FIN322 tutorial solutions CAPM PDF

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Risk and return Chapter 11: QP

a. What is the expected return on an equally weighted portfolio of these three stocks? Ans: To find the expected return of the portfolio, we need to find the return of the portfolio in each state of the economy. This portfolio is a special case since all three assets have the same weight. To find the expected return in an equally weighted portfolio, we can sum the returns of each asset and divide by the number of assets, so the expected return of the portfolio in each state of the economy is: Boom: Rp = (.06 + .16 + .33) / 3 Rp = .1833, or 18.33% Bust:

Rp = (.14 + .02 .06) / 3 Rp = .0333, or 3.33%

To find the expected return of the portfolio, we multiply the return in each state of the economy by the probability of that state occurring, and then sum. Doing this, we find: E(Rp) = .65(.1833) + .35(.0333) E(Rp) = .1308, or 13.08% b. What is the variance of a portfolio invested 20 percent each in A and B, and 60 percent in C? Ans: This portfolio does not have an equal weight in each asset. We still need to find the return of the portfolio in each state of the economy. To do this, we will multiply the return of each asset by its portfolio weight and then sum the products to get the portfolio return in each state of the economy. Doing so, we get: Boom: Rp=.20(.06) +.20(.16) + .60(.33) Rp =.2420, or 24.20% Bust:

Rp =.20(.13) +.20(.03) + .60(.06) Rp = –.0040, or –.40%

And the expected return of the portfolio is: E(Rp) = .65(.2420) + .35(.004) E(Rp) = .1559, or 15.59% To find the variance, we find the squared deviations from the expected return. We then multiply each possible squared deviation by its probability, and then add all of these up. The result is the variance. So, the variance of the portfolio is: p2 = .65(.2420 – .1559)2 + .35(.0040 – .1559)2  p2 = .013767

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Risk and return

24.Anal yz i ngaPor t f o l i o Yo uha v e$ 100, 000t oi n ve s ti napor t f ol i oc ont a i ni ngSt oc kXa ndSt o c kY. Yourgoa li st oc r e a t e apor t f ol i ot h a tha sa ne xpe c t e dr e t ur nof12. 9pe r c e nt . I fSt oc kXha sa ne xpe c t e dr e t ur nof11. 2 pe r c e nta n dabe t aof1 . 30a ndSt o c kYha sa ne xpe c t e dr e t ur nof7. 7pe r c e nta ndabe t aof. 80 , ho wmuc hmone ywi l ly oui n v e s ti nSt oc kY?Ho wdoy oui nt e r pr e ty ou ra ns we r ?Wh a ti st h e be t ao fy ourpor t f o l i o? Ans : Wea r egi v e nt h ee xpe c t e dr e t ur noft h ea s s e t si nt h epor t f ol i o . Wea l s okno wt h es u mo ft he we i ght sofe a c ha s s e tmu s tbee qua lt oone . Us i n gt h i sr e l a t i ons hi p, wec a ne xpr e s st hee xpe c t e d r e t ur noft h epor t f ol i oa s : E( Rp)=. 129=XX( . 112)+XY( . 077) . 1 29=XX( . 112)+( 1–XX) ( . 077) . 1 29=. 112XX+. 077–. 077XX . 0 52=. 035XX 4857 1 XX =1. Andt hewe i gh tofSt oc kYi s : XY =1–1. 485 71 4857 1 XY =–. Thea mo u ntt oi n v e s ti nSt oc kYi s : I n v e s t me nti nSt oc kY=– . 485 71( $1 00, 000) I n v e s t me nti nSt oc kY=–$48 , 571. 43 Ane g at i v epor t f ol i owe i ghtme a n st ha ty o us hor ts e l lt hes t oc k. I fy oua r enotf a mi l i a rwi t hs hor t s e l l i n g , i tme a n sy oub or r o was t o c kt oda ya nds e l li t .Youmus t t he npur c ha s et h es t oc ka tal a t e r da t et or e pa yt h ebor r o we ds t oc k. I fy ous ho r ts e l las t oc k, y ouma k eapr ofiti ft h es t oc k de c r e as e si nv a l ue . The beta of the portfolio is = WXx +WYY =1. 4 8571*1. 3+( –. 48 571)*0. 80=1. 54285 5

32. Beta and CAPM Suppose the risk-free rate is 4.7 percent and the market portfolio has an expected return of 11.2 percent. The market portfolio has a variance of .0382. Portfolio Z has a correlation coefficient with the market of .28 and a variance of .3285. According to the capital asset pricing model, what is the expected return on Portfolio Z? Ans: First, we need to find the standard deviation of the market and the portfolio, which are: M = (.0382)1/2 M = .1954, or 19.54% Z = (.3285)1/2 Z = .5731, or 57.31%

Now we can use the equation for beta to find the beta of the portfolio, which is:  Z) /  M Z = (Z,M)( Z = (.28)(.5731) / .1954 Z = .82

Now, we can use the CAPM to find the expected return of the portfolio, which is: 2

Risk and return

E(RZ) = Rf + Z[E(RM) – Rf] E(RZ) = .047 + .82(.112 – .047) E(RZ) = .1004, or 10.04%

Ans: The amount of systematic risk is measured by the  of an asset. Since we know the market risk premium and the risk-free rate, if we know the expected return of the asset we can use the CAPM to solve for the  of the asset. The expected return of Stock I is: E(RI) = .15(.11) + .55(.18) + .30(.08) = .1395, or 13.95% Using the CAPM to find the  of Stock I, we find: .1395 = .04 + .075I I = 1.33 The total risk of an asset is measured by its standard deviation, so we need to calculate the standard deviation of Stock I. Beginning with the calculation of the stock’s variance, we find: I2 = .15(.11 – .1395)2 + .55(.18 – .1395)2 + .30(.08 – .1395)2 I2 = .00209 I = (.00209)1/2 = .0458, or 4.58%

Using the same procedure for Stock II, we find the expected return to be: E(RII) = .15(–.25) + .55(.11) + .30(.31) = .1160 Using the CAPM to find the  of Stock II, we find: .1160 = .04 + .075II II = 1.01 And the standard deviation of Stock II is: II2 = .15(–.25 – .1160)2 + .55(.11 – .1160) 2 + .30(.31 – .1160)2 II2 = .03140  II = (.03140)1/2 = .1772, or 17.72%

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Risk and return 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. Chapter 12 QP

a. What is the systematic risk of the stock return? Ans: If m is the systematic risk portion of return, then: m = GDPΔGDP + InflationΔInflation + rΔInterest rates m = .0000734($17,863 – 17,034) – .90(2.60% – 2.80%) – .32(3.50% – 3.70%) m = 6.33% b. Suppose unexpected bad news about the firm was announced that causes the stock price to drop by 1.1 percent. If the expected return on the stock is 11.7 percent, what is the total return on this stock? Ans. The unsystematic return is the return that occurs because of a firm specific factor such as the bad news about the company. So, the unsystematic return of the stock is –1.1 percent. The total return is the expected return, plus the two components of unexpected return: the systematic risk portion of return and the unsystematic portion. So, the total return of the stock is: R= R +m+ R = 11.70% + 6.33% – 1.1% R = 16.93%

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Risk and return

Ans: The beta for a particular risk factor in a portfolio is the weighted average of the betas of the assets. This is true whether the betas are from a single factor model or a multi-factor model. So, the betas of the portfolio are: B1 = .20(1.55) + .20(.81) + .60(.73) B1 = .91 B2 = .20(.80) + .20(1.25) + .60(–.14) B2 = .33 B3 = .20(.05) + .20(–.20) + .60(1.24) B3 = .71 So, the expression for the return of the portfolio is: Ri = 3.2% + .91F1 + .33F2 – .71F3 Which means the return of the portfolio is: Ri = 3.2% + .91(4.90%) + .33(3.80%) – .71(5.30%) Ri = 5.11%

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