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Chapter 13 Return, Risk, and the Security Market Line - PDF - PDF by ybb83869

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									CHAPTER 13
RETURN, RISK, AND THE SECURITY MARKET LINE

Answers to Concepts Review and Critical Thinking Questions

1.   Some of the risk in holding any asset is unique to the asset in question. By investing in a variety of
     assets, this unique portion of the total risk can be eliminated at little cost. On the other hand, there are
     some risks that affect all investments. This portion of the total risk of an asset cannot be cost-lessly
     eliminated. In other words, systematic risk can be controlled, but only by a costly reduction in
     expected returns.

2.   If the market expected the growth rate in the coming year to be 2 percent, then there would be no
     change in security prices if this expectation had been fully anticipated and priced. However, if the
     market had been expecting a growth rate different than 2 percent and the expectation was incorpo-
     rated into security prices, then the government’s announcement would most likely cause security
     prices in general to change; prices would drop if the anticipated growth rate had been more than
     2 percent, and prices would rise if the anticipated growth rate had been less than 2 percent.

3.   a.    systematic
     b.    unsystematic
     c.    both; probably mostly systematic
     d.    unsystematic
     e.    unsystematic
     f.    systematic

4.   a.    a change in systematic risk has occurred; market prices in general will most likely
           decline.
     b.    no change in unsystematic risk; company price will most likely stay constant.
     c.    no change in systematic risk; market prices in general will most likely stay constant.
     d.    a change in unsystematic risk has occurred; company price will most likely decline.
     e.    no change in systematic risk; market prices in general will most likely stay constant.

5.   No to both questions. The portfolio expected return is a weighted average of the asset returns, so it
     must be less than the largest asset return and greater than the smallest asset return.

6.   False. The variance of the individual assets is a measure of the total risk. The variance on a well-
     diversified portfolio is a function of systematic risk only.

7.   Yes, the standard deviation can be less than that of every asset in the portfolio. However, βp cannot be
     less than the smallest beta because βp is a weighted average of the individual asset betas.

8.   Yes. It is possible, in theory, to construct a zero beta portfolio of risky assets whose return would be
     equal to the risk-free rate. It is also possible to have a negative beta; the return would be less than the
     risk-free rate. A negative beta asset would carry a negative risk premium because of its value as a
     diversification instrument.

9.   Such layoffs generally occur in the context of corporate restructurings. To the extent that the market
     views a restructuring as value-creating, stock prices will rise. So, it’s not layoffs per se that are being
     cheered on. Nonetheless, Bay Street does encourage corporations to takes actions to create value,
     even if such actions involve layoffs.




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10. Earnings contain information about recent sales and costs. This information is useful for projecting
    future growth rates and cash flows. Thus, unexpectedly low earnings often lead market participants
    to reduce estimates of future growth rates and cash flows; price drops are the result. The reverse is
    often true for unexpectedly high earnings.

Solutions to Questions and Problems

          Basic

1.   total value = 90($35) + 70($25) = $4,900
     weight1 = 90($35)/$4,900 = .6429 ; weight2 = 70($25)/$4,900 = .3571

2.   E[Rp] = ($700/$3,100)(0.11) + ($2,400/$3,100)(0.18) = .1642

3.   E[Rp] = .50(.10) + .30(.18) + .20(.13) = .1300

4.   E[Rp] = .135 = .15wX + .10(1 – wX); wX = 0.70
     investment in X = 0.70($10,000) = $7,000; investment in Y = (1 – 0.70)($10,000) = $3,000

5.   E[R] = .3(–.02) + .7(.34) = 23.2%

6.   E[R] = .4(–.05) + .5(.12) + .1(.25) = 6.5%

7.   E[RA] = .20(.06) + .60(.07) + .20(.11) = 7.60%
     E[RB] = .20(–.2) + .60(.13) + .20(.33) = 10.40%
     σA2 =.20(.06–.0760)2 + .60(.07–.0760)2 + .20(.11–.0760)2 = .000304; σA = [.000304]1/2 = .01744
     σB2 =.20(–.2–.1040)2 + .60(.13–.1040)2 + .20(.33–.1040)2 = .029104; σB = [.029104]1/2 = .17060

8.   E[Rp] = .20(.05) + .70(.16) + .1(.35) = 15.70%

9.   a.     boom: E[Rp] = (.07 + .15 + .33)/3 = .1833 ; bust: E[Rp] = (.13 + .03 −.06)/3 = .0333
            E[Rp] = .60(.1833) + .40(.0333) = .1233
     b.     boom: E[Rp]=.20(.07) +.20(.15) + .60(.33) =.2420
            bust: E[Rp] =.20(.13) +.20(.03) + .60(−.06) = –.004
            E[Rp] = .60(.2420) + .40(−.004) = .1436
            σp2 = .60(.2420 – .1436)2 + .40(−.004 – .1436)2 = .014524; σp = [.014524]1/2 = .1205

10. a.      boom: E[Rp] = .30(.3) + .40(.45) + .30(.33) = .3690
            good:     E[Rp] = .30(.12) + .40(.10) + .30(.15) = .1210
            poor:     E[Rp] = .30(.01) + .40(–.15) + .30(–.05) = –.0720
            bust:     E[Rp] = .30(–.06) + .40(–.30) + .30(–.09) = –.1650
            E[Rp] = .20(.3690) + .40(.1210) + .30(–.0720) + .10(–.1650) = .0841
     b.     σp2 = .20(.3690 – .0841)2 + .40(.1210 – .0841)2 + .30(–.0720 – .0841)2 + .10(–.1650 – .0841)2
            σp2 = .03029; σp = [.03029]1/2 = .174

11. σ2p = 0.0144 = (.4)2(.2)2 + .(.6)2σy2 + 2(.4)(.6)(.2)(σy)(.7)
        = 0.0144 = .0064 + .36σy2 + .0672σy
        .36σy2 + .0672σy – .008 = 0
    While this equation can be solved using the quadratic formula, we only consider the positive root for
    the standard deviation. We get: σy = 0.0825 or 8.25%.

12. A $1 rise associated with a $0.60 drop suggests that the correlation between the returns for the two
    stocks is –0.6. The portfolio variance becomes:
    σ2p = (.25)2(.0225) + .(.75)2(.0121) + 2(.25)(.75)(.15)(.11)(-.6) = 0.0045



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     σp = 0.0671.
     When correlation = 0.5 we get:
     σ2p = (.25)2(.0225) + .(.75)2(.0121) + 2(.25)(.75)(.15)(.11)(.5) = 0.0113
     σp = 0.1063.

     When correlation = 1.0 we get:
     σ2p = (.25)2(.0225) + .(.75)2(.0121) + 2(.25)(.75)(.15)(.11)(1) = 0.0144.
     σp = 0.12.
     As the correlation between returns increases, so too do the variance and standard deviation of the
     portfolio. The benefits of diversification are being reduced as the association between returns
     increases.

13. βp = .25(.9) + .20(1.4) + .15(1.1) + .40(1.8) = 1.39

14. βp = 1.0 = 1/3(0) + 1/3(.8) + 1/3(βX) ; βX = 2.2

15. E[Ri] = .05 + (.14 – .05)(1.5) = .185

16. E[Ri] = .13 = .05 + .07βi ; βi = 1.14

17. E[Ri] = .10 = .06 + (E[RM] – .06)(.9) ;   E[RM] = .1044

18. E[Ri] = .14 = Rf + (.11 – Rf)(1.6) ; Rf = .060

19. a.     E[Rp] = (.15 + .05)/2 = .10
    b.     βp = 0.6 = xS(1.1) + (1 – xS)(0) ; xS = 0.6/1.1 = .5455 ; xRf = 1 – .5455 = .4545
    c.     E[Rp] = .09 = .15xS + .05(1 – xS) ; xS = .4; βp = .4(1.1) + .6(0) = 0.44
    d.     βp = 2.2 = xS(1.1) + (1 – xS)(0) ; xS = 2.2/1.1 = 2 ; xRf = 1 – 2 = –1
           The portfolio is invested 200% in the stock and –100% in the risk-free asset. This represents
           borrowing at the risk-free rate to buy more of the stock.

20. ßp = xW(1.4) + (1 – xW)(0) = 1.4xW
    E[RW] = .17 = .04 + MRP(1.40) ; MRP = .13/1.4 = .0929
    E[Rp] = .04 + .0929βp ; slope of line = MRP = .0929 ; E[Rp] = .04 + .0929βp = .04 + .13xW
                xW        E[Rp]         ßp         xW        E[Rp]        βp

                0%        .0400         0         100%       .1700       1.40
               25         .0725       0.35        125        .2025       1.75
               50         .1050       0.70        150        .2350       2.10
               75         .1375       1.05

21. E[Rii] = .06 + .075βi
    .17 > E[RY] = .06 + .075(1.45) = .1688;          Y plots above the SML and is undervalued.
    reward-to-risk ratio Y = (.17 – .06) / 1.45 = .0759
    .12 < E[RZ] = .06 + .075(0.85) = .1238;          Z plots below the SML and is overvalued.
    reward-to-risk ratio Z = (.12 – .06) / .85 = .0706

22. [.17 – Rf]/1.45 = [.12 – Rf]/0.85 ; Rf = .0492

         Intermediate

23. For a portfolio that is equally invested in large-company stocks and long-term bonds:
    return = (10.29% + 9.01%)/2 = 9.65%
    For a portfolio that is equally invested in small stocks and Treasury bills:



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     return = (13.31% + 6.89%)/2 = 10.10%

24. (E[RA] – Rf)/βA = (E[RB] – Rf)/ßB
    RPA/βA = RPB/βB ; βB/βA = RPB/RPA

25. a.     boom: E[Rp] = .4(.20) + .4(.35) + .2(.60) = .34
           normal: E[Rp] = .4(.15) + .4(.12) + .2(.05) = .118
           bust:      E[Rp] = .4(.01) + .4(–.25) + .2(–.50) = –.196
           E[Rp] = .2(.34) + .5(.118) + .3(–.196) = .0682
           σ2p = .2(.34 – .0682)2 + .5(.118 – .0682)2 + .3(–.196 – .0682)2 = .036956
           σp = [.036956]1/2 = .1922
     b.    RPi = E[Rp] – Rf = .0682 – .038 = .0302
     c.    Approximate expected real return = .0682 – .035 = .0332
           (1 + E[Ri]) = (1 + h)(1 + e[ri]) ;
           Exact expected real return = e[ri] = [1.0682/1.035] – 1 = .0321
           Approximate expected real risk premium = 0.0302 – 0.0350 = - 0.0048
           Exact expected real risk premium = 1.0302 / 1.035 – 1 = - 0.0046

26. xA = $200,000 / $1,000,000 = .20;        xB = $250,000/$1,000,000 = .25 ; xC + xRf = 1 – xA – xB =
    .55
    βp = 1.0 = xA(.7) + xB(1.1) + xC(1.6) + xRf(0); xC = .365625, invest .365625($1,000,000) = $365,625 in C.
    xRf = 1 – .20 – .25 – .365625 = .184375
    invest .184375($1,000,000) = $184,375 in the risk-free asset.

27. E[Rp] = .125 = wX(.28) + wY(.16) + (1 – wX – wY)(.07)
    βp = .8 = wX(1.6) + wY(1.2) + (1 – wX – wY)(0)
    solving these two equations in two unknowns gives wX = –0.05555555 wY = 0.74074047
    wR = 0.3148148
    amount of stock X to sell short = –0.05555555($100,000) = $5,555.55

28. E[RI] = .2(.09) + .6(.42) + .2(.26) = .322 ; .322 = .04 + .10βI , βI = 2.82
    σI2 = .2(.09 – .322)2 + .6(.42 – .322)2 + .2(.26 – .322)2 = .017296;           σI = [.017296]1/2 = .1315
    E[RII] = .2(–.30) + .6(.12) + .2(.44) = .10 ; .10 = .04 + .10βII , βII = 0.60
    σII2 = .2(–.30 – .10)2 + .6(.12 – .10)2 + .2(.44 – .10)2 = .05536;   σII = [.05536]1/2 = .2353

     Although stock II has more total risk than I, it has much less systematic risk, since its beta is much
     smaller than I’s. Thus, I has more systematic risk, and II has more unsystematic and more total risk.
     Since unsystematic risk can be diversified away, I is actually the “riskier” stock despite the lack of
     volatility in its returns. Stock I will have a higher risk premium and a greater expected return.

29. E[RPete Corp.] = .20 = Rf + 1.3(RM – Rf);          E[RRepete Co.] = .14 = Rf + .8(RM – Rf)
    .20 = Rf + 1.3RM – 1.3Rf = 1.3RM – .3Rf;                     .14 = Rf + .8(RM – Rf) = Rf + .8RM – .8Rf
    Rf = (1.3RM – .20)/.3                              RM = (.14 – .2Rf)/.8 = .175 – .25Rf

     Rf = [1.3(.175 – .25Rf) – .20]/.3
     .625Rf = .0275
     Rf = .044

     .20 = .044 + 1.3(RM – .044); RM = .164            .14 = .044 + .8(RM – .044);         RM = .164




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