To help you analyze the pension fund manager's decision, let's first define the key data provided:

• Stock Fund (S):

  • Expected Return $E(R_S) = 16\%$
  • Standard Deviation $\sigma_S = 34\%$

• Bond Fund (B):

  • Expected Return $E(R_B) = 10\%$
  • Standard Deviation $\sigma_B = 25\%$

• T-bill Money Market Fund:

  • Return $R_{T-bill} = 5.5\%$ (This is a risk-free rate)
  • Standard Deviation $\sigma_{T-bill} = 0\%$

• Correlation between Stock and Bond Fund returns: $\rho_{S,B} = 0.17$

This setup involves two risky assets (the stock and bond funds) and one risk-free asset (the T-bill money market fund). The goal is often to analyze how to optimally combine these assets, especially the risky ones, to form a portfolio that provides the best return for a given level of risk (or the least risk for a given level of return).

Step 1: Calculating the Portfolio Standard Deviation for a Combination of the Stock and Bond Funds

If the manager decides to invest in a portfolio with both the stock and bond funds (ignoring the T-bill for a moment), we can calculate the portfolio's expected return and standard deviation using the following formulas:

1. Expected Return of the Portfolio: $E(R_P) = w_S E(R_S) + w_B E(R_B)$ Where $w_S$ and $w_B$ are the weights of the stock and bond funds in the portfolio.

2. Portfolio Standard Deviation: $\sigma_P = \sqrt{w_S^2 \sigma_S^2 + w_B^2 \sigma_B^2 + 2w_S w_B \sigma_S \sigma_B \rho_{S,B}}$ Where:

• $\sigma_S$ is the standard deviation of the stock fund,
• $\sigma_B$ is the standard deviation of the bond fund,
• $\rho_{S,B}$ is the correlation between the returns of the stock and bond funds.

You can adjust the weights $w_S$ and $w_B$ to explore different combinations of the stock and bond funds.

Step 2: Efficient Frontier and Optimal Portfolio

Once you calculate the expected return and risk (standard deviation) for various combinations of the stock and bond funds, you can generate the efficient frontier—a curve that shows the maximum return for each level of risk.

Next, you could combine these risky assets with the risk-free T-bill fund. If you wanted to analyze a portfolio with the T-bill included, the overall portfolio return and risk would be determined by combining the risk-free asset with the risky portfolio, which would shift the efficient frontier to include combinations of the T-bill.

Step 3: Optimal Portfolio and Sharpe Ratio

To find the optimal portfolio (the portfolio that maximizes return for a given level of risk), you can calculate the Sharpe ratio:

$\text{Sharpe Ratio} = \frac{E(R_P) - R_{T-bill}}{\sigma_P}$ Where $E(R_P)$ is the expected return of the portfolio, $R_{T-bill}$ is the risk-free rate (5.5%), and $\sigma_P$ is the standard deviation of the portfolio.

This ratio can help you determine which portfolio is most efficient for the manager's needs.

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Would you like to calculate the portfolio's expected return and standard deviation for different combinations of these funds? Or would you prefer an analysis on how to combine the stock and bond funds with the T-bill?