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1. Return and Risk: The Capital Asset
Pricing Model, Asset pricing
Theories
11-1

2. 11.1 Individual Securities
• The characteristics of individual securities that
are of interest are the:
– Expected Return
– Variance and Standard Deviation
– Covariance and Correlation (to another security or
index)
11-2

3. 11.2 Expected Return, Variance, and Covariance
Consider the following two risky asset
world. There is a 1/3 chance of each state of
the economy, and the only assets are a
shares fund and a bond fund.
Rate of Return
Scenario ProbabilityShares Fund Bond Fund
Recession 33.3% -7% 17%
Normal 33.3% 12% 7%
Boom 33.3% 28% -3%
11-3

4. Expected Return
Shares Fund Bond Fund
Rate of Squared Rate of Squared
Scenario Return Deviation Return Deviation
Recession -7% 0.0324 17% 0.0100
Normal 12% 0.0001 7% 0.0000
Boom 28% 0.0289 -3% 0.0100
Expected return 11.00% 7.00%
Variance 0.0205 0.0067
Standard Deviation 14.3% 8.2%
11-4

5. Expected Return
Stock Fund Bond Fund
Rate of Squared Rate of Squared
Scenario Return Deviation Return Deviation
Recession -7% 0.0324 17% 0.0100
Normal 12% 0.0001 7% 0.0000
Boom 28% 0.0289 -3% 0.0100
Expected return 11.00% 7.00%
Variance 0.0205 0.0067
Standard Deviation 14.3% 8.2%
E (rS ) = 1 × (−7%) + 1 × (12%) + 1 × (28%)
3 3 3
E (rS ) = 11%
11-5

6. Variance
Stock Fund Bond Fund
Rate of Squared Rate of Squared
Scenario Return Deviation Return Deviation
Recession -7% 0.0324 17% 0.0100
Normal 12% 0.0001 7% 0.0000
Boom 28% 0.0289 -3% 0.0100
Expected return 11.00% 7.00%
Variance 0.0205 0.0067
Standard Deviation 14.3% 8.2%
(−7% − 11%) = .0324
2
11-6

7. Variance
Stock Fund Bond Fund
Rate of Squared Rate of Squared
Scenario Return Deviation Return Deviation
Recession -7% 0.0324 17% 0.0100
Normal 12% 0.0001 7% 0.0000
Boom 28% 0.0289 -3% 0.0100
Expected return 11.00% 7.00%
Variance 0.0205 0.0067
Standard Deviation 14.3% 8.2%
1
.0205 = (.0324 + .0001 + .0289)
3
11-7

8. Standard Deviation
Stock Fund Bond Fund
Rate of Squared Rate of Squared
Scenario Return Deviation Return Deviation
Recession -7% 0.0324 17% 0.0100
Normal 12% 0.0001 7% 0.0000
Boom 28% 0.0289 -3% 0.0100
Expected return 11.00% 7.00%
Variance 0.0205 0.0067
Standard Deviation 14.3% 8.2%
14.3% = 0.0205
11-8

9. Covariance
Stock Bond
Scenario Deviation Deviation Product Weighted
Recession -18% 10% -0.0180 -0.0060
Normal 1% 0% 0.0000 0.0000
Boom 17% -10% -0.0170 -0.0057
Sum -0.0117
Covariance -0.0117
“Deviation” compares return in each state to the expected return.
“Weighted” takes the product of the deviations multiplied by the
probability of that state.
11-9

10. Correlation
Cov (a, b)
ρ=
σ aσ b
− .0117
ρ= = −0.998
(.143)(.082)
11-10

11. Diversification and Portfolio Risk
• 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.
11-11

12. 11.3 The Return and Risk for Portfolios
Stock Fund Bond Fund
Rate of Squared Rate of Squared
Scenario Return Deviation Return Deviation
Recession -7% 0.0324 17% 0.0100
Normal 12% 0.0001 7% 0.0000
Boom 28% 0.0289 -3% 0.0100
Expected return 11.00% 7.00%
Variance 0.0205 0.0067
Standard Deviation 14.3% 8.2%
Note that stocks have a higher expected return than bonds
and higher risk. Let us turn now to the risk-return tradeoff
of a portfolio that is 50% invested in bonds and 50%
invested in stocks. 11-12

13. Portfolios
Rate of Return
Scenario Stock fund Bond fund Portfolio squared deviation
Recession -7% 17% 5.0% 0.0016
Normal 12% 7% 9.5% 0.0000
Boom 28% -3% 12.5% 0.0012
Expected return 11.00% 7.00% 9.0%
Variance 0.0205 0.0067 0.0010
Standard Deviation 14.31% 8.16% 3.08%
The rate of return on the portfolio is a weighted average of
the returns on the stocks and bonds in the portfolio:
rP = wB rB + wS rS
5% = 50% × (−7%) + 50% × (17%) 11-13

14. Portfolios
Rate of Return
Scenario Stock fund Bond fund Portfolio squared deviation
Recession -7% 17% 5.0% 0.0016
Normal 12% 7% 9.5% 0.0000
Boom 28% -3% 12.5% 0.0012
Expected return 11.00% 7.00% 9.0%
Variance 0.0205 0.0067 0.0010
Standard Deviation 14.31% 8.16% 3.08%
The expected rate of return on the portfolio is a weighted
average of the expected returns on the securities in the
portfolio.
E (rP ) = wB E (rB ) + wS E (rS )
9% = 50% × (11%) + 50% × (7%) 11-14

15. Portfolios
Rate of Return
Scenario Stock fund Bond fund Portfolio squared deviation
Recession -7% 17% 5.0% 0.0016
Normal 12% 7% 9.5% 0.0000
Boom 28% -3% 12.5% 0.0012
Expected return 11.00% 7.00% 9.0%
Variance 0.0205 0.0067 0.0010
Standard Deviation 14.31% 8.16% 3.08%
The variance of the rate of return on the two risky assets
portfolio is
σ P = (wB σ B ) 2 + (wS σ S ) 2 + 2(wB σ B )(wS σ S )ρ BS
2
where ρBS is the correlation coefficient between the returns
on the stock and bond funds. 11-15

16. Portfolios
Rate of Return
Scenario Stock fund Bond fund Portfolio squared deviation
Recession -7% 17% 5.0% 0.0016
Normal 12% 7% 9.5% 0.0000
Boom 28% -3% 12.5% 0.0012
Expected return 11.00% 7.00% 9.0%
Variance 0.0205 0.0067 0.0010
Standard Deviation 14.31% 8.16% 3.08%
Observe the decrease in risk that diversification offers.
An equally weighted portfolio (50% in stocks and 50%
in bonds) has less risk than either stocks or bonds held
in isolation.
11-16

17. 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.
11-17

18. Risk: Systematic and Unsystematic
We can break down the total risk of holding a stock into
two components: systematic risk and unsystematic risk:
σ2
R = R +U
Total risk
becomes
ε R = R+m+ε
where
Nonsystematic Risk: ε
m is the systematic risk
Systematic Risk: m ε is the unsystematic risk
n 11-18

19. 12.2 Systematic Risk and Betas
• The beta coefficient, β, tells us the response
of the stock’s return to a systematic risk.
• In the CAPM, β measures the responsiveness
of a security’s return to a specific risk factor,
the return on the market portfolio.
Cov( Ri , RM )
βi =
σ ( RM )
2
• We shall now consider other types of systematic risk.
11-19

20. Systematic Risk and Betas
• For example, suppose we have identified three
systematic risks: inflation, GNP growth, and the
dollar-euro spot exchange rate, S($,€).
• Our model is:
R = R+m+ε
R = R + β I FI + βGNP FGNP + βS FS + ε
β I is the inflation beta
βGNP is the GNP beta
βS is the spot exchange rate beta
ε is the unsystematic risk 11-20

21. Systematic Risk and Betas: Example
R = R + βI FI + βGNP FGNP + βS FS + ε
• Suppose we have made the following
estimates:
βI = -2.30
βGNP = 1.50
βS = 0.50
• Finally, the firm was able to attract a
ε = 1%
“superstar” CEO, and this unanticipated
development contributes 1% .to × FS + 1%
R = R − 2.30 × FI + 1.50 × FGNP + 0 50 the return.
11-21

22. Systematic Risk and Betas: Example
R = R − 2.30 × FI + 1.50 × FGNP + 0.50 × FS + 1%
We must decide what surprises took place in the
systematic factors.
If it were the case that the inflation rate was
expected to be 3%, but in fact was 8% during the
time period, then:
FI = Surprise in the inflation rate = actual – expected
= 8% – 3% = 5%
R = R − 2.30 × 5% + 1.50 × FGNP + 0.50 × FS + 1%
11-22

23. Systematic Risk and Betas: Example
R = R − 2.30 × FI + 1.50 × FGNP + 0.50 × FS + 1%
We must decide what surprises took place in the
systematic factors.
If it were the case that the inflation rate was
expected to be 3%, but in fact was 8% during the
time period, then:
FI = Surprise in the inflation rate = actual – expected
= 8% – 3% = 5%
R = R − 2.30 × 5% + 1.50 × FGNP + 0.50 × FS + 1%
11-23

24. Systematic Risk and Betas: Example
R = R − 2.30 × 5% + 1.50 × (−3%) + 0.50 × FS + 1%
If it were the case that the dollar-euro spot
exchange rate, S($,€), was expected to
increase by 10%, but in fact remained
stable during the time period, then:
FS = Surprise in the exchange rate
= actual – expected = 0% – 10% = – 10%
R = R − 2.30 × 5% + 1.50 × (−3%) + 0.50 × (−10%) + 1%
11-24

25. Systematic Risk and Betas: Example
R = R − 2.30 × 5% + 1.50 × (−3%) + 0.50 × (−10%) + 1%
Finally, if it were the case that the expected
return on the stock was 8%, then:
R = 8%
R = 8% − 2.30 × 5% + 1.50 × (−3%) + 0.50 × (−10%) + 1%
R = −12%
11-25

26. Systematic Risk and Betas: Example
R = R − 2.30 × 5% + 1.50 × FGNP + 0.50 × FS + 1%
If it were the case that the rate of GNP growth
was expected to be 4%, but in fact was 1%,
then:
FGNP = Surprise in the rate of GNP growth
= actual – expected = 1% – 4% = – 3%
R = R − 2.30 × 5% + 1.50 × (−3%) + 0.50 × FS + 1%
11-26

27. Portfolio Risk and Number of Stocks
In a large portfolio the variance terms are
σ effectively diversified away, but the covariance
terms are not.
Diversifiable Risk;
Nonsystematic Risk;
Firm Specific Risk;
Unique Risk
Portfolio risk
Nondiversifiable risk;
Systematic Risk;
Market Risk
n 11-27

28. Risk: Systematic and Unsystematic
• A systematic risk is any risk that affects a large
number of assets, each to a greater or lesser degree.
• An unsystematic risk is a risk that specifically affects
a single asset or small group of assets.
• Unsystematic risk can be diversified away.
• Examples of systematic risk include uncertainty
about general economic conditions, such as GNP,
interest rates or inflation.
• On the other hand, announcements specific to a
single company are examples of unsystematic risk.
11-28

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.
11-29

30. Risk When Holding the Market Portfolio
• Researchers have shown that the best
measure of the risk of a security in a large
portfolio is the beta (β)of the security.
• Beta measures the responsiveness of a
security to movements in the market portfolio
(i.e., systematic risk).
Cov( Ri , RM )
βi =
σ ( RM )
2
11-30

31. The Formula for Beta
Cov ( Ri , RM ) σ ( Ri )
βi = =ρ
σ ( RM )
2
σ ( RM )
Clearly, your estimate of beta will
depend upon your choice of a proxy
for the market portfolio.
11-31

32. 11.9 Relationship between Risk and
Expected Return (CAPM)
• Expected Return on the Market:
R M = RF + Market Risk Premium
• Expected return on an individual security:
R i = RF + β i × ( R M − RF )
Market Risk Premium
This applies to individual securities held within well-
diversified portfolios.
11-32

33. Capital Asset Pricing Model (CAPM)
• The pure time value of money : As measured by the risk
free rate, Rf, this is the reward for merely waiting for your
money, without taking any risk.
• The reward for bearing systematic risk: As measured by
the market risk premium, E (RM) - Rf, this component is the
reward the market offers for bearing an average amount of
systematic risk in addition to waiting.
• The amount of systematic risk: As measured by βi, this is
the amount of systematic risk present in a particular assets
or portfolio, relative to that in an average asset.
11-33

34. Expected Return on a Security
• This formula is called the Capital Asset
Pricing Model (CAPM):
R i = RF + β i × ( R M − RF )
Expected
Risk- Beta of the Market risk
return on = + ×
free rate security premium
a security
• Assume βi = 0, then the expected return is RF.
• Assume βi = 1, then R i = R M
11-34