The optima mutual fund has an expected return of 20%, and a volatility of 20%. optima claims that no other portfolio offers a higher sharpe ratio. suppose this claim is true, and the risk-free interest rate is 5%.
a. what is optima’s sharpe ratio?
b. if ebay’s stock has a volatility of 40% and an expected return of 11%, what must be its correlation with the optima fund?
c. if the suboptima fund has a correlation of 80% with the optima fund, what is the sharpe ratio of the suboptima fund?

(a) 0.75

(b) 0.2

(c) 0.6

Explanation:

(a)Calculating Sharpe ratio-

Given-

Expected return = 20%,

Risk free rate of return = 5%,

Volatility = 20%

Sharpe ratio = (Mean portfolio return - Risk free return) ÷ Standard deviation of portfolio

Sharpe Ratio = (20% - 5%) ÷ 20%

                      = 0.75

(b) Given-

Standard deviation = 40%,

Portfolio return= 11%,

Risk free return will remain same as 5%

Sharpe Ratio of Ebay = (11% - 5%) ÷ 40%

Sharpe Ratio of Ebay = 0.15

Correlation of Ebay with Optima fund:

= Sharpe ratio of Ebay ÷ Sharpe ratio of Optima fund  

= 0.15 ÷ 0.75

= 0.2      

(c) Correlation of Sub-Optima fund with Optima fund = 80%,

Sharpe ratio of Optima = 0.75

Correlation of Sub-Optima fund with Optima fund:

= Sharpe ratio of Sub-Optima fund ÷ Sharpe ratio of Optima fund

0.80 = Sharpe ratio of Sub-Optima fund ÷  0.75

Sharpe ratio of Sub-Optima fund = 0.80 × 0.75

                                                       = 0.6        

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