Download Market Portfolio

Introduction
The next two chapters (together with Ch. 2 of Haugen)
will briefly examine the following aspects of capital
market theory that underlie quantitative investment
management:
• Modeling risk and return – CAPM & APT – theory,
testing, and extensions
• Estimating risk and return – the Single-Index Model
(SIM) and multiple-factor models for risk and
expected return
Copyright © 2000 by Harcourt, Inc. All rights reserved.
Lecture Presentation Software
to accompany
Investment Analysis and
Portfolio Management
Seventh Edition
by
Frank K. Reilly & Keith C. Brown
Chapters 8 & 9
Modeling Risk & Return
Part One:
• The Risk-Free Asset,
• Portfolio Separation, and
• The Capital Asset Pricing Model (CAPM)
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THE RISK-FREE ASSET
• WHAT IS A RISK FREE-ASSET?
– DEFINITION:
an asset whose terminal value
is certain
• variance of returns = 0,
• covariance with other assets = 0
i  0
If
then     
ij
ij i j
 0
THE RISK-FREE ASSET
• WHAT IS A RISK FREE-ASSET?
– DEFINITION:
an asset whose terminal value
is certain
•
•
•
•
•
An investment with NO risk
An asset with zero variance
Zero correlation with all other risky assets
Provides the risk-free rate of return (RFR)
Will lie on the vertical axis of a portfolio graph
THE RISK-FREE ASSET
• DOES A RISK-FREE ASSET EXIST?
– CONDITIONS FOR EXISTENCE:
• Fixed-income security
• No possibility of default
• No interest-rate risk
• No reinvestment risk
THE RISK-FREE ASSET
• DOES A RISK-FREE ASSET EXIST?
– Given the conditions, what qualifies?
• a U.S. Treasury security with a maturity matching
the investor’s horizon
Combining the Risk-Free Asset
with a Risky Portfolio
Portfolio expected return is a linear relationship
 the weighted average of the two returns
E(R port )  WRF (RFR)  (1 - WRF )E(R i )
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Combining the Risk-Free Asset
with a Risky Portfolio
Portfolio standard deviation is also a linear
relationship, equal to the weighted average of
the two standard deviations (zero for the riskfree asset and i for the risky portfolio)
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Combining a Risk-Free Asset
with a Risky Portfolio
Standard deviation
The expected variance for a two-asset portfolio is
E(
2
port
)  w   w   2w 1 w 2 r1,2 1 2
2
1
2
1
2
2
2
2
Substituting the risk-free asset for Security 1, and the risky
asset for Security 2, this formula would become
2
2
E( port
)  w 2RF RF
 (1  w RF ) 2  i2  2w RF (1 - w RF )rRF,i RF i
Since we know that the variance of the risk-free asset is
zero and the correlation between the risk-free asset and any
risky asset i is zero we can adjust the formula
2
E( port
)  (1  w RF ) 2  i2
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Combining a Risk-Free Asset
with a Risky Portfolio
Given the variance formula
the standard deviation is
2
E( port
)  (1  w RF ) 2  i2
E( port )  (1  w RF ) 2  i2
 (1  w RF )  i
Therefore, the standard deviation of a portfolio that
combines the risk-free asset with risky assets is the
linear proportion of the standard deviation of the risky
asset portfolio.
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Combining a Risk-Free Asset
with a Risky Portfolio
Example
Assume:
– E(RF) = 7%,
– E(RS&P) = 12%,
– S&P = 20%
• Expected Return on Combined Portfolio:
E RC    F RF  1   F E RP   0.27%  1  0.212%  11.0%
• Standard Deviation on Combined Portfolio:
 C  1  F  P  1  0.2 20 %   16 %
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Combining a Risk-Free Asset
with a Risky Portfolio
Since both the expected return and the standard
deviation of return for such a portfolio are linear
combinations, a graph of possible portfolio returns
and risks looks like a straight line between the two
assets.
Thus, the existence of a risk-free asset adds value to
investors by expanding the set of portfolios
available to them.
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Portfolio Possibilities Combining the Risk-Free Asset
and Risky Portfolios on the Efficient Frontier
E(R port )
Figure 9.1
D
P*
C
RFR
B
A
E( port )
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The Risk-Free Asset and
Portfolio Separation Theory
Assuming the investor can both lend (by buying
Treasury bonds) and borrow (by shorting the
bonds with full use of the proceeds) at the riskfree rate, this means that the investor now faces a
linear (rather than convex) efficient frontier:
 E  RP   RF 
E  RC   RF   C 

P


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The Risk-Free Asset and
Portfolio Separation Theory
This linear efficient frontier, comprising various
combinations of the risk-free asset and the risky
portfolio P*, dominates all other possible risky
portfolios within the original (Markowitz)
efficient frontier.
This fact led to the development of the Portfolio
Separation Theory (cf., James Tobin).
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Portfolio Separation Theory
Under the portfolio separation theory, the ideal risky
portfolio in which an investor should invest is the
same (P*), regardless of how aggressive or risk
averse the investor is.
– I.e., the point on the Markowitz efficient frontier at
which the investor will invest is independent of the
investor’s risk preferences.
Where risk preferences are reflected is in terms of
how much of his or her portfolio is allocated to P*
and how much is invested in the risk-free asset.
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Portfolio Separation Theory
Thus, in order to obtain his or her optimal portfolio,
there are two separate decisions for the investor
to make:
1. The investment decision
•
•
•
Which portfolio on the Markowitz efficient frontier to
choose?
This is determined by the point of tangency between
the Markowitz efficient frontier and a line extending
from the risk-free rate
This leads to the choice of portfolio P* as the optimal
risky portfolio for the investor, regardless of the
investor’s risk preferences
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Portfolio Separation Theory
2. The financing decision
•
•
•
This is where risk preferences come into the picture
If the investor is more risk averse, he or she will put
part of his or her money in P* and the rest in Treasury
bonds (this is known as a lending portfolio, because
the rest of the investor’s money is lent to the federal
government)
If the investor is more aggressive, he or she will
leverage up his or her holdings and invest in P* on
margin by borrowing at the risk-free rate (this is
known as a borrowing portfolio)
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Portfolio Possibilities Combining the Risk-Free Asset
and Risky Portfolios on the Efficient Frontier
E(R port )
P*
RFR
E( port )
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Capital Market Theory:
An Overview
• Question: What are the general implications for
security prices if investors act the way Markowitz
portfolio theory and portfolio separation theory
say they should? If such theories hold, what
would equilibrium in the capital markets entail?
• Capital market theory extends portfolio theory and
develops a model for pricing all risky assets
• The capital asset pricing model (CAPM) will
allow you to determine the required rate of return
(for use in discounting future cash flows) for any
risky asset
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Assumptions of
Capital Market Theory
1. All investors are Markowitz mean-variance
efficient investors who want to target points
on the efficient frontier.
– Also, they include all investable assets in their
estimation of the efficient frontier
– Not necessarily a realistic assumption!
• Most investors do not use Markowitz optimization
• Of those who do, they typically optimize w.r.t. alpha
and tracking error rather than mean and variance
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Assumptions of
Capital Market Theory
2. Investors can borrow or lend any amount of
money at the risk-free rate of return (RFR).
– This means that the conditions of portfolio separation
theory will hold, at least at the individual level.
– Note: it is always possible to lend money at the riskfree rate by buying securities such as T-bills, but (unless
you’re the government) it is not usually possible to
borrow at this risk-free rate.
– However, assuming a higher borrowing rate does not
change the general results (although it does change
their form a bit).
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Assumptions of
Capital Market Theory
3. All investors have homogeneous expectations;
that is, investors have identical estimates for the
probability distributions of future rates of return.
– This implies that all investors will estimate the efficient
frontier to be in the exact same location (including
using the same risk and expected return factor models),
and the optimal portfolio P* (i.e., the investment
decision from portfolio separation theory) will be the
same for all investors.
– This assumption can be relaxed, and as long as the
differences in expectations are not vast their effects will
be minor.
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Assumptions of
Capital Market Theory
4. All investors have the same one-period time
horizon such as one-month, six months, or one
year.
– Markowitz portfolio theory is a single-period model;
making the model dynamic requires additional
constraints, such as on portfolio turnover, in calculating
the efficient frontier.
– With regard to capital market theory, differences in
investors’ time horizons would require investors to
derive risk measures and risk-free assets that are
consistent with their time horizons.
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Assumptions of
Capital Market Theory
5. Capital markets are “frictionless,” i.e.:
– No taxes – true for many classes of investors
– No transactions costs – becoming more true over time,
but still can be an impediment
– Fixed supply of stocks – i.e., don’t have to worry about
incorporating IPO shares into the analysis
– Infinitely divisible supply of stocks – this assumption
allows us to discuss investment alternatives as
continuous curves. Changing it would have little
impact on the theory, and it is also becoming more true
over time.
– Information is costless and available to all investors
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Assumptions of
Capital Market Theory
6. Capital markets are in equilibrium.
– This means that we begin with all investments
properly priced in line with their risk levels.
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Assumptions of
Capital Market Theory
• Note that some of these assumptions are
unrealistic,
• But relaxing many of these assumptions would
have only minor influence on the model and
would not change its main implications or
conclusions;
• Moreover (as Milton Friedman argues), a theory
can be useful for helping to explain and predict
behavior, even if not all of its assumptions hold
true (e.g., many useful models in physics assume
the absence of any friction).
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Derivation of the
Capital Market Line
a) Homogeneous expectations (together with the
same investment horizon) means that investors
all face the same estimated efficient frontier.
b) Existence of a risk-free asset means that each
investor can mix the riskless asset with a risky
portfolio.
c) (a) and (b) imply that all investors choose the
same risky portfolio to hold in combination with
the risk-free asset.
•
Call this portfolio P*
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Derivation of the
Capital Market Line
d) In order to have equilibrium (supply = demand),
all risky assets must be included in P*. If this
were not the case, then some assets would not be
held at all.
e) In view of (d), the optimal portfolio P* is called
the Market Portfolio (M)
•
Value-weighted portfolio, with E(RM) and M
f) The line connecting RFR with M now represents
the market-wide opportunities for expected
return and risk. Thus, this line is called the:
•
Capital Market Line (CML)
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The Capital Market Line (CML)
E(R port )
M
Figure 9.2
RFR
 port
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The Market Portfolio
Because portfolio M lies at the point of tangency, it
has the highest portfolio possibility line
Everybody will want to invest in Portfolio M and
borrow or lend to be somewhere on the CML
Therefore this portfolio must include ALL RISKY
ASSETS (else there will be stocks out there on the
market the NO ONE owns!)
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The Market Portfolio
Because the market is in equilibrium, all risky
assets are included in this portfolio in
proportion to their market value
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The Market Portfolio
Because it contains all risky assets, it is a
completely diversified portfolio (once you
already own everything, you can’t diversify
any more!), which means that all the unique
risk of individual assets (unsystematic risk)
is diversified away (all the risk that’s left
over is, by definition, systematic risk)
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Systematic Risk
Only systematic risk remains in the market
portfolio
Systematic risk is the variability in all risky
assets caused by macroeconomic variables
Systematic risk can be measured by the
standard deviation of returns of the market
portfolio and can (and does) change over
time
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Examples of Macroeconomic
Factors that Affect Systematic Risk
• Variability in growth of money supply
• Interest rate volatility
• Variability in:
industrial production
corporate earnings
and cash flow
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The Market Portfolio and
How to Measure Diversification
All portfolios on the CML are perfectly positively
correlated with each other and with the completely
diversified market Portfolio M
A completely diversified portfolio would have a
correlation with the market portfolio of +1.00
Thus, can use regression R2 of portfolio’s returns
regressed on the “market” portfolio’s returns as a
measure of the extent of diversification
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The Capital Market Line (CML)
•
•
•
•
Describes the risk / return relationship for welldiversified portfolios (idiosyncratic risk has been
diversified away).
Portfolio standard deviation (Q) is the relevant
measure of risk, and the portfolio’s expected
return (E(RQ)) will be a direct linear function of
its risk:
 E  RM   R F 
E RQ   RF  
Q

M


To obtain higher expected returns, must accept
higher risk.
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The Security Market Line (SML)
•
•
Key Question: What is the relevant measure of
risk for an individual security when it is held as
part of a well diversified portfolio (i.e., the
Market portfolio, M)?
The Security Market Line describes the risk /
return relationship for an individual security.
– Also applies to non-diversified portfolios or any other
holding for which the total risk may include some
diversifiable or idiosyncratic risk.
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The Security Market Line (SML)
• The relevant risk measure for an individual risky
asset is its covariance with the market portfolio
(Covi,m)
• This is the risk measure for the SML, which
describes the relationship between risk and
expected return for all portfolios, whether welldiversified or not, as well as for all securities
• The return for the market portfolio should be
consistent with its own risk, which is the
covariance of the market with itself - or its
2
variance:

m
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Figure 9.5
Graph of Security Market Line
(SML)
E(R i )
SML
Rm
RFR

2
m
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Cov im
The Security Market Line (SML)
The equation for the risk-return line is
R M - RFR
E(R i )  RFR 
(Cov i,M )
2
M
 RFR 
We then redefine
Cov i,M

2
M
Cov i,M

(R M - RFR)
as beta
2
M
E(R i )  RFR   i (R M - RFR)
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( i )
Figure 9.6
Graph of SML with
Normalized Systematic Risk
E(R i )
SML
Rm
Negative
Beta
RFR
0
1.0
Beta( iM / )
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2
M
Determining the Expected
Rate of Return for a Risky Asset
E(R i )  RFR   i (R M - RFR)
The expected rate of return of a risk asset is determined
by the RFR plus a risk premium for the individual asset
The risk premium is determined by the systematic risk of
the asset () and the prevailing market risk premium
(RM-RFR)
In equilibrium, to obtain higher expected returns,
investors must accept higher “covariance” risk
In equilibrium, investors receive no compensation for
diversifiable (non-systematic or idiosyncratic) risk
Q: What is market is not in equilibrium?
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The Capital Asset Pricing Model:
Expected Return and Risk
• CAPM indicates what should be the expected or
required rates of return on risky assets
• This helps to value an asset by providing an
appropriate discount rate to use in dividend (or
other discounted cash flow) valuation models
• Conversely, you can compare an estimated rate of
return to the required rate of return implied by
CAPM – over / under valued ?
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Determining the Required
Rate of Return for a Risky Asset
Stock
Beta
A
B
C
D
E
0.70
1.00
1.15
1.40
-0.30
RFR = 6% (0.06)
RM = 12% (0.12)
Implied market risk premium = 6% (0.06)
Assume:
E(R i )  RFR   i (R M - RFR)
E(RA) = 0.06 + 0.70 (0.12-0.06) = 0.102 = 10.2%
E(RB) = 0.06 + 1.00 (0.12-0.06) = 0.120 = 12.0%
E(RC) = 0.06 + 1.15 (0.12-0.06) = 0.129 = 12.9%
E(RD) = 0.06 + 1.40 (0.12-0.06) = 0.144 = 14.4%
E(RE) = 0.06 + -0.30 (0.12-0.06) = 0.042 = 4.2%
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Determining the Required
Rate of Return for a Risky Asset
In equilibrium, all assets and all portfolios of assets
should plot on the SML
Any security with an estimated return that plots above
the SML is underpriced (or under-valued)
Any security with an estimated return that plots below
the SML is overpriced (or over-valued)
A superior investor must derive value estimates for
assets that are consistently superior to the consensus
market evaluation to earn better risk-adjusted rates
of return than the average investor
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Identifying Undervalued and
Overvalued Assets
Compare the required rate of return to the
expected rate of return for a specific risky
asset using the SML over a specific
investment horizon to determine if it is an
appropriate investment
Independent estimates of return for the
securities provide price and dividend
outlooks
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Price, Dividend, and
Rate of Return Estimates
Table 9.1
Current Price
Stock
A
B
C
D
E
(Pi )
25
40
33
64
50
Expected Dividend
Expected Price (Pt+1 )
27
42
39
65
54
(Dt+1 )
0.50
0.50
1.00
1.10
0.00
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Expected Future Rate
of Return (Percent)
10.0 %
6.2
21.2
3.3
8.0
Comparison of Required Rate of Return
to Estimated Rate of Return
Table 9.2
Stock
Beta
A
B
C
D
E
0.70
1.00
1.15
1.40
-0.30
Required Return
Estimated Return
E(Ri )
Minus E(R i )
Estimated Return
10.2%
12.0%
12.9%
14.4%
4.2%
10.0
6.2
21.2
3.3
8.0
-0.2
-5.8
8.3
-11.1
3.8
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Evaluation
Properly Valued
Overvalued
Undervalued
Overvalued
Undervalued
Plot of Estimated Returns
E(R i ) on SML Graph
Figure 9.7
E
-.40 -.20
Rm
.22
.20
.18
.16
.14
.12
Rm
.10
.08
.06
.04
.02
0
C
SML
A
B
D
.20
.40
.60
.80
1.0 1.20
1.40 1.60 1.80
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Beta
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