Question

1. The universe of available securities includes two risky stock funds, A and B, and T-blls. The data for the universe are as follows Expected Return Standard Deviation A 10% 20% В 30 60 T-bills 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 A 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 $1,940 $1,360 1.0 10% 40% A 0.2 В 2 30 1.7 17 50 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 B 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 A 13% 0.8 30% 1.2 18 40 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 B: 45 T-bills: 25

Answer 1

a. It can be shown that the weights associated with the optimal risky portfolio P can be determined using the following equations: Subscript D is for security A and E for security B and rf is the expected return on the risk-free T-bills LE(TO) - r;1052 – [E(rg) – r;COV (ro, rg) [E() - r;]0E2 + [E(rg) – r;]002 – [E(ro) – ry +E(ra) – rz]COV (rd,re) and wg = 1 - WD WD [(0.10 - 0.05) x 0.202 – (0.30 – 0.05) x -0.2] [(0.10 -0.05) x 0.202 + (0.30 -0.05) x 0.602 - ((0.10 -0.05) + (0.30 -0.05)] x -0.211 =0.052/0.152=0.342 We = 1-Wp = 1 -0.342 = 0.658 Determine the expected return and standard deviation of the optimal risky portfolio: E(rp) = 0.342 x 0.10 + 0.658 x 0.30 = 0.2316 Portfolio Variance = wA*SD (Ra) + w2, *SD (Rs) + 2*(w.)*(w.)*Cov(RAR) =0.3422 x 0.202 + 0.6582 x 0.602 + 2 x 0.342 x 0.658 (-0.2) = 0.070531 Op=v0.070531 = 0.2656

The CAL of the optimal risky portfolio Phas a slope of (23.16-5)/26.56=0.6837. This exceeds the slope of any portfolio consider thus far, and therefore produces the highest reward to variability ratio. The formula for the optimal weights (shown on the prior page) were determined by maximizing the function maxw.Sy = E(Tp) - 110 subject to the constraint that the portfolio weights sum to 1.0 (i.e. Wg + wp=1). By doing so, we end up with the weights that result in the risky portfolio with the highest reward-to variability ratio. C. Determine the optimal proportion of the complete portfolio to invest in the risky component. Using the following equation, y = EmpTI = (0.2316 - 0.05)/(5 x 0.070531) = 0.5149 , the investor would place 51.49% of his/her wealth in Portfolio P and 48.51% in T- Bills.