Question

1. You manage a risky portfolio that has an expected return of 9% and a standard deviation of returns of 12.5%. The T-bill rate is 3%.
   1. Your client wants to allocate her investment portfolio between your fund and T-bills to achieve an expected rate of return of 7%. What proportion of her portfolio should she allocate to the risky portfolio? What proportion to T-bills? What is the standard deviation of her portfolio?
   2. Another client wants to allocate his investment portfolio between your fund and T-bills in order to achieve the highest possible return subject to the constraint that the standard deviation of his portfolio is no more than 9%. What proportion of his portfolio should he allocate to the risky portfolio? How much to T-bills? What is the expected return of his portfolio?

Based on your answers above, which client is more risk averse

Answer 1

Part a

Let x% of the client's portfolio be allocated to the risky portfolio, then (1-x)% will be in the T-bills. The total return will be the weighted average of these which she wants to be equal to 7%. So we can write

x*9% + (1-x)*3% = 7%

0.09x+ 0.03 -0.03x = 0.07

0.06x = 0.04

x =0.04/0.06

=0.6667

So she should allocate 66.67 % of her portfolio to the risky portfolio and 33.33% to T-bills.

Standard deviation: T-bills have fixed returns, so their standard deviation can be considered zero. So, in this case, the standard deviation of her portfolio will simply be the standard deviation of risky portfolio times weight of it in her portfolio

So Standard deviation = 0.6667 * 12.5% = 8.33%

Part b

To get the highest return the standard deviation has to be the highest of her range which is 9%. If we represent the weight of risky portfolio by y then we can write:

y*12.5% = 9%

y=9%/12.5%

=72%

So she should allocate 72% to the risky portfolio and 28% to T-bills.

Based on this analysis, we can say that the first client is more risk-averse as she is opting for a portfolio with low standard deviation which implies low risk.