Problem 1 You are given the following information about stock X and the market portfolio, M: o Riskless Asset (f) Stock X Market Portfolio (M) ET 0.04 (4%) 0.00 ? 0.30 0.10 0.20 You are not given the expected return of stock X. The correlation of the returns on the stock X and the market portfolio is equal to 0.4. a) What is the beta (B) of stock X? b) Assuming the CAPM holds, what is the expected return on stock X? c) You have $1,000 to invest in some combination of the risk-free asset, stock X, and the market portfolio. You are thinking of investing $300 in the risk free asset, $400 in stock X, and $300 in the market portfolio. What is the overall expected return, standard deviation and beta of this portfolio? d) Instead of making the investment described in part c), you decide to be a little more sophis- ticated. You are willing to accept an overall standard deviation on your investment of up to 30%, so you decide to invest your $1,000 in whatever combination of the risk-free asset, stock X, and the market portfolio gives you the highest possible expected return, given a standard deviation of 30%? How much money do you invest in each of the three securities, and what is the expected return you achieve? You can assume that stock X is part of the market portfolio. Problem 2 The graphs below illustrate the properties of 4 assets: The riskless asset (f), an individual stock (stock A), a mutual fund (fund H) and the market portfolio of risky assets (M). 0.16 0.16 0.15 2 0.15 CML 0.14 0.14 0.13 0.13 0.12 0.12 Stock A Stock A 0.11 0.11 SML 0.10 0.10 0.09 Market, M 0.09 Market, M 0.08 0.08 Fund H Expected Return, E(T) 0.07 Fund H Expected Return, E(r) 0.07 0.06 0.06 0.05 0.05 0.04 0.04 f Riskless asset f 0.03 0.03 0.02 0.02 0.01 0.01 0.00 0.00 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 0.000 0.025 0.050 0.075 0.100 0.125 0.150 0.175 0.200+ 0.225 0.250 0.275 0.300 0.325 0.350 0.375 0.400 0.425+ 0.450 Beta, Standard Deviation, a) What is the Sharpe ratio of stock A? What is the Sharpe ratio of fund H? b) Suppose you could invest in either (1) a portfolio of the riskless asset and stock A, or (2) a portfolio of the riskless asset and fund H. What would you choose and why? c) What is the alpha (a) of stock A? What is the alpha (a) of fund H? Note on the variance of a portfolio of three assets When one the three assets is the risk free rate you can use the formula for the two risky assets, except that you still need to guarantee that wi + W2 + W3 = 1. Let me show you that. This is the formula of the variance for three risky assets: om = wc + wo+ wo + 2W1W20102P12 + 2w1W30103213 + 2w3w20302P32 But if asset 3 is the risk free rate then 03 = 0, which reduces the above formula to: om wc + wo + 2w1w20102012 But the weights still need to sum to one this way: Wi + W2 + W3 - 1