Chapter
Thirteen
Return, Risk, and the
Security Market Line
© 2003 The McGraw-Hill Companies, Inc. All rights reserved.
13.1
Key Concepts and Skills
•
•
•
•
•
•
Know how to calculate expected returns
Understand the impact of diversification
Understand the systematic risk principle
Understand the security market line
Understand the risk-return trade-off
Be able to use the Capital Asset Pricing Model
(CAPM)
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13.2
Chapter Outline
•
•
•
•
•
•
•
•
Expected Returns and Variances
Portfolios
Risk: Systematic and Unsystematic
Diversification and Portfolio Risk
Systematic Risk and Beta
The Security Market Line
The SML and the Cost of Capital: A Preview
Arbitrage Pricing Theory
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13.3
Expected Returns 13.1
• Expected returns are based on the probabilities
of possible outcomes
• In this context, “expected” means average if
the process is repeated many times
• The “expected” return does not even have to
be a possible return
•
n = total number of states, p = probability that state i occurs, R = return in state i
n
E ( R)   pi Ri
i 1
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13.4
Expected Returns – Example 1
• Suppose you have predicted the following
returns for stocks C and T in three possible
states of nature. What are the expected
returns?
–
–
–
–
State
Boom
Normal
Recession
Probability
0.3
0.5
0.2
C
0.15
0.10
0.02
T
0.25
0.20
0.01
• RC = .3(.15) + .5(.10) + .2(.02) = .099 = 9.9%
• RT = .3(.25) + .5(.20) + .2(.01) = .177 = 17.7%
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13.5
Variance and Standard Deviation
• Variance (σ²) and standard deviation (σ) still
measure the volatility of returns
• You can use unequal probabilities for the
entire range of possibilities
• Weighted average of squared deviations
n
σ 2   pi ( Ri  E ( R)) 2
i 1
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13.6
Variance and Standard Deviation – Example 1
• Consider the previous example. What is the variance
and standard deviation for each stock?
• Stock C
2 = .3(.15-.099)2 + .5(.1-.099)2 + .2(.02-.099)2 =
.002029
 = .045
• Stock T
2 = .3(.25-.177)2 + .5(.2-.177)2 + .2(.01-.177)2 =
.007441
 = .0863
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13.7
Portfolios 13.2
• A portfolio is a collection of assets
• An asset’s risk and return is important in how
it affects the risk and return of the portfolio
• The risk-return trade-off for a portfolio is
measured by the portfolio expected return and
standard deviation, just as with individual
assets
• The sum of risks of individual assets does not
equal the risk of the portfolio
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13.8
Example: Portfolio Weights
• Suppose you have $15,000 to invest and you
have purchased securities in the following
amounts. What are your portfolio weights in
each security?
–
–
–
–
$2000 of ABC
$3000 of DEF
$4000 of GHI
$6000 of JKL
•ABC: 2/15 = .133
•DEF: 3/15 = .2
•GHI: 4/15 = .267
•JKL: 6/15 = .4
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13.9
Portfolio Expected Returns
• The expected return of a portfolio is the weighted
average of the expected returns for each asset in the
portfolio
•
w = the weight of asset j in the portfolio
m
E ( RP )   w j E ( R j )
j 1
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13.10
Example: Expected Portfolio Returns
• Consider the portfolio weights computed
previously. If the individual stocks have the
following expected returns, what is the
expected return for the portfolio?
– ABC: 19.65%
– DEF: 8.96%
– GHI: 9.67%
– JKL: 8.13%
• E(RP) = .133(19.65) + .2(8.96) + .267(9.67) +
.4(8.13) = 10.24%
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13.11
Portfolio Variance
• Compute the portfolio return for each state:
RP = w1R1 + w2R2 + … + wmRm
• Compute the expected portfolio return using
the same formula as for an individual asset
• Compute the portfolio variance and standard
deviation using the same formulas as for an
individual asset
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13.12
Example: Portfolio Variance
• Consider the following information
–
–
–
–
Invest 50% of your money in Asset A
State Probability A
B
Portfolio
Boom .5
70%
10% 7.3%
Bust .5
-20%
30% 12.8%
• What is the expected return and standard
deviation for each asset?
• What is the expected return and standard
deviation for the portfolio?
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13.13
Portfolio Variance
Asset A
E( R)= 0.5 (0.7) + 0.5 (-0.2) = 0.35 – 0.1 = 0.25
σ²= 0.5 (0.7 - 0.25)² + 0.5 (-0.2 - 0.25)² = 0.2025
σ= 0.45
Asset B
E( R)= 0.5 (0.1) + 0.5 (0.3) = 0.05 + 0.15 = 0.2
σ²= 0.5 (0.1 - 0.2)² + 0.5 (0.3 - 0.2)² = 0.01
σ= 0.1
Portfolio
E( R)= 0.5 (0.25) + 0.5 (0.2) = 0.125 + 0.1 = 0.225
σ²= 0.5 (0.25 – 0.225)² + 0.5 (0.2 – 0.225)² = 0.000625
σ= 0.025
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13.14
Another Way to Calculate Portfolio Variance
• Portfolio variance can also be calculated using the
following formula:
 P2  xL2 L2  xU2  U2  2 xL xU CORR L ,U  L U
Assuming that the correlatio n between A and B is - 1.00, we have
 P2  0.52  0.2025  0.52  0.01  2  0.5  0.5  -1.00 0.45  0.1
 P2  0.030625
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13.15
Figure 13.1 – Different Correlation Coefficients
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13.16
Figure 13.1 – Different Correlation Coefficients
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13.17
Figure 13.1 – Different Correlation Coefficients
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13.18
Figure 13.2 – Graphs of Possible Relationships
Between Two Stocks
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13.19
Diversification
• There are benefits to diversification whenever
the correlation between two stocks is less than
perfect (p < 1.0)
• Figure 13.4
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13.20
Terminology
• Feasible set (also called the opportunity set) –
the curve that comprises all of the possible
portfolio combinations
• Efficient set – the portion of the feasible set
that only includes the efficient portfolio
(where the maximum return is achieved for a
given level of risk, or where the minimum risk
is accepted for a given level of return)
• Minimum Variance Portfolio – the possible
portfolio with the least amount of risk
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13.21
Quick Quiz II
• Consider the following information
–
–
–
–
State
Boom
Normal
Recession
Probability
.25
.60
.15
X
15%
10%
5%
Z
10%
9%
10%
• What is the expected return and standard
deviation for a portfolio with an investment of
$6000 in asset X and $4000 in asset Y?
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13.22
Systematic Risk 13.4
• Risk that cannot be diversified away through
portfolio formation (rewarded with return)
• Risk factors that affect a large number of
assets
• Also known as non-diversifiable risk or
market risk
• Includes such things as changes in GDP,
inflation, interest rates, etc.
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13.23
Unsystematic Risk
• Risk that can be diversified away through
portfolio formation (no reward of return)
• Risk factors that affect a limited number of
assets
• Also known as diversifiable risk, unique risk
and asset-specific risk
• Includes such things as labor strikes,
shortages, etc.
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13.24
Diversification 13.5
• Portfolio diversification is the investment in
several different asset classes or sectors
• Diversification is not just holding a lot of
assets
• For example, if you own 50 internet stocks,
you are not diversified
• However, if you own 50 stocks that span 20
different industries, then you are diversified
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13.25
Table 13.8 – Portfolio Diversification
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13.26
The Principle of Diversification
• Diversification can substantially reduce the
variability of returns without an equivalent
reduction in expected returns
• This reduction in risk arises because worse
than expected returns from one asset are offset
by better than expected returns from another
• However, there is a minimum level of risk that
cannot be diversified away and that is the
systematic portion
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13.27
Figure 13.6 – Portfolio Diversification
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13.28
Diversifiable (Unsystematic) Risk
• The risk that can be eliminated by combining
assets into a portfolio
• Synonymous with unsystematic, unique or
asset-specific risk
• If we hold only one asset, or assets in the same
industry, then we are exposing ourselves to
risk that we could diversify away
• The market will not compensate investors for
assuming unnecessary risk
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13.29
Total Risk
• Total risk = systematic risk + unsystematic
risk
• The standard deviation of returns is a measure
of total risk
• For well diversified portfolios, unsystematic
risk is very small
• Consequently, the total risk for a diversified
portfolio is essentially equivalent to the
systematic risk
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13.30
Systematic Risk Principle 13.6
• There is a reward for bearing risk
• There is not a reward for bearing risk
unnecessarily
• The expected return on a risky asset depends
only on that asset’s systematic risk since
unsystematic risk can be diversified away
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13.31
Measuring Systematic Risk
• How do we measure systematic risk?
– We use the beta coefficient to measure systematic
risk. i is the percent change in asset i’s return for
a 1% change in the market portfolio’s return.
• What does beta tell us?
– A beta of 1 implies the asset has the same
systematic risk as the overall market
– A beta < 1 implies the asset has less systematic
risk than the overall market
– A beta > 1 implies the asset has more systematic
risk than the overall market
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13.32
Figure 13.7 – High and Low Betas
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13.33
Table 13.10 – Beta Coefficients for Selected
Companies
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13.34
Example: Portfolio Betas
• Consider the previous example with the
following four securities
– Security
– ABC
– DEF
– GHI
– JKL
Weight
.133
.2
.267
.4
Beta
3.69
0.64
1.64
1.79
• What is the portfolio beta?
• .133(3.69) + .2(.64) + .267(1.64) + .4(1.79) =
1.773
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13.35
Beta and the Risk Premium 13.7
• Remember that the risk premium = expected
return – risk-free rate
• The higher the beta, the greater the risk
premium should be
• Can we define the relationship between the
risk premium and beta so that we can estimate
the expected return?
– YES!
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13.36
Figure 13.8A – Portfolio Expected Returns and Betas
Rf
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13.37
Reward-to-Risk Ratio: Definition and Example
• The reward-to-risk ratio is the slope of the line
illustrated in the previous example
– Slope = (E(RA) – Rf) / (A – 0)
– Reward-to-risk ratio for previous example =
(20 – 8) / (1.6 – 0) = 7.5
– f = 0
• What if an asset has a reward-to-risk ratio of 8
(implying that the asset plots above the line)?
• What if an asset has a reward-to-risk ratio of 7
(implying that the asset plots below the line)?
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13.38
Market Equilibrium
• In equilibrium, all assets and portfolios must
have the same reward-to-risk ratio and they all
must equal the reward-to-risk ratio for the
market
E ( RA )  R f
A

E ( RM  R f )
M
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13.39
Security Market Line
• The security market line (SML) is the
representation of market equilibrium
• The slope of the SML is the reward-to-risk
ratio: (E(RM) – Rf) / M
• But since the beta for the market is ALWAYS
equal to one, the slope can be rewritten
• Slope = E(RM) – Rf = market risk premium
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13.40
Figure 13.11 – Security Market Line
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13.41
The Capital Asset Pricing Model (CAPM)
• The capital asset pricing model defines the
relationship between risk and return
• E(RA) = Rf + A(E(RM) – Rf)
• If we know an asset’s systematic risk, we can
use the CAPM to determine its expected return
• This is true whether we are talking about
financial assets or physical assets
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13.42
Factors Affecting Expected Return
• Pure time value of money – measured by the
risk-free rate
• Reward for bearing systematic risk – measured
by the market risk premium
• Amount of systematic risk – measured by beta
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13.43
Example - CAPM
• Consider the betas for each of the assets given earlier.
If the risk-free rate is 4.5% and the market risk
premium is 8.5%, what is the expected return for
each?
Security
Beta
Expected Return
ABC
3.69
4.5 + 3.69(8.5) = 35.865%
DEF
.64
4.5 + .64(8.5) = 9.940%
GHI
1.64
4.5 + 1.64(8.5) = 18.440%
JKL
1.79
4.5 + 1.79(8.5) = 19.715%
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13.44
Example
• Calculate the market risk premium on the asset i, and
the expected return on asset i using the following
information: i = 1.5, E(RM )= 15%, Rf =5%
• Market risk premium = 15% - 5% = 10%
• Risk premium on asset i = 1.5(15% - 5%) = 15%
• Expected return on asset i = 5% + 1.5(15% - 5%) =
20%
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13.45
Arbitrage Pricing Theory (APT) 13.9
• The major advantage of the APT model is that it can handle
multiple factors not included in CAPM.
• Like CAPM, the APT model assumes that stock returns
depend on both expected and unexpected returns.
• Unlike CAPM, the unexpected return in the APT model is
related to several market factors.
• Assuming those factors are unanticipated changes in inflation,
GNP, and interest rates, the expected return would be written
as:
E( R)  RF  E( R1  RF )  1  E( R2  RF )  2  ...  E( RK  RF )   K
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13.46
Summary 13.10
• There is a reward for bearing risk
• Total risk has two parts: systematic risk and
unsystematic risk
• Unsystematic risk can be eliminated through
diversification
• Systematic risk cannot be eliminated and is rewarded
with a risk premium
• Systematic risk is measured by the beta
• The equation for the SML is the CAPM, and it
determines the equilibrium required return for a
given level of risk
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