Final answer:
To calculate the Sharpe Ratio, subtract the risk-free rate from the expected return of the market portfolio and divide by the standard deviation of the portfolio's returns. To achieve a return of 8%, allocate 75% to the market portfolio and 25% to the risk-free asset. The resulting portfolio has a 15% standard deviation as its risk.
Step-by-step explanation:
The question at hand involves calculating the Sharpe Ratio of a market portfolio, determining the capital allocation between the market portfolio and the risk-free asset, and assessing the resulting portfolio's risk using standard deviation.
a. Calculating the Sharpe Ratio:
The Sharpe Ratio is calculated using the formula:
Sharpe Ratio = (p - rf) / std(rp)
Where 'p' is the expected return of the market portfolio, 'rf' is the risk-free rate, and 'std(rp)' is the standard deviation of the market portfolio's returns. Plugging in the numbers:
Sharpe Ratio = (10% - 2%) / 20% = 0.4
b. Allocation between the market portfolio and risk-free asset:
To achieve a return of rp = 8%, we need to find the weights for the market portfolio (w) and the risk-free asset (1-w) that achieve this return:
8% = w * 10% + (1-w) * 2%
Solving for w, we get:
w = (8% - 2%) / (10% - 2%) = 0.75
Therefore, you should allocate 75% to the market portfolio and 25% to the risk-free asset.
c. Risk of the new portfolio:
The risk of the new portfolio, which is a combination of the market portfolio and the risk-free asset, is the weighted standard deviation of the market portfolio since the risk-free asset has no risk (0% standard deviation). Thus, the risk is:
Risk (std) = w * std(rp) = 0.75 * 20% = 15%