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 in the previous problem decides to invest in your risky portfolio a proportion ( y) of his total investment budget so that his overall portfolio will have an expected rate of return of 15%. (LO 5-3)

a. What is the proportion y?

b. What are your client’s investment proportions in your three stocks and the T-bill fund?

c. What is the standard deviation of the rate of return on your client’s portfolio?

Short answer

a. 80%

b.

Stock A: 21.6%

Stock B: 26.4%

Stock C: 32%

T-Bill: 20%

c. 21.6%

Step-by-step solution

Target expected return rate:

When an investor fixes a value of return rate which he perceives in the upcoming period and aligns their investment activity accordingly is termed the target rate of return. It is regarded as the investment objective achieved with financial plans' help.

a. Calculation of proportion of y:

As y is the proportion of risky portfolio, the proportion of T-bill would be 1-y. Solving the below equation will yield the value of y.

$\begin{array}{rcl}\text{Expected rate of return} & =& \left(\text{Weight of risky portfolio} \times \text{Rate of return of risky portfolio}\right) \\ & & + \left(\text{Weight of T - bill} \times \text{Rate of return of T - bill}\right)\\ 15\%& =& \left(y\times 17\%\right)+\left(\left(1-y\right)\times 7\%\right)\\ 0.15& =& 0.17y+0.07-0.07y\\ 0.08& =& 0.10y\\ y& =& 0.8\\ & & \end{array}$

So, the proportion of y is 0.8 i.e., 80%.

b. Calculation of proportion of three stocks and T-bill:

Based on the proportion of 80% for risky portfolio, the proportion for each stock is computed.

Particulars	Weight
Stock A	27% * 80% = 21.6%
Stock B	33% * 80% = 26.4%
Stock C	40% * 80% = 32%
T-Bill	20%

c. Calculation of the standard deviation of client’s portfolio:

Since the T-bill is a riskless investment, its standard deviation of T-bill rate is 0.

$\begin{array}{rcl}& & S\tan darddeviaitionoftheclient\text{'}sportfolio\\ & =& \sqrt{\left[{\left(Weighofinvestmentmadeinriskyportfolio\right)}^2\times {\left(S\tan darddeviation\right)}^2\times {\left(S\tan darddeviation\right)}^2\right]+\left[{\left(WeightofinvestmentmadeinT-Bill\right)}^2\times {\left(S\tan darddeviation\right)}^2\right]}\\ & =& \sqrt{\left[{\left(80\%\right)}^2\times {\left(27\%\right)}^2\right]+\left[{\left(20\%\right)}^2\times {\left(0\right)}^2\right]}\\ & =& \sqrt{0.04665600+0}\\ & =& 0.216\\ & =& 21.6\%\end{array}$

So, the standard deviation of the client’s portfolio is 21.6%.