Annual return

Year Market 1926 11.62% 1927 37.49% 1928 43.61% 1929 -8.42% 1930 -24.90% 1931 -43.34% 1932 -8.19% 1933 53.99% 1934 -1.44% 1935 47.67% 1936 33.92% 1937 -35.03% 1938 31.12% 1939 -0.41% 1940 -9.78% 1941 -11.59% 1942 20.34% 1943 25.90% 1944 19.75% 1945 36.44% 1946 -8.07% 1947 5.71% 1948 5.50% 1949 18.79% 1950 31.71% 1951 24.02% 1952 18.37% 1953 -0.99% 1954 52.62% 1955 31.56% 1956 6.56% 1957 -10.78% 1958 43.36% 1959 11.96% 1960 0.47% 1961 26.89% 1962 -8.73% 1963 22.80% 1964 16.48% 1965 12.45% 1966 -10.06% 1967 23.98% 1968 11.06% 1969 -8.50% 1970 4.01% 1971 14.31% 1972 18.98% 1973 -14.66% 1974 -26.47% 1975 37.20% 1976 23.84% 1977 -7.18% 1978 6.56% 1979 18.44% 1980 32.42% 1981 -4.91% 1982 21.41% 1983 22.51% 1984 6.27% 1985 32.16% 1986 18.47% 1987 5.23% 1988 16.81% 1989 31.49% 1990 -3.17% 1991 30.55% 1992 7.67% 1993 9.99% 1994 1.31% 1995 37.43% 1996 23.07% 1997 33.36% 1998 28.58% 1999 21.04% 2000 -9.11% 2001 -11.88% 2002 -22.10% 2003 28.70% 2004 10.87% 2005 4.91% 2006 16.10% 2007 6.43% 2008 -36.03% 2009 25.25% 2010 13.48% 2011 2.56% 2012 16.01% 2013 32.14% 2014 13.55% 2015 1.89%

Use the mean return and standard deviation of return (using the 90 annual returns) as the proxy for the risky portfolios 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 markets return with the risk-free return = 0.

1a) 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 + 0.5A²)