Question

You manage a risky portfolio with an expected rate of return of 18% and a standard deviation of 28%. The T-bill rate is 8%.
A. Your client chooses to invest 70% of a portfolio in you fund and 30% in an essentially risk-free money market fund. What is the expected value and standard deviation of the rate of return on his portfolio?
B. Suppose that your risky portfolio includes the following investments in the given proportions:
Stock A 25%
Stock B 32%
Stock C 43%
What are the investment proportions of your client's overall portfolio, including the position in T-bills?
C. What is the reward-to-volatility (Sharpe) ratio (S) of you risky portfolio? Your clients?
D. Draw the CAL of your portfolio on an expected return-standard deviation diagrm. What is the slop of the CAL? Show the position of your client on your fund's CAL.

Answer 1

Answer:

a) the expected value (rate of return) is 15%  and standard deviation of the rate of return on his portfolio is 19.6%

b)

17.5% in Stock A,

22.4% in Stock B,

30.1% in Stock C.

The fraction invested in T-bills is 30%.

c)

The reward to visibility ratio = 35.7 %

client's portfolio offers the same reward to visibility ratio = 35.7%

d)

the CAL is drawn in the image uploaded along this answer,

the slope of the CAL is the reward to visibility ratio, i.e 35.7%. The client's portfolio is the one providing an expected return E[rc] and standard deviation σc.

Explanation:

a)  

Let  c represent client’s portfolio and f represent money-market fund while p represent the risky portfolio.

so

E[rc] = E[70%rp + 30%rf ] = 70%Erp + 30%rf  

= (70/100  × 0.18) + (30/100  × 0.08) = 15%

∴ the expected value (rate of return) is 15%

Since σf = 0,

standard deviation of  client’s portfolio is calculated as;  

σc = 0.7σp

σc = 0.7 × 0.28 = 19.6%

∴ standard deviation of the rate of return on his portfolio is 19.6%

b)

we know that portfolio c is 70% invested in p, which means

70/100 × 0.25 = (0.175)  17.5% in Stock A,

70/100  × 0.32 = (0.224) 22.4% in Stock B,

70/100  × 0.43 = (0.301) 30.1% in Stock C.

The fraction invested in T-bills is 30%.

c)

The reward to visibility ratio of fund is given as

(E(rp) - rf) / σp = (0.18 - 0.08) / 0.28 = ( 0.357) = 35.7 %

Now the client's portfolio offers the same reward to visibility ratio

(E(rc) - rf) / σc = (0.15 -0.08) / 0.196 = ( 0.357) = 35.7 %

d)

the CAL is drawn in the image uploaded along this answer,

the slope of the CAL is the reward to visibility ratio, i.e 35.7%. The client's portfolio is the one providing an expected return E[rc] and standard deviation σc.