Capital Asset Pricing Model Homework Problems
Portfolio weights and expected return
1. Consider a portfolio of 300 shares of firm A worth $10/share and 50 shares of firm B
worth $40/share. You expect a return of 8% for stock A and a return of 13% for stock B.
(a) What is the total value of the portfolio, what are the portfolio weights and what is
the expected return?
(b) Suppose firm A’s share price goes up to $12 and firm B’s share price falls to $36.
What is the new value of the portfolio? What return did it earn? After the price
change, what are the new portfolio weights?
2. Consider a portfolio of 250 shares of firm A worth $30/share and 1500 shares of firm B
worth $20/share. You expect a return of 4% for stock A and a return of 9% for stock B.
(a) What is the total value of the portfolio, what are the portfolio weights and what is
the expected return?
(b) Suppose firm A’s share price falls to $24 and firm B’s share price goes up to $22.
What is the new value of the portfolio? What return did it earn? After the price
change, what are the new portfolio weights?
Portfolio volatility
3. For the following problem please refer to Table 1 (Table 11.3, p. 336 in Corporate Finance
by Berk and DeMarzo).
(a) What is the covariance between the returns for Alaskan Air and General Mills?
(b) What is the volatility of a portfolio with
i. equal amounts invested in these two stocks?
ii. 20% invested in Alaskan Air and 80% invested in General Mills?
iii. 80% invested in Alaskan Air and 20% invested in General Mills?
4. Suppose the historical volatility (standard deviation) of the return of a mid-cap stock is
50% and the correlation between the returns of mid-cap stocks is 30%.
(a) What is the average variance AvgV ar of a mid-cap stock?
(b) What is the average covariance AvgCov of a mid-cap stock?
(c) Consider a portfolio of n mid-cap stocks. What is an estimate of the volatility of
such a portfolio when n = 10? n = 20? n = 40? What is the limiting volatility?
Table 1: Historical Annual Volatilities and Correlations for Selected Stocks (based
on monthly returns, 1996-2008).
Volatility (StDev)
Correlation with:
Microsoft
Dell
Alaska Air
Southwest Airlines
Ford Motor
General Motors
General Mills
Microsoft
37%
Dell
50%
Alaskan
Air
38%
Southwest
Airlines
31%
Ford
Motor
42%
General
Motors
41%
General
Mills
18%
1.00
0.62
0.25
0.23
0.26
0.23
0.10
0.62
1.00
0.19
0.21
0.31
0.28
0.07
0.25
0.19
1.00
0.30
0.16
0.13
0.11
0.23
0.21
0.30
1.00
0.25
0.22
0.20
0.26
0.31
0.16
0.25
1.00
0.62
0.07
0.23
0.28
0.13
0.22
0.62
1.00
0.02
0.10
0.07
0.11
0.20
0.07
0.02
1.00
5. Consider a portfolio of two stocks. Data shown in Table 2. Let x denote the weight on
Stock A and 1 − x denote the weight on Stock B. Correlation coefficient equals ρAB .
(a) Write down a mathematical expression for the portfolio’s mean return and volatility
(standard deviation) as a function of x.
(b) What is the portfolio’s mean return and volatility when x = 0.4 if ρAB = 0? ρAB =
+1? ρAB = −1?
(c) Suppose ρAB = −1? Are there portfolio weights that will result in a portfolio with
no volatility? If so, what are the weights?
Table 2:
Stock
Stock A
Stock B
Expected Return
15%
7%
Volatility
40%
30%
Minimum variance portfolio
6. Consider the data shown in Table 2. The risk-free rate is rf = 3%.
(a) What is the minimum variance portfolio when ρAB = 0? What is its expected return
and volatility?
(b) What is the minimum variance portfolio when ρAB = 0.4? What is its expected
return and volatility?
(c) What is the minimum variance portfolio when ρAB = −0.4? What is its expected
return and volatility?
7. Consider two stocks, A and B, such that σA = 0.30, σB = 0.80, R̄A = 0.10, R̄B = 0.06
and rf = 0.02.
(a) What is the minimum variance portfolio when ρAB = 0 and what is its volatility?
(b) What is the minimum variance portfolio when ρAB = 0.6 and what is its volatility?
(c) What is the minimum variance portfolio when ρAB = −0.6 and what is its volatility?
8. Consider three risky assets whose covariance matrix Σ is


0.09 0.045 0.01


Σ =  0.045 0.25 0.06 ,
0.01 0.06 0.04
(1)
and whose expected returns are R̄1 = 0.08, R̄2 = 0.10, R̄3 = 0.16. The risk-free rate is
rf = 0.03. The inverse of the covariance matrix is


Σ−1
12.2137 −2.2901
0.3817


6.6794 −9.4466 .
=  −2.2901
0.3817 −9.4466 39.0744
(2)
What is the minimum variance portfolio and what is its volatility?
9. Consider three risky assets whose covariance matrix Σ is


2 1 0


Σ =  1 2 1 .
0 1 2
(3)
The expected returns are R̄1 = 0.11, R̄2 = 0.09, R̄3 = 0.05. The risk-free rate is rf = 0.02.
Solve for the minimum variance portfolio using the first-order optimality conditions, i.e.,
without computing the inverse of the covariance matrix. What is the minimum variance?
(Suggestion: By symmetry x∗1 = x∗3 . )
Tangent portfolio
10. For the data of problem 6 determine the tangent portfolios and their respective mean
returns and volatilities.
11. For the data of problem 7 determine the tangent portfolios and their respective mean
returns and volatilities.
12. For the data of problem 8 determine the tangent portfolio and its mean return and
volatility.
13. For the data of problem 9 determine the tangent portfolio and its mean return and
volatility.
14. Suppose the expected return on the tangent portfolio is 10% and its volatility is 40%.
The risk-free rate is 2%.
(a) What is the equation of the Capital Market Line (CML)?
(b) What is the standard deviation of an efficient portfolio whose expected return of
8%? How would you allocate $1,000 to achieve this position?
15. Suppose the expected return on the tangent portfolio is 12% and its volatility is 30%.
The risk-free rate is 3%.
(a) What is the equation of the Capital Market Line (CML)?
(b) What is the standard deviation of an efficient portfolio whose expected return of
16.5%? How would you allocate $3,000 to achieve this position?
Security market line
16. Suppose the market premium is 9%, market volatility is 30% and the risk-free rate is 3%.
(a) What is the equation of the SML?
(b) Suppose a security has a beta of 0.6. According to the CAPM, what is its expected
return?
(c) A security has a volatility of 60% and a correlation with the market portfolio of 25%.
According to the CAPM, what is its expected return?
(d) A security has a volatility of 80% and a correlation with the market portfolio of
-25%. According to the CAPM, what is its expected return?
17. Stock A has a beta of 1.20 and Stock B has a beta of 0.8. Suppose rf = 2% and R̄M = 12%.
(a) According to the CAPM, what are the expected returns for each stock?
(b) What is the expected return of an equally weighted portfolio of these two stocks?
(c) What is the beta of an equally weighted portfolio of these two stocks?
(d) How can you use your answer to part (c) to answer part (b)?
18. Suppose you estimate that stock A has a volatility of 32% and a beta of 1.42, whereas
stock B has a volatility of 68% and a beta of 0.75.
(a) Which stock has more total risk?
(b) Which stock has more market risk?
(c) Suppose the risk-free rate is 2% and you estimate the market’s expected return as
10%. Which firm has a higher cost of equity capital?
19. Consider a world with only two risky assets, A and B, and a risk-free asset. The two risky
assets are in equal supply in the market, i.e., the market portfolio M = 0.5A + 0.5B. It
is known that R̄M = 11%, σA = 20%, σB = 40% and ρAB = 0.75. The risk-free rate is
2%. Assume CAPM holds.
(a) What is the beta for each stock?
(b) What are the values for R̄A and R̄B ?
20. Consider a world with only two risky assets, A and B, and a risk-free asset. Stock A
has 200 shares outstanding, a price per share of $3.00, an expected return of 16% and
a volatility of 30%. Stock B has 300 shares outstanding, a price per share of $4.00, an
expected return of 10% and a volatility of 15%. The correlation coefficient ρAB = 0.4.
Assume CAPM holds.
(a) What is expected return of the market portfolio?
(b) What is volatility of the market portfolio?
(c) What is the beta of each stock?
(d) What is the risk-free rate?
21. Suppose you group all stocks into two mutually exclusive portfolios of growth or value
stocks. Suppose the growth stock portfolio and value stock portfolio have equal size in
terms of total value. Furthermore, suppose that the expected return of the value stocks
is 13% with a volatility of 12%, whereas the expected return of the growth stocks is 17%
with a volatility of 25%. The correlation of the returns of these two portfolios is 0.50.
The risk-free rate is 2%.
(a) What is the expected return and volatility of the market portfolio (which is a 50-50
combination of the two portfolios)?
(b) Does CAPM hold in this economy?
Improving the Sharpe ratio
22. Suppose portfolio P ’s expected return is 14%, its volatility is 30% and the risk-free rate
is 2%. Suppose further that a particular mix of asset i and P yields a portfolio P 0 with
an expected return of 22% and a volatility of 40%. Will adding asset i to portfolio P be
beneficial? Explain how.
23. Assume the risk-free rate is 4%. You are a financial advisor and your client has decided to
invest in exactly one of two risky funds, A and B. She comes to you for advice. Whichever
fund you recommend she will combine it with the risk-free asset. Expected returns are
R̄A = 13% and R̄B = 18%. Volatilities are σA = 20% and σB = 30%. Without knowing
your client’s tolerance for risk, which fund would you recommend?
24. You are currently invested in Fund F. It has an expected return of 14% with a volatility of
20%. The risk-free rate is 3.8%. Your broker suggests you add Stock B to your portfolio
with a positive weight. Stock B has an expected return of 20%, a volatility of 60% and a
correlation of 0 with Fund F.
(a) Is your broker right?
(b) You follow your broker’s advice and make a substantial investment in Stock B so that
now 60% is in Fund F and 40% is in Stock B. You tell your finance professor about
your investment and he says you made a mistake and should reduce your investment
in Stock B. Is your finance professor right?
(c) You decide to follow your finance professor’s advice and reduce your exposure to
Stock B. Now Stock B represents only 15% of your risky portfolio with the rest
invested in Fund F. Is the correct amount to hold of Stock B?
25. In addition to risk-free securities, you are currently invested in the Jones Fund, a broadbased fund with an expected return of 12% and a volatility of 25%. The risk-free rate is
4%. Your broker suggests you add a venture capital (VC) fund to your current portfolio.
The VC fund has an expected return of 20%, a volatility of 80% and a correlation of 0.2
with the Jones Fund.
(a) Is your broker right?
(b) Suppose you follow your broker’s advice and put 50% of your money in the VC fund.
(You sell 50% of your value of the Jones Fund.) What is the Sharpe ratio of your
new portfolio?
(c) What is the optimal fraction of your wealth to invest in the VC fund?
Capital Asset Pricing Model Homework Solutions
1. Portfolio value = 300($10) + 50($40) = $5,000. Portfolio weights are
xA = 300($10)/$5, 000 = 60% and xB = 50($40)/$5, 000 = 40%.
Expected return = 0.60(8%) + 0.40(13%) = 10%.
New portfolio value = 300($12) + 50($36) = $5,400.
Return is ($5,400 - $5,000)/$5,000 = 8% or, equivalently,
0.60[($12 - $10)/$10] + 0.40[($36 - $40)/$40] = 0.60(20%) + 0.40(-10%) = 8%.
New portfolio weights are
xA = 300($12)/$5, 400 = 66.6̄% and xB = 50($36)/$5, 400 = 33.3̄%.
2. Portfolio value = 250($30) + 1500($20) = $37,500. Portfolio weights are
xA = 250($30)/$37, 500 = 20% and xB = 1500($20)/$37, 500 = 80%.
Expected return = 0.20(4%) + 0.80(9%) = 8%.
New portfolio value = 250($24) + 1500($22) = $39,000.
Return is ($39,000 - $37,500)/$37,500 = 4% or, equivalently,
0.20[($24 - $30)/$30] + 0.80[($22 - $20)/$20] = 0.20(-20%) + 0.80(10%) = 4%.
New portfolio weights are
xA = 250($24)/$39, 000 = 15.38% and xB = 1500($22)/$39, 000 = 84.62%.
3. Cov = (0.11)(0.38)(0.18)
= 0.007524.
p
50:50: StDev = p(0.5)2 (0.38)2 + (0.5)2 (0.18)2 + 2(0.5)(0.5)(0.007524) = 21.90%.
20:80: StDev = p(0.2)2 (0.38)2 + (0.8)2 (0.18)2 + 2(0.2)(0.8)(0.007524) = 17.01%.
80:20: StDev = (0.8)2 (0.38)2 + (0.2)2 (0.18)2 + 2(0.8)(0.2)(0.007524) = 31.00%.
4. AvgV p
ar = 0.502 = 0.25. AvgCov = (0.50)(0.50)(0.30)
= 0.075.
p
σn = AvgV ar/n + (1 − 1/n) ∗ AvgCov = 0.25/n + (1 − 1/n)(0.075).
σ10 = 30.41%. σ20 = 28.94%. σ40 = 28.17%. σ∞ = 27.39%.
5. E[R(x)] = 0.15x
p + 0.07(1 − x) = 0.07 + 0.08x.
StDev(x)
=
(0.40)2 x2 + (0.30)2 (1 − x)2 + 2(0.40)(0.30)ρAB x(1 − x)
p
= (0.25 − 0.24ρAB )x2 + (0.24ρAB − 0.18)x + 0.09.
When ρAB = 0, σ = 24.08%.
When ρAB = +1, σ = 0.4(0.4) + 0.6(0.3) = 34%.
When ρAB = −1, σ = | 0.4(0.4) − 0.6(0.3) | = 2%,
and the portfolio xA = σB /(σA + σB ) = 3/7 and xB = 4/7 will have no volatility.
6. In general, xA =
2 −σ
σB
AB
2
2 −2σ
σA +σB
AB
.
ρAB = 0 =⇒ σAB = 0 and xA = 0.09/(0.16
+ 0.09) = 36%. Expected return =
p
0.36(15%) + 0.64(7%) = 9.88% and σ = (0.36)2 (0.40)2 + (0.64)2 (0.30)2 = 24%.
ρAB = 0.4 =⇒ σAB = (0.4)(0.3)(0.4) = 0.048 and xA = (0.09 − 0.048)/[(0.09 +
0.16) − 2(0.048)]
= 27.27%. Expected return = 0.2727(15%) + 0.7273(7%) = 9.18%
p
and σ = (0.2727)2 (0.40)2 + (0.7273)2 (0.30)2 + 2(0.048)(0.2727)(0.7273) = 28.03%.
ρAB = −0.4 =⇒ σAB = (0.4)(0.3)(−0.4) = −0.048 and xA = (0.09 + 0.048)/[(0.09 +
0.16)p+ 2(0.048)] = 39.88%. Expected return = 0.3988(15%) + 0.6012(7%) = 10.19% and
σ = (0.3988)2 (0.40)2 + (0.6012)2 (0.30)2 + 2(−0.048)(0.3988)(0.6012) = 18.70%.
σ 2 −σ
AB
B
7. In general, xA = σ2 +σ
.
2
A
B −2σAB
ρAB = 0 =⇒ σAB = 0 and
xA =p0.64/(0.09 + 0.64) = 87.67%.
σ = (0.8767)2 (0.30)2 + (0.1233)2 (0.80)2 = 28.09%.
ρAB = 0.6 =⇒ σAB = (0.3)(0.8)(0.6) = 0.144 and
xA =p(0.64 − 0.144)/[(0.09 + 0.64) − 2(0.144)] = 112.22%.
σ = (1.1222)2 (0.30)2 + (−0.1222)2 (0.80)2 + 2(0.144)(1.1222)(−0.1222) = 28.88%.
ρAB = −0.6 =⇒ σAB = (0.3)(0.8)(−0.6) = −0.144 and
xA =p(0.64 + 0.144)/[(0.09 + 0.64) + 2(0.144)] = 77.01%.
σ = (0.7701)2 (0.30)2 + (0.2299)2 (0.80)2 + 2(−0.144)(0.7701)(0.2299) = 19.03%.
8. Minimum variance portfolio is proportional to Σ−1 e = (10.3053, −5.0573, 30.0095).
Since eT Σe = 10.3053 − 5.0573 + 30.0095 = 35.2575,
minimum variance portfolio = (0.2923, -0.1434, 0.8511),
minimum variance = 1/(eT Σ−1p
e) = 1/35.2575
and the minimum volatility = 1/35.2575) = 16.84%.
9. The first-order optimality conditions are Σx = λe and x1 + x2 + x3 = 1. In equation form,
we have: (i) 2x1 + x2 = λ, (ii) x1 + 2x2 + x3 = λ and (iii) x2 + 2x3 = λ. Equations (i)
and (iii) imply that x1 = x3 . Using this fact, equations (ii) and (iii) imply that x2 = 0.
Since the sum of the xi = 1, it follows that x1 = x3 = 0.5, x2 = 0 and λ = 1. Minimum
variance = λ = 1.
10. R̂ = (0.15 − 0.03, 0.07 − 0.03) = (0.12, 0.04).
When ρAB = 0:
Σ =
Σ
−1
=
0.16 0.00
0.00 0.09
1
(0.16)(0.09)
!
,
0.09 0.00
0.00 0.16
!
=
6.25 0.00
0.00 11.1̄
!
.
x∗ ∝ Σ−1 R̂ = (0.75, 0.44̄),
eT Σ−1 R̂ = 0.75 + 0.44̄ = 1.194̄,
0.75 0.44̄
x∗ =
,
= (0.628, 0.372)
1.194̄ 1.194̄
E[RP ] = 0.628(0.15) + 0.372(0.07) = 0.1202,
R̂0 = E[RP ] − rf = 0.0902,
V ar(RP ) =
R̂0
eT Σ−1 R̂
=
√
0.0902
= 0.0755 =⇒ σP = 0.0755 = 27.48%.
1.194̄
When ρAB = 0.4:
0.16 0.048
0.048 0.09
Σ =
Σ
−1
!
,
0.09 −0.048
−0.048
0.16
1
(0.16)(0.09) − (0.048)2
=
!
=
7.4405 −3.9683
−3.9683 13.2275
x∗ ∝ Σ−1 R̂ = (0.7341, 0.0529),
eT Σ−1 R̂ = 0.7341 + 0.0529 = 0.7870,
0.7341 0.0529
∗
x =
= (0.9328, 0.0672)
,
0.7870 0.7870
E[RP ] = 0.9328(0.15) + 0.0672(0.07) = 0.1446,
R̂0 = E[RP ] − rf = 0.1146,
R̂0
V ar(RP ) =
eT Σ−1 R̂
=
√
0.1146
= 0.1456 =⇒ σP = 0.1456 = 38.16%.
0.7870
When ρAB = −0.4:
0.16 −0.048
−0.048
0.09
Σ =
−1
Σ
=
!
,
1
(0.16)(0.09) − (0.048)2
0.09 0.048
0.048 0.16
!
=
7.4405 3.9683
3.9683 13.2275
x∗ ∝ Σ−1 R̂ = (1.0516, 1.0053),
eT Σ−1 R̂ = 1.0516 + 1.0053 = 2.0569,
1.0516 1.0053
∗
x =
,
= (0.5113, 0.4887)
2.0569 2.0569
E[RP ] = 0.5113(0.15) + 0.4887(0.07) = 0.1109,
R̂0 = E[RP ] − rf = 0.0809,
V ar(RP ) =
R̂0
eT Σ−1 R̂
=
√
0.0809
= 0.0393 =⇒ σP = 0.0393 = 19.83%.
2.0569
11. R̂ = (0.10 − 0.02, 0.06 − 0.02) = (0.08, 0.04).
When ρAB = 0:
Σ =
Σ
−1
=
0.09 0.00
0.00 0.64
1
(0.09)(0.64)
!
,
0.64 0.00
0.00 0.09
x∗ ∝ Σ−1 R̂ = (0.88̄, 0.0625),
eT Σ−1 R̂ = 0.88̄ + 0.0625 = 0.9514,
!
=
11.1̄
0.00
0.00 1.5625
!
.
!
.
!
.
0.88̄ 0.0625
= (0.9343, 0.0657)
,
0.9514 0.9514
E[RP ] = 0.9343(0.10) + 0.0657(0.06) = 0.0974,
x∗ =
R̂0 = E[RP ] − rf = 0.0774,
V ar(RP ) =
R̂0
eT Σ−1 R̂
=
√
0.0774
= 0.0814 =⇒ σP = 0.0814 = 28.53%.
0.9514
When ρAB = 0.6:
0.09 0.144
0.144 0.64
Σ =
Σ
−1
=
!
,
0.64 −0.144
−0.144
0.09
1
(0.09)(0.64) − (0.144)2
!
=
17.3611 −3.9063
−3.9063
2.4414
x∗ ∝ Σ−1 R̂ = (1.2326, −0.2148),
eT Σ−1 R̂ = 1.2326 − 0.2148 = 1.0178,
1.2326 −0.2148
∗
x =
,
= (1.211, −0.211)
1.0178 1.0178
E[RP ] = 1.211(0.10) + −0.211(0.06) = 0.1084,
R̂0 = E[RP ] − rf = 0.0884,
V ar(RP ) =
R̂0
eT Σ−1 R̂
=
√
0.0884
= 0.0869 =⇒ σP = 0.0366 = 29.48%.
1.0178
When ρAB = −0.6:
Σ =
−1
Σ
=
0.09 −0.144
−0.144
0.64
!
,
1
(0.09)(0.64) − (0.144)2
0.64 0.144
0.144 0.09
!
=
17.3611 3.9063
3.9063 2.4414
x∗ ∝ Σ−1 R̂ = (1.5451, 0.4102),
eT Σ−1 R̂ = 1.5451 + 0.4102 = 1.9553,
1.5451 0.4102
∗
x =
,
= (0.7902, 0.2098)
1.9553 1.9553
E[RP ] = 0.7902(0.10) + 0.2098(0.06) = 0.0916,
R̂0 = E[RP ] − rf = 0.0716,
V ar(RP ) =
R̂0
eT Σ−1 R̂
=
√
0.0716
= 0.0366 =⇒ σP = 0.0366 = 19.14%.
1.9553
12. R̂ = (0.08 − 0.03, 0.10 − 0.03, 0.16 − 0.03) = (0.05, 0.07, 0.13).
x∗ ∝ Σ−1 R̂ = (0.50, −0.875, 4.4375),
!
.
!
.
eT Σ−1 R̂ = 0.50 + −0.875 + 4.4375 = 4.0625,
0.50 −0.875 4.4375
∗
x =
= (0.1231, −0.2154, 1.0923)
,
,
4.0625 4.0625 4.0625
E[RP ] = 0.1231(0.08) − 0.2154(0.10) + 1.0923(0.16) = 0.1631,
R̂0 = E[RP ] − rf = 0.1331,
R̂0
V ar(RP ) =
eT Σ−1 R̂
=
√
0.1331
= 0.0328 =⇒ σP = 0.0328 = 18.11%.
4.0625
13. R̂ = (0.11 − 0.02, 0.09 − 0.02, 0.05 − 0.02) = (0.09, 0.07, 0.03).


Σ−1
0.75 −0.50
0.25


1.00 −0.50 
=  −0.50
0.25 −0.50
0.75
x∗ ∝ Σ−1 R̂ = (0.04, 0.01, 0.01),
eT Σ−1 R̂ = 0.04 + 0.01 + 0.01 = 0.06,
0.04 0.01 0.01
x∗ =
= (2/3, 1/6, 1/6)
,
,
0.06 0.06 0.06
E[RP ] = (2/3)(0.11) + (1/6)(0.09) + (1/6)(0.05) = 0.096̄,
R̂0 = E[RP ] − rf = 0.076̄,
R̂0
V ar(RP ) =
eT Σ−1 R̂
=
√
0.076̄
= 1.27̄ =⇒ σP = 1.27̄ = 113.04%.
0.06
14. R̄P = 0.02 + 0.10−0.02
σP = 0.02 + 0.20σP . For an efficient portfolio whose expected
0.40
return is 8%, we have 0.08 = 0.02 + 0.20σP =⇒ σP = 30%. Allocate $750 to the tangent
portfolio and $250 to the risk-free asset.
15. R̄P = 0.03 + 0.12−0.03
σP = 0.03 + 0.30σP . For an efficient portfolio whose expected
0.30
return is 16%, we have 0.165 = 0.03 + 0.30σP =⇒ σP = 45%. Allocate $4,500 to the
tangent portfolio and borrow $1,500 at the risk-free asset.
16. R̄i = 0.03 + 0.09βi .
R̄i = 0.03 + 0.09(0.6) = 8.4%.
βi = 0.25(0.6)/(0.3) = 0.5 =⇒ R̄i = 0.03 + 0.09(0.5) = 7.5%.
βi = −0.25(0.8)/(0.3) = −2/3 =⇒ R̄i = 0.03 + 0.09(−2/3) = −3%.
17. R̄A = 0.02 + 1.20(0.10) = 14%. R̄B = 0.02 + 0.80(0.10) = 10%.
R̄P = 0.5(14%) + 0.5(10%) = 12%.
βP = 0.5(1.20) + 0.5(0.80) = 1.
R̄P = 0.02 + 1(0.10) = 12%.
18. Stock B has more total risk. Stock A has more market risk.
R̄A = 0.02 + 1.42(0.08) = 13.36%. R̄B = 0.02 + 0.75(0.08) = 8%. Firm A has the higher
cost of equity capital.
19.
Cov(RA , RB ) = (0.2)(0.4)(0.75) = 0.06
Cov(RA , RM ) = Cov(RA , 0.5RA + 0.5RB )
2
= 0.5σA
+ 0.5σAB = 0.5(0.2)2 + 0.5(0.06) = 0.05
V ar(RM ) = (0.5)2 (0.2)2 + (0.5)2 (0.4)2 + 2(0.5)(0.5)(0.06) = 0.08
βA = 0.05/0.08 = 0.625 =⇒ R̄A = 0.02 + 0.625(0.11 − 0.02) = 7.625%.
The market’s beta of 1 equals 0.5βA + 0.5βB . Since βA = 0.625, this implies that βB =
1.375. (You can verify this quantity via the formula for βB .) The market’s expected return
of 11% must equal 0.5R̄A + 0.5R̄B . Since R̄A = 7.625%, this implies that R̄B = 14.375%.
(You can verify this quantity via the SML.)
20. Value of the market portfolio = 200($3) + 300($4) = $1,800.
Portfolio weights are xA = 1/3 and xB = 2/3.
R̄M
= 1/3(16%) + 2/3(10%) = 12%
σAB = σA σB ρAB = (0.3)(.15)(0.4) = 0.018
Cov(RA , RM ) = Cov(RA , 1/3RA + 2/3RB )
2
= 1/3(σA
) + 2/3σAB = 1/3(0.3)2 + 2/3(0.018) = 0.042
V ar(RM ) = (1/3)2 (0.3)2 + (2/3)2 (0.15)2 + 2(1/3)(2/3)(0.018) = 0.028
√
StDev(RM ) =
0.028 = 16.73%
βA = 0.042/0.028 = 1.5 =⇒ 16 = rf + 1.5(12 − rf ) =⇒ rf = 4%.
The market’s beta of 1 equals 1/3βA +2/3βB . Since βA = 1.5, this implies that βB = 0.75.
You can verify that the SML holds for security B (as it should if the market portfolio is
efficient). You could then use security B to determine that the risk-free rate is 4%, too.
21. Sharpe ratios of the value and growth portfolios are (0.13 − 0.02)/0.12 = 0.916̄ and
(0.17 − 0.02)/0.25 = 0.6, respectively. R̄M = 0.5(13%) + 0.5(17%) = 15%.
σM
=
q
(0.5)2 (0.12)2 + (0.5)2 (0.25)2 + 2(0.5)(0.5)(0.12)(0.25)(0.5) = 16.3%
Thus, the Sharpe ratio for M is (0.15 − 0.02)/0.163 ≈ 0.8. Since the Sharpe ratio for M
is less than the Sharpe ratio for the value portfolio, the market portfolio is not efficient.
According to CAPM, investors could reallocate their investments to improve the Sharpe
ratio so that they could achieve a higher expected return for the same level of volatility
or, alternatively, they could reduce their volatility and still achieve the same expected
return.
22. Sharpe ratio of P is (0.14 − 0.02)/0.30 = 0.40 whereas the Sharpe ratio of P 0 is (0.22 −
0.02)/0.40 = 0.50. A portfolio of 25% in the risk-free asset and 75% in portfolio P 0 will
have a volatility of 0.75(0.40) = 30% (the same as σP ) yet have a higher expected return
of 17%.
23. Sharpe ratio of A is (0.13 − 0.04)/0.20 = 0.45 whereas the Sharpe ratio of B is (0.18 −
0.04)/0.30 = 0.46̄. You should recommend fund B.
24. Since βBF = 0, the required return for stock B to compensate for its risk to fund F is the
risk-free rate of 3.8%. Since stock B’s expected return is higher, it will pay to add stock
B to fund F with a positive weight.
The new portfolio P has an expected return of 0.4(20%) + 0.6(14%) = 16.4%. We also
have that
σBP
2
= Cov(RB , 0.4RB + 0.6RF ) = 0.4σB
= 0.144
V ar(RP ) = (0.4)2 (0.6)2 + (0.6)2 (0.2)2 = 0.072
0.144
P
βB
=
= 2 =⇒ R̄B = 3.8% + 2(16.4% − 3.8%) = 29%.
0.072
Since the actual expected return for stock B is 20% < 29%, you can increase the Sharpe
ratio by reducing the weight of B in the portfolio.
The new portfolio P has an expected return of 0.15(20%) + 0.85(14%) = 14.9%. We also
have that
σBP
2
= Cov(RB , 0.15RB + 0.6RF ) = 0.15σB
= 0.054
V ar(RP ) = (0.15)2 (0.6)2 + (0.85)2 (0.2)2 = 0.037
0.054
P
βB
=
= 1.459 =⇒ R̄B = 3.8% + 1.459(14.9% − 3.8%) = 20%
0.037
Since the actual expected return for stock B is 20%, this is the correct weight.
Note: The formula derived in the handout shows that the optimum value for x (the dollar
amount to invest in F per dollar invested in fund F) is
x∗ =
σF2 R̂B − σBF R̂F
2 R̂ − σ
σB
F
BF R̂B
=
(0.20)2 (0.162) − 0
= 0.17647.
(0.6)2 (0.102) − 0
The portfolio weight on stock B is x/(1 + x) = 0.17647/1.17647 = 15%.
25. We have that
Cov(RV C , RJ ) = σV C σJ ρV CJ = (0.8)(0.25)(0.2) = 0.04
βVJ C
= 0.04/(0.25)2 = 0.64
R̄V C
= 4% + 0.64(12% − 4%) = 9.12%.
Since the actual expected return of the VC fund is 20% > 9.12%, you can increase the
Sharpe ratio by adding the VC fund to the Jones Fund with a positive weight.
The Sharpe ratio of the Jones Fund is (0.12 − 0.04)/0.25 = 0.32. With a 50-50 mix,
new Sharpe ratio = p
[0.5(0.20) + 0.5(0.12)] − 0.04
= 0.2713.
(0.5)2 (0.8)2 + (0.5)2 (0.25)2 + 2(0.5)(0.5)(0.04)
The 50% weight on the VC Fund is too large; it should be reduced. As a function of x,
the weight to allocate to the VC fund, the Sharpe ratio S(x) is
S(x) = p
0.20x + 0.12(1 − x) − 0.04
.
+ (1 − x)2 (0.20)2 + 2x(1 − x)(0.04)
x2 (0.80)2
Enumeration (in increments of 1%) yields the optimum weight on the VC fund is 13%.
Note: The formula derived in the handout shows that the optimum value for x (the dollar
amount to invest in the VC fund per dollar invested in the Jones Fund) is
x∗ =
σJ2 R̂V C − σV CJ R̂J
σV2 C R̂J − σV CJ R̂V C
=
(0.25)2 (0.16) − (0.040)(0.08)
= 0.15179.
(0.8)2 (0.08) − (0.04)(0.16)
The portfolio weight on stock B is x/(1 + x) = 0.15179/1.15179 = 13.18%.