CHAPTER 8

Risk and Return

 

 

Answers to Practice Questions

 

1.         a.         False – investors demand higher expected rates of return on stocks with more nondiversifiable risk.

b.                  False – a security with a beta of zero will offer the risk-free rate of return.

c.                  False – the beta will be: (1/3)´(0) + (2/3)´(1) = 0.67

d.                  True.

e.                  True.

 

 

2.                  In the following solution, security one is Coca-Cola and security two is Reebok.  Then:

                        r1 = 0.10         s1 = 0.315

                        r2 = 0.20         s2 = 0.585

            Further, we know that for a two-security portfolio:

                        rp = x1r1 + x2r2

                        sp2 = x12s12 + 2x1x2s1s2r12 + x22s22

            Therefore, we have the following results:

 

            x1

 

x2

 

        rp

sp1

when r = 0

      sp1

when r = 1

        sp1

when r = -1

1.0

0.0

0.10

0.315

0.315

0.315

0.9

0.1

0.11

0.289

0.342

0.225

0.8

0.2

0.12

0.278

0.369

0.135

0.7

0.3

0.13

0.282

0.396

0.045

0.6

0.4

0.14

0.301

0.423

0.045

0.5

0.5

0.15

0.332

0.450

0.135

0.4

0.6

0.16

0.373

0.477

0.225

0.3

0.7

0.17

0.420

0.504

0.315

0.2

0.8

0.18

0.472

0.531

0.405

0.1

0.9

0.19

0.527

0.558

0.495

0.0

0.0

0.20

0.585

0.585

0.585

 



 


 


 


3.                  a.

Portfolio

r

s

 

1

10.0%

5.1%

2

9.0

4.6

3

11.0

6.4

 

b.                  See the figure below.  The set of portfolios is represented by the curved line. The five points are the three portfolios from Part (a) plus the two following two portfolios: one consists of 100% invested in X and the other consists of 100% invested in Y.

 

c.                  See the figure below.   The best opportunities lie along the straight line.  From the diagram, the optimal portfolio of risky assets is portfolio 1, and so Mr. Harrywitz should invest 50 percent in X and 50 percent in Y.-+

Text Box:

 

 


4.                  a.         Expected return = (0.6 ´ 15) + (0.4 ´ 20) = 17%

                        Variance = (0.6)2 ´(20)2 + (0.4)2 ´ (22)2 + 2(0.6)(0.4)(0.5)(20)(22) = 327

                        Standard deviation = (327)(1/2) = 18.1%

b.                  Correlation coefficient = 0 ̃ Standard deviation = 14.9%

                        Correlation coefficient = -0.5 ̃ Standard deviation = 10.8%

c.                  His portfolio is better.  The portfolio has a higher expected return and a lower standard deviation.

 


 

5.                  Internet exercise; answers will vary depending on time period.

 

6.                  Internet exercise; answers will vary depending on time period.

 

7.                 
Text Box:

a.

 


b.                  Market risk premium = rm - rf = 0.12 - 0.04 = 0.08 = 8.0%

 

c.                  Use the security market line:

                                    r = rf + b(rm - rf)

                                    r = 0.04 + [1.5´(0.12 - 0.04)] = 0.16 = 16.0%

d.         For any investment, we can find the opportunity cost of capital using the security market line.  With b = 0.8, the opportunity cost of capital is:

 

                                    r = rf + b(rm - rf)

                                    r = 0.04 + [0.8´(0.12 - 0.04)] = 0.104 = 10.4%

                        The opportunity cost of capital is 10.4 percent and the investment is expected to earn 9.8 percent.  Therefore, the investment has a negative NPV.

 

            e.         Again, we use the security market line:

                                    r = rf + b(rm - rf)

                                    0.112 = 0.04 + b(0.12 - 0.04)  ̃  b = 0.9

8.                  Internet exercise; answers will vary depending on time period.

 

9.         Internet exercise; answers will vary.

 

 

10.       a.         Percival’s current portfolio provides an expected return of 9 percent with an annual standard deviation of 10 percent.  First we find the portfolio weights for a combination of Treasury bills (security 1: standard deviation = 0 percent) and the index fund (security 2: standard deviation = 16 percent) such that portfolio standard deviation is 10 percent.  In general, for a two security portfolio:

 

                                    sP2 = x12s12 + 2x1x2s1s2r12 + x22s22

                                    (0.10)2 = 0 + 0 + x22(0.16)2

                                    x2 = 0.625 ̃ x1 = 0.375

                        Further:

                                    rp = x1r1 + x2r2

                                    rp = (0.375 ´ 0.06) + (0.625 ´ 0.14) = 0.11 = 11.0%

                        Therefore, he can improve his expected rate of return without changing the risk of his portfolio.

 

b.                  With equal amounts in the corporate bond portfolio (security 1) and the index fund (security 2), the expected return is:

 

                                    rp = x1r1 + x2r2

                                    rp = (0.5 ´ 0.09) + (0.5 ´ 0.14) = 0.115 = 11.5%

                                    sP2 = x12s12 + 2x1x2s1s2r12 + x22s22

                                    sP2 = (0.5)2(0.10)2 + 2(0.5)(0.5)(0.10)(0.16)(0.10) + (0.5)2(0.16)2

                                    sP2 = 0.0097

                                    sP = 0.985 = 9.85%

Therefore, he can do even better by investing equal amounts in the corporate bond portfolio and the index fund.  His expected return increases to 11.5% and the standard deviation of his portfolio decreases to 9.85%.

 

 

11.             No.  Every stock has unique risk in addition to market risk.  The unique risk reflects uncertain events that are unrelated to the return on the market portfolio.  The Capital Asset Pricing Model does not predict these events.  If the events are favorable, the stock will do better than the model predicts.  If the events are unfavorable, the stock will do worse.

 

12.       a.         True

b.                  True

c.                  True

13.       a.         True.  By definition, the factors represent macro-economic risks that cannot be eliminated by diversification.

b.                  False.  The APT does not specify the factors.

c.                  True.  Investors will not take on nondiversifiable risk unless it entails a positive risk premium.

d.                  True.  Different researchers have proposed and empirically investigated different factors, but there is no widely accepted theory as to what these factors should be.

            e.         True.  To be useful, we must be able to estimate the relevant parameters.  If this is impossible, for whatever reason, the model itself will be of theoretical interest only.

 

 

14.       For Stock P      ̃ r = (1.0)´(6.4%) + (-2.0)´(-0.6%) + (-0.2)´(5.1%) = 6.58%

For Stock P2     ̃ r = (1.2)´(6.4%) + (0)´(-0.6%) + (0.3)´(5.1%) = 9.21%

For Stock P3     ̃ r = (0.3)´(6.4%) + (0.5)´(-0.6%) + (1.0)´(5.1%) = 6.72%

 

15.       a.         Factor risk exposures:

                        b1(Market) = (1/3)´(1.0) + (1/3)´(1.2) + (1/3)´(0.3) = 0.83

                        b2(Interest rate) = (1/3)´(-2.0) +(1/3)´(0) + (1/3)´(0.5) = -0.50

                        b3(Yield spread) = (1/3)´(-0.2) + (1/3)´(0.3) + (1/3)´(1.0) = 0.37

b.                  rP = (0.83)´(6.4%) + (-0.50)´(-0.6%) + (0.37)´(5.1%) = 7.5%

 

16.       rCC = 3.5% + (0.82 ´ 8.8%) + (-0.29 ´ 3.1%) + (0.24 ´ 4.4%) = 10.87%

            rXON = 3.5% + (0.50 ´ 8.8%) + (0.04 ´ 3.1%) + (0.27 ´ 4.4%) = 9.21%

            rP = 3.5% + (0.66 ´ 8.8%) + (-0.56 ´ 3.1%) + (-0.07 ´ 4.4%) = 7.26%

            rR = 3.5% + (1.17 ´ 8.8%) + (0.73 ´ 3.1%) + (1.14 ´ 4.4%) = 21.08%


Challenge Questions

 

1.                  [NOTE: In the first printing of the seventh edition of the text, footnote 4 states that, for the minimum risk portfolio, the investment in Reebok is 21.4%.  This figure is incorrect.  The correct figure is 16.96%, as shown below.]

 

            In general, for a two-security portfolio:

                        sp2 = x12s12 + 2x1x2s1s2r12 + x22s22

            and:

                        x1 + x2 = 1

            Substituting for x2 in terms of x1 and rearranging:

                        sp2 = s12x12 + 2s1s2r12(x1 - x12) + s22(1 - x1)2

            Taking the derivative of sp2  with respect to x1, setting the derivative equal to zero and rearranging:

 

                        x1(s12 - 2s1s2r12 + s22) + (s1s2r12 - s22) = 0

            Let Coca-Cola be security one (s1 = 0.315) and Reebok be security two (s2 = 0.585).  Substituting these numbers, along with r12 = 0.2, we have:

 

                        x1 = 0.8304

            Therefore:

                        x2 = 0.1696

 

 

2.         a.         The ratio (expected risk premium/standard deviation) for each of the four portfolios is as follows:

 

                                          Portfolio A:           (34.6 – 10.0)/110.6 = 0.222

                                          Portfolio B:           (21.6 – 10.0)/30.8 = 0.377

                                          Portfolio C:           (19.0 – 10.0)/23.7 = 0.380

                                          Portfolio D:           (13.4 – 10.0)/14.6 = 0.233

                        Therefore, an investor should hold Portfolio C.


 

b.                  The beta for Amazon relative to Portfolio C is equal to the ratio of the risk premium of Amazon to the risk premium of the portfolio times the beta of the portfolio:

 

                              [(34.6% - 10.0%)/(19% - 10%)] ´ 1.0 = 2.733

            Similarly, the betas for the remainder of the holdings are as follows:

bAmazon

= 2.733

bBoeing

= 0.333

bDell

= 1.800

bEX-M

= 0.200

bGE

= 0.889

bMcD

= 0.444

bPfizer

= 0.533

bReebok

= 1.111

 

 

c.                  If the interest rate is 5%, then Portfolio C remains the optimal portfolio, as indicated by the following calculations:

 

                                          Portfolio A:           (34.6 – 5.0)/110.6 = 0.268

                                          Portfolio B:           (21.6 – 5.0)/30.8 = 0.539

                                          Portfolio C:           (19.0 – 5.0)/23.7 = 0.591

                                          Portfolio D:           (13.4 – 5.0)/14.6 = 0.575

                        The betas for the holdings in Portfolio C become:

bAmazon

= 2.114

bBoeing

= 0.571

bDell

= 1.514

bEX-M

= 0.486

bGE

= 0.929

bMcD

= 0.643

bPfizer

= 0.700

bReebok

= 1.071

 

 


 

3                    Whether the APT can be used to make money is a question related to competition in the financial markets.  Given sufficient competition, no widely‑known model will provide an advantage (i.e., enable someone to make a return above that expected, given the level of risk undertaken).  So, whether an economic model enables one to make money is not relevant to the validity of that model.  To put this somewhat differently, the validity of an economic model hinges on whether the model enables us to better identify and understand relationships among key parameters, not whether the model can be used to make money.

 

 

4.                  Let rx be the risk premium on investment X, let xx be the portfolio weight of X (and similarly for Investments Y and Z, respectively).

 

a.         rx = (1.75)´(0.04) + (0.25)´(0.08) = 0.09 = 9.0%

            ry = (-1.00)´(0.04) + (2.00)´(0.08) = 0.12 = 12.0%

            rz = (2.00)´(0.04) + (1.00)´(0.08) = 0.16 = 16.0%

b.                  This portfolio has the following portfolio weights:

            xx = 200/(200 + 50 - 150) = 2.0

            xy = 50/(200 + 50 - 150) = 0.5

            xz = -150/(200 + 50 - 150) = -1.5

            The portfolio’s sensitivities to the factors are:

                        Factor 1:         (2.0)´(1.75) + (0.5)´(-1.00) – (1.5)´(2.00) = 0

                        Factor 2:         (2.0)´(0.25) + (0.5)´(2.00) – (1.5)´(1.00) = 0

                        Because the sensitivities are both zero, the expected risk premium is zero.

c.                  This portfolio has the following portfolio weights:

            xx = 80/(80 + 60 - 40) = 0.8

            xy = 60/(80 + 60 - 40) = 0.6

            xz = -40/(80 + 60 - 40) = -0.4

                        The sensitivities of this portfolio to the factors are:

                        Factor 1:         (0.8)´(1.75) + (0.6)´(-1.00) – (0.4)´(2.00) = 0

                        Factor 2:         (0.8)´(0.25) + (0.6)´(2.00) – (0.4)´(1.00) = 1.0

            The expected risk premium for this portfolio is equal to the expected risk premium for the second factor, or 8 percent.


 

d.                  This portfolio has the following portfolio weights:

                        xx = 160/(160 + 20 - 80) = 1.6

                        xy = 20/(160 + 20 - 80 ) = 0.2

                        xz = -80/(160 + 20 - 80) = -0.8

                        The sensitivities of this portfolio to the factors are:

                        Factor 1:         (1.6)´(1.75) + (0.2)´(-1.00) - (0.8)´(2.00) = 1.0

                        Factor 2:         (1.6)´(0.25) + (0.2)´(2.00) – (0.8)´(1.00) = 0

            The expected risk premium for this portfolio is equal to the expected risk premium for the first factor, or 4 percent.

 

e.                  The sensitivity requirement can be expressed as:

Factor 1:         (xx)(1.75) + (xy)(-1.00) + (xz)(2.00) = 0.5

            In addition, we know that:

                                    xx + xy + xz = 1

With two linear equations in three variables, there is an infinite number of solutions.  Two of these are:

 

            1.         xx = 0               xy = 0.5           xz = 0.5

            2.         xx = (6/11)       xy = (5/11)       xz = 0

            The risk premiums for these two funds are:

            r1 = 0´[(1.75 ´ 0.04) + (0.25 ´ 0.08)]

                    + (0.5)´[(-1.00 ´ 0.04) + (2.00 ´ 0.08)]

                                + (0.5)´[(2.00 ´ 0.04) + (1.00 ´ 0.08)] = 0.14 = 14.0%

                        r2 = (6/11)´[(1.75 ´ 0.04) + (0.25 ´ 0.08)]

                                +(5/11)´[(-1.00 ´ 0.04) + (2.00 ´ 0.08)]

                                +0 ´ [(2.00 ´ 0.04) + (1.00 ´ 0.08)] = 0.104 = 10.4%

            These risk premiums differ because, while each fund has a sensitivity of 0.5 to factor 1, they differ in their sensitivities to factor 2.


 

f.                    Because the sensitivities to the two factors are the same as in Part (b), one portfolio with zero sensitivity to each factor is given by:

 

                                    xx = 2.0            xy = 0.5           xz = -1.5

                        The risk premium for this portfolio is:

                                    (2.0)´(0.08) + (0.5)´(0.14) – (1.5)´(0.16) = -0.01

            Because this is an example of a portfolio with zero sensitivity to each factor and a nonzero risk premium, it is clear that the Arbitrage Pricing Theory does not hold in this case.

 

                        A portfolio with a positive risk premium is:

                                    xx = -2.0          xy = -0.5          xz = 1.5