capital asset pricing model homework

7/21/2019 Capital Asset Pricing Model Homework

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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 Bworth $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 isthe 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 pricechange, what are the new portfolio weights?

2. Consider a portfolio of 250 shares of firm A worth $30/share and 1500 shares of firm Bworth $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 isthe 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 pricechange, 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 Financeby 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 (basedon monthly returns, 1996-2008).

Alaskan Southwest Ford General GeneralMicrosoft Dell Air Airlines Motor Motors Mills

Volatility (StDev) 37% 50% 38% 31% 42% 41% 18%Correlation with:

Microsoft 1.00 0.62 0.25 0.23 0.26 0.23 0.10Dell 0.62 1.00 0.19 0.21 0.31 0.28 0.07

Alaska Air 0.25 0.19 1.00 0.30 0.16 0.13 0.11Southwest Airlines 0.23 0.21 0.30 1.00 0.25 0.22 0.20

Ford Motor 0.26 0.31 0.16 0.25 1.00 0.62 0.07General Motors 0.23 0.28 0.13 0.22 0.62 1.00 0.02

General Mills 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 onStock 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 Expected Return Volatility

Stock A 15% 40%Stock B 7% 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 returnand volatility?



(b) What is the minimum variance portfolio when ρAB = 0.4? What is its expectedreturn 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 = 0.06and 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

Σ =

0.09 0.045 0.010.045 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 isrf = 0.03. The inverse of the covariance matrix is

Σ−1 =

12.2137 −2.2901 0.3817−2.2901 6.6794 −9.4466

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

Σ =

2 1 0

1 2 10 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 meanreturns and volatilities.



11. For the data of problem 7 determine the tangent portfolios and their respective meanreturns 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 andvolatility.

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 expectedreturn?

(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, whereasstock 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 as10%. 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 riskyassets are in equal supply in the market, i.e., the market portfolio M = 0.5A + 0.5B. Itis known that R̄M = 11%, σA = 20%, σB = 40% and ρAB = 0.75. The risk-free rate is2%. 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 Ahas 200 shares outstanding, a price per share of $3.00, an expected return of 16% anda volatility of 30%. Stock B has 300 shares outstanding, a price per share of $4.00, anexpected 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 interms of total value. Furthermore, suppose that the expected return of the value stocksis 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 50-50combination 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 rateis 2%. Suppose further that a particular mix of asset i and P yields a portfolio P with



an expected return of 22% and a volatility of 40%. Will adding asset i to portfolio P bebeneficial? 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. Whicheverfund you recommend she will combine it with the risk-free asset. Expected returns areR̄A = 13% and R̄B = 18%. Volatilities are σA = 20% and σB = 30%. Without knowingyour 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 portfoliowith a positive weight. Stock B has an expected return of 20%, a volatility of 60% and acorrelation 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 thatnow 60% is in Fund F and 40% is in Stock B. You tell your finance professor aboutyour investment and he says you made a mistake and should reduce your investmentin Stock B. Is your finance professor right?

(c) You decide to follow your finance professor’s advice and reduce your exposure toStock B. Now Stock B represents only 15% of your risky portfolio with the restinvested 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 broad-based fund with an expected return of 12% and a volatility of 25%. The risk-free rate is4%. 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 0.2with 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 yournew 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 arexA = 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 arexA = 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 arexA = 250($24)/$39, 000 = 15.38% and xB = 1500($22)/$39, 000 = 84.62%.

3. Cov = (0.11)(0.38)(0.18) = 0.007524.50:50: StDev =

(0.5)2(0.38)2 + (0.5)2(0.18)2 + 2(0.5)(0.5)(0.007524) = 21.90%.

20:80: StDev =

(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 ar = 0.502 = 0.25. AvgCov = (0.50)(0.50)(0.30) = 0.075.

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 + 0.07(1 − x) = 0.07 + 0.08x.StDev(x) =

(0.40)2x2 + (0.30)2(1− x)2 + 2(0.40)(0.30)ρABx(1 − x)

=

(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 = σ2

B−σAB

σ2A+σ2

B−2σAB .

ρAB = 0 =⇒ σAB = 0 and xA = 0.09/(0.16 + 0.09) = 36%. Expected return =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%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) + 2(0.048)] = 39.88%. Expected return = 0.3988(15%) + 0.6012(7%) = 10.19% andσ =

(0.3988)2(0.40)2 + (0.6012)2(0.30)2 + 2(−0.048)(0.3988)(0.6012) = 18.70%.

7. In general, xA = σ2

B−σAB

σ2A+σ2

B−2σAB

.

ρAB = 0 =⇒ σAB = 0 andxA = 0.64/(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 andxA = (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 andxA = (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−1e = (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−1e) = 1/35.2575and 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. Minimumvariance = λ = 1.

10. R̂ = (0.15 − 0.03, 0.07 − 0.03) = (0.12, 0.04).When ρAB = 0:

Σ =

0.16 0.000.00 0.09

,

Σ−1 = 1

(0.16)(0.09)

0.09 0.000.00 0.16

=

6.25 0.000.00 11.1̄

.

x∗ ∝ Σ−1 R̂ = (0.75, 0.44̄),

eT Σ−1 R̂ = 0.75 + 0.44̄ = 1.194̄,

x∗ =

0.751.194̄

, 0.44̄1.194̄

= (0.628, 0.372)

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

1.194̄ = 0.0755 =⇒ σP =

√ 0.0755 = 27.48%.



When ρAB = 0.4:

Σ =

0.16 0.0480.048 0.09

,

Σ−1 = 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 = (0.7341, 0.0529),

eT Σ−1 R̂ = 0.7341 + 0.0529 = 0.7870,

x∗ =

0.7341

0.7870, 0.0529

0.7870

= (0.9328, 0.0672)

E [RP ] = 0.9328(0.15) + 0.0672(0.07) = 0.1446,

R̂0 = E [RP ] − rf = 0.1146,

V ar(RP ) =

R̂0

eT Σ−1 R̂ =

0.1146

0.7870 = 0.1456 =⇒ σP = √ 0.1456 = 38.16%.

When ρAB = −0.4:

Σ =

0.16 −0.048−0.048 0.09

,

Σ−1 = 1

(0.16)(0.09) − (0.048)2

0.09 0.0480.048 0.16

=

7.4405 3.96833.9683 13.2275

.

x∗ ∝ Σ−1 R̂ = (1.0516, 1.0053),

eT Σ−1 R̂ = 1.0516 + 1.0053 = 2.0569,

x∗ =

1.05162.0569

, 1.00532.0569

= (0.5113, 0.4887)

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

2.0569 = 0.0393 =⇒ σP =

√ 0.0393 = 19.83%.

11. R̂ = (0.10 − 0.02, 0.06 − 0.02) = (0.08, 0.04).When ρAB = 0:

Σ = 0.09 0.00

0.00 0.64

,

Σ−1 = 1

(0.09)(0.64)

0.64 0.000.00 0.09

=

11.1̄ 0.000.00 1.5625

.

x∗ ∝ Σ−1 R̂ = (0.88̄, 0.0625),

eT Σ−1 R̂ = 0.88̄ + 0.0625 = 0.9514,



x∗ =

0.88̄

0.9514, 0.0625

0.9514

= (0.9343, 0.0657)

E [RP ] = 0.9343(0.10) + 0.0657(0.06) = 0.0974,

ˆR0 = E [RP ] − rf = 0.0774,

V ar(RP ) =R̂0

eT Σ−1 R̂=

0.0774

0.9514 = 0.0814 =⇒ σP =

√ 0.0814 = 28.53%.

When ρAB = 0.6:

Σ =

0.09 0.1440.144 0.64

,

Σ−1 = 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 = (1.2326,−0.2148),

eT Σ−1 R̂ = 1.2326 − 0.2148 = 1.0178,

x∗ =

1.2326

1.0178,−0.2148

1.0178

= (1.211,−0.211)

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

1.0178 = 0.0869 =⇒ σP =

√ 0.0366 = 29.48%.

When ρAB = −0.6:

Σ = 0.09

−0.144

−0.144 0.64

,

Σ−1 = 1

(0.09)(0.64) − (0.144)2

0.64 0.1440.144 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,

x∗ =

1.5451

1.9553, 0.4102

1.9553

= (0.7902, 0.2098)

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

1.9553 = 0.0366 =⇒ σP = √ 0.0366 = 19.14%.

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,

x∗ =

0.50

4.0625,−0.875

4.0625 ,

4.4375

4.0625

= (0.1231,−0.2154, 1.0923)

E [RP ] = 0.1231(0.08) − 0.2154(0.10) + 1.0923(0.16) = 0.1631,R̂0 = E [RP ] − rf = 0.1331,

V ar(RP ) =R̂0

eT Σ−1 R̂=

0.1331

4.0625 = 0.0328 =⇒ σP =

√ 0.0328 = 18.11%.

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

−0.50 1.00 −0.500.25

−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,

x∗ =

0.04

0.06, 0.01

0.06, 0.01

0.06

= (2/3, 1/6, 1/6)

E [RP ] = (2/3)(0.11) + (1/6)(0.09) + (1/6)(0.05) = 0.096̄,

R̂0 = E [RP ] − rf = 0.076̄,

V ar(RP ) =R̂0

eT Σ−1 R̂=

0.076̄

0.06 = 1.27̄ =⇒ σP =

√ 1.27̄ = 113.04%.

14. ¯RP = 0.02 +

0.10−0.020.40

σP = 0.02 + 0.20σP . For an efficient portfolio whose expected

return is 8%, we have 0.08 = 0.02 + 0.20σP =⇒ σP = 30%. Allocate $750 to the tangentportfolio and $250 to the risk-free asset.

15. R̄P = 0.03 +0.12−0.03

0.30

σP = 0.03 + 0.30σP . For an efficient portfolio whose expected

return is 16%, we have 0.165 = 0.03 + 0.30σP =⇒ σP = 45%. Allocate $4,500 to thetangent portfolio and borrow $1,500 at the risk-free asset.

16. R̄i = 0.03 + 0.09β i.R̄i = 0.03 + 0.09(0.6) = 8.4%.β i = 0.25(0.6)/(0.3) = 0.5 =⇒ R̄i = 0.03 + 0.09(0.5) = 7.5%.β i =

−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%.β P = 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 highercost of equity capital.

19.

Cov(RA, RB) = (0.2)(0.4)(0.75) = 0.06

Cov(RA, RM ) = Cov(RA, 0.5RA + 0.5RB)

= 0.5σ2A + 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

β A = 0.05/0.08 = 0.625 =⇒ R̄A = 0.02 + 0.625(0.11 − 0.02) = 7.625%.

The market’s beta of 1 equals 0.5β A + 0.5β B. Since β A = 0.625, this implies that β B =1.375. (You can verify this quantity via the formula for β B .) The market’s expected return

of 11% must equal 0.5 ¯RA + 0.5

¯RB. Since

¯RA = 7.625%, this implies that

¯RB = 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 = σAσBρAB = (0.3)(.15)(0.4) = 0.018

Cov(RA, RM ) = Cov(RA, 1/3RA + 2/3RB)

= 1/3(σ2A) + 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%β A = 0.042/0.028 = 1.5 =⇒ 16 = rf + 1.5(12 − rf ) =⇒ rf = 4%.

The market’s beta of 1 equals 1/3β A+2/3β B. Since β A = 1.5, this implies that β B = 0.75.You can verify that the SML holds for security B (as it should if the market portfolio isefficient). 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)/0.1 2 = 0.916̄ and(0.17 − 0.02)/0.25 = 0.6, respectively. R̄M = 0.5(13%) + 0.5(17%) = 15%.

σM =

(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 Mis 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 Sharperatio so that they could achieve a higher expected return for the same level of volatilityor, alternatively, they could reduce their volatility and still achieve the same expectedreturn.



22. Sharpe ratio of P is (0.14 − 0.02)/0.30 = 0.40 whereas the Sharpe ratio of P is (0.22 −0.02)/0.40 = 0.50. A portfolio of 25% in the risk-free asset and 75% in portfolio P willhave 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 β BF = 0, the required return for stock B to compensate for its risk to fund F is therisk-free rate of 3.8%. Since stock B’s expected return is higher, it will pay to add stockB 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 alsohave that

σBP = Cov(RB, 0.4RB + 0.6RF ) = 0.4σ2B = 0.144

V ar(RP ) = (0.4)2(0.6)2 + (0.6)2(0.2)2 = 0.072

β P B = 0.144

0.072 = 2 =⇒ R̄B = 3.8% + 2(16.4% − 3.8%) = 29%.

Since the actual expected return for stock B is 20% < 29%, you can increase the Sharperatio 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 alsohave that

σBP = Cov(RB, 0.15RB + 0.6RF ) = 0.15σ2B = 0.054

V ar(RP ) = (0.15)2(0.6)2 + (0.85)2(0.2)2 = 0.037

β P B = 0.054

0.037 = 1.459 =⇒ R̄B = 3.8% + 1.459(14.9% − 3.8%) = 20%

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 dollaramount to invest in F per dollar invested in fund F) is

x∗ = σ2F R̂B − σBF R̂F

σ2BR̂F − σBF R̂B

= (0.20)2(0.162) − 0

(0.6)2(0.102) − 0 = 0.17647.

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

β J V 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 theSharpe 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 = [0.5(0.20) + 0.5(0.12)] − 0.04

(0.5)2(0.8)2 + (0.5)2(0.25)2 + 2(0.5)(0.5)(0.04)= 0.2713.

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) = 0.20x + 0.12(1 − x) − 0.04

x2(0.80)2 + (1 − x)2(0.20)2 + 2x(1− x)(0.04).

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∗ = σ2

J R̂V C − σV CJ R̂J

σ2V C R̂J − σV CJ R̂V C

= (0.25)2(0.16) − (0.040)(0.08)

(0.8)2(0.08) − (0.04)(0.16) = 0.15179.

The portfolio weight on stock B is x/(1 + x) = 0.15179/1.15179 = 13.18%.

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