FBE555_Assignment4_sol

# FBE555_Assignment4_sol - University of Southern California...

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University of Southern California Marshall School of Business FBE 555: Investments - Fall 2007 Professor Joe Chen Solutions to Homework Assignment #4 1. Suppose that there are two assets, A and B, that are perfectly correlated ( ρ AB =1.0). Suppose further that E(R A )=10%, σ A =20%, E(R B )=15%, σ B =30%, and R f =5%. There exists an arbitrage opportunity here. Find the arbitrage strategy: state very carefully which asset you will buy, which you will sell, and how much (in what proportion). The first step is to find the risk-free portfolio using just the risky assets. So we want to find a portfolio P with weight ω in asset A and (1- ω ) in asset B such that σ p =0. So, we need to solve ω A 2 σ A 2 + (1 ω A ) 2 σ B 2 + 2 ω A (1 ω A ) σ A σ B = 0 , or equivalently, ω A σ A + (1 ω A ) σ B ( ) 2 = 0 . This has the solution ω A = σ B σ B σ A = 0.30 0.30 0.20 = 3 , and ω B =1- ω A =-2, which is a short position. The expected return of this portfolio is E[r p ] = 3.0 *10% + -2.0 * 15% = 0.00%. Which is less than the risk-free rate. (gets a bit tricky here….) – so what we have is a riskless investment opportunity that yields less than the riskfree rate. So we would want to borrow at the low rate and deposit at the higher rate. Therefore, the correct strategy is to short this portfolio and deposit the proceeds at the risk-free rate of 5%. The correct strategy is to short-sell \$300 worth of asset A, buy \$200 worth of asset B, deposit \$100 worth in the risk-free rate. The resulting portfolio will produce riskless return of \$5 – now, repeat as much as possible.

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2. The expected return on the Market Portfolio M is E(R M )=15%, the standard deviation is σ M =25% and the risk-free rate is R f =5%. The CAPM is assumed to hold. a. Draw a diagram of the Capital Market Line derived from the above data. Make sure to clarify the intercept and the slope. The Capital Market Line is a straight line on the mean-standard deviation graph (expected return on the Y-axis and standard deviation on the X-axis). The intercept is the risk-free rate of 5% and the slope is the Sharpe ratio of the Market portfolio: SR M = E [ R M ] R f σ M = 0.15 0.05 0.25 = 0.4 b. Draw a diagram of the Security Market Line. Identify where individual stocks with beta of one and stocks with beta of zero would lie. The Security Market Line is a straight line in the mean-beta graph (expected return on the Y-axis and beta on the X-axis). The intercept is the risk-free rate of 5% and the slope is the Market Risk Premium of 15%-5%=10%. c. According to CAPM, what is the expected rate of return of a zero-beta security? Risk-free rate of return d. Compute the expected return of two well-diversified portfolios (i.e. portfolios on the CML), one with st.dev of 18%, and the other with st.dev of 30%. From the CML, E [ R p ] = R f + E [ R M ] R f σ M σ P = 0.05 + 0.4 σ P So a well-diversified portfolio with standard deviation of 18% has an expected return of 0.05+0.4(.18)=12.2% and a portfolio with standard deviation of 30% has an expected return of 0.05+0.4(.30)=17%.
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