Question

Assume that you manage a risky portfolio with an expected rate of return of 17% and a standard deviation of 33%. The T-bill rate is 6%. Your client chooses to invest 75% of a portfolio in your fund and 25% in a T-bill money market fund.

a. What is the expected return and standard deviation of your client's portfolio? (Round your answers to 2 decimal places.)

Expected return % per year
Standard deviation % per year

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

Stock A 30%
Stock B 35%
Stock C 35%

What are the investment proportions of your client’s overall portfolio, including the position in T-bills? (Round your answers to 2 decimal places.)

Security     Investment
  Proportions
T-Bills %
Stock A %
Stock B %
Stock C %

c. What is the reward-to-volatility ratio (S) of your risky portfolio and your client's overall portfolio? (Round your answers to 4 decimal places.)

Reward-to-Volatility Ratio
Risky portfolio
Client’s overall portfolio

Answer 1

Expected return and standard deviation of a portfolio are given by the mathematical relations below:

where ? is the correlation coefficient

Expected Return = (75% * 17%) + (25% * 6%) = 12.75% + 1.5% = 14.25%

Correlation between portfolio and T-bill fund and standard deviation of T-bill would be 0 (since T-bill is a risk free fund).

Hence,

expected standard deviation = 0.2475 = 24.75%

Investment weights/proportions:

Risky portfolio fund = 75%, T-bill fund = 25%.

Hence, investment in Stock A = 30% * 75% = 22.5%

investment in Stock B = 35% * 75% = 26.25%

investment in Stock C = 35% * 75% = 26.25%

T-bill fund = 25%

Reward to volatility portfolio is also referred to as Sharpe's ratio, which is a way to examine the performance of an investment by adjusting for its risk.Mathematically, it is expressed as:

Risk free rate = return on T-bill fund = 6%

So, now for Risky portfolio, Expected Return = 17%, Standard deviation = 33%

Sharpe's Ratio (risky portfolio) = (17% - 6%)/33% = 0.333

For client's overall portfolio, Expected Return = 14.25%, Standard deviation = 24.75%

Sharpe's Ratio (risky portfolio) = (14.25% - 6%)/24.75% = 0.333

Hence, Sharpe's ratio is same for both these portfolios.