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%

Your client chooses to invest 70% of a portfolio in your fund and 30% in a T-bill money market fund. (LO 5-3)

a. What is the expected return and standard deviation of your client’s portfolio?

b. Suppose your risky portfolio includes the following investments in the given proportions:

Stock A 27%

Stock B 33%

Stock C 40%

What are the investment proportions of your client’s overall portfolio, including the position in T-bills?

c. What is the reward-to-volatility ratio (S) of your risky portfolio and your client’s overall portfolio?

d. Draw the CAL of your portfolio on an expected return/standard deviation diagram. What is the slope of the CAL? Show the position of your client on your fund’s CAL

Short answer

a.

Expected return: 14%

Standard deviation: 18.9%

b.

Stock A: 18.9%

Stock B: 23.1%

Stock C: 28%

T-Bill: 30%

c. 0.37

d. Slope: 0.37

Step-by-step solution

Weight of investment:

Weight of investment refers to the ratio in which several investments are distributed in a portfolio. Based on these weights only, the overall returns of the portfolio can be determined.

a. Calculation of the expected return and the standard deviation of portfolio:

The expected return for the client’s portfolio can be found by the multiplication of the rate of return and weight of each component in their portfolio.

Particulars	Rate of return (a)	Weight (b)	Expected rate of return (a*b)
Risky Portfolio	17%	70%	11.9%
T-Bill	7%	30%	2.1%
Total			14%

Hence, the expected rate of return is 14%.

Since there is no default in T-bill, the standard deviation of it will be 0. Using the weights and the standard deviation of 27 for portfolio, the standard deviation of the client’s portfolio is computed.

$\begin{array}{rcl}S\tan darddeviaitionoftheclient\text{'}sportfolio& =& \sqrt{\left[{\left(Weightofinvestmentmadeonriskyportfolio\right)}^2\times {\left(S\tan darddiviation\right)}^2\right]+\left[{\left(WeightofinvestmentmadeinT-Bill\right)}^2\times {\left(S\tan darddeviation\right)}^2\right]}\\ & =& \sqrt{\left[{\left(70\%\right)}^2\times {\left(27\right)}^2\right]+\left[{(30\%)}^2\times {\left(0\right)}^2\right]}\\ & =& \sqrt{357.21+0}\\ & =& 18.9\end{array}$

b. Calculation of investment proportions:

Since 70% of the investment is made in the risky portfolio, each stock in it must be multiplied by 70%.

Particulars	Weight
Stock A	27% * 70% = 18.9%
Stock B	33% * 70% = 23.1%
Stock C	40% * 70% = 28%
T-Bill	30%

c. Reward to volatility ratio:

Sharpe ratio, also referred to as the reward to volatility ratio, is calculated by applying changes in an investment's prior results to offer an estimationof its upcoming performance.

$\begin{array}{rcl}Sharperatio& =& \frac{Expectedrateofreturnforriskyportfolio-ExpectedrateofreturnforT-Bill}{S\tan darddeviation}\\ & =& \frac{17\%-7\%}{27\%}\\ & =& 0.37\end{array}$

So, the Sharpe ratio is 0.37.

d. CAL

The slope of the CAL is nothing but the Sharpe ratio, which is 0.37.

The client portfolio’s is at the point (0.189, 0.17)