Investment Analysis and Portfolio Management Christmas Worksheet 2009 1. Assume there are two risky stocks available. Stock A has an expected return of 12% and a standard deviation of 8%. Stock B has an expected return of 10% and a standard deviation of 5%. a. Assuming the returns on the two stocks are uncorrelated, calculate the expected return and deviation of the portfolios consisting of: i. X A .5 , ii. X A 0 , iii. X A 0.25 , iv. X A 0.5 , v. X A 0.75 vi. X A 1 . b. Using the answers to (a), sketch the frontier of returns, the set of efficient portfolios and mark on the minimum variance portfolio. c. If all investors are risk averse, will short sales of either stock be observed? 2. Now introduce borrowing and lending at a risk-free rate of interest of 5%. a. Illustrate the new efficient frontier. b. Determine the structure of the market portfolio. c. How will an increase in the rate of interest affect the percentage of stock A in the market portfolio? 3. i. What is the “market model” and how is the beta of a stock interpreted? ii. How are historic betas calculated? Assume there are two stocks, A and B, with A 1.2 and B 0.9 , and indiosyncratic variations eA 4, eB 2 . iii. If assets A and B are the only risky assets available, what must be their proportions in the market portfolio? iv. Now assume that there are other risky assets available and that the risk-free rate of return is 5%. If the mean return on the market portfolio is rM 8 % and 2 the variance is M 25 , calculate the mean return and variance of a portfolio consisting of 70% of stock A and 30% of stock B. v. What is the mean return and variance of the portfolio in (iv) if a. It is 50% financed by borrowing? b. It is financed entirely by borrowing? 4. i. Assume that there are available two risky assets. Describe the efficient set when the correlation of their returns is: a. + 1 b. – 1 c. 0. ii. If the correlation is zero, show how the efficient set is modified when a riskfree asset is available. ii. For case (ii), describe the portfolio choice of a risk-averse investor. iii. How are the answers to (ii) and (iii) modified when the rate of interest for borrowing is greater than that for lending? 5. i. Define the single index model. How does it differ from the CAPM? ii. You have estimated a beta of 1.25 for M&P Inc. What factors would you take into account in adjusting this value? Consider the information on stocks A, B and C. Asset A B C beta 1.5 0.9 0.7 Expected return 13 9.4 8.2 Variance of return 30 16 9 iii. If the variance of the return on the market is 12, find the idiosyncratic error for each stock. Hence calculate the variance of a portfolio consisting of £100 of stock A, £300 of stock B and a short sale of £200 of stock C. iv. If the CAPM model applies, what are the expected return on the market and the risk-free rate? Hence find the expected return and variance of return if the portfolio in (iii) is 20% financed by borrowing. 6. i. Using the binomial pricing model calculate the value of a put option on a stock that currently sells for £50 but may rise to £55 or fall to £45 when there is 1 year to expiry, the risk free rate of return is 5% and the exercise price is £57.50. ii. Repeat (i) under the assumption that the year is split into i. 2 periods and ii. 3 periods but retaining the assumption that the highest price is achieved £55 and the lowest £45.