Question

1. The universe of available securities includes two risky stock funds, A and B and T-bills. The data for the universe are as follows: Expected Return Standard Deviation 109 20 Tbilis The correlation coefficient between funds A and B is -0.2. a. Find the optimal risky portfolio, P. and its expected return and standard deviation b. Find the slope of the CAL supported by T-bills and portfolio P. c. How much will an investor with 4-5 invest in funds A and B and in T-bills? 2. The data below describe a three-stock financial market that satisfies the single-index model Mean Excess Standard Stock Capitalization Beta Return Deviation $3,000 1.0 10% 4096 $1.940 2 $1,360 02 17 The standard deviation of the market index portfolio is 25% a. What is the mean excess return of the index portfolio b. What is the covariance between stock A and stock B? c. What is the covariance between stock and the index? d. Break down the variance of stock B into its systematic and firm-specific components. 3. The following are estimates for two stocks. Stock Expected Return Beta Firm-Specific Standard Deviation 30% 13% 18 12 The market index has a standard deviations of 22% and the risk-free rate is 8%. a. What are the standard deviations of stocks A and B? b. Suppose that we were to construct a portfolio with proportions: Stock A: 30 Stock : 45 T-bills: 25 Compute the expected return, standard deviation, beta, and nonsystematic standard deviation of the portfolio 4. Assume that the total market that the owner of this portfolio

Answer 1

1)

T Bill rate	5.00%
Correlation ρ	-0.20
	Expected Return (R )	Standard Deviation (σ)
Stock Fund (S)	10%	20%
Bond Fund (B)	30%	60%

Cov (B,S) =	ρ *σS *σB = -0.2*0.2*0.6
Cov (B,S) =	-0.024

Proportion of Stocks in optimal risky portfolio
Ws=	(Rs-rf)*σ2B - (RB-rf)Cov (B,S) /
	(RB-rf)*σ2S + (RS-rf)σ2B -(RS-rf+RB-rf) (Cov (B,S)

Ws = [(0.1-0.05)*0.6² - (0.3-0.05)*0.2² *-0.024] / [(0.3-0.05)*0.2² + (0.1-0.05)*0.6² - (0.1+0.3-0.05-0.05)*(-0.024)]

Ws=	68.1818182%

Proportion of Bonds in optimal risky portfolio

Wb=1-Ws = 1-0.6818 = 0.31818

Portfolio Return E (R )=

Ws*Rs+WB*Rb = 0.6818*0.1+0.3181*0.3 =

16.3636%

Portfolio Std Dev σp

√(W2sσ2S + W2Bσ2B + 2WsWBCov (B,S)) = √ (0.68182 * 0.22 + 0.31812 * 0.62 + 2*0.6818*0.31818*(-0.024)

21.1254%

Sharpe Ratio (Risk to Reward)	(E(Rp)-rf) / σp	E(Rp) is return of optimal risky portfolio
	(16.3636%-5%) / 21.1254%	σp is std dev of optimal risky portfolio

SR=

0.5379

Thus slope of CAL is 0.5379

c)

If A is given

y= (E(Rp)-rf)/ (A*σ2p)

where y is proportion invested in T bills

y= (16.3636%-5%) / (5*21.1254² ) = 0.5093

1-y = 0.4907

Proportion of stocks in complete portfolio =y*Ws = 0.5093*0.6818 =0.3472 =34.72%

Proportion of Bonds in complete portfolio = y*Wb = 0.5093*0.31818 = 0.162 = 16.2%

Investment in T bill =0.4907 =49.07%