INVESTMENT PORTFOLIO MANAGEMENT QUESTION SOLUTIONS, Assignments of Investment Management and Portfolio Theory

Download INVESTMENT PORTFOLIO MANAGEMENT QUESTION SOLUTIONS and more Assignments Investment Management and Portfolio Theory in PDF only on Docsity! INVESTMENT PORTFOLIO ANALYSIS 1 CHAPTER 6 Question 1 Consider a risky portfolio. The end-of-year cash flow derived from the portfolio will be either $70,000 or $200,000 with equal probabilities of .5. The alternative risk-free investment in T-bills pays 6% per year. 1. If you require a risk premium of 8%, how much will you be willing to pay for the portfolio? Solution Typically, an investment needs yield an expected return equivalent to the risk free rate plus the risk premium to enable someone to invest in it. Using the utility function → U (r )=E (r )− 1 2 A σ2 (E [R]=6%+8%=14%) E [ R ]=14%= E [ Pt+1 ]−Pt Pt ¿ 0.5× $ 70000+0.5 × $ 200000 Pt −1 1.14= $ 135000 Pt Pt= $ 135000 1.14 Pt=$ 118,421.05 2. Suppose that the portfolio can be purchased for the amount you found in (a). What will be the expected rate of return on the portfolio? Solution The expected rate of return on the portfolio will be (E [ R ]=6 %+8 % 2 2. Suppose that your risky portfolio includes the following investments in the given proportions: Stock A 25% Stock B 32% Stock C 43% What are the investment proportions of your client’s overall portfolio, including the position in T-bills? Solution Investments Proportions Stock A 0.7 × 25% 17.5% Stock B 0.7× 32% 22.4% Stock C 0.7 × 43% 30.1% T-Bills - 30% Total 100% 3. What is the reward-to-volatility ratio (S) of your risky portfolio? Your client’s? Solution The Reward to volatility Ratio of my risky portfolio ¿ 18−8 28 ¿0.3571 The Reward to volatility Ratio of my client’s risky portfolio¿ 15−8 19.6 ¿0.3571 5 Clients P CAL(slope=0.3571) 4. Draw the CAL of your portfolio on an expected return–standard deviation diagram. What is the slope of the CAL? Show the position of your client on your fund’s CAL. Solution 5. Suppose that your client decides to invest in your portfolio a proportion y of the total investment budget so that the overall portfolio will have an expected rate of return of 16%. Solution a. What is the proportion y? The expected return of portfolio ¿ t f +( r p−r f ) × y=8+10 y Given that the expected return of portfolio ¿16% 6 25 50 Therefore y=0.8 implying an 80% investment in risk fund and remaining 20% in T-bills. b. What are your client’s investment proportions in your three stocks and the T-bill fund? Investments Proportions Stock A 0.8×25% 20% Stock B 0.8 ×32% 25.6% Stock C 0.8× 43% 34.4% T-Bills - 20% Total 100% a. What is the standard deviation of the rate of return on your client’s portfolio? The Standard Deviation ¿0.8 ×28% ¿22.4 6. Suppose that your client prefers to invest in your fund a proportion y that maximizes the expected return on the complete portfolio subject to the constraint that the complete portfolio’s standard deviation will not exceed 18%. Solution a. What is the investment proportion, y? Since Standard deviation ¿ y ×28 % <18% y= 18 28 ¿64.29 % b. What is the expected rate of return on the complete portfolio? expected rate of return ¿8+10 y=14.43% 7 Solution 1. What are the investment proportions in the minimum-variance portfolio of the two risky funds, and what is the expected value and standard deviation of its rate of return? σ 2 p=w s 2 σb 2 +2 w s wb cov (B , S) wb=1−w s, σ s 2 =900 σ b 2 =225 cov ( B , S )=p σb σ s=45 minσ2 p , ws=0.1739, wb=0.8261 Expected return ¿0.1739 ×20+0.8261×12=13.38 % Standard Deviation ¿13.92 % 2. Tabulate and draw the investment opportunity set of the two risky funds. Use investment proportions for the stock fund of zero to 100% in increments of 20%. Stock (%) Bond (%) Expected return Variance 0 100 12 15 17.39 82.61 13.39 13.92 20 80 13.6 13.94 40 60 15.2 15.70 45.16 54.84 15.61 16.54 60 40 16.80 19.53 80 20 18.40 24.48 100 0 20 30 3. Draw a tangent from the risk-free rate to the opportunity set. What does your graph show for the expected return and standard deviation of the optimal portfolio? Max slope proportion of stock ¿4516 and bond ¿0.5484 Expected return of optimal portfolio ¿0.4516 ×20+0,5484 ×12=15.61 10 Standard variance ¿16.54 4. Solve numerically for the proportions of each asset and for the expected return and standard deviation of the optimal risky portfolio. wE ¿ =(E [ RD ]−R f )Var ( RE )−(E [ RE ]−R f )Cov ¿¿ ¿ (0.12−0.08 ) 0.302−(0.20−0.08 ) 0.0045 (0.12−0.08 )0.302+(0.20−0.08 ) 0.152−( 0.12+0.20−2 ×0.8 )0.0045 ¿ 0.0036−00054 0.0036+0.0027−0.00072 ¿0.5484 E [ RRP ]=wE ¿ E [ RE ]+(1−wE ¿ ) E [RD ]=16.4 % σ ( RRP )=¿ ¿18.4% 5. What is the Sharpe ratio of the best feasible CAL? SP ¿ = E [R p ]−R f σ ( Rp ) ¿ 0.164−0.08 0.184 ¿0.4565 6. You require that your portfolio yield an expected return of 14%, and that it be efficient, on the best feasible CAL. a. What is the standard deviation of your portfolio? E (rc )=r f +E ( r p )−rf E (r p )=15.61 % , σ p=16.54 % No required return = 14%, then standard variance = 13.04% E (rc )=(1− y ) r f + yE (r p ) , y=0,7884, 1− y=0.2116 E (rc )=14 % , E (r p )=16.54 % 11 b. What is the proportion invested in the T-bill fund and each of the two risky funds? Investment Proportions T-Bill 1−0.4762 0.5238 Stock Fund 0.4762 ×0.5484 0.2611 Bond Fund 0.4762 × (1−0.5484 ) 0.2151 7. If you were to use only the two risky funds, and still require an expected return of 14%, what would be the investment proportions of your portfolio? Compare its standard deviation to that of the optimized portfolio in Problem 9. What do you conclude? If the complete portfolio just includes two risky securities, then 14%=20% w s+12% (1−w s ) , w s=0.25, w b=0.75, σ p=14.13% In comparison, it is clear that this is not the optimal portfolio. 12