Question

# 1. The goal of this exercise is to understand how to construct the optimal complete portfolio, which is the

combination of risk-free asset and an optimal risky portfolio.

Download the annual market return (proxied by return on large-cap stocks) from Blackboard Learn for the 90 year period, 1926-2015. Use the mean return and standard deviation of return (using the 90 annual returns) as the proxy for the risky portfolio's future expected return and risk. It turns out to be 12.0% and 19.9%.

Assume the latest annualized risk-free rate of return = 4%. Assume that the market portfolio is the proxy for the optimal risky portfolio. Assume correlation of market's return with the risk-free return = 0.

a) Compute the risk and expected return of the complete portfolios (various portfolios in the investment opportunity set). Start with 0% in the risky asset and go on to 200% in the risky asset (borrow $1 for every $1 of your own investment and invest $2 in risky asset) in increments of 5%.

b) Graph the investment opportunity set (capital allocation line). In the graph, make sure you mark clearly (i) the X-axis and Y-axis, (ii) the risky portfolio, and (iii) the risk-free portfolio.

c) What is the slope of CAL? How is it related to the Sharpe Ratio of the market portfolio?

d) Compute the utility for all portfolios on the CAL for 2 different investors, say, U2 and U4, whose coefficients of risk aversion are 2 and 4 respectively. Based on eye-balling these estimated utilities, what proportion of investment in the risky portfolio maximizes the utility of these two investors? Give your answer in a 10% range (say, between 20%-30%) because the portfolio weights increase in increments of 5%. Also, in a single graph plot the utility for both investors as a function of the fraction invested in risky portfolio to identify the utility maximizing optimal mix. In the graph, clearly mark (i) the X-axis and Y-axis and (ii) the risk-aversion of the investors whose lines are represented.

e) Using the analytical expression, compute the fraction of their wealth that the two investors should invest in the risky portfolio. (Now, you should be able to pinpoint the exact point within the range identified in the earlier question.) Whose optimal portfolio (A = 2 or A = 4 investor) has a higher weight on the risky portfolio? Are the results consistent with your expectation? Explain.

f) What is the expected return and standard deviation of the complete portfolio (the optimal mix of market portfolio and risk-free asset)? The answer will be different for the two investors because the optimal 'y' will be different. What is the maximal utility for each investor given their optimal capital allocation? Graphically, show that the optimal mix is the point at which CAL is tangential to the indifference curves. That is, draw two indifference curves (one for each investor) such that the CAL is exactly tangential to the indifference curves at the optimal mix of Market and risk-free asset. (Hint: To draw the indifference curves, compute the expected return for various portfolios given these portfolios' standard deviation and given U = U_{max}; also, note E(r) = U + 0.5Aσ^{2})

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