Download Chapter 1 Financial Returns: Basic Concepts and Returns

Chapter 1
Financial Returns: Basic Concepts and
Returns
Financial Econometrics
Summer Term 2012
Nikolaus Hautsch
Humboldt-Universität zu Berlin
1. Introduction and Motivation
Outline I
1. Introduction and Motivation
2. Financial Time Series
2.1 Illustrations
2.2 Returns on Financial Assets
3. Distributional Properties
3.1 Conditional & Marginal Distrib’s
3.2 Evaluating Marginal Distributions
3.3 Empirical Evidence
3.4 Distributions for Returns
Financial Econometrics – Chapter 1
2 | 55
1. Introduction and Motivation
3 | 55
Financial Econometrics - What is it?
I Combining finance theory with statistical theory.
I Transferring a theoretical (financial) model into a (financial)
econometric model.
I Modelling financial data.
I Predicting financial variables as well as relations thereof.
Financial Econometrics – Chapter 1
1. Introduction and Motivation
Financial Econometrics – Chapter 1
4 | 55
1. Introduction and Motivation
4 | 55
Why to Study Financial Econometrics?
I Financial econometrics has become one of the most active
areas of research in econometrics.
I Financial econometrics attracted substantial attention in
recent years in both academia as well as financial practice.
I 2003: Nobel prize for Robert F. Engle.
I 2003: Foundation of the ”Journal of Financial Econometrics”.
I 2007: Foundation of the ”Society for Financial Econometrics
(SoFiE)”.
Financial Econometrics – Chapter 1
1. Introduction and Motivation
5 | 55
Financial Econometrics - Why is it
important?
I ”Financial economics is a highly empirical discipline, perhaps
the most empirical among the branches of economics and even
among the social sciences in general.”
(Campbell/Lo/MacKinlay, 1997)
I Financial theory without measurement is ”l’art pour l’art”.
I Financial econometrics bridges the gap between financial
economics, statistics and mathematical finance.
I The demand for quantitatively well educated people in the
financial industry is permanently increasing – also after the
financial crisis!
Financial Econometrics – Chapter 1
1. Introduction and Motivation
Themes in Financial Econometrics
I Asset price dynamics
I Time-varying volatility and correlation
I Prediction
I Information processing
I Valuation
I Liquidity
I High-frequency finance
Financial Econometrics – Chapter 1
6 | 55
1. Introduction and Motivation
7 | 55
Goal of the Course
I Covering main (however, not all) parts of the spectrum of
modern financial econometrics.
I Empirical application of financial theory using econometric
techniques.
I Discussion of important results in the empirical finance
literature.
I Doing own empirical work using econometric software and real
financial data.
I Raising the sensitivity for problems and pitfalls in empirical
studies. Interpretation of empirical results.
Financial Econometrics – Chapter 1
1. Introduction and Motivation
Course Outline
1. Financial Returns: Basic Concepts and Properties
2. Foundations in Time Series Analysis
3. Modelling Time-Varying Volatility
4. Estimating and Testing Asset Pricing Models
5. Modelling High-Frequency Financial Data
Financial Econometrics – Chapter 1
8 | 55
1. Introduction and Motivation
9 | 55
Books
I Taylor, S. J. (2005): ”Asset Price Dynamics, Volatility, and
Prediction”, Princeton University Press.
I Tsay, R. S. (2005): ”Analysis of Financial Time Series:
Financial Econometrics”, Wiley, 2nd edition.
I Cochrane, J. H. (2005): ”Asset Pricing”, revised edition,
Princeton University Press.
I Campbell, J. Y., A. W. Lo, and A. C. MacKinlay (1997): ”The
Econometrics of Financial Markets”, Princeton University
Press.
I Hamilton, J. D. (1994): ”Time Series Analysis”, Princeton
University Press.
Financial Econometrics – Chapter 1
1. Introduction and Motivation
10 | 55
Books ctd.
I Hautsch, N. (2012): ”Econometrics of Financial
High-Frequency Data”, Springer.
I Hasbrouck, J. (2007): ”Empirical Market Microstructure: The
Institutions, Economics and Econometrics of Securities
Trading”, Oxford University Press.
I Härdle, W., Hautsch, N., and Overbeck L. (2008): ”Applied
Quantitative Finance”, 2nd ed., Springer
I Franses, P. H., and D. van Dijk (2000): ”Non-Linear Time
Series Models in Empirical Finance”, Cambridge University
Press.
Financial Econometrics – Chapter 1
1. Introduction and Motivation
Organizational Issues
I Lectures:
. Monday 10:15-11:45, SPA 1, 203
. Monday 12:00-13:30, SPA 1, 203
I Tutorials:
. Partly directly imbedded in lecture
. Partly in room 025 (will be announced)
. Computer problems
. Theoretical problems
I Course website: Moodle
Financial Econometrics – Chapter 1
11 | 55
2. Financial Time Series
Outline I
1. Introduction and Motivation
2. Financial Time Series
2.1 Illustrations
2.2 Returns on Financial Assets
3. Distributional Properties
3.1 Conditional & Marginal Distrib’s
3.2 Evaluating Marginal Distributions
3.3 Empirical Evidence
3.4 Distributions for Returns
Financial Econometrics – Chapter 1
12 | 55
2. Financial Time Series | 2.1 Illustrations
13 | 55
Daily Prices, S&P500, 1980-2007
800
200
400
600
Price
1000
1200
1400
S&P 500 Index, Daily
1980
1985
1990
1995
Time
Financial Econometrics – Chapter 1
2000
2005
2. Financial Time Series | 2.1 Illustrations
14 | 55
Daily Returns, S&P500, 1980-2007
−0.05
−0.10
−0.20
−0.15
Log−return
0.00
0.05
S&P 500 Log−return, Daily
1980
1985
1990
1995
Time
Financial Econometrics – Chapter 1
2000
2005
2. Financial Time Series | 2.1 Illustrations
15 | 55
Quarterly Dividends, IBM, 1980-2007
0.30
IBM Stock Dividend
●●●●●●●●●●●●●●●
●●●●
0.25
●●●●●●●●●●●●●●●●●●●
●●●●●
0.20
●●●●
●●●●
●●●●
0.15
Dividend
●●●●●●●●●●●●●
●●●●
●●●●
●●
●●●●
●●●●
0.10
●●●●
●●●
●●●●
●●●●●●●●●●●
1980
1982
1984
Time
Financial Econometrics – Chapter 1
1986
1988
2. Financial Time Series | 2.1 Illustrations
16 | 55
Daily Volumes, S&P500, 1980-2007
2e+09
0e+00
1e+09
Trading Volume
3e+09
4e+09
S&P 500 Trading Volume, Daily
1980
1985
1990
1995
Time
Financial Econometrics – Chapter 1
2000
2005
2. Financial Time Series | 2.1 Illustrations
17 | 55
Daily 3-Month Interest Rates, Germany, 1975-2007
8
2
4
6
Interest Rate
10
12
14
German Three−month Market Rate, Daily
1975
1980
1985
1990
Time
Financial Econometrics – Chapter 1
1995
2000
2005
2. Financial Time Series | 2.1 Illustrations
18 | 55
Daily FX Rates U.S. Dollar vs. EURO, 1975-2007
1.2
0.8
1.0
Exchange Rate
1.4
Daily Exchange Rate, USD/EURO
1975
1980
1985
1990
Time
Financial Econometrics – Chapter 1
1995
2000
2005
2. Financial Time Series | 2.1 Illustrations
Data Frequencies
I Yearly
I Quarterly
I Monthly
I Weekly
I Daily
I Intradaily (1h, 10min, 5min, 1min etc.)
I Transaction level
I Order level
Financial Econometrics – Chapter 1
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2. Financial Time Series | 2.1 Illustrations
20 | 55
Yearly Returns, S&P500, 1980-2007
0.1
0.0
−0.2
−0.1
Log−return
0.2
0.3
S&P 500 Log−return, Yearly
1980
1985
1990
1995
Time
Financial Econometrics – Chapter 1
2000
2005
2. Financial Time Series | 2.1 Illustrations
21 | 55
Quarterly Returns, S&P500, 1980-2007
0.0
−0.2
−0.1
Log−return
0.1
0.2
S&P 500 Log−return, Quarterly
1980
1985
1990
1995
Time
Financial Econometrics – Chapter 1
2000
2005
2. Financial Time Series | 2.1 Illustrations
22 | 55
Monthly Returns, S&P500, 1980-2007
Log−return
−0.2
−0.1
0.0
0.1
S&P 500 Log−return, Monthly
1980
1985
1990
1995
Time
Financial Econometrics – Chapter 1
2000
2005
2. Financial Time Series | 2.1 Illustrations
23 | 55
Weekly Returns, S&P500, 1980-2007
0.00
−0.05
−0.10
Log−return
0.05
S&P500 Log−return, Weekly
1980
1985
1990
1995
Time
Financial Econometrics – Chapter 1
2000
2005
2. Financial Time Series | 2.1 Illustrations
24 | 55
Daily Returns, S&P500, 1980-2007
−0.05
−0.10
−0.20
−0.15
Log−return
0.00
0.05
S&P 500 Log−return, Daily
1980
1985
1990
1995
Time
Financial Econometrics – Chapter 1
2000
2005
2. Financial Time Series | 2.2 Returns on Financial Assets
25 | 55
Simple Returns
I Asset price at t: Pt
I Dividend paid at the end of period t: Dt .
I One-period simple return:
Pt + Dt
1 + Rt :=
Pt−1
Pt + Dt
Pt − Pt−1 + Dt
Rt :=
−1=
Pt−1
Pt−1
simple gross return
simple net return
I k-period simple return (for Dt−k = ... = Dt = 0):
Pt
1 + Rt (k) :=
Pt−k
Pt − Pt−k
Rt (k) :=
Pt−k
Financial Econometrics – Chapter 1
2. Financial Time Series | 2.2 Returns on Financial Assets
26 | 55
Annualized Returns
I Annualized returns for short-term periods, which are available
over many years: Assume mt equi-distant periods
(0, 1], (1, 2], . . . , (mt−1 , mt ] within year t and k years. Then:

1/k
k−1
Y
a
Rta (k, m) :=  (1 + Rt−j
(mt−j )) − 1
j=0

k−1
Y
=
j=0
mt−j
Y
1 + Rit−j
!1/k

− 1,
i=1
where m := (mt , mt−1 , . . . , mt−k+1 ) and Rit denotes the
simple return in period i in year t.
Financial Econometrics – Chapter 1
2. Financial Time Series | 2.2 Returns on Financial Assets
27 | 55
Log Returns
I Continuously compounded (gross) return (log return):
rt := ln(1 + Rt ) = ln
Pt + Dt
= ln(Pt + Dt ) − ln Pt−1 .
Pt−1
I In case of no dividends, the log return is easily computed as
rt = pt − pt−1 ,
where pt := ln Pt .
I Multi-period continuously compounded return:
rt (k) := ln(1 + Rt (k)) = ln [(1 + Rt )(1 + Rt−1 ) · · · (1 + Rt−k+1 )]
= ln(1 + Rt ) + . . . + ln(1 + Rt−k+1 )
= rt + rt−1 + . . . + rt−k+1
Financial Econometrics – Chapter 1
2. Financial Time Series | 2.2 Returns on Financial Assets
28 | 55
Continuous Compounding
I EUR 1000 Deposit for one year
Interest payment
10% p.a.
5% semiannually
2.5% quarterly
0.1
m % every m-th period
continuously
I General: limm→∞ (1 +
r m
m)
Financial Econometrics – Chapter 1
Net value of deposit
1100
1102.5
1103.81
m
1000·(1 + 0.1
m )
1000 · limm→∞ (1 + mr )m = 1000 · exp(r )
= exp(r )
2. Financial Time Series | 2.2 Returns on Financial Assets
Log Returns vs. Simple Returns
I Return on a portfolio of N assets with weights wi ,
i = 1, . . . , N:
Rp,t :=
N
X
wi Rit
i=1
P
⇒ Note: rp,t = ln(1 + Rp,t ) 6= N
i=1 wi rit .
PN
⇒ But if rit ≈ 0, then rp,t ≈ i=1 wi rit .
Financial Econometrics – Chapter 1
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3. Distributional Properties |
Outline I
1. Introduction and Motivation
2. Financial Time Series
2.1 Illustrations
2.2 Returns on Financial Assets
3. Distributional Properties
3.1 Conditional & Marginal Distrib’s
3.2 Evaluating Marginal Distributions
3.3 Empirical Evidence
3.4 Distributions for Returns
Financial Econometrics – Chapter 1
30 | 55
3. Distributional Properties | 3.1 Conditional & Marginal Distrib’s
31 | 55
Conditional and Marginal Distributions
I Consider N assets at time t with (log) return
rit , t = 1, . . . , T ; i = 1, . . . , N.
I Joint distribution function:
Fr (r11 , . . . , rN1 ; r12 , . . . , rN2 ; . . . ; r1T , . . . , rNT ; θ),
where θ are parameters.
I Marginal distribution functions:
Fi· (ri1 , . . . , riT ; θ)
marginal distribution of return i
F·t (r1t . . . , rNt ; θ)
cross-sectional distribution of all returns at t
Financial Econometrics – Chapter 1
3. Distributional Properties | 3.1 Conditional & Marginal Distrib’s
32 | 55
Conditional Distribution Functions
I Conditional distribution functions:
Fi·|X (ri1 , . . . , riT |X1 , . . . , XT ; θ)
cond. distribution of i given X
Fi·|j· (ri1 , . . . , riT |rj1 , . . . , rjT ; θ)
cond. distribution of i given j
I (Dynamic) partitioning of distributions:
Fi· (ri1 , . . . , riT ; θ)
= Fi1 (ri1 ) · Fi2 (ri2 |ri1 ) · Fi3 (ri3 |ri2 , ri1 ) · · · FiT (riT |ri,T −1 , . . . , ri1 )
Financial Econometrics – Chapter 1
3. Distributional Properties | 3.2 Evaluating Marginal Distributions
Sample Statistics
µ
bi :=
T
1 X
rit
T
sample mean
t=1
T
σ
bi2
1 X
:=
(rit − µbi )2
T −1
Sbi :=
Kbi :=
1
T σbi 3
1
T σbi 4
t=1
T
X
(rit − µbi )3
sample variance
sample skewness
t=1
T
X
(rit − µbi )4
t=1
Financial Econometrics – Chapter 1
sample kurtosis
33 | 55
3. Distributional Properties | 3.2 Evaluating Marginal Distributions
34 | 55
Kolmogorov-Smirnov Test
I H0 : The true distribution of x = rit is FX (x).
I Empirical distribution for n i.i.d. observations xi is given by
n
1X
l1{Xi ≤xi } ,
n
i=1
(
1 if Xi ≤ xi ,
=
0 otherwise.
Fn (xi ) =
where l1{Xi ≤xi }
I The Kolmogorov-Smirnov test statistic is given by
Dn = sup |Fn (x) − F (x)|.
x
I Critical values are tabulated.
Financial Econometrics – Chapter 1
3. Distributional Properties | 3.2 Evaluating Marginal Distributions
35 | 55
Pearson’s χ2 -Goodness-of-Fit Test
I H0 : The true distribution of rit is F (rit , θ), where θ is a vector
of (distribution) parameters.
I Idea: Categorizing the realizations of rit and comparing the
empirical frequencies in each category with their theoretical
counterparts under F (rit , θ).
I Number of categories of rit : k
I Number of observations in the j-th category: Nj , j = 1, . . . , k.
Financial Econometrics – Chapter 1
3. Distributional Properties | 3.2 Evaluating Marginal Distributions
36 | 55
I Estimated probability for an observation in category j:
b
c it ∈ kj |F (rit ; θ))
pj = Pr(r
I Number of parameters to be estimated (the dimension of θ): s
I Then, an asymptotic χ2 -test is given by
k
X
(Nj − Npj )2 a 2
∼ χk−s−1;1−α ,
χ =
Npj
2
j=1
where α denotes the significance level.
Financial Econometrics – Chapter 1
3. Distributional Properties | 3.2 Evaluating Marginal Distributions
37 | 55
Probability Integral Transform
I H0 : The true distribution of rit is F (rit , θ), where θ is a vector
of (distribution) parameters.
I Probability integral transform:
Z rit
qit :=
f (s)ds = F (rit ; θ)
−∞
I Rosenblatt (1952): Under the null hypothesis,
qit ∼ i.i.d. U[0; 1].
I Sequence of qit ’s is tested against the i.i.d. uniform
distribution.
Financial Econometrics – Chapter 1
3. Distributional Properties | 3.2 Evaluating Marginal Distributions
38 | 55
Further Tests ...
I Jarque-Bera test against normality:
1
T −s
2
2
Si + (Ki − 3)
∼ χ22 ,
JB :=
6
4
where s denotes the number of estimated parameters.
I Quantile-quantile-plots (QQ-plots): Plotting the theoretical
quantiles against the empirical quantiles.
Financial Econometrics – Chapter 1
3. Distributional Properties | 3.3 Empirical Evidence
39 | 55
Daily DAX and FTSE 100 log returns
2800
Series: LOG_RETURN
Sample 2 11015
Observations 11014
2400
2000
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
1600
1200
800
400
Jarque-Bera
Probability
0
-0.05
0.00
0.000288
0.000000
0.075527
-0.070394
0.011284
-0.053655
7.652943
9940.776
0.000000
0.05
3200
Series: LOG_RETURN
Sample 2 7602
Observations 7601
2800
2400
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
2000
1600
1200
800
400
Jarque-Bera
Probability
0
-0.050
-0.025
0.000
Financial Econometrics – Chapter 1
0.025
0.050
0.000352
0.000000
0.067883
-0.065073
0.009454
-0.007661
9.544347
13564.21
0.000000
3. Distributional Properties | 3.3 Empirical Evidence
40 | 55
Daily Dow Jones and S&P 500 log returns
5000
Series: LOG_RETURN
Sample 2 14667
Observations 14666
4000
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
3000
2000
0.000288
6.04E-05
0.061547
-0.074549
0.008620
-0.129662
8.148105
1000
0
-0.075
Jarque-Bera
Probability
-0.050
-0.025
0.000
0.025
16236.62
0.000000
0.050
3500
Series: LOG_RETURN
Sample 2 11277
Observations 11276
3000
2500
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
2000
1500
1000
500
0
-0.075
Jarque-Bera
Probability
-0.050
-0.025
0.000
Financial Econometrics – Chapter 1
0.025
0.050
0.000283
0.000100
0.055732
-0.071127
0.008974
-0.066483
7.329116
8813.567
0.000000
3. Distributional Properties | 3.3 Empirical Evidence
QQ-plots of daily DAX and FTSE 100 log returns
4
3
Normal Quantile
2
1
0
-1
-2
-3
-4
-.08
-.04
.00
.04
.08
.04
.08
LOG_RETURN
4
3
Normal Quantile
2
1
0
-1
-2
-3
-4
-.08
-.04
.00
LOG_RETURN
Financial Econometrics – Chapter 1
41 | 55
3. Distributional Properties | 3.3 Empirical Evidence
QQ-plots of daily Dow Jones and S&P 500 log returns
6
Normal Quantile
4
2
0
-2
-4
-6
-.08
-.04
.00
.04
.08
.04
.08
LOG_RETURN
4
3
Normal Quantile
2
1
0
-1
-2
-3
-4
-.08
-.04
.00
LOG_RETURN
Financial Econometrics – Chapter 1
42 | 55
3. Distributional Properties | 3.3 Empirical Evidence
43 | 55
Monthly DAX and FTSE 100 log returns
120
Series: LRETURN
Sample 2 507
Observations 506
100
80
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
60
40
20
Jarque-Bera
Probability
0
-0.3
-0.2
-0.1
0.0
0.1
0.005202
0.007277
0.193738
-0.293327
0.056199
-0.679271
5.870356
212.6166
0.000000
0.2
100
Series: LRETURN
Sample 2 350
Observations 349
80
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
60
40
0.007514
0.011992
0.134771
-0.301699
0.046952
-1.175222
8.467757
20
Jarque-Bera
Probability
0
-0.3
-0.2
-0.1
Financial Econometrics – Chapter 1
0.0
0.1
515.0796
0.000000
3. Distributional Properties | 3.3 Empirical Evidence
44 | 55
Monthly DJ and S&P 500 log returns
200
Series: LRETURN
Sample 2 675
Observations 674
160
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
120
80
0.005819
0.009913
0.123256
-0.270986
0.040354
-0.637989
6.549530
40
Jarque-Bera
Probability
0
-0.2
-0.1
0.0
399.5494
0.000000
0.1
160
Series: LRETURN
Sample 2 519
Observations 518
140
120
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
100
80
60
40
20
Jarque-Bera
Probability
0
-0.2
-0.1
0.0
Financial Econometrics – Chapter 1
0.1
0.005659
0.008650
0.151043
-0.245428
0.042494
-0.614376
5.785449
200.0464
0.000000
3. Distributional Properties | 3.3 Empirical Evidence
QQ-plots of monthly DAX and FTSE 100 log returns
4
3
Normal Quantile
2
1
0
-1
-2
-3
-4
-.3
-.2
-.1
.0
.1
.2
LRETURN
4
3
Normal Quantile
2
1
0
-1
-2
-3
-4
-.4
-.3
-.2
-.1
LRETURN
Financial Econometrics – Chapter 1
.0
.1
.2
45 | 55
3. Distributional Properties | 3.3 Empirical Evidence
46 | 55
QQ-plots of monthly Dow Jones and S&P 500 log returns
4
3
Normal Quantile
2
1
0
-1
-2
-3
-4
-.3
-.2
-.1
.0
.1
.2
.1
.2
LRETURN
4
3
Normal Quantile
2
1
0
-1
-2
-3
-4
-.3
-.2
-.1
.0
LRETURN
Financial Econometrics – Chapter 1
3. Distributional Properties | 3.3 Empirical Evidence
47 | 55
Stylized Facts
I Overkurtosis: K > 3.
I Fat tails compared to the standard normal distribution: Large
returns occur more often than expected.
I More probability mass in the middle of the distribution
compared to the the standard normal distribution.
I For lower aggregation levels (e.g. daily data) (log-)normality
assumption clearly rejected.
I Aggregated returns tend to normality.
I Left-skewness: S < 0 ⇒ large returns are often negative.
Financial Econometrics – Chapter 1
3. Distributional Properties | 3.4 Distributions for Returns
48 | 55
Log-Normal Distribution
I If rit ∼ N(µi , σi2 ), then e rit = 1 + Rit is log normally
distributed with density function
(ln z − µi )2
1 1
√
exp −
,
f1+Rit (z) =
z 2πσi
2σi2
where the mean and the variance are given by
E[Ri,t ] = e µi +
σi2
2
− 1;
2
2
V[Ri,t ] = e 2µi +σi (e σi − 1).
I If Ri,t ∼ N(mi , si2 ), then



E[ri,t ] = ln 
mi + 1 
2  ;
si
1 + mi +1
Financial Econometrics – Chapter 1
V[ri,t ] = ln 1 +
si
mi + 1
2 !
.
3. Distributional Properties | 3.4 Distributions for Returns
Fat-Tailed Distributions
I Mixtures of normal distributions
I Student-t-distributions
I Hyperbolic distributions
I Scale mixtures of normal distributions
I Gamma distributions
I Extreme value distributions
I Stable Paretian distributions
Financial Econometrics – Chapter 1
49 | 55
3. Distributional Properties | 3.4 Distributions for Returns
50 | 55
Stable Distributions
I Generalizations of the normal distribution.
I Stability: Distribution family does not depend on the time
interval over which the returns are measured.
I Higher order moments often do not exist.
Example: Cauchy distribution: frit (z) =
Financial Econometrics – Chapter 1
γ
1
π γ 2 +(z−δ)2 ,
γ > 0.
3. Distributional Properties | 3.4 Distributions for Returns
51 | 55
The t-Distribution
I The t-distribution has the density
1
frit (z) =
n 1 √
B 2, 2
n
where B(n, m) :=
Γ(n)Γ(m)
Γ(n+m)
z2
1+
n
and Γ(n) :=
− n+1
2
R∞
0
,
x n−1 e −x dx, n > 0.
I Properties:
. E[rit ] = 0, (n > 2),
n
. V[rit ] = n−2
(n > 2),
. for n → ∞: convergence to the normal distribution.
Financial Econometrics – Chapter 1
3. Distributional Properties | 3.4 Distributions for Returns
52 | 55
A Mixture of Normal Distributions
I rit originates with probability (1 − α) from X ∼ N(µ1 , σ12 ) and
with probability α from Y ∼ N(µ2 , σ22 ) with X and Y
independent.
I Then:
rit ∼ (1 − α)N(µ1 , σ12 ) + αN(µ2 , σ22 ),
0≤α≤1
with
.
.
.
.
E[rit ] = (1 − α)µ1 + αµ2 ,
V[rit ] = (1 − α)σ12 + ασ22 ,
K (rit ) ≥ 3,
K (rit ) large for low values of α and large values of σ22 .
Financial Econometrics – Chapter 1
3. Distributional Properties | 3.4 Distributions for Returns
53 | 55
Scale Mixtures
I Idea: Variances are stochastic and follow some distribution:
rt |ωt ∼ N(µ, f (ωt )),
where σt2 = f (ωt ), ωt denotes the mixing variable and f is
some function.
I Clark (1973):
r t = σ t εt
εt ∼ N(0, 1)
ln(σt2 ) ∼ N(0, σσ2 ),
where εt and σt are independent.
I If σt2 follows an inverse gamma distribution, then the resulting
returns are t-distributed.
Financial Econometrics – Chapter 1
3. Distributional Properties | 3.4 Distributions for Returns
QQ-plots of Daily Dax Index Returns
... against the normal and mixed normal (α = 0.05, σ22 = 10,
V[rt ] = 1)
4
3
Normal Quantile
2
1
0
-1
-2
-3
-4
10 -8
-6
-4
-2
0
2
4
6
8
4
6
8
Quantile of MIXNORM
LOG_RETURN
5
0
-5
-10
-8
-6
-4
-2
0
2
Quantile of LOG_RETURN_ST
Financial Econometrics – Chapter 1
54 | 55
3. Distributional Properties | 3.4 Distributions for Returns
55 | 55
QQ-plots of Daily Dow Jones Index Returns
... against the normal and mixed normal (α = 0.05, σ22 = 10,
V[rt ] = 1)
6
Normal Quantile
4
2
0
-2
-4
-6
12-12
-8
Quantile of MIXNORM
-4
0
4
8
4
8
LOG_RETURN
8
4
0
-4
-8
-12
-12
-8
-4
0
Quantile of LOG_RETURN
Financial Econometrics – Chapter 1