Question

For Problems 12–16, assume that you manage a risky portfolio with an expected rate of return of 17% and a standard deviation of 27%. The T-bill rate is 7%.

Suppose the same client as in the previous problem prefers to invest in your portfolio a proportion ( y) that maximizes the expected return on the overall portfolio subject to the constraint that the overall portfolio’s standard deviation will not exceed 20%. (LO 5-3)

a. What is the investment proportion, y?

b. What is the expected rate of return on the overall portfolio?

Short answer

a. 74.07%

b. 15%

Step-by-step solution

Maximization of expected return

Maximization of the expected return occurs when investors targets to earn the highest profit by splitting their investment in various securities without exceeding their risk tolerance.

a. Calculation of y:

Here, the target standard deviation is $\le 20\%$. The current standard deviation is 27%. The value of y can be found in the below manner.

$\begin{array}{rcl}y\times 27\%& \le & 20\%\\ y& \le & \frac{20\%}{27\%}\\ y& \le & 0.7407\end{array}$

Hence, the proportion of y must be 0.7407 i.e., 74.07%.

b. Calculation of expected rate of return:

Since the proportion of y, i.e., the proportion of the risky portfolio, is 74.07%, the proportion of the T-bill rate would be

Propotion of T-bill rate = 1 - y

= 1 - 74.07%

= 0.2593

= 25.93%

Hence, the proportion of the T-bill rate is 20%. With these weights, the expected return rate can be computed.

$\begin{array}{rcl}Expectedrateofreturn& =& \left(Weightofriskyportfolio\times Rateofretunofriskyportfolio\right)\\ & & +\left(WeighofT-bill\times RateofreetunofT-bill\right)\\ & =& \left(74.07\%\times 17\%\right)+\left(25.93\times 7\%\right)\\ & =& 12.59\%+1.82\%\\ & =& 14.41\%\end{array}$

So, the expected rate of return is 14.41%.