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]  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 = αi - βi Rm
RA = ‑.609 + 1.183(3.005) = 2.946 RB = 6.032 RC = 3. 556
From the single‑index model the variance is: σ2 i = βi 2 σ2 m + σei2 σ2 A = (1.183)2(20.908) + (4.677)2 = 51.14 σ2 B = 46.62 σ2 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 σ2 A = 51.15 RB = 6.031 σ2 B = 46.61 RC = 3. 554 σ2 C = 265.0

c. (i ) Under the single‑index model covariance: cov(i j) = βi βj σm2
CovAB = (1.183)(1.021)(20.908) = 25.254 CovAC = 57.433 CovBC = 49.568
(ii) From the historic data itself: cov(i j) = Σ (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 = αi + βi 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) = σp = [(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) = σp = [(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) j which gives 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 “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] 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: k = .04/(.2)2 = 1.0
The systematic risk of a portfolio is a weighted average of asset’s ‘s. If “y” is the percent of MF, P = (1 - y) F+ y 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  of .8
5. a) Using E(RP) = Rf + [E(Rm) - Rf] P to solve for P=2.2
b)We know that efficient portfolios have no unsystematic risk. The total risk is
2P= 2P 2m + 2 and since the unsystematic risk of an efficient portfolio, 2 is zero,
P = P m = 2.2 (.18) = .396 or 39.6%
c)The definition of correlation is CorrJ m = cov (RJ,Rm) J m
To find cov(Rj,Rm), use the definition of j = cov(Rj,Rm) 2 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) J , solving which gives J= .1.125
If the rate of return covariance with the market

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