Question

Suppose the risk free rate is 4%, the expected rate of return is 8% and the variance of a portfolio is estimated as 500%. What is the slope of the Capital Allocation Line? Interpret the meaning of your estimation of the slope. [see p.168 of your text]

Imagine that the expected returns that are associated with an investment are 0.06, 0.08, 0.094, 0.12 and 0.134. If the standard deviations that are associated with the returns are 0, 0.022, 0.066, 0.154, and 0.176 respectively, what is the investor’s utility function? [see p. 159, pp.170-171 of your text]

Plot the utility function of the investor on a scale of 0, 0.1, 0.2, 0.3, and 0.4 for additional credit.

Answer 1

Slope of the Capital Allocation Line(CAL) is Sharpe ratio and the formula is as below:-

Sharpe ratio = (Expected rate of return - Risk free rate)/ (Square Root of (Variance of the portfolio) or standard deviation) = (8% -4%)/ SQRT(500%) = 0.018

The sloe of the CAL indicates that for every unit of excess portfolio return, one needs to take 0.018 units of risk.

The investor's utiltiy function can be used comparatively for a comparative measure of risk tolerance and not an absolute tolerance. The utility formula is shown below:-

Utility Score = Expected Return – 0.005 X Variance × Risk Aversion Coefficient

The risk aversion coefficient is a number proportionate to the amount of risk aversion of the investor and is usually set to integer values less than 6, and 0.005 is a normalizing factor to reduce the size of the variance, ?², which is the square of the standard deviation (?), a measure of the volatility of the investment and therefore its risk.

In the current example, let us assume the risk aversion coefficient be equal to 3. If an investor is more risk averse, we might use 4 instead of 3 to indicate the investor's greater aversion to risk.

Expected return	Standard Deviation	Variance	Utility Score
6%	0.0%	0.0%	6.00%
8%	2.2%	0.0%	8.00%
9.40%	6.6%	0.4%	9.39%
12%	15.4%	2.4%	11.96%
13.40%	17.6%	3.1%	13.35%

Assuming he risk free rate is 4%, the expected rate of return is 8% and the variance of a portfolio is estimated as 500%, utility score assuming risk aversion coefficient of  0, 0.1, 0.2, 0.3, and 0.4 is as below:-

Scale	Utility Score
0.0	4.00%
0.1	3.75%
0.2	3.50%
0.3	3.25%
0.4	3.00%