13
Return, Risk, the
Security Market Line
and CAPM
McGraw-Hill/Irwin
Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved.
Chapter Outline
 Expected Returns and Variances
 Portfolios
 Risk: Systematic and Unsystematic
 Diversification and Portfolio Risk
 Systematic Risk and Beta
 The Security Market Line
 The Capital Asset Pricing Model (CAPM)
13-1
Expected Returns and Variances
 Expected returns are based on the probabilities
of possible outcomes
n
E ( R )   pi Ri
i 1
 Variance & Standard deviation (STD) measure
the volatility of returns
 Measure total risk
n
2
2
σ   pi (Ri  E(R))
i 1
13-2
Expected Returns and Variances
 Suppose you have predicted the following returns for
stocks C and T in three possible states of nature.
What are the expected returns, variance and STD?
 State
 Boom
 Normal
 Recession
Probability
0.3
0.5
?
C (%)
15
10
2
T (%)
25
20
1
 RC = .3(15) + .5(10) + .2(2) = 9.9%
 RT = .3(25) + .5(20) + .2(1) = 17.7%
 2(C) = .3(15-9.9)2 + .5(10-9.9)2 + .2(2-9.9)2 =20.29
 = 4.5  CV(C) = / RT = 4.5/9.9 = 0.45
 2(T) = .3(25-17.7)2 +.5(20-17.7)2 +.2(1-17.7)2 =74.41
13-3
  = 8.63  CV(T) = / RT = 8.63/17.7 = 0.48
Risk Analysis

Expected value E(Ai)
n
E ( Ai )   ( R ij * P j )
j 1

Standard deviation: is a
measure of the dispersion of
a set of data from its
expected value E(Ai)

Coefficient of
variation (CV)
is the ratio of the standard
deviation to the mean.
n
 ( Ai )  (Rij  E( Ai ))2 * Pj
j 1
CV is a standardized measure
of risk that shows the risk per
unit of return.
CV 
 ( Ai )
E ( Ai )
The higher the coefficient of
variation, the greater the level
of dispersion around the mean
13-4
 Higher risk!
Another Example
 Consider the following information:
 State
 Boom
 Normal
 Slowdown
 Recession
Probability
.25
.50
.15
.10
ABC, Inc. (%)
15
8
4
-3
 What is the expected return?
 E(R) = …
 What is the variance?
 Variance = …
 What is the standard deviation? STD = …
13-5
Portfolios
 A portfolio is a collection of assets
 The expected return of a portfolio:
 is the weighted average of the expected returns
of the respective assets in the portfolio.
m
E (RP )   w j E (R j )
j 1
 You can also find the expected return by finding the
portfolio return in each possible state and computing
the expected value as we did with individual
securities
13-6
Portfolio Variance
 Portfolio Variance / Standard deviation:
• Compute the portfolio return for each state
RP = w1R1 + w2R2 + … + wmRm
• Compute the expected portfolio return
• Compute the portfolio variance and standard
deviation using the same formulas as for an
individual asset
2
n
σ p   pi ( R p  E ( R p )) 2
i 1
13-7
Example: Portfolio Variance
 Consider the following information
 Invest 50% of your money in Asset A, 50% in B.
 State
Probability
Boom
Bust
.4
.6
A
30%
-10%
B
-5%
25%
Portfolio
12.5%
7.5%
 What are the expected return and STD for each asset?
 What are the expected return and STD for the portfolio?
Ans: A: 6%,19.6%; B: 13%, 14.7%; Portfolio: 9.5% & 2.45%
 CV(A) = 3.2; CV(B) = 1.1; CV(P) = 0.2
 Difference in return & STD between each stock and portfolio?
13-8
Portfolio Variance
 Portfolio Variance / Standard deviation:
n
2
σ p   pi ( R p  E ( R p )) 2
i 1
n
σ X ,Y   pi [ Xi  E( X )][Yi  E(Y )]
i 1
σ 2p  w 12 σ 12  w 22 σ 22  2w 1 w 2 σ 1 σ 2 ρ 1,2
.  w 12 σ 12  w 22 σ 22  2w 1 w 2 σ 1,2
Với: w1 là trọng số tài sản 1
1,2 là correlation coefficient
1,2 là covariance
13-9
Another Example
 Consider the following information
 State
 Boom
 Normal
 Recession
Probability
.25
.60
.15
X
15%
10%
5%
Z
10%
9%
10%
 What are the expected return and standard
deviation for a portfolio with an investment of
$6,000 in asset X and $4,000 in asset Z?
 Expected return = …
 Standard deviation = …
 COV(X,Z) = …
13-10
Expected return - Risk
Expected return
4
2
3
1
Variance or STD
• (2) vs. (1): …?
• (2) vs. (3): …?
• (4) vs. (3): …?
13-11
Returns
 Total Return = expected return +
unexpected return
 Unexpected return = systematic portion +
unsystematic portion
 Therefore, total return can be expressed
as follows:
 Total Return = expected return +
systematic portion + unsystematic portion
13-12
Systematic Risk
 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.
13-13
Unsystematic Risk
 Risk factors that affect a limited
number of assets
 Also known as unique risk and
asset-specific risk (diversifiable risk)
 Includes such things as labor strikes,
part shortages, etc.
13-14
What risk should be priced?
 Total risk =
Systematic risk
+ Unsystematic risk
 Relationship between Diversification
and Risk
13-15
Diversification
 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
13-16
Diversification & Risk
 Diversification can substantially reduce
the variability of returns (risk) 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
13-17
Table 13.7
13-18
Figure 13.1
13-19
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
13-20
Systematic Risk Principle
 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
13-21
Measuring Systematic Risk
 How do we measure systematic risk?
 We use the beta coefficient to measure
systematic risk
 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
13-22
Total versus Systematic Risk
 Consider the following information:
Standard Deviation
 Security C
 Security K
20%
30%
Beta
1.25
0.95
 Which security has more total risk?
 Which security has more systematic risk?
 Which security should have the higher expected
return?
13-23
Example: Portfolio Betas
 Consider the previous example with the
following four securities
 Security
 DCLK
 KO
 INTC
 KEI
Weight
.133
.2
.267
.4
 What is the portfolio beta?
Beta
2.685
0.195
2.161
2.434
m
beta P   w jbeta j
j1
 Beta(p) = .133(2.685) + .2(.195) + .267(2.161)
+ .4(2.434) = 1.947
13-24
Beta and the Risk Premium
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!
13-25
Example: Portfolio Expected
Returns and Betas
30%
Expected Return
25%
E(RA)
20%
15%
10%
Slope = (E(RA) – Rf) / A =Reward-to-Risk ratio
Rf
5%
0%
0
0.5
1
1.5
A
2
2.5
3
Beta
13-26
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
 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)?
13-27
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 )  Rf
A

E(RM  Rf )
M
Rf: Risk-free rate
RM: Market return
RA: Return of asset A
A: Systematic risk of A
M: Systematic risk of market (=1)
13-28
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 (M =1), the slope can be
rewritten: Slope = E(RM) – Rf
 Slope = E(RM) – Rf = market risk premium
13-29
Figure 13.4
13-30
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
13-31
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 ()
13-32
Example - CAPM
 Consider the betas for each of the assets
given earlier. If the risk-free rate is 2.13% and
the market risk premium is 8.6%, what is the
expected return for each?
Security
DCLK
KO
INTC
KEI
Beta
2.685
0.195
2.161
2.434
Expected Return
2.13 + 2.685(8.6) = 25.22%
2.13 + 0.195(8.6) = 3.81%
2.13 + 2.161(8.6) = 20.71%
2.13 + 2.434(8.6) = 23.06%
13-33
13
End of chapter
McGraw-Hill/Irwin
Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved.