Chapter 10: Return and Risk: The Capital-Asset-Pricing Model (CAPM)
10.1 a.
R
b.
σ2
σ
10.2 a.
b.
RA
RB
σA2
σA
σB2
σB
c.
10.3 a.
b.
c.
= 0.1 (– 4.5%) + 0.2 (4.4%) + 0.5 (12.0%) + 0.2 (20.7%)
= 10.57%
= 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%
= (6.3 + 10.5 + 15.6) / 3 = 10.8%
= (-3.7 + 6.4 + 25.3) / 3 = 9.3%
= {(0.063 – 0.108)2 + (0.105 – 0.108)2 +{(0.156 – 0.108)2} / 3
= 0.001446
= (0.001446)1/2 = 0.0380 = 3.80%
= {(– 0.037 – 0.093)2 +(0.064 – 0.093)2 +(0.253 – 0.093)2} / 3
= 0.014447
= (0.014447)1/2 = 0.1202 = 12.02%
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
R HB = 0.25 (–2.0) + 0.60 (9.2) + 0.15 (15.4)
= 7.33%
R SB = 0.25 (5.0) + 0.60 (6.2) + 0.15 (7.4)
= 6.08%
σHB2 = 0.25 (– 0.02 – 0.0733)2 + 0.60 (0.092 – 0.0733)2
+ 0.15 (0.154 – 0.0733)2
= 0.003363
σHB = (0.003363)1/2 = 0.05799 = 5.80%
σSB2 = 0.25 (0.05 – 0.0608)2 + 0.60 (0.062 – 0.0608)2
+ 0.15 (0.074 – 0.0608)2
= 0.000056
σSB = (0.000056)1/2 = 0.00749 = 0.75%
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
10.4 Holdings of Atlas stock = 120 × $50 = $6,000
Holdings of Babcock stock = 150 × $20 = $3,000
1
Weight of Atlas stock = $6,000 / $9,000 = 2 / 3
Weight of Babcock stock = $3,000 / $9,000 = 1 / 3
10.5 a.
b.
RP
σP 2
σP
10.6 a.
b.
c.
= 0.3 (0.12) + 0.7 (0.18) = 0.162 = 16.2%
= 0.32 (0.09)2 + 0.72 (0.25)2 + 2 (0.3) (0.7) (0.09) (0.25) (0.2)
= 0.033244
= (0.033244)1/2 = 0.1823 = 18.23%
RP
σP 2
= 0.4 (0.15) + 0.6 (0.25) = 0.21 = 21%
= 0.42 (0.1)2 + 0.62 (0.2)2 + 2 (0.4) (0.6) (0.1) (0.2) (0.5)
= 0.0208
σP
= (0.0208)1/2 = 0.1442 = 14.42%
σP 2 = 0.42 (0.1)2 + 0.62 (0.2)2 + 2 (0.4) (0.6) (0.1) (0.2) (-0.5)
= 0.0112
σP
= (0.0112)1/2 = 0.1058 = 10.58%
As the stocks are more negatively correlated, the standard deviation of the portfolio
decreases.
10.7 Macrosoft: 100 × $80 = $8,000
Intelligent: 300 × $40 = $12,000
Weight:
a.
b.
10.8 a.
b.
2
Macrosoft: $8,000 / $20,000 = 0.4
Intelligent: $12,000 / $20,000 = 0.6
R P = 0.4 (0.15) + 0.6 (0.20) = 0.18 = 18%
σP 2 = 0.42 (0.08)2 + 0.62 (0.2)2 + 2 (0.4) (0.6) (0.38) (0.08) (0.20)
= 0.0183424
σP
= (0.0183424)1/2 = 0.1354 = 13.54%
New weight:
Macrosoft: $8,000 / $12,000 = 0.667
Intelligent: $4,000 / $12,000 = 0.333
R P = 0.667 (0.15) + 0.333 (0.20) = 0.1666 = 16.66%
σP 2 = 0.6672 (0.08)2 + 0.3332 (0.2)2 + 2 (0.667) (0.333) (0.38) (0.08) (0.20)
= 0.009984
σP
= (0.009984)1/2 = 0.09992 = 9.99%
RU
RV
σU2
σV2
= 7%
= 0.2 (-0.05) + 0.5 (0.10) + 0.3 (0.25) = 0.115 = 11.5%
= σU = 0
= 0.2 (-0.05 - 0.115)2 + 0.5 (0.10 - 0.115)2 + 0.3 (0.25 - 0.115)2
= 0.0110
σV
= (0.0110)1/2 = 0.105 = 10.5%
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
c.
10.9 a.
b.
c.
Corr (RU, RV) = 0
R P = 0.5 (0.115) + 0.5 (0.07) = 0.0925 = 9.25%
σP 2 = 0.52 (0.0110) = 0.00275
σP
= (0.00275)1/2 = 0.0524 = 5.24%
R P = 0.3 (0.10) + 0.7 (0.20) = 0.17 = 17.0%
σP 2 = 0.32 (0.05)2 + 0.72 (0.15)2 = 0.01125
σP
= (0.01125)1/2 = 0.10607 = 10.61%
R P = 0.9 (0.10) + 0.1 (0.20) = 0.11 = 11.0%
σP 2 = 0.92 (0.05)2 + 0.12 (0.15)2 = 0.00225
σP
= (0.00225)1/2 = 0.04743 = 4.74%
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.
10.10 The 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.
10.11 a.
b.
c.
RA
RB
σA2
= 0.4 (0.03) + 0.6 (0.15) = 0.102 = 10.2%
= 0.4 (0.065) + 0.6 (0.065) = 0.065 = 6.5%
= 0.4 (0.03 - 0.102)2 + 0.6 (0.15 - 0.102)2
= 0.003456
σA
= (0.003456)1/2 = 0.05878 = 5.88%
σB2 = σB = 0
XA = $2,500 / $6,000 = 0.417
XB = 1 - 0.417 = 0.583
R P = 0.417 (0.102) + 0.583 (0.065) = 0.0804 = 8.04%
σP 2 = XA2 σA2 = 0.0006
σP
= (0.0006)1/2 = 0.0245 = 2.45%
Amount borrowed = -40 × $50 = -$2,000
XA = $8,000 / $6,000 = 4 / 3
XB = 1 - XA = -1 / 3
R P = (4 / 3) (0.102) + (-1 / 3) (0.065) = 0.1143 = 11.43%
σP 2 = (4 / 3)2 (0.003456) = 0.006144
σP
= (0.006144)1/2 = 0.07838 = 7.84%
10.12 The 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
3
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.
10.13 a.
R1
R2
R3
= 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
0.25
0.20
0.15
0.10
R1 - R 1
0.075
0.025
-0.025
-0.075
(R1 - R 1 )2
0.005625
0.000625
0.000625
0.005625
Variance
P × (R1 - R 1 )2
0.0005625
0.0002500
0.0002500
0.0005625
0.0016250
(R2 - R 2 )2
0.005625
0.000625
0.000625
0.005625
Variance
P × (R2 - R 2 )2
0.0005625
0.0002500
0.0002500
0.0005625
0.0016250
(R3 - R 3 )2
0.005625
0.000625
0.000625
0.005625
Variance
P × (R3 - R 3 )2
0.0005625
0.0002500
0.0002500
0.0005625
0.0016250
Standard deviation = 0.001625
= 0.0403
R2 if
State occurs
0.25
0.15
0.20
0.10
R2 - R 2
0.075
-0.025
0.025
-0.075
Standard deviation = 0.001625
= 0.0403
R3 if
State occurs
0.10
0.15
0.20
0.25
R3 - R 3
-0.075
-0.025
0.025
0.075
Standard deviation = 0.001625
= 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
4
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
σ = 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
σ=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
σ = 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.
10.14 a.
b.
RP
10.15 a.
RP
σP 2
b.
c.
State
Return on A
Return on B
Probability
1
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
= 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%
= ∑ R i / N = {N (0.10)} / N = 0.10 = 10%
= ∑∑ Cov (Ri, Rj) / N2 + ∑ σi2 / N2
= N (N - 1) (0.0064) / N2 + N (0.0144) / N2
= (0.0064) (N - 1) / N + (0.0144) / N
As N → ∞, σP 2 → 0.0064 = Cov (Ri, Rj)
The covariance of the returns of the securities is the most important factor to consider
when placing securities in a well-diversified portfolio.
5
10.16 The 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.
10.17 The 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.
10.18 If 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.
10.19 Note: 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 σA2 + XB2 σB2 + 2 XA XB Cov(RA , RB)}
subject to XA + XB = 1
Let XA = 1 - XB. Then,
Min {(1 - XB)2 σA2 + XB2 σB2+ 2(1 - XB) XB Cov (RA, RB)}
Take a derivative with respect to XB.
d{•} / dXB = (2 XB - 2) σA2 + 2 XB σB2 + 2 Cov(RA, RB) - 4 XB Cov(RA, RB)
Set the derivative equal to zero, cancel the common 2 and solve for XB.
XB σA2 - σA2 + XB σB2 + Cov(RA, RB) - 2 XB Cov(RA, RB) = 0
XB = {σA2 - Cov(RA, RB)} / {σA2 + σB2 - 2 Cov(RA, RB)}
and
XA = {σB2 - Cov(RA, RB)} / {σA2 + σB2 - 2 Cov(RA, RB)}
Using the data from the problem yields,
XA = 0.8125 and
XB = 0.1875.
a.
b.
6
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%
Using the formula derived above, the weights are
XA = 2 / 3 and
XB = 1 / 3
c.
The variance of this portfolio is zero.
σP 2 = XA2 σA2 + XB2 σB2 + 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.
10.20 The slope of the capital market line is
( R M - Rf) / σM = (12 - 5) / 10 = 0.7
a.
R P = 5 + 0.7 (7) = 9.9%
b.
σP = ( R P - Rf) / 0.7 = (20 - 5) / 0.7 = 21.4%
10.21 The slope of the characteristic line of Fuji is
R Fuji (Bull ) − R Fuji (Bear)
= (12.8 - 3.4) / (16.3 - 2.5)
R M (Bull ) − R M (Bear )
a.
b.
= 0.68
Beta = slope of the characteristic line = 0.68
The responsiveness to the market = 0.68
Slope = 0.68 = {12.8 - R Fuji ( Bear ) } / {16.3 - (-4.0)}
Thus, R Fuji ( Bear ) = 12.8 - 0.68 (16.3 + 4.0)
= -1.00%
10.22 Polonius’ portfolio will be the market portfolio. He will have no borrowing or lending in his
portfolio.
10.23 a.
b.
c.
R P = (0.10 + 0.14 + 0.20) / 3 = 0.1467 = 14.67%
βP
= (0.7 + 1.2 + 1.8) / 3 = 1.23
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.
10.24 Expected Return For Alpha = 6% + 1.2×8.5% =16.2%
10.25 Expected return for Ross = 6% + (0.8×8.5%) = 12.8%
10.26 Expected Return in Jordan = 8% + (1.5×7%) =18.5%
10.27 14.2%= 3.7%+β(7.5%)
10.28
⇒ β = 1.4
0.25 = Rf + 1.4 [RM – Rf]
0.14 = Rf + 0.7 [RM – Rf]
(I)
(II)
(I) – (II)=0.11 = 0.7 [RM – Rf] (III)
[RM – Rf ]= 0.1571
7
Put (III) into (I)
0.25 = Rf + 1.4[0.1571]
Rf = 3%
[RM – Rf ]= 0.1571
RM = 0.1571 + 0.03
= 18.71%
10.29 a.
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
b.
(I)
E(RA) = Rf + βA [E(RM) – Rf] = 0.075
(II)
E(RB) = Rf + βB [E(RM) – Rf] = 0.05
(I) – (II)=0.025 = (βA - βB)[E(RM) – Rf]
0.025 = 0.25[E(RM) – Rf],
so the market risk premium = [E(RM) – Rf ]= 10%
10.30 a.
E(R)
E(Ri) = 7% + βi (5%)
22%
20%
C
12%
7%
B
4%
2%
βi
-1.4
b.
i.
ii.
c.
i.
ii.
-1
0
1
See point B on the graph in part a.
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.
See point C on the graph in part a.
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.
10.31 Expected return = 0.05 + 1.8 (0.08) = 0.194 = 19.4%
The analyst expects only 18%, so he is pessimistic.
8
3
10.32 a.
b.
R = 6.4 + 1.2 (13.8 - 6.4) = 15.28%
R = 3.5 + 1.2 (13.8 - 3.5) = 15.86%
10.33 Market excess return = E(RM) - Rf
= 20% - 5% = 15%
Portfolio excess return = E(RE) - Rf
= 25% - 5% = 20%
Portfolio beta = βE = 20% / 15% = 4 / 3
βE = {Corr(RE, RM) σ(RE)} / σ(RM)
Therefore,
4 / 3 = (1 × 4%) / σ(RM)
σ(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,
βS = {Corr(RS, RM) σ(RS)} / σ(RM)
= (0.5 × 2%) / 3%
= 0.3333
Thus, E(RS) = 5% + βS (15%)
= 5% + (0.3333) (15%)
= 10%
10.34 a.
b.
c.
The risk premium = R M - Rf
Potpourri stock return:
16.7 = 7.6 + 1.7 ( R M - Rf)
R M - Rf = (16.7 - 7.6) / 1.7 = 5.353%
R Mag = 7.6 + 0.8 (5.353) = 11.88%
XPot βPot + XMag βMag = 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.
R P = 7.6 + 1.07 (5.353) = 13.33%
Note: The other way to calculate R P is
R P = 0.3 (16.7) + 0.7 (11.88) = 13.33%
10.35
R Z = Rf + βZ ( R M - Rf)
βZ = Cov(RZ, RM) / σM 2 = Corr(RZ, RM) σZ σM / σM 2
= Corr(RZ, RM) σZ / σM
σZ = (0.0169)1/2 = 0.13
σM = (0.0121)1/2 = 0.11
Thus, βZ = 0.45 (0.13) / 0.11 = 0.5318
9
RZ
10.36 a.
b.
= 0.063 + 0.5318 (0.148 - 0.063)
= 0.1082
= 10.82%
R i = 4.9% + βi (9.4%)
βD = Cov(RD, RM) / σM 2 = 0.0635 / 0.04326 = 1.468
R D = 4.9 + 1.468 (9.4) = 18.70%
10.37 CAPM:
Johnson
19 = Rf + 1.7 ( R M - Rf)
Williamson 14 = Rf + 1.2 ( R M - Rf)
5 = 0.5 ( R M - Rf)
Thus ( R M - Rf) = 10%
19 = Rf + 1.7 (10)
Rf = 2%
R M = 12%
10.38 The 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 + βi {E(RM) - Rf)}
If βi < 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
R P = 4 + 1.293 (15 - 4) = 18.22%
10.40 a.
(i)
(ii)
(iii)
(iv)
(v)
(vi)
10
βA
= ρA,M σA / σM
ρA,M = βA σM / σA
= (0.9) (0.10) / 0.12
= 0.75
σB = βB σM / ρB,M
= (1.10) (0.10) / 0.40
= 0.275
βC
= ρC,M σC / σM
= (0.75) (0.24) / 0.10
= 1.80
ρM,M = 1
βM = 1
σf
=0
(vii) ρf,M
(viii) βf
b.
=0
=0
SML:
E(Ri) = Rf + βi {E(RM) - Rf}
= 0.05 + (0.10) βi
Security
A
B
C
βi
0.90
1.10
1.80
Ri
0.13
0.16
0.25
E(Ri)
0.14
0.16
0.23
Security A performed worse than the market, while security C performed better than
the market. Security B is fairly priced.
c.
10.41 a.
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).
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
Recession
Normal
Expansion
Probability
0.1
0.8
0.1
Return on A*
-0.20
0.10
0.20
* Since security A pays no dividend, the return on A is simply (P1 / P0) - 1.
R A = 0.1 (-0.20) + 0.8 (0.10) + 0.1 (0.20)
= 0.08
R B = 0.09 This was given in the problem.
Risk:
RA - R A
-0.28
0.02
0.12
(RA - R A )2
0.0784
0.0004
0.0144
Variance
P × (RA - R A )2
0.00784
0.00032
0.00144
0.00960
Standard deviation (RA) = 0.0980
βA = {Corr(RA, RM) σ(RA)} / σ(RM)
= 0.8 (0.0980) / 0.10
= 0.784
11
βB = {Corr(RB, RM) σ(RB)} / σ(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.
RP
σP 2
σP
c.
12
= (0.7) R A + (0.3) R B
= (0.7) (0.8) + (0.3) (0.09)
= 0.083
= 0.72 σA2 + 0.32 σB2 + 2 (0.7) (0.3) Corr(RA , RB) σA σB
= (0.49) (0.0096) + (0.09) (0.0144) + (0.42) (0.6) (0.0980) (0.12)
= 0.0089635
= 0.0089635
= 0.0947
The beta of a portfolio is the weighted average of the betas of the components of the
portfolio.
βP
= (0.7) βA + (0.3) βB
= (0.7) (0.784) + (0.3) (0.240)
= 0.621