Answer:
Step-by-step explanation:
To calculate the Sharpe and Treynor ratios, we first need to find the average excess return of the fund and the beta of the fund.
1. **Average Excess Return (\(R_i - R_f\)):**
- Calculate the excess return for each year by subtracting the risk-free rate from the fund's return.
- Find the average of these excess returns.
\[ \text{Average Excess Return} = \frac{\sum_{i=1}^{n} (R_{\text{Fund},i} - R_f)}{n} \]
2. **Beta (\(\beta\)):**
- Beta measures the sensitivity of the fund's returns to market returns. It is calculated using the covariance and variance of the fund's returns with respect to the market returns.
- The formula for beta is: \[ \beta = \frac{\text{Covariance}(\text{Fund, Market})}{\text{Variance}(\text{Market})} \]
3. **Sharpe Ratio:**
- The Sharpe ratio is the ratio of the average excess return to the standard deviation of excess returns.
- The formula for the Sharpe ratio is: \[ \text{Sharpe Ratio} = \frac{\text{Average Excess Return}}{\text{Standard Deviation of Excess Returns}} \]
4. **Treynor Ratio:**
- The Treynor ratio is the ratio of the average excess return to the beta of the fund.
- The formula for the Treynor ratio is: \[ \text{Treynor Ratio} = \frac{\text{Average Excess Return}}{\beta} \]
Now, let's calculate these values:
```plaintext
Year Fund Market Risk-Free Excess Return
2018 -19.4% -37.5% 1% -20.4%
2019 25.1 20.8 4 21.1
2020 13.7 13.3 2 11.7
2021 7.2 8.4 6 1.2
2022 -1.98 -4.2 2 -3.98
Average Excess Return = ( -20.4 + 21.1 + 11.7 + 1.2 - 3.98 ) / 5 = 1.328%
Covariance(Fund, Market) = 0.97 * ( -20.4 * -37.5 + 21.1 * 20.8 + 11.7 * 13.3 + 1.2 * 8.4 - 3.98 * -4.2 ) / 5
= 511.9456
Variance(Market) = ( ( -37.5 )^2 + 20.8^2 + 13.3^2 + 8.4^2 - ( -4.2 )^2 ) / 5
= 576.42
Beta = 511.9456 / 576.42 = 0.888
Standard Deviation of Excess Returns = sqrt( Var( -20.4, 21.1, 11.7, 1.2, -3.98 ) ) = 12.8692
Sharpe Ratio = 1.328 / 12.8692 = 0.103
Treynor Ratio = 1.328 / 0.888 = 1.4978
```
Therefore, rounding to 4 decimal places:
- Sharpe Ratio = 0.1030
- Treynor Ratio = 1.4978