Question

3. You have a risky portfolio that yields an expected rate of return of 15% with a standard deviation of 25%. Draw the CAL for an expected return/standard deviation diagram if the risk free rate is 5%.
a. What is the slope of the CAL?
b. If your coefficient of risk aversion is 5, how much should you invest in the risky portfolio?

4. A pension fund manager is considering three mutual funds. The first is a stock fund, the second is a long-term government and corporate bond fund, and the third is a T-bill money market fund that yields a sure rate of 5%. The probability distributions of the risky funds are:

	Expected Return	Standard Deviation
Stock fund (S)	15%	32%
Bond fund (B)	10%	23%

The correlation between the fund returns is 0.25.
a. Tabulate and draw the investment opportunity set of the two risky funds. Use investment proportions for the stock fund of 0% to 100% in increments of 10%. What expected return and standard deviation does your graph show for the minimum-variance portfolio? Use the Excel template I post on Blackboard.

b. Draw a tangent from the risk-free rate to the opportunity set. What does your graph show for the expected return and standard deviation of the optimal risky portfolio?

c. What is the Sharpe ratio of the best feasible CAL?

5. Continue with above problem. Suppose now that your portfolio must yield an expected return of 12% and be efficient, that is, on the best feasible CAL.
a. What is the standard deviation of your portfolio?
b. What is the proportion invested in the T-bill fund and each of the two risky funds?
c. If you were to use only the two risky funds and your coefficient of risk aversion A = 3.91, what would be the investment proportions of your portfolio? That is how would you invest in each of the risky assets and how much would you invest in the T-bill?

Answer 1

Ans. 3

Given in the problem is one risky portfolio comprising of two risky assets and one risk-free asset.

• Expected return on risky portfolio is 15%
• Standard deviation on risky portfolio is 25%
• Risk-free return is 5%

a) Slope of the CAL: ?

CAL stands for Capital Allocation Line. It's a line passing through the risk-free rate of return on the vertical axis and through the Portfolio frontier of the two risky assets. The slope of CAL can simply be calculated by dividing the difference of the expected return of the portfolio of risky assets (E(rp) and risk-free(rf) asset by the standard deviation of the risky portfolio. This slope is also known as Sharpe ratio.

Let us understand this by plotting a graph in Excel sheet. Kindly refer to the excel file images below:

Capital Allocation Line (CAL) Risky portfolio 15% Rate of return Standard deviation 15% 25% 5% Risky portfolio Risk-free asset 10% Slope = (15% - 5% e 25% = 0.04 CAL 25%- Expected rate of return 15%-5% Risk-free rate 0% 5% 25% 10% 15% Standard Deviation 20% 25% 30%

b. Slope of CAL is:

(E(rp) - rf) Stope - Standard deviation of risky portfolio

(15%-5%)/25% = 0.04

Slope =- (15% - 5%) 10% 25% -0 = 25% = 0.04

c. Risk-aversion coefficient is 5. The optimal weight of investment in the portfolio of risky assets is:

= [E(rp) - rf]/A*(Standard deviation of portfolio of risky assets)^2 = 32%