Question

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?

Answer 1

Optima Fund's Sharpe ratio is 0.75, indicating a return of 0.75 units above the risk-free rate for each unit of risk. There is insufficient information to calculate the correlation between eBay's stock and the Optima Fund or the Sharpe ratio of the SubOptima Fund without their respective expected returns and volatilities.

Step-by-step explanation:

The Sharpe ratio is calculated as the difference between the portfolio return and the risk-free rate divided by the portfolio volatility (standard deviation of excess return). For the Optima Fund:

Sharpe Ratio = (Expected Return - Risk-Free Rate) / Volatility

Sharpe Ratio of Optima = (0.20 - 0.05) / 0.20 = 0.75

This means that for every unit of risk taken, there is a return of 0.75 units above the risk-free rate.

Correlation of eBay's Stock with Optima Fund

There is not enough information to determine the correlation between eBay's stock and the Optima Fund as correlation measures the relationship in return movements between two assets and is independent of their individual volatilities and expected returns.

Sharpe Ratio of the SubOptima Fund

Without the expected return and volatility of the SubOptima Fund, it is not possible to calculate its Sharpe ratio. The correlation tells us about the relationship between the funds' returns but not their individual risk-adjusted performances.

The Sharpe ratio requires specific data on expected returns and volatility, and with a correlation of 0.80 with Optima Fund, if SubOptima's returns are not known, we cannot compute its Sharpe ratio.

Answer 2

(a) 0.75

(b) 0.2

(c) 0.6

Step-by-step 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