Question

Refer the table below on the average risk premium of the S&P 500 over T-bills and the standard deviation of that risk premium. Suppose that the S&P 500 is your risky portfolio. Period 1926-2015 1992–2015 1970–1991 1948-1969 1926-1947 Average Annual Returns S&P 500 1-Month Portfolio T-Bills 11.77 3.47 10.79 2.66 12.87 7.54 14.14 2.70 9.25 0.91 S&P 500 Portfolio Risk Standard Premium Deviation 8.30 20.59 8.13 18.29 5.33 18.20 11.44 17.67 8.33 27.99 Sharpe Ratio 0.40 0.44 0.29 0.65 0.30 a. If your risk-aversion coefficient is A= 2.5 and you believe that the entire 1926–2015 period is representative of future expected performance, what fraction of your portfolio should be allocated to T-bills and what fraction to equity? Assume your utility function is U= EU) - 0.5 ~ Ao. (Do not round intermediate calculations. Round your answers to 2 decimal places.) T-bills Equity b. If your risk-aversion coefficient is A= 2.5 and you believe that the entire 1970–1991 period is representative of future expected performance, what fraction of your portfolio should be allocated to T-bills and what fraction to equity? (Do not round intermediate calculations. Round your answers to 2 decimal places.) T-bills Equity

Answer 1

Capital allocation between Optimal risky portfolio and risk free assets can be computed with following equation.

y = Erp) - rf A*02

where,

y = weight of risky portfolio

(1-y) = weight of risk free assets

rf = risk free rate

A = Coefficient of Risk Aversion

$\large \sigma _p$ = Standard deviation of risky portfolio

This is formula is derived from Utility function (given in question) of Investor.

Please refer to below spreadsheet for calculation and answer. Cell reference also provided.

А в C D E F Period S&P 500 Return 11.77% T-bills rate 3.47% S&P 500 Risk Premium 8.30% S&P 500 Std. Deviation 20.59% 1926-2015 2.5 Capital Allocation T-bills Equity 21.69% 78.31% Period S&P 500 Return 12.87% T-bills rate | 7.54% S&P 500 Risk Premium 5.33% S&P 500 Std. Deviation 18.20% 1970-1991 2.5 Capital Allocation T-bills Equity 35.64% 64.36%

Cell reference -

B D E F Period S&P 500 Return T-bills rate S&P 500 Risk Premium =C3-D3 S&P 500 Std. Deviation 0.2059 1926-2015 0.1177 0.0347 A 2.5 Capital Allocation T-bills Equity =1-09 =E3/(C5*F3^2) Period S&P 500 Return T-bills rate 13 b. 14 S&P 500 Risk Premium =C14-D14 S&P 500 Std. Deviation 0.182 1970-1991 0.1287 0.0754 2.5 Capital Allocation T-bills Equity =1-C20 =E14/(C16*F14^2)

Hope this will help, please do comment if you need any further explanation. Your feedback would be highly appreciated.