1. 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 be costlessly eliminated. In other words, systematic risk can be controlled, but only by a costly reduction in expected returns.
2. 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.
3. a. systematic
b. unsystematic
c. both; probably mostly systematic
d. unsystematic
e. unsystematic
f. systematic
4. 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.
5. No to both questions. The portfolio expected return is a weighted average of the asset returns, so it must be less than the largest asset return and greater than the smallest asset return.
6. False. The variance of the individual assets is a measure of the total risk. The variance on a well-diversified portfolio is a function of systematic risk only.
7. Yes, the standard deviation can be less than that of every asset in the portfolio. However, bp cannot be less than the smallest beta because bp is a weighted average of the individual asset betas.
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.
9. Such layoffs generally occur in the context of corporate restructurings. To the extent that the market views a restructuring as value-creating, stock prices will rise. So, it’s not layoffs per se that are being cheered on. Nonetheless, Wall Street does encourage corporations to takes actions to create value, even if such actions involve layoffs.
10. Earnings contain information about recent sales and costs. This information is useful for projecting future growth rates and cash flows. Thus, unexpectedly low earnings often lead market participants to reduce estimates of future growth rates and cash flows; price drops are the result. The reverse is often true for unexpectedly high earnings.
Solutions to Questions and Problems
NOTE: All end of chapter problems were solved using a spreadsheet. Many problems require multiple steps. Due to space and readability constraints, when these intermediate steps are included in this solutions manual, rounding may appear to have occurred. However, the final answer for each problem is found without rounding during any step in the problem.
Basic
1. The portfolio weight of an asset is total investment in that asset divided by the total portfolio value. First, we will find the portfolio value, which is:
Total value = 70($40) + 110($22) = $5,220
The portfolio weight for each stock is:
WeightA = 70($40)/$5,220 = .5364
WeightB = 110($22)/$5,220 = .4636
2. The expected return of a portfolio is the sum of the weight of each asset times the expected return of each asset. The total value of the portfolio is:
Total value = $1,200 + 1,900 = $3,100
So, the expected return of this portfolio is:
E(Rp) = ($1,200/$3,100)(0.11) + ($1,900/$3,100)(0.16) = .1406 or 14.06%
3. The expected return of a portfolio is the sum of the weight of each asset times the expected return of each asset. So, the expected return of the portfolio is:
E(Rp) = .50(.11) + .30(.17) + .20(.14) = .1340 or 13.40%
4. Here we are given the expected return of the portfolio and the expected return of each asset in the portfolio, and are asked to find the weight of each asset. We can use the equation for the expected return of a portfolio to solve this problem. Since the total weight of a portfolio must equal 1 (100%), the weight of Stock Y must be one minus the weight of Stock X. Mathematically speaking, this means:
E(Rp) = .122 = .14wX + .09(1 – wX)
We can now solve this equation for the weight of Stock X as:
.122 = .14wX + .09 – .09wX
.032 = .05wX
wX = 0.64
So, the dollar amount invested in Stock X is the weight of Stock X times the total portfolio value, or:
Investment in X = 0.64($10,000) = $6,400
And the dollar amount invested in Stock Y is:
Investment in Y = (1 – 0.64)($10,000) = $3,600
5. 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 the asset is:
E(R) = .3(–.08) + .7(.28) = .1720 or 17.20%
6. 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 the asset is:
E(R) = .2(–.05) + .5(.12) + .3(.25) = .1250 or 12.50%
7. 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 asset is:
E(RA) = .10(.06) + .60(.07) + .30(.11) = .0810 or 8.10%
E(RB) = .10(–.2) + .60(.13) + .30(.33) = .1570 or 15.70%
To calculate the standard deviation, we first need to calculate the variance. To find the variance, we find the squared deviations from the expected return. We then multiply each possible squared deviation by its probability, then add all of these up. The result is the variance. So, the variance and standard deviation of each stock is:
sA2 =.10(.06 – .0810)2 + .60(.07–.0810)2 + .30(.11 – .0810)2 = .00037
sA = (.00037)1/2 = .0192 or 1.92%
sB2 =.10(–.2 – .1570)2 + .60(.13–.1570)2 + .30(.33 – .1570)2 = .02216
sB = (.022216)1/2 = .1489 or 14.89%
8. The expected return of a portfolio is the sum of the weight of each asset times the expected return of each asset. So, the expected return of the portfolio is:
E(Rp) = .20(.08) + .70(.15) + .1(.24) = .1450 or 14.50%
If we own this portfolio, we would expect to get a return of 14.50 percent.
9. a. To find the expected return of the portfolio, we need to find the return of the portfolio in each state of the economy. This portfolio is a special case since all three assets have the same weight. To find the expected return in an equally weighted portfolio, we can sum the returns of each asset and divide by the number of assets, so the expected return of the portfolio in each state of the economy is:
Boom: E(Rp) = (.07 + .15 + .33)/3 = .1833 or 18.33%
Bust: E(Rp) = (.13 + .03 -.06)/3 = .0333 or 3.33%
To find the expected return of the portfolio, we multiply the return in each state of the economy by the probability of that state occurring, and then sum. Doing this, we find:
E(Rp) = .70(.1833) + .30(.0333) = .1383 or 13.83%
b. This portfolio does not have an equal weight in each asset. We still need to find the return of the portfolio in each state of the economy. To do this, we will multiply the return of each asset by its portfolio weight and then sum the products to get the portfolio return in each state of the economy. Doing so, we get:
Boom: E(Rp)=.20(.07) +.20(.15) + .60(.33) =.2420 or 24.20%
Bust: E(Rp) =.20(.13) +.20(.03) + .60(-.06) = –.0040 or –0.40%
And the expected return of the portfolio is:
E(Rp) = .70(.2420) + .30(-.004) = .1682 or 16.82%
To calculate the standard deviation, we first need to calculate the variance. To find the variance, we find the squared deviations from the expected return. We then multiply each possible squared deviation by its probability, than add all of these up. The result is the variance. So, the variance and standard deviation of the portfolio is:
sp2 = .70(.2420 – .1682)2 + .30(-.0040 – .1682)2 = .012708
sp = (.012708)1/2 = .1781 or 17.81%
10. a. This portfolio does not have an equal weight in each asset. We first need to find the return of the portfolio in each state of the economy. To do this, we will multiply the return of each asset by its portfolio weight and then sum the products to get the portfolio return in each state of the economy. Doing so, we get:
Boom: E(Rp) = .30(.3) + .40(.45) + .30(.33) = .3690 or 36.90%
Good: E(Rp) = .30(.12) + .40(.10) + .30(.15) = .1210 or 12.10%
Poor: E(Rp) = .30(.01) + .40(–.15) + .30(–.05) = –.0720 or –7.20%
Bust: E(Rp) = .30(–.06) + .40(–.30) + .30(–.09) = –.1650 or –16.50%
And the expected return of the portfolio is:
E(Rp) = .30(.3690) + .40(.1210) + .25(–.0720) + .05(–.1650) = .1329 or 13.29%
b. To calculate the standard deviation, we first need to calculate the variance. To find the variance, we find the squared deviations from the expected return. We then multiply each possible squared deviation by its probability, than add all of these up. The result is the variance. So, the variance and standard deviation of the portfolio is:
sp2 = .30(.3690 – .1329)2 + .40(.1210 – .1329)2 + .25 (–.0720 – .1329)2 + .05(–.1650 – .1329)2
sp2 = .03171
sp = (.03171)1/2 = .1781 or 17.81%
11. 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:
bp = .25(.6) + .20(1.7) + .15(1.15) + .40(1.34) = 1.20
12. 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:
bp = 1.0 = 1/3(0) + 1/3(1.9) + 1/3(bX)
Solving for the beta of Stock X, we get:
bX = 1.10
63013. CAPM states the relationship between the risk of an asset and its expected return. CAPM is:
E(Ri) = Rf + [E(RM) – Rf] × bi
Substituting the values we are given, we find:
E(Ri) = .05 + (.14 – .05)(1.3) = .1670 or 16.70%
14. We are given the values for the CAPM except for the b of the stock. We need to substitute these values into the CAPM, and solve for the b 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) = .14 = .04 + .06bi
bi = 1.67
15. 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) = .11 = .055 + [E(RM) – .055](.85)
E(RM) = .1197 or 11.97%
16. Here we need to find the risk-free rate using the CAPM. Substituting the values given, and solving for the risk-free rate, we find:
E(Ri) = .17 = Rf + (.11 – Rf)(1.9)
.17 = Rf + .209 – 1.9Rf
Rf = .0433 or 4.33%
17. a. Again we have a special case where the portfolio is equally weighted, so we can sum the returns of each asset and divide by the number of assets. The expected return of the portfolio is:
E(Rp) = (.16 + .05)/2 = .1050 or 10.50%
b. We need to find the portfolio weights that result in a portfolio with a b of 0.75. We know the b of the risk-free asset is zero. We also know the weight of the risk-free asset is one minus the weight of the stock since the portfolio weights must sum to one, or 100 percent. So:
bp = 0.75 = wS(1.2) + (1 – wS)(0)
0.75 = 1.2wS + 0 – 0wS
wS = 0.75/1.2
wS = .6250
And, the weight of the risk-free asset is:
wRf = 1 – .6250 = .3750
c. We need to find the portfolio weights that result in a portfolio with an expected return of 8 percent. We also know the weight of the risk-free asset is one minus the weight of the stock since the portfolio weights must sum to one, or 100 percent. So:
E(Rp) = .08 = .16wS + .05(1 – wS)
.08 = .16wS + .05 – .05wS
wS = .2727
So, the b of the portfolio will be:
bp = .2727(1.2) + (1 – .7273)(0) = 0.327
d. Solving for the b of the portfolio as we did in part a, we find:
bp = 2.4 = wS(1.2) + (1 – wS)(0)
wS = 2.4/1.2 = 2
wRf = 1 – 2 = –1
The portfolio is invested 200% in the stock and –100% in the risk-free asset. This represents borrowing at the risk-free rate to buy more of the stock.
18. First, we need to find the b of the portfolio. The b of the risk-free asset is zero, and the weight of the risk-free asset is one minus the weight of the stock, the b of the portfolio is:
ßp = wW(1.3) + (1 – wW)(0) = 1.3wW
So, to find the b of the portfolio for any weight of the stock, we simply multiply the weight of the stock times its b.
Even though we are solving for the b 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) = .16 = .05 + MRP(1.30)
MRP = .11/1.3 = .0846 or 8.46%
So, now we know the CAPM equation for any stock is:
E(Rp) = .05 + .0846bp
The slope of the SML is equal to the market risk premium, which is 0.0846. Using these equations to fill in the table, we get the following results:
wW E(Rp) ßp
0% .0500 0
25 .0775 0.325
50 .1050 0.650
75 .1325 0.975
100 .1600 1.300
125 .1875 1.625
150 .2150 1.950
19. There are two ways to correctly answer this question. We will work through both. First, we can use the CAPM. Substituting in the value we are given for each stock, we find:
E(RY) = .055 + .075(1.50) = .1675 or 16.75%
It is given in the problem that the expected return of Stock Y is 17 percent, but according to the CAPM, the return of the stock based on its level of risk, the expected return should be 16.75 percent. This means the stock return is too high, given its level of risk. Stock Y plots above the SML and is undervalued. In other words, its price must increase to reduce the expected return to 16.75 percent. For Stock Z, we find:
E(RZ) = .055 + .075(0.80) = .1150 or 11.50%
The return given for Stock Z is 10.5 percent, but according to the CAPM the expected return of the stock should be 11.50 percent based on its level of risk. Stock Z plots below the SML and is overvalued. In other words, its price must decrease to increase the expected return to 11.50 percent.
We can also answer this question using the reward-to-risk ratio. All assets must have the same reward-to-risk ratio. The reward-to-risk ratio is the risk premium of the asset divided by its b. We are given the market risk premium, and we know the b of the market is one, so the reward-to-risk ratio for the market is 0.075, or 7.5 percent. Calculating the reward-to-risk ratio for Stock Y, we find:
Reward-to-risk ratio Y = (.17 – .055) / 1.50 = .0767
The reward-to-risk ratio for Stock Y is too high, which means the stock plots above the SML, and the stock is undervalued. Its price must increase until its reward-to-risk ratio is equal to the market reward-to-risk ratio. For Stock Z, we find:
Reward-to-risk ratio Z = (.105 – .055) / .80 = .0625
The reward-to-risk ratio for Stock Z is too low, which means the stock plots below the SML, and the stock is overvalued. Its price must decrease until its reward-to-risk ratio is equal to the market reward-to-risk ratio.
20. We need to set the reward-to-risk ratios of the two assets equal to each other, which is:
(.17 – Rf)/1.50 = (.105 – Rf)/0.80
We can cross multiply to get:
0.80(.17 – Rf) = 1.50(.105 – Rf)
Solving for the risk-free rate, we find:
0.136 – 0.80Rf = 0.1575 – 1.50Rf
Rf = .0307 or 3.07%
Intermediate
21. For a portfolio that is equally invested in large-company stocks and long-term bonds:
Return = (12.4% + 6.2%)/2 = 9.3%
For a portfolio that is equally invested in small stocks and Treasury bills:
Return = (17.5% + 3.8%)/2 = 10.65%
22. We know that the reward-to-risk ratios for all assets must be equal. This can be expressed as:
[E(RA) – Rf]/bA = [E(RB) – Rf]/ßB
The numerator of each equation is the risk premium of the asset, so:
RPA/bA = RPB/bB
We can rearrange this equation to get:
bB/bA = RPB/RPA
If the reward-to-risk ratios are the same, the ratio of the betas of the assets is equal to the ratio of the risk premiums of the assets.
23. a. We need to find the return of the portfolio in each state of the economy. To do this, we will multiply the return of each asset by its portfolio weight and then sum the products to get the portfolio return in each state of the economy. Doing so, we get:
Boom: E(Rp) = .4(.20) + .4(.35) + .2(.60) = .3400 or 34.00%
Normal: E(Rp) = .4(.15) + .4(.12) + .2(.05) = .1180 or 11.80%
Bust: E(Rp) = .4(.01) + .4(–.25) + .2(–.50) = –.1960 or –19.60%
And the expected return of the portfolio is:
E(Rp) = .4(.34) + .4(.118) + .2(–.196) = .1440 or 14.40%
To calculate the standard deviation, we first need to calculate the variance. To find the variance, we find the squared deviations from the expected return. We then multiply each possible squared deviation by its probability, than add all of these up. The result is the variance. So, the variance and standard deviation of the portfolio is:
s2p = .4(.34 – .1440)2 + .4(.118 – .1440)2 + .2(–.196 – .1440)2
s2p = .03876
sp = (.03876)1/2 = .1969 or 19.69%
b. The risk premium is the return of a risky asset, minus the risk-free rate. T-bills are often used as the risk-free rate, so:
RPi = E(Rp) – Rf = .1440 – .038 = .1060 or 10.60%
c. The approximate expected real return is the expected nominal return minus the inflation rate, so:
Approximate expected real return = .1440 – .035 = .1090 or 10.90%
To find the exact real return, we will use the Fisher equation. Doing so, we get:
1 + E(Ri) = (1 + h)[1 + e(ri)]
1.1440 = (1.0350)[1 + e(ri)]
e(ri) = (1.1440/1.035) – 1 = .1053 or 10.53%
The approximate real risk premium is the expected return minus the risk-free rate, so:
Approximate expected real risk premium = .1440 – .038 = .1060 or 10.60%
The exact expected real risk premium is the approximate expected real risk premium, divided by one plus the inflation rate, so:
Exact expected real risk premium = .1060/1.035 = .1024 or 10.24%
24. Since the portfolio is as risky as the market, the b of the portfolio must be equal to one. We also know the b of the risk-free asset is zero. We can use the equation for the b of a portfolio to find the weight of the third stock. Doing so, we find:
bp = 1.0 = wA(.8) + wB(1.3) + wC(1.5) + wRf(0)
Solving for the weight of Stock C, we find:
wC = .343333
So, the dollar investment in Stock C must be:
Invest in Stock C = .343333($1,000,000) = $343,333
We know the total portfolio value and the investment of two stocks in the portfolio, so we can find the weight of these two stocks. The weights of Stock A and Stock B are:
wA = $200,000 / $1,000,000 = .20
wB = $250,000/$1,000,000 = .25
We also know the total portfolio weight must be one, so the weight of the risk-free asset must be one minus the asset weight we know, or:
1 = wA + wB + wC + wRf = 1 – .20 – .25 – .34333 – wRf
wRf = .206667
So, the dollar investment in the risk-free asset must be:
Invest in risk-free asset = .206667($1,000,000) = $206,667
25. We are given the expected return and b of a portfolio and the expected return and b of assets in the portfolio. We know the b of the risk-free asset is zero. We also know the sum of the weights of each asset must be equal to one. So, the weight of the risk-free asset is one minus the weight of Stock X and the weight of Stock Y. Using this relationship, we can express the expected return of the portfolio as:
E(Rp) = .135 = wX(.31) + wY(.20) + (1 – wX – wY)(.07)
And the b of the portfolio is:
bp = .8 = wX(1.8) + wY(1.3) + (1 – wX – wY)(0)
We have two equations and two unknowns. Solving these equations, we find that:
wX = –0.0833333
wY = 0.6538462
wRf = 0.4298472
The amount to invest in Stock X is:
Investment in stock X = –0.0833333($100,000) = –$8,333.33
A negative portfolio weight means that your short sell the stock. If you are not familiar with short selling, it means you borrow a stock today and sell it. You must then purchase the stock at a later date to repay the borrowed stock. If you short sell a stock, you make a profit if the stock decreases in value.
26. The amount of systematic risk is measured by the b 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 b of the asset. The expected return of Stock I is:
E(RI) = .15(.09) + .70(.42) + .15(.26) = .3465 or 34.65%
Using the CAPM to find the b of Stock I, we find:
.3465 = .04 + .10bI
bI = 3.07
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:
sI2 = .15(.09 – .3465)2 + .70(.42 – .3465)2 + .15(.26 – .3465)2
sI2 = .01477
sI = (.01477)1/2 = .1215 or 12.15%
Using the same procedure for Stock II, we find the expected return to be:
E(RII) = .15(–.30) + .70(.12) + .15(.44) = .1050
Using the CAPM to find the b of Stock II, we find:
.1050 = .04 + .10bII
bII = 0.65
And the standard deviation of Stock II is:
sII2 = .15(–.30 – .105)2 + .70(.12 – .105)2 + .15(.44 – .105)2
sII2 = .04160
sII = (.04160)1/2 = .2039 or 20.39%
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.
27. Here we have the expected return and beta for two assets. We can express the returns of the two assets using CAPM. Now we have two equations and two unknowns. Going back to Algebra, we can solve the two equations. We will solve the equation for Pete Corp. to find the risk-free rate, and solve the equation for Repete Co. to find the expected return of the market. We next substitute the expected return of the market into the equation for Pete Corp., then solve for the risk-free rate. Now that we have the risk-free rate, we can substitute this into either original CAPM expression and solve for expected return of the market. Doing so, we get:
E(RPete Corp.) = .23 = Rf + 1.3(RM – Rf); E(RRepete Co.) = .13 = Rf + .6(RM – Rf)
.23 = Rf + 1.3RM – 1.3Rf = 1.3RM – .3Rf; .13 = Rf + .6(RM – Rf) = Rf + .6RM – .6Rf
Rf = (1.3RM – .23)/.3 RM = (.13 – .4Rf)/.6
RM = .217 – .667Rf
Rf = [1.3(.217 – .667Rf) – .23]/.3
1.167Rf = .0521
Rf = .0443 or 4.43%
.23 = .0443 + 1.3(RM – .0443) .13 = .0443 + .6(RM – .0443)
RM = .1871 or 18.71% RM = .1871 or 18.71%
