Solution Assignment 3 Essay

Solution Assignment 3 Essay Solution Assignment 3 Essay Solutions to Assignment 3 1. a. using excel Stock A Stock B i. alpha -.609 2.964 ii. beta 1.183 1.021 iii. standard deviation of Residuals 4.676 4.983 iv. correlation with Market .757 .684 v. Average of the Market = 3.005 vi. Variance of the Market = 20.908 vii. First, we need Rf from SML 2.964 = RF + 1.183 [ 3.005 – RF] ïÆ'ž solving for RF = 3.352 Therefore for next year E(R A) = 3.352 + 1.183 (5 – 3.352) = 5.3 b . (i ) From single‑index model use: Rj = ÃŽ ±i - ÃŽ ²i Rm RA = ‑.609 + 1.183(3.005) = 2.946 RB = 6.032 RC = 3. 556 From the single‑index model the variance is: ÏÆ'2 i = ÃŽ ²i 2 ÏÆ'2 m + ÏÆ'ei2 ÏÆ'2 A = (1.183)2(20.908) + (4.677)2 = 51.14 ÏÆ'2 B = 46.62 ÏÆ'2 C = 265. 0 *The answers should be identical whichever way means and variances are computed. Any slight differences are due to rounding errors in the calculations. (ii) RA = 2.946 ÏÆ'2 A = 51.15 RB = 6.031 ÏÆ'2 B = 46.61 RC = 3. 554 ÏÆ'2 C = 265.0 c. (i ) Under the single‑index model covariance: cov(i j) = ÃŽ ²i ÃŽ ²j ÏÆ'm2 CovAB = (1.183)(1.021)(20.908) = 25.254 CovAC = 57.433 CovBC = 49.568 (ii) From the historic data itself: cov(i j) = ÃŽ £ (1/T-1)(Ri ‑ Ri)(Rj ‑ Rj) CovAB = 18.462 CovAC = 61.618 CovBC = 54.085 The calculations of covariances are different because the single‑index model computes covariances as if the correlation between residuals from the equation Ri = ÃŽ ±i + ÃŽ ²i Rm + ei are zero [cov(ei ej) = 0]. While computing covariance from historic data is equivalent to incorporating the historic level of cov(ei ej) into the measurement of covariance. d. For a portfolio made up of one‑half stocks A and B: (i) Expected return and standard deviation under the single‑index model: Rp = 1/2(2.946) + 1/2(6.032) = ÏÆ'p = [(1/2)2(51.14) + (1/2)2(46.62) + 2(1/3)2(25.25)+2(1/3)2(57.43) = (ii) Expected return and standard deviation using historical data: Rp = 1/2(2.946) + 1/2(6.031) = ÏÆ'p = [(1/2)2(51.15) + (1/2)2(46.61) + +2(1/3)2(18.46) = 2. a)We know by the CAPM:.18 = .04 + (.11 - .06) ï  ¢j which gives ï  ¢j = 2 The CAPM assumes that the market is in equilibrium and that investors hold efficient portfolios, i.e., that all portfolios lie on the security market line. b) Let â€Å"y† be the percent invested in the risk-free asset. Portfolio return is the point on the market line where 18% = y (4%) + (1 - y) (11%) and y = -1. Therefore, (1-y) = 2, i.e., the individual should put 200% of his portfolio into the market portfolio. 3. Assuming that the company pays no dividends, the one period expected rate of return, E(Rj) = [E(P1) - P0 ] / P0 where E(P1) = $179. Using the CAPM, we have E(Rj) = Rf + [E(Rm) - Rf] ï  ¢j = [E(P1) - P0 ] / P0 Substituting in the appropriate numbers and solving for P0, we have .08 + [.18 - .08]2.0 = [$100 - P0]/ P0 and solving for P0 = $154.3 4. Using the definition of the correlation coefficient, we have .8 = and cov (K, M) = .8(.25) (.2) = .04 Using the definition of Beta, we can calculate the systematic risk of MF: ï  ¢k = .04/(.2)2 = 1.0 The systematic risk of a portfolio is a weighted average of asset’s ï  ¢Ã¢â‚¬Ëœs. If â€Å"y† is the percent of MF, ï  ¢P = (1 - y) ï  ¢F+ y ï  ¢K or .8 = (1 - y ) 0 + y 1.0 or y =80% In this case the investor would invest an amount equal to 80 percent of his wealth in MF in order to obtain a portfolio with a ï  ¢ of .8 5. a) Using E(RP) = Rf + [E(Rm) - Rf] ï  ¢P to solve for ï  ¢P=2.2 b)We know that efficient portfolios have no unsystematic risk. The total risk is ï  ³2P= ï  ¢2P ï  ³2m + ï  ³2ï  ¥ and since the unsystematic risk of an efficient portfolio, ï  ³2ï  ¥ is zero, ï  ³P = ï  ¢P ï  ³m = 2.2 (.18) = .396 or 39.6% c)The definition of correlation is CorrJ m = cov (RJ,Rm) ï  ³J ï  ³m To find cov(Rj,Rm), use the definition of ï  ¢j = cov(Rj,Rm) ï  ³2 m Solving, we get Corr J m = 1.0, which indicates that the efficient portfolios are perfectly correlated with the market (and with each other). 6. We know from the CAPM : .13 = .04 + (.08)ï  ¢ J , solving which gives ï  ¢J= .1.125 If the rate of return covariance with the market