Chap011 - Chapter 11 An Alternative View of Risk and Return The Arbitrage Pricing Theory 11.1 Real GNP was higher than anticipated Since returns

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Unformatted text preview: Chapter 11: An Alternative View of Risk and Return: The Arbitrage Pricing Theory 11.1 Real GNP was higher than anticipated. Since returns are positively related to the level of GNP, returns should rise based on this factor. Inflation was exactly the amount anticipated. Since there was no surprise in this announcement, it will not affect Lewis-Striden returns. Interest Rates are lower than anticipated. Since returns are negatively related to interest rates, the lower than expected rate is good news. Returns should rise due to interest rates. The President's death is bad news. Although the president was expected to retire, his retirement would not be effective for six months. During that period he would still contribute to the firm. His untimely death mean that those contributions would not be made. Since he was generally considered an asset to the firm, his death will cause returns to fall. The poor research results are also bad news. Since Lewis-Striden must continue to test the drug as early as expected. The delay will affect expected future earnings, and thus it will dampen returns now. The research breakthrough is positive news for Lewis Striden. Since it was unexpected, it will cause returns to rise. The competitor's announcement is also unexpected, but it is not a welcome surprise. this announcement will lower the returns on Lewis-Striden. Systematic risk is risk that cannot be diversified away through formation of a portfolio. Generally, systematic risk factors are those factors that affect a large number of firms in the market. Note those factors do not have to equally affect the firms. The systematic factors in the list are real GNP, inflation and interest rates. Unsystematic risk is the type of risk that can be diversified away through portfolio formation. Unsystematic risk factors are specific to the firm or industry. Surprises in these factors will affect the returns of the firm in which you are interested, but they will have no effect on the returns of firms in a different industry and perhaps little effect on other firms in the same industry. For LewisStriden, the unsystematic risk factors are the president's ability to contribute to the firm, the research results and the competitor. 11.2 a. b. c. 11.3 a. b. c. 11.4 a. Systematic Risk = 0.042(4,480 4,416) 1.4(4.3% 3.1%) 0.67(11.8% 9.5%) = 0.53% Unsystematic Risk = 2.6% Total Return = 9.5% 0.53% 2.6% = 6.37% Systematic Risk = 2.04( 4.8% - 3.5% ) - 1.90(15.2% - 14.0% ) = 0.372% Unsystematic Return = 0.36( 27 - 23) = 1.44% Total Return = 10.0 + 0.37 + 1.44 = 11.81% Stock A: R A = RA + A ( R m - Rm ) + A = 10.5% + 1.2( R m - 14.2%) + A Answers to End-of-Chapter Problems B-115 Stock B: R B = R B + ( R m - R m ) + B = 13.0% + 0.98( R m - 14.2%) + B Stock C: R C = RC + C ( R m - R m ) + C = 15.7% + 1.37( R m - 14.2%) + C b. R P = 0.30R A + 0.45R B + 0.25R C = 0.30[10.5% + 1.2( R m - 14.2% ) + A ] + 0.45[13.0% + 0.98( R m - 14.2% ) + B ] + 0.25[15.7% + 1.37( R m - 14.2% ) + c ] = 0.30(10.5% ) + 0.45(13% ) + 0.25(15.7% ) + [ 0.30(1.2 ) + 0.45( 0.98) + 0.25(1.37 ) ] ( R m - 14.2% ) = 12.925% + 1.1435( R m - 14.2% ) + 0.30 A + 0.45 B + 0.25 C c. i. + 0.30 A + 0.45 B + 0.25 c R A = 10.5% + 1.2(15% - 14.2%) = 11.46% R B = 13% + 0.98(15% - 14.2%) = 13.7% R C = 15.7% + 1.37( 15% - 14.2%) = 16.8% R P = 12.925% + 11435(15% - 14.2%) . = 138398% . ii. 11.5 a. Since five stocks have the same expected returns and the same betas, the portfolio also has the same expected return and beta. R p = 11.0 + 0.84 F1 + 1.69F2 + b. 1 ( E1 + E 2 + E 3 + E 4 + E 5 ) 5 B-116 Answers to End-of-Chapter Problems R p = 11.0 + 0.84 F1 + 1.69F2 + As N , E E1 E 2 + +...+ N N N N 1 0, but E js are finite, N Thus, R p = 11.0 + 0.84 F1 + 1.69F2 11.6 To determine which investment investor would prefer, you must compute the variance of portfolios created by many stocks from either market. Note, because you know that diversification is good, it is reasonable to assume that once an investor chose the market in which he or she will invest, he or she will buy many stocks in that market. Known: Assume: E F = 0 and = 0.1 E = 0 and i = 0.2 for all i. The weight of each stock is 1/N; that is, X i = 1 / N for all i. If a portfolio is composed of N stocks each forming 1/N proportion of the portfolio, the return on the portfolio is 1/N times the sum of the returns on the N stocks. Recall that the return on each stock is 0.1+F+ . R P = (1/N ) R i = ( (1/N) ( 0.1 + F + ) ) = 0.1 + F + (1/N ) E ( R P ) = E [ 0.1 + F + (1/N ) ] = 0.1 + E( F) + (1/N ) E ( ) = 0.1 + ( 0 ) + (1/N ) 0 = 0.1 2 2 Var ( R P ) = E [ R P - E ( R P ) ] = E [ 0.1 + F + ( 0.1/N ) - 0.1] = E [ F + (1/N ) ] 2 = E 2 F 2 + 2F(1/N ) + 1/N 2 () 2 = 2 s 2 + (1/N ) s 2 i + (1 - 1/N ) Cov ( i , j ) In the limit as N , the variance is [ ( ) = 2 s 2 + Cov ( i , j ) = 0.01 2 + 0.04Corr ( i , j ) Answers to End-of-Chapter Problems B-117 Thus, R1i = 0.10 + 1.5F + 1i ( ) Var( R ) = 0.0225 + 0.04Corr( Var ( R ) = 0.0025 + 0.04Corr( E R 1p = E( R 2 P ) = 0.1 1p 2P R 2 i = 0.10 + 0.5f + 2 i 1i , 1 j 2i ) , 2 j ) a. Corr 1i , 1 j = Corr 2 i , 2 j = 0 ( ) Var( R ) = 0.00225 Var R1p = 0.0225 2p ( ) ( ) Since Var R1p Var R 2p , a risk averse investor will prefer to invest in the second market. b. Corr 1i , 1 j = 0.9 and Corr 2i , 2 j = 0 1p ( ) ( ) ( ) Var( R ) = 0.0585 Var( R ) = 0.0025 2p ( ) Since Var R 1p Var R 2p , a risk averse investor will prefer to invest in the second market. c. ( ) ( ) Corr 1i , 1 j = 0 and Corr 2i , 2 j = 0.5 ( ) Var( R ) = 0.0225 Since Var( R ) = Var( R ) , a risk averse investor will be indifferent Var R1p = 0.0225 2p 1p 2p ( ) ( ) between investing in the two market. d. Indifference implies that the variances of the portfolio in the two markets are equal. Var R1p = Var R 2 p ( ) ( ( 0.0225 + 0.04Corr 1i , 1 j = 0.0025 + 0.04Corr 2 i , 2 j Corr 2 i , 2 j = Corr 1i , 1 j + 0.5 B-118 ) ( ) ( ) ) ( ) Answers to End-of-Chapter Problems This is exactly the relationship used in part c. 11.7 a. Var( R j ) = i Var( R m ) + Var( i ) 2 s 2 A = 0.7 2 (1.21) +1.00 = 1.5929% s 2 B = 1.2 2 (1.21) +1.44 = 3.1824% s C = 1.5 2 (1.21) + 2.25 = 4.9725% 2 s A = 1.5929/100 = 12.62% s B = 3.1824/100 = 0.1784 = 17.84% s C = 4.9725/100 = 0.2230 = 22.30% b. i. As N , Var ( j ) /N 0 Var( R i ) = i Var( R m ) 2 2 s A = 0.7 2 (1.21) = 0.5929% s 2 = 1.2 2 (1.21) = 1.7424% B 2 s C = 1.5 2 (1.21) = 2.7225% ii. APT Model: R i = R F + ( R m - R F ) i R A = 3.3 + (10.6 - 3.3)(0.7) = 8.41% R B = 3.3 + (10.6 - 3.3)(1.2) = 12.06% R C = 3.3 + (10.6 - 3.3)(1.5) = 14.25% APT Model shows that assets A & B are accurately priced but asset C is overpriced. Thus, rational investors will not hold asset C. iii. opportunity until the return 11.8 a. If short selling is allowed, all rational investors will sell short asset C so that the price of asset C will decrease until no arbitrage exists. In other words, price of asset C should decrease become 14.25%. Let X= the proportion of security of one in the portfolio and (1-X) = the proportion of security two in the portfolio. R pt = XR 1t + (1 - X ) R 2 t = x[ E ( R 1t ) + 11 F1t + 12 F2 t ] + (1 - x ) [ E( R 2 t ) + 21 F1t + 22 F2 t ] The condition that the return of the portfolio does not depend on F1 implies: Answers to End-of-Chapter Problems B-119 X11 + (1 - X ) 21 = 0 X + (1 - X)0.5 = 0 Thus, P=(-1,2); i.e. sell short security one and buy security two. p 2 = ( - 1)(1.5) + 2( 2 ) = 2.5 b. E ( R p ) = ( - 1) 20% + 2( 20% ) = 20% Follow the same logic as in part a, we have X 31 + (1 - X ) 41 = 0 X + (1 - X )1.5 = 0 X=3 Where X is the proportion of security three in the portfolio. Thus, sell short security four and buy security three. p 2 = 3 ( 0.5) - 2 ( 0.75) = 0 this is a risk free portfolio! c. E ( R p ) = 3 (10% ) + ( - 2 ) (10% ) = 10% The portfolio in part b provides a risk free return of 10% which is higher than the 5% return provided by the risk free security. To take advantage of this opportunity, borrow at the risk free rate of 5% and invest the funds in a portfolio built by selling short security four and buying security three with weights (3,-2). Assuming that the risk free security will not change. The price of security four ( that everyone is trying to sell short) will decrease and the price of security three ( that everyone is trying to buy ) will increase. Hence the return of security four will increase and the return of security three will decrease. The alternative is that the prices of securities three and four will remain the same, and the price of the risk-free security drops until its return is 10%. Finally, a combined movement of all security prices is also possible. The prices of security four and the risk-free security will decrease and the price of security four will increase until the opportunity disappears. E Rj 20% 10% 5% d. ( ) 12 ( i1 = 0) 0 2.5 B-120 Answers to End-of-Chapter Problems ...
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This note was uploaded on 05/07/2010 for the course FIN 302 taught by Professor Corporationfinance during the Spring '10 term at Uni Potsdam.

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