chapter 23 - answer

Solutions Manual

CHAPTER 23

CAPITAL ASSET PRICING MODEL AND MODERN PORTFOLIO THEORY

SUGGESTED ANSWERS TO THE REVIEW QUESTIONS AND PROBLEMS

I. Questions

1. Systematic risk is that part of a security’s risk caused by factors affecting the market as a whole. Systematic risk is undiversifiable and is measured by beta while unsystematic risk is that part of a security’s risk caused by factors unique to a particular firm or industry. Unsystematic risk is diversifiable.

2. Beta is computed by regressing a security’s returns of a broad-based market index. The slope of the regression line, also called the characteristics line, is an estimate of the security’s beta. A security’s or portfolio’s beta is interpreted relative to the market beta of 1.0. A beta may be positive or negative. The returns of a security with a positive beta move in the same direction as the return of the market. The returns of a security with a negative beta move in the opposite direction of the market. Negative betas are uncommon:

If b > 1.0, the security is more volatile or risky than the market.

If b = 1.0, the security has the same volatility or risk as the market.

If b < 1.0, the security is less volatile or risky than the market.

If b = 0.0, the security is uncorrelated with market movements and is riskless.

3. The standard deviation can be used as a measure of the amount of absolute risk associated with an outcome while beta is a measure of the sensitivity of a security’s return relative to the returns of a broad-based market portfolio of securities.

4. Some of the risk in holding any asset is unique to the asset in question. By investing in a variety of assets, this unique portion of the total risk can be eliminated at little cost. On the other hand, there are some risks that affect all investments. This portion of the total risk of an asset cannot

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Chapter 23 Capital Asset Pricing Model and Modern Portfolio Theory

be costlessly eliminated. In other words, systematic risk can be controlled, but only by a costly reduction in expected returns.

5. If the market expected the growth rate in the coming year to be 2 percent, then there would be no change in security prices if this expectation had been fully anticipated and priced. However, if the market had been expecting a growth rate other than 2 percent and the expectation was incorporated into security prices, then the government’s announcement would most likely cause security prices in general to change; prices would drop if the anticipated growth rate had been more than 2 percent, and prices would rise if the anticipated growth rate had been less than 2 percent.

6. (a) systematic

(b) unsystematic

(c) both; probably mostly systematic

(d) unsystematic

(e) unsystematic

(f) systematic

7. (a) A change in systematic risk has occurred; market prices in general will most likely decline.

(b) No change in unsystematic risk; company price will most likely stay constant.

(c) No change in systematic risk; market prices in general will most likely stay constant.

(d) A change in unsystematic risk has occurred; company price will most likely decline.

(e) No change in systematic risk; market prices in general will most likely stay constant.

8. Yes. It is possible, in theory, to construct a zero beta portfolio of risky assets whose return would be equal to the risk-free rate. It is also possible to have a negative beta; the return would be less than the risk-free rate. A negative beta asset would carry a negative risk premium because of its value as a diversification instrument.

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Capital Asset Pricing Model and Modern Portfolio Theory Chapter 23

II. Multiple Choice Questions

1. A 4. B 7. C 10. D2. C 5. A 8. B3. A 6. D 9. C

III. Problems

Problem 1

(a) The required rate of return is:

ri = 0.07 + (1.2) (0.12 − 0.07)

= 0.13 or 13% (b) The market risk premium is the return on the market portfolio (rm) less

the risk-free rate (rf).

rm − rf = 0.12 − 0.07

= 0.05 or 5%

Problem 2

The risk-free rate consists of a real rate and an inflation premium. If the inflation premium increased from 4 to 6 percent, the risk-free rate would increase from 7 to 9 percent. Inflation would also lead investors to expect a higher return on the market portfolio which would increase the return on the market portfolio, rm, from 12 to 14 percent. The required rate of return under the new inflationary expectations would be:

ri = 0.09 + (1.2) (0.14 − 0.09) = 0.15 or 15%

Thus, inflationary expectations would increase the required return for Dimension Industries and other ordinary equity shares.

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Problem 3

(a) The risk premiums are shown below:

StockRisk Premium

bi (rm − rf)

1 (1.5) (0.14 − 0.08) = 0.09 or 9 percent 2 (1.0) (0.14 − 0.08) = 0.06 or 6 percent3 (0.8) (0.14 − 0.08) = 0.048 or 4.8 percent4 (2.0) (0.14 − 0.08) = 0.12 or 12 percent5 (0.3) (0.14 − 0.08) = 0.018 or 1.8 percent6 (1.2) (0.14 − 0.08) = 0.072 or 7.2 percent

(b) The four stocks that provide the lowest risk portfolio are those with the lowest betas and therefore the lowest risk premiums, namely Stocks 2, 3, 5, and 6.

(c) The required rate of return on the portfolio of Stocks 2, 3, 5, and 6 is determined in two steps:

First, assuming equal weights of 25 percent for each stock, the portfolio’s beta is found by substituting these weights and the stock’s beta.

bp = (0.25) (1.0) + (0.25) (0.8) + (0.25) (0.3) + (0.25) (1.2)

= 0.825

Next, using the capital asset pricing model produces the required rate of return.

rp = 0.08 + (0.825) (0.14 − 0.08)

= 0.1295 or 12.95%

Problem 4

If we compute the reward-to-risk ratios, we get (22% − 7%) / 1.8 = 8.33% for Earth, Inc. versus 8.4% for Fire Company. Relative to that of Earth, Fire’s expected return is too high, so its price is too low.

If they are correctly priced, then they must offer the same reward-to-risk ratio. The risk-free rate would have to be such that:

(22% − Rf) / 1.8 = (20.44% − Rf) / 1.6

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With a little algebra, we find that the risk-free rate must be 8 percent:

22% − Rf = (20.44% − Rf) (1.8/1.6)22% − 20.44% x 1.125 = Rf − Rf x 1.125

Rf = 8%

Problem 5

Because the expected return on the market is 16 percent, the market risk premium is 16% − 8% = 8%.

The first stock has a beta of .7, so its expected return is:

8% + .7 x 8% = 13.6%

For the second stock, notice that the risk premium is:

24% − 8% = 16%

Because this is twice as large as the market risk premium, the beta must be exactly equal to 2. We can verify this using the CAPM:

E (Ri) = Rf + [E (RM) − Rf] x βi

24% = 8% + (16% − 8%) x βi

βi = 16%/8%

= 2.0

Problem 6

The beta of a portfolio is the sum of the weight of each asset times the beta of each asset. So, the beta of the portfolio is:

p = .25(.84) + .20(1.17) + .15(1.11) + .40(1.36)

= 1.15

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Problem 7

The beta of a portfolio is the sum of the weight of each asset times the beta of each asset. If the portfolio is as risky as the market it must have the same beta as the market. Since the beta of the market is one, we know the beta of our portfolio is one. We also need to remember that the beta of the risk-free asset is zero. It has to be zero since the asset has no risk. Setting up the equation for the beta of our portfolio, we get:

p = 1.0 = 1/3 (0) + 1/3 (1.38) + 1/3 (X)

Solving for the beta of Stock X, we get:

X = 1.62

Problem 8

CAPM states the relationship between the risk of an asset and its expected return. CAPM is:

E (Ri) = Rf + [E (RM) – Rf] × i

Substituting the values we are given, we find:

E (Ri) = .052 + (.11 – .052) (1.05)

= .1129 or 11.29%

Problem 9

We are given the values for the CAPM except for the of the stock. We need to substitute these values into the CAPM, and solve for the of the stock. One important thing we need to realize is that we are given the market risk premium. The market risk premium is the expected return of the market minus the risk-free rate. We must be careful not to use this value as the expected return of the market. Using the CAPM, we find:

E (Ri) = .102 = .045+ .085i

i = 0.67

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Problem 10

Here we need to find the expected return of the market using the CAPM. Substituting the values given, and solving for the expected return of the market, we find:

E (Ri) = .135 = .055 + [E (RM) – .055] (1.17)

E(RM) = .1234 or 12.34%

Problem 11

First, we need to find the of the portfolio. The of the risk-free asset is zero, and the weight of the risk-free asset is one minus the weight of the stock, the of the portfolio is:

ßp = wW (1.25) + (1 – wW) (0) = 1.25wW

So, to find the of the portfolio for any weight of the stock, we simply multiply the weight of the stock times its .

Even though we are solving for the and expected return of a portfolio of one stock and the risk-free asset for different portfolio weights, we are really solving for the SML. Any combination of this stock, and the risk-free asset will fall on the SML. For that matter, a portfolio of any stock and the risk-free asset, or any portfolio of stocks, will fall on the SML. We know the slope of the SML line is the market risk premium, so using the CAPM and the information concerning this stock, the market risk premium is:

E (RW) = .152 = .053 + MRP (1.25)

MRP = .099/1.25

= .0792 or 7.92%

So, now we know the CAPM equation for any stock is:

E (Rp) = .053 + .0793p

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The slope of the SML is equal to the market risk premium, which is 0.0792. Using these equations to fill in the table, we get the following results:

wW E (Rp) ßp

0.00% 5.30% 0.000

25.00% 7.78% 0.313

50.00% 10.25% 0.625

75.00% 12.73% 0.938

100.00% 15.20% 1.250

125.00% 17.68% 1.563

150.00% 20.15% 1.875

Problem 12

The amount of systematic risk is measured by the of an asset. Since we know the market risk premium and the risk-free rate, if we know the expected return of the asset we can use the CAPM to solve for the of the asset. The expected return of Stock I is:

E (RI) = .25 (.11) + .50 (.29) + .25 (.13)

= .2050 or 20.50%

Using the CAPM to find the of Stock I, we find:

.2050 = .04 + .08I

I = 2.06

The total risk of the asset is measured by its standard deviation, so we need to calculate the standard deviation of Stock I. Beginning with the calculation of the stock’s variance, we find:

I2 = .25 (.11 – .2050) 2 + .50 (.29 – .2050) 2 + .25 (.13 – .2050) 2

I2 = .00728

I = (.00728) 1/2

= .0853 or 8.53%

Using the same procedure for Stock II, we find the expected return to be:

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E (RII) = .25 (–.40) + .50 (.10) + .25 (.56)

= .0900

Using the CAPM to find the of Stock II, we find:

.0900 = .04 + .08II

II = 0.63

And the standard deviation of Stock II is:

II2 = .25 (–.40 – .0900) 2 + .50 (.10 – .0900) 2 + .25 (.56 – .0900) 2

II2 = .11530

II = (.11530)1/2

= .3396 or 33.96%

Although Stock II has more total risk than I, it has much less systematic risk, since its beta is much smaller than I’s. Thus, I has more systematic risk, and II has more unsystematic and more total risk. Since unsystematic risk can be diversified away, I is actually the “riskier” stock despite the lack of volatility in its returns. Stock I will have a higher risk premium and a greater expected return.

Problem 13

Here we have the expected return and beta for two assets. We can express the returns of the two assets using CAPM. If the CAPM is true, then the security market line holds as well, which means all assets have the same risk premium. Setting the risk premiums of the assets equal to each other and solving for the risk-free rate, we find:

(.132 – Rf)/1.35 = (.101 – Rf)/.80

.80(.132 – Rf) = 1.35(.101 – Rf)

.1056 – .8Rf = .13635 – 1.35Rf

.55Rf = .03075

Rf = .0559 or 5.59%

Now using CAPM to find the expected return on the market with both stocks, we find:

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.132 = .0559 + 1.35 (RM – .0559)

RM = .1123 or 11.23%

.101 = .0559 + .80 (RM – .0559)

RM = .1123 or 11.23%

Problem 14

(a) The expected return of an asset is the sum of the probability of each return occurring times the probability of that return occurring. So, the expected return of each stock is:

E (RX) = .15 (–.08) + .70 (.13) + .15 (.48)

= .1510 or 15.10%

E (RY) = .15 (–.05) + .70 (.14) + .15 (.29)

= .1340 or 13.40%

(b) We can use the expected returns we calculated to find the slope of the Security Market Line. We know that the beta of Stock X is .25 greater than the beta of Stock Y. Therefore, as beta increases by .25, the expected return on a security increases by .017 (= .1510 – .1340). The slope of the security market line (SML) equals:

SlopeSML = Rise / Run

SlopeSML = Increase in expected return / Increase in beta

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SlopeSML = (.1510 – .1340) / .25

SlopeSML = .0680 or 6.80%

Since the market’s beta is 1 and the risk-free rate has a beta of zero, the slope of the Security Market Line equals the expected market risk premium. So, the expected market risk premium must be 6.8 percent.

We could also solve this problem using CAPM. The equations for the expected returns of the two stocks are:

E (RX) = .151 = Rf + (B + .25) (MRP)

E (RY) = .134 = Rf + B (MRP)

We can rewrite the CAPM equation for Stock X as:

.151 = Rf + B (MRP) + .25(MRP)

Subtracting the CAPM equation for Stock Y from this equation yields:

.017 = .25MRP

MRP = .068 or 6.8%

which is the same answer as our previous result.

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chapter 23 - answer

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