Question

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 risky portfolio includes the following investments in the given proportions:

Stock A	27%
Stock B	33%
Stock C	40%

Your client decides to invest in your risky portfolio a proportion (y) of his total investment budget with the remainder in a T-bill money market fund so that his overall portfolio will have an expected rate of return of 15%.

a. What is the proportion y? (Round your answer to 1 decimal place.)

Proportion y            

b. What are your client's investment proportions in your three stocks and the T-bill fund? (Round your answers to 1 decimal place.)

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

c. What is the standard deviation of the rate of return on your client's portfolio? (Round your answer to 1 decimal place.)

Standard deviation             % per year

Answer 1

^{a) If the weight of risky portfolio is 'y' then weight of T-bill would be (1-y).}

^{Expected return on clients portfolio = weight of risky portfolio x return on risky portfolio + weight of T-bill x return on T-bill}

^{or, 15% = y x 17% + (1 - y) x 7%}

^{or, y = 0.8}

^{weight of risky portfolio = 0.8, weight of T-bill = 0.2}

^b)

Security	Investment Proportions
T-bill	20% (from part a)
Stock A	80% x 0.27 = 21.6%
Stock B	80% x 0.33 = 26.4%
Stock C	80% x 0.40 = 32%
Total	100%

c) Standard deviation of a portfolio is computed as follows -

$\sigma = \sqrt{(W^{2}_{risky} *\sigma^{2}_{risky})+ (W^{2}_{T-bill} *\sigma^{2}_{T-bill})+ 2*W_{risky} *\sigma_{risky}*W_{T-bill} *\sigma_{T-bill}*r_{riskyAndT-bill} }$

Now, standard deviation of T-bill would be zero as it is risk free. So, our formula would become

$\sigma = \sqrt{W^{2}_{risky} *\sigma^{2}_{risky}} = \sqrt {(0.8)^{2} * (27)^{2}} = 21.6$%