Solutions to Practice Problems – Chapters 7, 8

1 a) Using the CAPM equation R_i = R_f + ß_i RP_m, we have .085 = .025 + .75 * RP_m

So  RP_m = (.085 - .025)/.75 = 8%

b) The risk premium on the stock is RP_i = R_i - R_f = .085 - .025 = 6%

c) The risk premium on the market portfolio increases to 11%.  The new required return for the stock is:

R_i = .025 + .75 * .11 = 10.75%

(Alternatively, increase in required return = .75 * .03 = 2.25%, so new required return = .085 + .0225 = 10.75%)

2 a)  0.09266 = .023 + 0.86* RP_m    =>   RP_m = (.09266 - .023)/.86 = 8.1%

E(R_m) = R_f + RP_m = .023 + .081 = 10.4%

b) The extra or unexpected return on the market portfolio = .06 - .104 = -0.044

As the stock moves down with the market, the extra return on the stock will be (-0.044) * ß_i = -3.784%

The return now expected from the stock is the ex-post expected return = .09266 - .03784 = 5.482%  

[Or the ex-post expected return can be computed in one step:

            Ex Post E(R) =  R_f + ß_i *(Actual R_m - R_f) = .023 + .086*(.06 - .023) = 5.482%]

3 a)  0.1355 = .033 + ß_i (0.115 - 0.033)    =>   ß_i = (.1355 - .033)/(.115 - .033) = .1025/.082 = 1.25

b) The extra or unexpected return on the market portfolio = .15 - .115 = .035

As the stock moves with the market, the extra return on the stock will be .035 * ß_i = 4.375%

The return expected from the stock would now be .1355 + .04375 = 17.925%

4 a) Applying the SML equation to XYZ's stock:

        .1048 =  R_f + b_i * RP_m = .022 +  b_i * .09  

=>   b_i = (.1048 - .022)/.09  =   0.92%

b)  Risk premium for the stock = E(R_i) - R_f = .1048 - .022 = 8.28%

c)  Expected return on the market portfolio = RP_m + R_f  = .09 + .022 = 11.2%

d) The extra return on the market is .08 - .112 = -3.2%.  

The extra return on the stock will be 0.92 * (-.032) = - 2.944%.  

The return we now expect from the stock  = .1048 - .02944 = 7.536%

5 a) Applying the SML equation to BBC's stock:

        .10525 = R_f + 0.86 [.1175 -  R_f ]     =>    R_f (1 - 0.86)  =   .10525 - 0.86 * .1175

=>      R_f = .0042)/0.14 = 3%

b)  Risk premium for the market portfolio = E(R_m) -  R_f = .1175 - .03 = 8.75%

c)  Risk premium for BBC's stock= E(R_i) -  R_f = .10525 - .03 = 7.525%

d) The extra return on the market is -.05 - .1175 = -16.75%.  The extra return on the stock will be 0.86 * (-.1675) = -14.405%.  This means the return we now expect from the stock  = .10525 - .14405 = -3.88%

6 a) We are going to have two unknown variables, R_f and RP_m. So we need two equations to solve for two unknowns  => apply the CAPM equation twice, once to PPI's stock and once to the stock portfolio:

            0.1119 = R_f + 1.15* RP_m  

            0.0473 = R_f + 0.3* RP_m  

Subtracting the second equation from the first gives:

           .0646 = 0.85 * RP_m =>  RP_m = .0646/.85 = 7.6%

Substituting this back in the first equation:

           0.1119 = R_f + 1.15* .076   =>   R_f = .1119 - 1.15 * .076 = 2.45%

b) E(R_m) = RP_m + R_f = .076 + .0245 = 10.05%

c) The extra or unexpected return on the market portfolio = .16 - .1005 = 0.0595

As the stock moves with the market, the extra return on the stock will be 0.0595 * 1.15 = 6.8425%

The return now expected from the stock, the ex post E(R), = .1119 + 0.068425 = 18.0325%  

PPI’s abnormal return = actual return - ex post E(R) = .15 - .180325 = -3.0325%

(Here's how we would interpret the numbers.  If there was no economy-wide news or firm-specific news, the stock would have produced a return of 11.19%.  The effect of economy-wide news was to move the stock up by 6.8425%, taking us from 11.19% to 18.0325%.  The effect of firm-specific news was to drive the stock down by 3.0325%, taking us from 18.0325% to 15%.  The market as a whole did well, but the stock did badly.) 

7) Stock's abnormal return  = stock's actual return - stock's ex-post expected return 

                                            = 0.11 - [0.03 + 1.2 * (0.10 - .03)] = 0.11 - 0.114 = -0.4%

(Moving with the market, we'd expect the stock to go up 11.4%.  Since it went up only 11%, it did worse than expected, producing a negative abnormal return.  Although economy-wide news was positive over the year, firm-specific news was negative to the tune of 0.4%.)

8) Stock's abnormal return  = stock's actual return - stock's ex-post expected return 

                                            = (-0.10) - [0.02 + 0.8 * (-0.16 - .02)] = (-0.10) - (-0.124) = 2.4%

(Moving with the market, we'd expect the stock to fall by 12.4%.  Since it fell by only 10%, it did better than expected, producing a positive abnormal return of 2.4%.  Although economy-wide news was negative over the year, firm-specific news was positive to the tune of 2.4%.)

9 a)  A risk averse investor is one who will invest in riskfree assets rather than hold risky assets and bear risk. F She is willing to hold risky assets as long as she is compensated for the risk she bears (through a higher expected return).

b)   A higher beta necessarily means a higher required return. T Required return depends only on systematic risk, which is measured by beta. 

c)   If the beta of a stock doubles, the required return on the stock will double. F The risk premium will double but not the required return.

d)   Firm-specific news constitutes diversifiable risk only because the firm-specific news of different firms is uncorrelated. T Because it is uncorrelated, on a given day randomly half the firms move up due to good news and half move down due to bad news, and these movements cancel out in a large portfolio.

e)   On a given day, you cannot predict whether economy-wide news will be randomly positive or negative.  If it is randomly positive or negative, it should cancel out in a large portfolio and constitute diversifiable risk. F What matters is not whether we can predict if it will be positive or negative, but how firms react to whatever news arrives.  If the firms reacted randomly, then the movements would cancel out and the risk would be diversifiable.  Since most firms move in the same direction (up when there is good news, down when there is bad news), the movements don't cancel out and the risk is not diversifiable. 

f)    Beta is used to measure the systematic risk of a stock because the numerator of beta measures the marginal risk the stock contributes to the market portfolio.  T Since investors hold the market portfolio, this represents the marginal risk the stock imposes on investors.

g)   Risk averse investors will not be willing to hold a risky stock if the expected return is less than the riskless rate. F They are willing to hold negative beta stocks even though they offer expected returns less than the riskless rate.  The reason is that negative beta stocks reduce the risk of the market portfolio, i.e. the risk investors actually bear.

h)   If two stocks have the same standard deviation of returns, the one which has a lower correlation with the market portfolio always has the smaller beta.  T This follows from the formula b_i  = (s_i/s_m) r_im .

i)    If the beta of the market portfolio increases, E(R_m) will increase. F Trick question.  Beta of the market portfolio is always 1 by definition. 

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