Question

1. You are given the option of choosing between three well diversified portfolios to use as the optimal portfolio in a single index model.    Information on them is given below:

Portfolio	Expected Return	Expected Standard Deviation
A	.1	.05
B	.12	.07
C	.14	.08

1. (20 points) If the risk-free rate is .04, which portfolio would you choose? Why?

2. (20 points) How, if at all, does your answer change if the risk-free rate is .06? Explain.

Answer 1

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A) When the risk-free rate is 0.4

Parameters provided in the question

Portfolio	Expected Return	Expected Standard Deviation
A	.1	0.05
B	.12	0.07
C	.14	0.08

The choice of the portfolio to use as the optimal portfolio in a single index model can be done using Sharpe Ratio. The Sharpe ratio is used to help investors understand the return of an investment compared to its risk. A portfolio with the highest Sharpe ratio must be chosen as that gives the maximum return keeping risk in check.

Sharpe ratio =( Expected return of the portfolio - Risk-free rate) / Expected Standard Deviation

Now, Sharpe Ratios of the different portfolio are

A) Sharpe ration of A = ( 0.1 - 0.04) / 0.05

= 1.20

B) Sharpe ration of B = ( 0.12 - 0.04) / 0.07

=1.142

C) Sharpe ration of C = ( 0.14 - 0.04) / 0.08

= 1.25

Since Portfolio C has the highest Sharpe ratio, it must be chosen because it shows a higher risk-adjusted return as discussed before

Part B) When the risk-free rate is 0.6

Sharpe Ratios of the different portfolio are

A) Sharpe ration of A = ( 0.1 - 0.06) / 0.05

= 0.80

B) Sharpe ration of B = ( 0.12 - 0.06) / 0.07

=0.8571

C) Sharpe ration of C = ( 0.14 - 0.06) / 0.08

= 1

Since Portfolio C has the highest Sharpe ratio, it must be chosen because. The result for portfolio selection hasn't changed as still, the optimal portfolio is portfolio C.However the value of the Sharpe ratio has decreased to 1 from 1.25 because the risk-free rate has increased to 0.06.This shows that the risk-adjusted return will decrease if for scenario 2.