Chapter 10: Return and Risk: The Capital-Asset-Pricing Model (CAPM)
10.1a.= 0.1 (– 4.5%) + 0.2 (4.4%) + 0.5 (12.0%) + 0.2 (20.7%)
= 10.57%
b.2= 0.1 (–0.045 – 0.1057)2 + 0.2 (0.044 – 0.1057)2 + 0.5 (0.12 – 0.1057)2
+ 0.2 (0.207 – 0.1057)2
= 0.0052
= (0.0052)1/2 = 0.072 = 7.20%
10.2a.= (6.3 + 10.5 + 15.6) / 3 = 10.8%
= (-3.7 + 6.4 + 25.3) / 3 = 9.3%
b.A2= {(0.063 – 0.108)2 + (0.105 – 0.108)2 +{(0.156 – 0.108)2} / 3
= 0.001446
A= (0.001446)1/2 = 0.0380 = 3.80%
B2= {(– 0.037 – 0.093)2 +(0.064 – 0.093)2 +(0.253 – 0.093)2} / 3
= 0.014447
B= (0.014447)1/2 = 0.1202 = 12.02%
c.Cov(RA,RB) = [(.063 – .108) (– .037 – .093) + (.105 – .108) (.064 – .093)
+ (.156 – .108) (.253 – .093) ] / 3
= .013617 / 3 = .004539
Corr(RA,RB) = .004539 / (.0380 x .1202) = .9937
10.3a.= 0.25 (–2.0) + 0.60 (9.2) + 0.15 (15.4)
= 7.33%
= 0.25 (5.0) + 0.60 (6.2) + 0.15 (7.4)
= 6.08%
b.HB2= 0.25 (– 0.02 – 0.0733)2 + 0.60 (0.092 – 0.0733)2
+ 0.15 (0.154 – 0.0733)2
= 0.003363
HB= (0.003363)1/2 = 0.05799 = 5.80%
SB2= 0.25 (0.05 – 0.0608)2 + 0.60 (0.062 – 0.0608)2
+ 0.15 (0.074 – 0.0608)2
= 0.000056
SB= (0.000056)1/2 = 0.00749 = 0.75%
c.Cov (RHB, RSB)
= 0.25 (– 0.02 – 0.0733) (0.05 – 0.0608)
+ 0.60 (0.092 – 0.0733) (0.062 – 0.0608)
+ 0.15 (0.154 – 0.0733) (0.074 – 0.0608)
= 0.000425286
Corr (RHB, RSB)
= 0.000425286 / (0.05799 0.00749)
= 0.9791
10.4Holdings of Atlas stock = 120 $50 = $6,000
Holdings of Babcock stock = 150 $20 = $3,000
Weight of Atlas stock = $6,000 / $9,000 = 2 / 3
Weight of Babcock stock = $3,000 / $9,000 = 1 / 3
10.5a.= 0.3 (0.12) + 0.7 (0.18) = 0.162 = 16.2%
b.P 2= 0.32 (0.09)2 + 0.72 (0.25)2 + 2 (0.3) (0.7) (0.09) (0.25) (0.2)
= 0.033244
P= (0.033244)1/2 = 0.1823 = 18.23%
10.6a.= 0.4 (0.15) + 0.6 (0.25) = 0.21 = 21%
P 2= 0.42 (0.1)2 + 0.62 (0.2)2 + 2 (0.4) (0.6) (0.1) (0.2) (0.5)
= 0.0208
P= (0.0208)1/2 = 0.1442 = 14.42%
b.P 2= 0.42 (0.1)2 + 0.62 (0.2)2 + 2 (0.4) (0.6) (0.1) (0.2) (-0.5)
= 0.0112
P= (0.0112)1/2 = 0.1058 = 10.58%
c.As the stocks are more negatively correlated, the standard deviation of the portfolio decreases.
10.7Macrosoft: 100 $80 = $8,000
Intelligent: 300 $40 = $12,000
Weight:Macrosoft: $8,000 / $20,000 = 0.4
Intelligent: $12,000 / $20,000 = 0.6
a.= 0.4 (0.15) + 0.6 (0.20) = 0.18 = 18%
P 2= 0.42 (0.08)2 + 0.62 (0.2)2 + 2 (0.4) (0.6) (0.38) (0.08) (0.20)
= 0.0183424
P= (0.0183424)1/2 = 0.1354 = 13.54%
b.New weight:
Macrosoft: $8,000 / $12,000 = 0.667
Intelligent: $4,000 / $12,000 = 0.333
= 0.667 (0.15) + 0.333 (0.20) = 0.1666 = 16.66%
P 2= 0.6672 (0.08)2 + 0.3332 (0.2)2 + 2 (0.667) (0.333) (0.38) (0.08) (0.20)
= 0.009984
P= (0.009984)1/2 = 0.09992 = 9.99%
10.8a.= 7%
= 0.2 (-0.05) + 0.5 (0.10) + 0.3 (0.25) = 0.115 = 11.5%
U2= U = 0
V2= 0.2 (-0.05 - 0.115)2 + 0.5 (0.10 - 0.115)2 + 0.3 (0.25 - 0.115)2
= 0.0110
V= (0.0110)1/2 = 0.105 = 10.5%
b.Cov (RU, RV)
= 0.2 (-0.05 - 0.115) (0.07 - 0.07)
+ 0.5 (0.10 - 0.115) (0.07 - 0.07)
+ 0.3 (0.25 - 0.115) (0.07 - 0.07)
= 0
Corr (RU, RV) = 0
c.= 0.5 (0.115) + 0.5 (0.07) = 0.0925 = 9.25%
P 2= 0.52 (0.0110) = 0.00275
P= (0.00275)1/2 = 0.0524 = 5.24%
10.9a.= 0.3 (0.10) + 0.7 (0.20) = 0.17 = 17.0%
P 2= 0.32 (0.05)2 + 0.72 (0.15)2 = 0.01125
P= (0.01125)1/2 = 0.10607 = 10.61%
b.= 0.9 (0.10) + 0.1 (0.20) = 0.11 = 11.0%
P 2= 0.92 (0.05)2 + 0.12 (0.15)2 = 0.00225
P= (0.00225)1/2 = 0.04743 = 4.74%
c.No, I would not hold 100% of stock A because the portfolio in b has higher expected return but less standard deviation than stock A.
I may or may not hold 100% of stock B, depending on my preference.
10.10The expected return on any portfolio must be less than or equal to the return on the stock with the highest return. It cannot be greater than this stock’s return because all stocks with lower returns will pull down the value of the weighted average return.
Similarly, the expected return on any portfolio must be greater than or equal to the return of the asset with the lowest return. The portfolio return cannot be less than the lowest return in the portfolio because all higher earning stocks will pull up the value of the weighted average.
10.11a.= 0.4 (0.03) + 0.6 (0.15) = 0.102 = 10.2%
= 0.4 (0.065) + 0.6 (0.065) = 0.065 = 6.5%
A2= 0.4 (0.03 - 0.102)2 + 0.6 (0.15 - 0.102)2
= 0.003456
A= (0.003456)1/2 = 0.05878 = 5.88%
B2 = B = 0
b.XA= $2,500 / $6,000 = 0.417
XB= 1 - 0.417 = 0.583
= 0.417 (0.102) + 0.583 (0.065) = 0.0804 = 8.04%
P 2= XA2 A2 = 0.0006
P= (0.0006)1/2 = 0.0245 = 2.45%
c.Amount borrowed = -40 $50 = -$2,000
XA = $8,000 / $6,000 = 4 / 3
XB = 1 - XA = -1 / 3
= (4 / 3) (0.102) + (-1 / 3) (0.065) = 0.1143 = 11.43%
P 2= (4 / 3)2 (0.003456) = 0.006144
P= (0.006144)1/2 = 0.07838 = 7.84%
10.12The wide fluctuations in the price of oil stocks do not indicate that oil is a poor investment. If oil is purchased as part of a portfolio, what matters is only its beta. Since the price captures beta plus idiosyncratic risks, observing price volatility is not an adequate measure of the appropriateness of adding oil to a portfolio. Remember, total variability should not be used when deciding whether or not to put an asset into a large portfolio.
10.13a.= 0.1 (0.25) + 0.4 (0.20) + 0.4 (0.15) + 0.1 (0.10)
= 0.175 = 17.5%
= 0.1 (0.25) + 0.4 (0.15) + 0.4 (0.20) + 0.1 (0.10)
= 0.175 = 17.5%
= 0.1 (0.10) + 0.4 (0.15) + 0.4 (0.20) + 0.1 (0.25)
= 0.175 = 17.5%
R1 if
State occurs / R1 - / (R1 -)2 / P (R1 -)20.25 / 0.075 / 0.005625 / 0.0005625
0.20 / 0.025 / 0.000625 / 0.0002500
0.15 / -0.025 / 0.000625 / 0.0002500
0.10 / -0.075 / 0.005625 / 0.0005625
Variance / 0.0016250
Standard deviation=
= 0.0403
R2 if
State occurs / R2 - / (R2 -)2 / P (R2 -)20.25 / 0.075 / 0.005625 / 0.0005625
0.15 / -0.025 / 0.000625 / 0.0002500
0.20 / 0.025 / 0.000625 / 0.0002500
0.10 / -0.075 / 0.005625 / 0.0005625
Variance / 0.0016250
Standard deviation=
= 0.0403
R3 if
State occurs / R3 - / (R3 -)2 / P (R3 -)20.10 / -0.075 / 0.005625 / 0.0005625
0.15 / -0.025 / 0.000625 / 0.0002500
0.20 / 0.025 / 0.000625 / 0.0002500
0.25 / 0.075 / 0.005625 / 0.0005625
Variance / 0.0016250
Standard deviation=
= 0.0403
b. Cov(1,2) = .10 (.25 - .175) (.25 - .175) + .40 (.20 - .175) (.15 - .175) + .40 (.15 - .175) (.20 - .175) + .10 (.10 - .175) (.10 - .175)
= 0.000625
Cov(1,3) = .10 (.25 - .175) (.10 - .175) + .40 (.20 - .175) (.15 - .175) + .40 (.15 - .175) (.20 - .175) + .10 (.10 - .175) (.25 - .175)
= - 0.001625
Cov(2,3) = .10 (.25 - .175) (.10 - .175) + .40 (.15 - .175) (.15 - .175) + .40 (.20 - .175) (.20 - .175) + .10 (.10 - .175) (.25 - .175)
= - 0.000625
Corr(1,2) = 0.000625 / (0.0403 x 0.0403) = 0.385
Corr(1,3) = - 0.001625 / (0.0403 x 0.0403) = - 1
Corr(2,3) = - 0.000625 / (0.0403 x 0.0403) = - 0.385
c.E(R) = .5 x .175 + .5 x .175 = .175
Var = .5 x .5 x .0403 x .0403 + .5 x .5 x .0403 x .0403 + 2 x .5 x .5 x .000625
= 0.0011245
= 0.0335
d.E(R) = .5 x .175 + .5 x .175 = .175
Var = .5 x .5 x .0403 x .0403 + .5 x .5 x .0403 x .0403 + 2 x .5 x .5 x (-.001625)
= 0
= 0
e.E(R) = .5 x .175 + .5 x .175 = .175
Var = .5 x .5 x .0403 x .0403 + .5 x .5 x .0403 x .0403 + 2 x .5 x .5 x (-.000625)
= 0.0004995
= 0.0224
f.Portfolio with negatively correlated stocks can achieve higher degree of diversification than portfolio with positively correlated stocks, holding expected return for each stock constant. Applying proper weights on perfectly negatively correlated stocks can reduce portfolio variance to 0. As long as the correlation is not 1, there is benefit of diversification.
10.14a.
State / Return on A / Return on B / Probability1 / 15% / 35% / 0.4 0.5 = 0.2
2 / 15% / -5% / 0.4 0.5 = 0.2
3 / 10% / 35% / 0.6 0.5 = 0.3
4 / 10% / -5% / 0.6 0.5 = 0.3
b.= 0.2 [0.5 (0.15) + 0.5 (0.35)] + 0.2[0.5 (0.15) + 0.5 (-0.05)]
+ 0.3 [0.5 (0.10) + 0.5 (0.35)] + 0.3 [0.5 (0.10) + 0.5 (-0.05)]
= 0.135
= 13.5%
10.15a.= / N = {N (0.10)} / N = 0.10 = 10%
P 2= Cov (Ri, Rj) / N2 + i2 / N2
= N (N - 1) (0.0064) / N2 + N (0.0144) / N2
= (0.0064) (N - 1) / N + (0.0144) / N
b.As N , P 2 0.0064 = Cov (Ri, Rj)
c.The covariance of the returns of the securities is the most important factor to consider when placing securities in a well-diversified portfolio.
10.16The statement is false. Once the stock is part of a well-diversified portfolio, the important factor is the contribution of the stock to the variance of the portfolio. In a well-diversified portfolio, this contribution is the covariance of the stock with the rest of the portfolio.
10.17The covariance is a more appropriate measure of risk in a well-diversified portfolio because it reflects the effect of the security on the variance of the portfolio. Investors are concerned with the variance of their portfolios and not the variance of the individual securities. Since covariance measures the impact of an individual security on the variance of the portfolio, covariance is the appropriate measure of risk.
10.18If we assume that the market has not stayed constant during the past three years, then the low volatility of Southern Co.’s stock price only indicates that the stock has a beta which is very near to zero. The high volatility of Texas Instruments’ stock price does not imply that the firm’s beta is high. Total volatility (the price fluctuation) is a function of both systematic and unsystematic risk. The beta only reflects the systematic risk. Observing price volatility does not indicate whether it was due to systematic factors, or firm specific factors. Thus, if you observe a high price volatility like that of TI, you cannot claim that the beta of TI’s stock is high. All you know is that the total risk of TI is high.
10.19Note: The solution to this problem requires calculus.
Specifically, the solution is found by minimizing a function subject to a constraint. Calculus ability is not necessary to understand the principles behind a minimum variance portfolio.
Min { XA2 A2 + XB2B2 + 2 XA XB Cov(RA , RB)}
subject to XA + XB = 1
Let XA = 1 - XB. Then,
Min {(1 - XB)2A2 + XB2B2+ 2(1 - XB) XB Cov (RA, RB)}
Take a derivative with respect to XB.
d{} / dXB = (2 XB - 2) A2 + 2 XB B2 + 2 Cov(RA, RB) - 4 XB Cov(RA, RB)
Set the derivative equal to zero, cancel the common 2 and solve for XB.
XBA2 - A2 + XB B2 + Cov(RA, RB) - 2 XB Cov(RA, RB) = 0
XB = {A2 - Cov(RA, RB)} / {A2 + B2 - 2 Cov(RA, RB)}
and
XA = {B2 - Cov(RA, RB)} / {A2 + B2 - 2 Cov(RA, RB)}
Using the data from the problem yields,
XA = 0.8125 and
XB = 0.1875.
a.Using the weights calculated above, the expected return on the minimum variance portfolio is
E(RP) = 0.8125 E(RA) + 0.1875 E(RB)
= 0.8125 (5%) + 0.1875 (10%)
= 5.9375%
b.Using the formula derived above, the weights are
XA = 2 / 3 and
XB = 1 / 3
c.The variance of this portfolio is zero.
P 2= XA2 A2 + XB2B2 + 2 XA XB Cov(RA , RB)
= (4 / 9) (0.01) + (1 / 9) (0.04) + 2 (2 / 3) (1 / 3) (-0.02)
= 0
This demonstrates that assets can be combined to form a riskfree portfolio.
10.20The slope of the capital market line is
( - Rf) / M = (12 - 5) / 10 = 0.7
a. = 5 + 0.7 (7) = 9.9%
b.P = ( - Rf) / 0.7 = (20 - 5) / 0.7 = 21.4%
10.21The slope of the characteristic line of Fuji is
= (12.8 - 3.4) / (16.3 - 2.5)
= 0.68
a.Beta = slope of the characteristic line = 0.68
The responsiveness to the market = 0.68
b.Slope = 0.68 = {12.8 - } / {16.3 - (-4.0)}
Thus, = 12.8 - 0.68 (16.3 + 4.0)
= -1.00%
10.22Polonius’ portfolio will be the market portfolio. He will have no borrowing or lending in his portfolio.
10.23a.= (0.10 + 0.14 + 0.20) / 3 = 0.1467 = 14.67%
b.P= (0.7 + 1.2 + 1.8) / 3 = 1.23
c.To be in equilibrium, three securities should be located on a straight line (the Security Market Line).
Check the slopes.
Slope between A & B = (0.14 - 0.10) / (1.2 - 0.7) = 0.08
Slope between B & C = (0.20 - 0.14) / (1.8 - 1.2) = 0.10
Since the slopes are different, these securities are not in equilibrium.
10.24Expected Return For Alpha = 6% + 1.28.5% =16.2%
10.25Expected return for Ross = 6% + (0.88.5%) = 12.8%
10.26Expected Return in Jordan = 8% + (1.57%) =18.5%
10.27 14.2%= 3.7%+(7.5%) = 1.4
10.280.25 = Rf + 1.4 [RM – Rf](I)
0.14 = Rf + 0.7 [RM – Rf](II)
(I) – (II)=0.11 = 0.7 [RM – Rf](III)
[RM – Rf ]= 0.1571
Put (III) into (I)0.25 = Rf + 1.4[0.1571]
Rf = 3%
[RM – Rf ]= 0.1571
RM = 0.1571 + 0.03
= 18.71%
10.29a.E(RA) = (0.25)(-0.1) + (0.5)(0.1) + (0.25)(0.2) = 0.075
E(RB) = (0.25)(-0.3) + (0.5)(0.05) + (0.25)(0.4)= 0.05
(I)E(RA) = Rf + A [E(RM) – Rf] = 0.075
(II)E(RB) = Rf + B [E(RM) – Rf] = 0.05
(I) – (II)=0.025 = (A - B)[E(RM) – Rf]
0.025 = 0.25[E(RM) – Rf],
so the market risk premium = [E(RM) – Rf]= 10%
10.30a.
b.i.See point B on the graph in part a.
ii.There does exist a mispricing of the security. According to the SML, this asset should have a return of 2% [= 7% + (-1) (5%)]. Since the return is too high, the price of this asset must be too low. (Remember, asset prices and rates of return are inversely related!) Since the asset is underpriced, you should buy it.
c.i.See point C on the graph in part a.
ii.There does exist a mispricing of the security. According to the SML, this asset should have a return of 22% [= 7% + (3) (5%)]. Since the return is too low, the price of this asset must be too high. Since the asset is overpriced, you should sell it.
10.31Expected return = 0.05 + 1.8 (0.08) = 0.194 = 19.4%
The analyst expects only 18%, so he is pessimistic.
10.32a. = 6.4 + 1.2 (13.8 - 6.4) = 15.28%
b. = 3.5 + 1.2 (13.8 - 3.5) = 15.86%
10.33Market excess return = E(RM) - Rf
= 20% - 5% = 15%
Portfolio excess return = E(RE) - Rf
= 25% - 5% = 20%
Portfolio beta = E = 20% / 15% = 4 / 3
E = {Corr(RE, RM) (RE)} / (RM)
Therefore,
4 / 3 = (1 4%) / (RM)
(RM) = 3%
Note: Corr(RE, RM) = 1 because this portfolio is a combination of the riskless asset and the market portfolio.
For the security with Corr(RS, RM) = 0.5,
S = {Corr(RS, RM) (RS)} / (RM)
= (0.5 2%) / 3%
= 0.3333
Thus, E(RS) = 5% + S (15%)
= 5% + (0.3333) (15%)
= 10%
10.34a.The risk premium = - Rf
Potpourri stock return:
16.7 = 7.6 + 1.7 ( - Rf)
- Rf = (16.7 - 7.6) / 1.7 = 5.353%
b. = 7.6 + 0.8 (5.353) = 11.88%
c.XPot Pot + XMag Mag = 1.07
1.7 XPot + 0.8 (1 - XPot) = 1.07
0.9 XPot = 0.27
XPot = 0.3
XMag = 0.7
Thus invest $3,000 in Potpourri stock and $7,000 in Magnolia.
= 7.6 + 1.07 (5.353) = 13.33%
Note: The other way to calculate is
= 0.3 (16.7) + 0.7 (11.88) = 13.33%
10.35 = Rf + Z ( - Rf)
Z= Cov(RZ, RM) / M 2 = Corr(RZ, RM) Z M / M 2
= Corr(RZ, RM) Z / M
Z = (0.0169)1/2 = 0.13
M = (0.0121)1/2 = 0.11
Thus, Z = 0.45 (0.13) / 0.11 = 0.5318
= 0.063 + 0.5318 (0.148 - 0.063)
= 0.1082
= 10.82%
10.36a. = 4.9% + i (9.4%)
b.D = Cov(RD, RM) / M 2 = 0.0635 / 0.04326 = 1.468
= 4.9 + 1.468 (9.4) = 18.70%
10.37CAPM:
Johnson19 = Rf + 1.7 ( - Rf)
Williamson14 = Rf + 1.2 ( - Rf)
5 = 0.5 ( - Rf)
Thus( - Rf) = 10%
19 = Rf + 1.7 (10)
Rf = 2%
= 12%
10.38The statement is false. If a security has a negative beta, investors would want to hold the asset to reduce the variability of their portfolios. Those assets will have expected returns that are lower than the risk free rate. To see this, examine the SML equation.
E(Ri) = Rf + i {E(RM) - Rf)}
If i < 0, E(Ri) < Rf.
10.39 Weights:
XA = 5 / 30 = 0.1667
XB = 10 / 30 = 0.3333
XC = 8 / 30 = 0.2667
XD = 1 - XA - XB - XC = 0.2333
Beta of portfolio
= 0.1667 (0.75) + 0.3333 (1.10) + 0.2667 (1.36) + 0.2333 (1.88)
= 1.293
= 4 + 1.293 (15 - 4) = 18.22%
10.40a.(i)A= A,MA / M
A,M = A M / A
= (0.9) (0.10) / 0.12
= 0.75
(ii)B= B M / B,M
= (1.10) (0.10) / 0.40
= 0.275
(iii)C= C,MC / M
= (0.75) (0.24) / 0.10
= 1.80
(iv)M,M= 1
(v)M= 1
(vi)f= 0
(vii)f,M= 0
(viii)f= 0
b.SML:
E(Ri) = Rf + i {E(RM) - Rf}
= 0.05 + (0.10) i
Security / / i / E(Ri)A / 0.13 / 0.90 / 0.14
B / 0.16 / 1.10 / 0.16
C / 0.25 / 1.80 / 0.23
Security A performed worse than the market, while security C performed better than the market. Security B is fairly priced.
c.According to the SML, security A is overpriced while security C is under-priced. Thus, you could invest in security C while sell security A (if you currently hold it).
10.41a.The typical risk-averse investor seeks high returns and low risks. To assess the two stocks, find the risk and return profiles for each stock.
Returns:
State of economy / Probability / Return on A*Recession / 0.1 / -0.20
Normal / 0.8 / 0.10
Expansion / 0.1 / 0.20
* Since security A pays no dividend, the return on A is simply (P1 / P0) - 1.
= 0.1 (-0.20) + 0.8 (0.10) + 0.1 (0.20)
= 0.08
= 0.09 This was given in the problem.
Risk:
RA - / (RA -)2 / P (RA -)2-0.28 / 0.0784 / 0.00784
0.02 / 0.0004 / 0.00032
0.12 / 0.0144 / 0.00144
Variance / 0.00960
Standard deviation (RA) = 0.0980
A = {Corr(RA, RM) (RA)} / (RM)
= 0.8 (0.0980) / 0.10
= 0.784
B = {Corr(RB, RM) (RB)} / (RM)
= 0.2 (0.12) / 0.10
= 0.24
The return on stock B is higher than the return on stock A. The risk of stock B, as measured by its beta, is lower than the risk of A. Thus, a typical risk-averse investor will prefer stock B.
b.= (0.7) + (0.3)
= (0.7) (0.8) + (0.3) (0.09)
= 0.083
P 2= 0.72 A2 + 0.32B2 + 2 (0.7) (0.3) Corr(RA , RB) A B
= (0.49) (0.0096) + (0.09) (0.0144) + (0.42) (0.6) (0.0980) (0.12)
= 0.0089635
P=
= 0.0947
c.The beta of a portfolio is the weighted average of the betas of the components of the portfolio.
P= (0.7) A + (0.3) B
= (0.7) (0.784) + (0.3) (0.240)
= 0.621
1