Risk Ruler
5. Do you have any children? [chil]
(a) None
(b) One or Two
(c) More than two
6. What educational degree did you accomplish? [degr]
(a) Highschool
(b) Professional degree
(c) BA / MA
(d) PhD or higher
7. Please name your profession: [prof ]
(a) Managerial, Executive, Professional
(b) Technical, Sales, Administrative
(c) Operator, Fabricator, Laborer
8. When answering the following question please consider that you are investing
only that part of your wealth that is not bound to finance planned projects
(such as a house, car etc.). Consider investing that part of your wealth which
does not have to be liquidated within the next 5-10 years. Below you are given
the distributions of the yearly returns of 6 different investment opportunities.
Please select >the one< that you think fits your risk tolerance level best.
[risk]
Chapter H. Practical implementation of a Risk Ruler
271
9. Which describes best your attitude regarding fluctuations of your portfolio
value: [fluc]
(a) I couldn’t accept a daily loss of 1%, monthly loss of 5% or a yearly loss
of 16% of my portfolio value.
(b) I couldn’t accept a daily loss of 1.5%, monthly loss of 9% or a yearly
loss of 32% of my portfolio value.
(c) I couldn’t accept a daily loss of 2%, monthly loss of 12% or a yearly loss
of 40% of my portfolio value.
(d) I could accept any of the above knowing I’m able to wait out even longer
periods of a downmarket.
10. Concerning Gains: Imagine you could invest part of your portfolio, say $10’000,
in one of the following lotteries that after 3 years result in the given additional
payoffs. Which lottery would you choose? (All have the same expected value).
You >will not lose< your principal of $10’000. You will get that amount back
at the end of 3 years! [lot1]
(a) 50% probability for a gain of 4’000 and 50% probability for a gain of
3’000
(b) 30% probability for a gain of 9’100 and 70% probability for a gain of
1’100
(c) 20% probability for a gain of 17’500 and 80% probability for no gain
11. Concerning Losses: Imagine you could invest yet another part of your portfolio, say $10’000, in one of the following lotteries that after 3 years result in
the given additional payoffs. Which lottery would you choose? You will ¿not¡
lose your principal of $10’000!You will get it back after the 3 years! [lot2]
(a) 50% probability for a gain of 5’500 and 50% probability for a loss of
-3’900
(b) 40% probability for a gain of 4’000 and 60% probability for a loss of
-1’400
(c) 30% probability for a gain of 2’500 and 70% probability for no loss
12. What percentage of your total financial assets do you currently have invested
in stocks (or riskier assets such as options, hedge funds)? Your total financial
assets consist of all your liquid wealth:checking and saving accounts, bonds,
stocks, funds, options etc. [curr]
272
H.2. Example of an interactive Risk Ruler
(a) 0% - 20%
(b) 21% - 40%
(c) 41% - 60%
(d) 61% - 80%
(e) 81% - 100%
13. How would you rate the current risk level of your portfolio: [levl]
(a) safe
(b) moderate risk
(c) considerable risk
(d) very high-risk
14. What percentage of stocks of your total financial assets do you consider optimal for yourself for the next 5-10 years? [opt]
(a) 0% - 20%
(b) 21% - 40%
(c) 41% - 60%
(d) 61% - 80%
(e) 81% - 100%
15. According to you: what will be the path of development for the economy of
your country over the next 3-5 years? [econ]
(a) considerable higher growth than during the last few years
(b) growth will be a bit better than during the last few years
(c) there will be no additional growth
(d) growth will slow down
(e) growth will slow down considerably
16. Since how many years do you trace the development of the financial markets or
since how many years do you carry out financial market transactions? [finex]
(a) less than a year
(b) 1 - 2 years
(c) 3 - 4 years
Chapter H. Practical implementation of a Risk Ruler
273
(d) 3 - 4 years
(e) 5 - 10 years
(f) more than 10 years
17. Imagine that after allocating all your financial assets, your portfolio - after an
initial gain - lost considerably in value. What do you do? [aftls]
(a) I switch to safer assets. As I check the prices of my investments at least
several times a month, I can sell quickly if they begin to lose money.
(b) Daily losses in the value of my investments make me uncomfortable
but do not cause me to sell immediately. If my investments suffer a
substantial loss over a full quarter, however, I am likely to sell.
(c) I realize there may be substantial day-to-day changes in the value of
my investments. Although I focus on quarterly performance trends, I
usually wait an entire year before making any changes.
(d) If my investments suffered significant losses over a given year (in a down
market), I would continue to follow a consistent long-term investment
plan and maintain my asset mix.
18. The primary goal of my investment strategy is ... [invgol]
(a) ... mainly to protect conservatively the principal value of my investments. I accept lower returns of a conservative strategy in order to
minimize the danger of a loss.
(b) ... long-term protection of the principal value. Daily temporary fluctuations of my portfolio value are acceptable, as I achieve higher returns
with a little more risk.
(c) ... long-term growth through higher exposure in the stock market. Moderate fluctuations of my portfolio value are a daily routine for me and
do not bother me.
(d) ... aggressive growth through maximum exposure in the stock market.
Considerable fluctuations of my portfolio value are a daily routine for
me and I accept them.
(e) ... speculation: I constantly track gains and losses of my position trying
to sell winners before a downward movement gathers momentum and
trying to convert losses into money before they become substantial.
19. What role does income tax play within your investment decision? [tax]
274
H.2. Example of an interactive Risk Ruler
(a) taxes play a central role in deciding over my asset allocation.
(b) taxes need to be considered, however they are not as important as other
factors (such as safety of my investments).
(c) taxes play a neglectable or no role for my asset allocation.
20. What is your primary source of information when making investment decisions? [isrc]
(a) friends, family, relatives, own research
(b) media:magazines, newspapers, advertisements
(c) specialists’ journals, investment seminars or clubs
(d) banks, brokers, financial advisors
21. Do you think you have the temperament and time to sit through a long
downturn - sometimes maybe years of losses - without selling? [tprt]
(a) no, I would sell my stocks and invest in fixed-income assets. I wouldn’t
have the patience to wait.
(b) yes, I wouldn’t sell my stocks even if a down market prevailed for as long
as five years
22. Do you tend to sell stocks if they have performed well and made you the
profit you wanted and at the same time keep the stocks that have already
experienced a loss in the hope that they’d come up again? [sellos]
(a) yes, I tend to keep stocks that performed badly and sell stocks that made
a good profit.
(b) no, my decision over buying or selling the stocks of a specific company
depends solely on the economic situation of the company, analyst reports
and the overall economic outlook.
23. How long could you live from your financial assets, if you stopped receiving
any income from working today? [dep]
(a) less than 3 months
(b) 3 - 6 months
(c) 6 - 12 months
(d) 1 - 5 years
(e) more than 5 years
Chapter H. Practical implementation of a Risk Ruler
275
24. In planning your saving and spending, which of the time periods listed below
is most important to you? [savhor]
(a) next few months
(b) next year
(c) next few years
(d) next 5 - 10 years
(e) longer than 10 years
25. How high do you think are the odds of beating a money-market account with
a pure stock investment in any given year from now? [perf ]
(a) < 50%
(b) 50%
(c) > 50%
H.2.1
Small sample Internet survey
An internet survey on the above questionnaire resulted in the small sample
printed on the pages 276-278. The following explanations apply to all tables
and figures in this section: Question 14, the optimal stock ratio class, was
chosen as the dependent variable. The variable ‘nationality’, with the key
‘[nat]’, was coded as: 1 = Switzerland, 2 = Germany, 3 = USA/Canada, 4
= Japan, 5 = India, 6 = UK, 7 = Australia, 8 = China, 9 = Portugal. The
codes for all other questions correspond to the letter of the subchoice, e.g.
‘a)’ for example translates into ‘1’.
H.2.2
Calculating the predicted choice
In the Evaluation Sheet on page 281 all answers need to be multiplied with
the rounded coefficients and summed up vertically. These vertical sums are
the logits βjk · xik of each risk class j (where the number of individuals i =
1, 2, ...79, the number of risk classes j = 0, 1, ..., 5 and the number of factors
k = 0, 1, ...14; k = 0 depicts the alternative specific constant). The predicted
choice is the one with the highest calculated probability P rob[Y = j]:
exp(β x)
Prob[Y=j] = J
j=0 exp(β x)
276
H.2. Example of an interactive Risk Ruler
Figure H.1: Sample Data on above questionnaire collected by an Internet
survey.
Chapter H. Practical implementation of a Risk Ruler
277
278
H.2. Example of an interactive Risk Ruler
Chapter H. Practical implementation of a Risk Ruler
279
Table H.1: Setting 2b - Results for Multinomial Logit model, Internet
Survey for Questionnaire on Page 269, in-sample-regression. The standard
errors of the regression coefficients are given in brackets. The names of the
factors refer to the [keys] given in the questionnaire. The significance levels
are abbreviated with asterisks: ‘*’ and ‘**’ are significant at the 5, and 1
percent level. Few estimates prove significant due to the small sample size.
Observations = 79, Parameters = 15, Log-L = -40.74,
Constant
NAT
AGE
SEX
ECON
FINEX
AFTLS
INVGOL
TAX
ISRC
TPRT
SELLOS
DEP
SAVHOR
PERF
Prob[Y=1]
-2.3393
(3.2555)
0.1163
(0.486)
1.6714 *
(0.8231)
1.1775
(1.1415)
-0.5189
(0.4126)
0.4082
(0.411)
-0.9862
(0.7167)
1.2263
(1.0289)
0.4674
(0.8801)
-0.6322
(0.4618)
0.1585
(1.0306)
-0.6461
(0.9436)
0.437
(0.4084)
-0.7542
(0.4348)
-0.0854
(0.6832)
Prob[Y=2]
-63.0035
(33.8814)
-0.2131
(1.0249)
3.8266 *
(1.5819)
7.2082 *
(3.1556)
1.8349
(1.6637)
-1.9749
(1.1658)
-3.4778
(1.8582)
10.4711
(6.4943)
0.5292
(1.3102)
-0.3862
(0.6645)
0.7303
(1.6275)
9.5108
(5.7451)
1.2171
(0.6751)
1.3851
(1.1408)
-2.9827
(1.9595)
Prob[Y=3]
-8.0629
(9.4166)
-0.4891
(0.752)
2.3133
(1.3083)
-0.6264
(2.797)
-2.0392
(1.1158)
0.7545
(1.1107)
0.564
(1.5494)
4.3067
(2.8606)
-1.6319
(2.3037)
-1.4987
(0.9294)
-0.6025
(2.2954)
-1.2642
(2.4853)
1.4342
(0.9864)
-1.9625
(1.0817)
2.9755
(1.9896)
Prob[Y=4]
-1266
(97150376)
8.0704
(3811863)
-84.4998
(16528327)
180.1105
(27177377)
31.4971
(4708341)
-8.5671
(6745442)
-12.3785
(3892892)
88.3713
(8490158)
19.8785
(8144898)
77.7727
(5824206)
69.0922
(17522556)
-2.6697
(11264384)
75.1689
(5896293)
31.9328
(4581164)
-9.2831
(8964300)
Prob[Y=5]
-196.3973
(443980640)
9.8495
(7162340)
22.8121
(13715327)
-13.5277
(101743810)
-5.3666
(16386961)
7.0477
(21673740)
0.4784
(25293093)
22.3252
(47691625)
10.672
(98016705)
-2.5162
(34863411)
1.1108
(39984903)
-18.534
(47807819)
-14.0191
(23613952)
8.2517
(16895304)
8.3509
(17744862)
280
H.2. Example of an interactive Risk Ruler
The previously estimated coefficients produced the following
Classification Table for the small internet sample:
Predicted
0
1
2
3
23
2
1
1
5
10
2
1
1
0
12
1
1
1
1
6
0
0
0
0
0
0
0
0
30
13
16
9
Percent correct: 78.48%
Actual
0
1
2
3
4
5
4
0
0
0
0
9
0
9
5
0
0
0
0
0
2
2
27
18
14
9
9
2
79
Chapter H. Practical implementation of a Risk Ruler
281
Figure H.2: Evaluation Sheet for Answers on Internet Questionnaire of
Page 269, for explanations please see the previous pages.
282
H.2. Example of an interactive Risk Ruler
Appendix I
Empirical performance of
risk classes
Descriptive Statistics - Monthly time series data for the time period 1975-2000 on the
Swiss market produced the following moment estimates for the 6 buy-and-hold strategies
motivated in Figure 2.1. As the risky asset the MSCI Switzerland Total Return Index and
as the riskfree asset the 1M EUROCHF rate was used:
0% Stocks
20% Stocks
40% Stocks
60% Stocks
80% Stocks
100% Stocks
Range
Statistic
0.00956
0.08345
0.16163
0.23981
0.31799
0.39617
0% Stocks
20% Stocks
40% Stocks
60% Stocks
80% Stocks
100% Stocks
Std. Dev.
Statistic
0.00224
0.00952
0.01891
0.02838
0.03788
0.04738
Minimum
Statistic
0.00008
-0.05105
-0.10437
-0.15769
-0.21100
-0.26432
Maximum
Statistic
0.00964
0.03240
0.05726
0.08212
0.10698
0.13185
Mean
Statistic
3.45E-03
4.81E-03
6.17E-03
7.52E-03
8.88E-03
1.02E-02
Skewness
Statistic
0.660
-1.121
-1.127
-1.120
-1.115
-1.111
Std. Err.
0.14
0.14
0.14
0.14
0.14
0.14
Kurtosis
Statistic
-0.469
5.300
5.153
5.040
4.973
4.930
283
Std. Err.
1.29E-04
5.48E-04
1.09E-03
1.63E-03
2.18E-03
2.73E-03
Std. Err.
0.28
0.28
0.28
0.28
0.28
0.28
Std.Dev.
Statistic
2.24E-03
9.52E-03
1.89E-02
2.84E-02
3.79E-02
4.74E-02
284
Investment performance for 6 buy-and-hold strategies exhibiting linearly increasing risk. For the risky asset the Swiss Stock Market represented by the MSCI
Switzerland Total Return Index over the period 1975-2000 was chosen. The riskless
asset is the 1M EUROCHF rate. All time series in monthly data for the time period
1975-2000.
The second column depicts the continuously compounded monthly returns which
give the multiplicators in the third column. Lastly, the terminal wealth for an
initial investment of 100 in 1975 is given.
0% Stocks
20% Stocks
40% Stocks
60% Stocks
80% Stocks
100% Stocks
Sum of ccmr
1.0433
1.4528
1.8622
2.2716
2.6810
3.0904
Multiplicator
2.83868
4.27489
6.43775
9.69490
14.59999
21.98678
Total after 25 years
284
427
644
969
1460
2199
The multiplicator is simply: exp(sum of returns). The multiplicators mpf of the
mixed strategies (20% - 80% stocks) can be calculated according to
mpf
=
exp(smm · pctmm ) + exp(ssm · pctsm )
=
exp(1.0433 · pctmm ) + exp(3.0904 · pctsm )
smm stands for the sum of returns of the money market investment, pctmm represents the percentage of investment in the money market, while ssm stands for sum
of returns of the stock market investment and pctsm represents the percentage of
investment in the stock market.
Chapter I. Empirical performance of risk classes
285
Figure I.1: Distribution of returns for portfolios with stock ratios of 0% to
40%: Swiss Market, yearly data, 1925-1997, Source: Pictet.
Panel A: Stock Ratio=0%
Expected return=4.4% p.a.,
Max.values: -4% and 15% p.a.,
Volatility=3.5% p.a.,
Probability for loss=3%,
Growth over last 25 years: from 100
to 283.
Panel B: Stock Ratio=20%
Expected return=5.2% p.a.,
Max.values: -6.5% and 20% p.a.,
Volatility=5.5% p.a.,
Probability for loss=14%,
Growth over last 25 years: from 100
to 420.
Panel C: Stock Ratio=40%
Expected return=5.9% p.a.,
Max.values: -15% and 25% p.a.,
Volatility=8.5% p.a.,
Probability for loss=19%,
Growth over last 25 years: from 100
to 607.
286
Figure I.2: Distribution of returns for portfolios with stock ratios of 60% to
100%: Swiss Market, yearly data, 1925-1997, Source: Pictet.
Panel D: Stock Ratio=60%
Expected return=6.6% p.a.,
Max.values: -23% and 31% p.a.,
Volatility=11.9% p.a.,
Probability for loss=26%,
Growth over last 25 years: from 100
to 850.
Panel E: Stock Ratio=80%
Expected return=7.4% p.a.,
Max.values: -32% and 40% p.a.,
Volatility=15.3% p.a.,
Probability for loss=26%,
Growth over last 25 years: from 100
to 1’160.
Panel F: Stock Ratio=100%
Expected return=8.1% p.a.,
Max.values: -40% and 48% p.a.,
Volatility=18.8% p.a.,
Probability for loss=27%,
Growth over last 25 years: from 100
to 1’530.
Appendix J
Survey of Consumer
Finances: Details
The following explanations on the procedures and statistical measures used
for the development of the Survey of Consumer Finances were taken from
Kennickell, Starr-McCluer, and Sunden (1997):
Since 1989, the questionnaires for the SCF have changed only slightly.
Generally, changes have been introduced to gather additional information
needed to understand other data in the survey - for example, the 1995 SCF
introduced a question on uses of funds for mortgages that were taken out after
the time a primary residence was purchased. Also, the major aspects of the
sample design have been fixed over this time. Thus, the information obtained
by the survey is comparable over 1989-95.
The survey is intended to provide an adequate descriptive basis for the
analysis of family assets and liabilities. To address this requirement, the SCF
combines two types of samples. First, a standard multi-stage area-probability
design is selected to provide good coverage of characteristics, such as home
ownership, that are broadly distributed in the population. Second, a special
list sample is included to oversample wealthy families, who hold a disproportionately large share of such assets as noncorporate businesses and tax-exempt
bonds. This list sample is drawn from a sample of tax records made available for this purpose under strict rules governing confidentiality, the rights
287
288
of potential respondents to refuse participation in the survey, and the types
of information from the interview that can be made generally available. Of
the 3,906 completed interviews in the 1992 SCF, 2,456 families were from the
area-probability sample and 1,450 were from the list sample; the comparable
figures for the 4,299 interviews completed in 1995 are 2,780 families from the
area-probability sample and 1,519 from the list sample.
A very important factor in the ability to conduct surveys is the generosity
of the public in giving their time for an interview. In the 1995 SCF, the average
interview required 90 minutes. However, for some particularly complicated
cases, the amount of time needed was substantially more than two hours.
Data for the 1992 and 1995 surveys were collected by the National Opinion
Research Center at the University of Chicago (NORC) between the months of
June and December in each of the two years. The great majority of interviews
were conducted in person, although interviewers were allowed to conduct telephone interviews if that was a better arrangement for the respondent. In the
1995 survey, one important change was the introduction of laptop computers
for use in administering the questionnaire. This change increased the length
of the interview a little, and it may also have had some effects on the quality
of information collected. Nonetheless, the effects of the change in the mode
of questionnaire administration appear to be fairly small.
Errors may be introduced into survey results at many stages. Sampling
error, the variability expected to occur in estimates based on a sample instead
of a census, is a particularly important source of error. Such error may be
reduced either by increasing the size of the sample or by designing the sample
to reduce important types of variability; the latter course has been chosen for
the SCF. Estimation of sampling error in the SCF is described further below.
Interviewers may introduce errors, though SCF interviewers are given
lengthy project-specific training to minimize this problem. In addition, computer control of the 1995 survey greatly reduced technical errors made by
interviewers. Respondents may introduce errors by understanding a question
in a sense different from that which was intended by the survey designers. For
the SCF, extensive pretesting and other review of questions tend to reduce
this source of error.
Nonresponse - either complete nonresponse to the survey or nonresponse
to selected items within the survey - may be another important source of
Chapter J. Survey of Consumer Finances: Details
289
error. As noted in more detail below, the SCF uses weighting adjustments to
compensate for complete nonresponse. To deal with missing information on
individual items, the SCF uses statistical methods to impute missing data.
Response rates differ markedly in the two parts of the SCF sample. In
both 1992 and 1995, about 70 percent of families selected for the area probability sample actually completed interviews. The overall response rate in
the list sample was about 34 percent. Detailed analysis of the data suggests
that the tendency to refuse participation in an interview is highly correlated
with wealth. The response rates for both samples are low by the standards
of other major government surveys. However unlike other surveys, which almost certainly also have differential nonresponse by wealthy families, the SCF
sample frame provides a basis for adjusting for nonresponse by such families.
To provide a measure of the frequency with which families similar to the sample families could be expected to be found in the population of all families,
analysis weights are computed for each case to account for both the systematic properties of the design and for nonresponse. A major part of research
by SCF staff is devoted to adjustments for nonresponse through the analysis
weights for the survey.
One possibility to make the key findings of a study more reliable is the
trimming of the weights: weights can be adjusted to decrease the possibility
that the results are overly affected by a small number of observations. Such
influential observations can be detected using a graphica1 technique to inspect
the underlying data. Most of the cases to be found will be holders of an
unusual asset or liability or members of demographic groups for which such
holdings are rare.
290
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Index
Active strategies, 11
Adaptive aspiration, 121
Akaike Information Criterion, 83
Allais, Maurice, 5
Allianz Anleger Analyse, 4, 263
Arbitrage Pricing Theory, 10
Asset allocation
intertemporal, 12, 41
Objective function, 163
Optimal three-moment, 169
Optimal two-moment, 167
Assets
free, 15
tied, 15
Attributes
Choice-, 77
Investors’, 77
Distribution
Compactness of, 113
Lognormal, 114
lognormal, 142
logWeibull,EVD, 78, 171
Normal, 113
Draw-down criterion, 42
Empirical Analysis
Data sample, 53
Discretizing data, 76
Hypotheses, 55
Independent factors, 16, 251,
252
Results, 85, 193, 247
Structure, 68, 179
Equity-premium puzzle, 7
Error Distance, 82
Bernoulli principle, 4, 46
Bounded rationality, 6, 44
testing for, 58
Buy-and-hold, 23
Fallacy of large numbers, 38
Gambles, 4
critique of, 5
fair, 26
Capital Market Line, 22
CAPM, 21
Certainty equivalent, 27, 130
Classification table, 81
Conditional Logit model, 77, 78
Habit models, 9
Herding, 45
Hypotheses
306
INDEX
Testing, 84
Independence from irrelevant alternatives (IIA), 78
Insurance and gambling, 125
Internet survey, 268
Self-test, 269
Intuitive Risk Rulers, 3
Joint estimation, 105, 158
Least Squares
Ordinary, 73
Weighted, 73
Likelihood ratio test, 83, 247
LIMDEP7.0, 70, 83
Mean reversion, 10, 13, 41
Mean-variance approach
Critique, 111
Moment trade-offs, 133
Multinomial Logit
model, 77
Utility maximization, 171
Myopic loss aversion, 47
Nested Logit model, 80
Likelihood function, 175
Objective function, 70
Ordered Logit model, 75
Predictive power, 16, 97, 99
Preferences
observed, 68
stated, 68
Prospect theory, 120
307
Prudence, 132
Random walk, 12
Rebalancing, 23
Returns
predictable, 11
skewed, 14
Risk aversion
Empirical findings, 7
absolute, 30, 43
Assessment example, 136
relative, 30, 43
Two-moment, 25, 71, 103
Risk classes
Empirical performance, 283
Risk premium
Markowitz, 28, 32
Pratt-Arrow, 29, 32
Three-moment, 129
Risk Quota by Fidelity, 3, 262
Safety-first, 16
Settings ‘1-4’, 68
Shortfall-approach, 15
Skewness, 115
Fisher, 109
Implementation, 141
Preference, 117, 119, 131, 135
Strategies
Enhancement, 115
Strategy
balanced, 35
Covered-Call, 153
Protective Put, 143
switching, 35
Survey of Consumer Finances, 51
308
INDEX
Details, 287
Time diversification, 38, 41
Time horizon effect, 12, 38
Timing, 24
Tobit model, 74
Transferability of results, 99
Two-moment risk aversion, 167
Utility
cubic, 123
isoelastic, 28
quadratic, 112
Utility function
Von Neumann-Morgenstern, 45
Wealth
free, 15
reserved, 15
Curriculum Vitae
Name:
Born:
Nationalities:
Fabian Wenner
1.2.1972 in Frankfurt/Main, Germany
Swiss and German
Education
1999 - 2000
1992 - 1998
1996 - 1998
1996
1982 - 1991
Visiting Fellow at the Department of Economics,
Stanford University, CA, USA.
Supervisor: Prof. Takeshi Amemiya.
Studies of Finance and Capital Markets at the
University of St.Gallen, degree: lic.oec. HSG.
CEMS Studies and exams, degree: CEMS Master.
Exchange Program at Copenhagen Business School, Denmark
Humboldt-Gymnasium, Ulm
Practical Experience
1997 - 1999
1995
1994
1993
1991 - 1992
Research assistant for Prof. Dr. K. Spremann at the
Swiss Institute for Banking and Finance, St.Gallen
Mercedes Benz Italia SpA, Rome, Marketing department
UBS, Zurich, FX Dealing and Treasury
Deutsche Shell AG, Hamburg, law and controlling department
Qualified radio operator (10wpm), German Air Force.