The maximum Sharpe ratio of a portfolio is calculated using the formula:

$\text{Sharpe Ratio} = \frac{E(R_p) - R_f}{\sigma_p}$

Where:

• $E(R_p)$ is the expected return of the portfolio.
• $R_f$ is the risk-free rate.
• $\sigma_p$ is the portfolio's standard deviation.

We are given the following:

• Risk-free asset $F$: Return $R_f = 0.03$, Standard Deviation = 0 (as it's risk-free).
• Asset $A$: Expected Return $E(R_A) = 0.17$, Standard Deviation $\sigma_A = 0.50$.
• Asset $B$: Expected Return $E(R_B) = 0.09$, Standard Deviation $\sigma_B = 0.31$.
• The correlation between $A$ and $B$ is $\rho_{A,B} = -0.17$.

To calculate the maximum Sharpe ratio, we need to:

1. Determine the optimal portfolio of $A$ and $B$.
2. Use the combined portfolio return and standard deviation to compute the Sharpe ratio.

Step 1: Portfolio Expected Return

The expected return of a portfolio is:

$E(R_p) = w_A E(R_A) + w_B E(R_B)$

Where $w_A$ and $w_B$ are the weights of assets $A$ and $B$, respectively.

Step 2: Portfolio Standard Deviation

The standard deviation of a two-asset portfolio is:

$\sigma_p = \sqrt{w_A^2 \sigma_A^2 + w_B^2 \sigma_B^2 + 2w_Aw_B \sigma_A \sigma_B \rho_{A,B}}$

We will assume the weights $w_A$ and $w_B$ are optimized to maximize the Sharpe ratio. Let's compute the optimal Sharpe ratio for this portfolio.The maximum possible Sharpe ratio, rounded to three decimal places, is 0.335.

Would you like further details on any part of the calculation, or do you have any other questions?

Here are 5 related questions for further exploration:

1. How would the Sharpe ratio change if the correlation between $A$ and $B$ were 0?
2. What happens to the portfolio's expected return and risk if you add a third asset?
3. How can changing the risk-free rate $R_f$ impact the optimal portfolio weights?
4. What is the effect of a negative Sharpe ratio on portfolio performance?
5. Can the Sharpe ratio alone determine the best investment strategy?

Tip: A higher correlation between assets increases portfolio risk, potentially lowering the Sharpe ratio, unless well diversified.