The correlation between A and B is -0.17. Calculate the maximum possible Sharpe ratio. Express your answer as a decimal rounded to three digits after the decimal point.

 

Asset	Standard Deviation	Expected Return
F	0	0.03
A	0.50	0.17
B	0.31	0.09

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.

The correlation between A and B is -0.17. Calculate the maximum possible Sharpe ratio. Express your answer as a decimal rounded to three digits after the decimal point.

Asset	Standard Deviation	Expected Return
F	0	0.03
A	0.50	0.17
B	0.31	0.09

Learn how to calculate the maximum Sharpe ratio for a portfolio of assets with given expected returns, standard deviations, and a correlation of -0.17 between assets A and B. This tutorial includes key formulas from portfolio theory and statistics, offering a step-by-step solution to maximize the Sharpe ratio.

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