Question

3. John is an investor with treasury bill (T) and stock market (S) on his portfolio. If he puts all the money in treasury bill, he is expected to receive S1000 after 12 months with no risk. On the other hand, if he puts all the money in stock market, he is expected to receive $5000 with standard deviation of 3 after 12 months. a. What is the price of risk? b. Please draw John's investment indifference curves and his budget line, illustrate his utility-maximizing investment portfolio. c. How does his portfolio change if the stock market becomes more volatile than before?

Answer 1

(A) The price of risk is the risk of a decline of the value of stocks or other assets relative to the market. It is another name for Sharpe Ratio and can be calculated as -  

Price of Risk = Mean (Return) / S.D = 5000 / 3 = 1666.67 (Ans)

(B) The investment indifference curves can be best explained by using a risk- return trade off function where risk (Standard deviation ) is measured on the x-axis and return is measured along the Y axis. The indifference curves are upward sloping. The more risk averse John is, steeper will be the utility curve and it flattens if John is less risk averse. Now, An investor tries to reduce his risk by diversification and does so by choosing an optimum portfolio of both Treasury bills and stocks. The budget line represents combinations of risk and return that can be obtained with the given funds from a mixed portfolio of treasury bills and stocks. The return portfolio = R_p = W₁ R₁ + W₂R₂ ( W₁ + W₂ = 1) = > Rp = W1 (5000) + W2 (1000). Now, given his budget frontier, the combination of treasury bills and stocks that fit exactly into his budget gives the utility maximizing portfolio.

U1 Budget Frontier p- 1.0 Risk (o) 0.5

The above figure shows the combinations of different investment IC's and the budget frontier. The one where, slope of the budget frontier = slope of the investment IC gives the utility maximizing portfolio.

The budget frontier gives us = Risk on bills * Return + Risk on stocks * Return = 0 * 1000 + 3* 5000 = 15000

The IC gives us = W(1) * 1000 + W(2) * 5000

Utility max = 5000 W 1 + 1000 W 2 = 15000 ( W 1 + W 2 = 1)

Solving this gives us, W 1 = 3.5 and W 2 = -2.5 (Short selling)

(C) If stock markets become more volatile, it means that S.D increases so if SD increases then return on the portfolio increases. This implies that John would invest more in stocks compared to treasury bills.