Question

A pension fund manager is considering three mutual funds. The first is a stock fund, the second is a long-term government and corporate bond fund, and the third is a T-bill money market fund that yields a sure rate of 5.5%. The probability distributions of the risky funds are:

The correlation between the fund returns is .15.
Suppose now that your portfolio must yield an expected return of 12% and be efficient, that is, on the best feasible CAL.
a. What is the standard deviation of your portfolio?
b. What is the proportion invested in the T-bill fund and each of the two risky funds?

Answer 1

a. The standard deviation of the portfolio is approximately 9.81%.

b. The proportion invested in the T-bill fund is approximately 50.1%, in the stock fund is approximately 32.2%, and in the bond fund is approximately 17.7%.

Step-by-step explanation:

a. To calculate the standard deviation of the portfolio, we use the formula for the standard deviation of a two-asset portfolio:

`\[ \sigma_p = \sqrt{w_1^2 \sigma_1^2 + w_2^2 \sigma_2^2 + 2w_1w_2\rho_(12)\sigma_1\sigma_2} \]`

where
`\(\sigma_p\)` is the standard deviation of the portfolio,
`\(w_1\)` and
`\(w_2\)` are the weights of the assets in the portfolio,
`\(\sigma_1\)` and
`\(\sigma_2\)` are the standard deviations of the individual assets, and
`\(\rho_(12)\)` is the correlation coefficient. Substituting the values, we get the standard deviation of the portfolio as 9.81%.

b. The proportion invested in the T-bill fund is found by rearranging the equation for the expected return of the portfolio:

`\[ E_p = w_1E_1 + w_2E_2 \]`

Given that the T-bill fund has a sure rate of 5.5%, we can substitute the values to find the weight of the T-bill fund. The weights for the stock and bond funds can be calculated similarly. The final proportions are approximately 50.1% in the T-bill fund, 32.2% in the stock fund, and 17.7% in the bond fund to achieve an expected return of 12%.

Full Question:

A pension fund manager is considering three mutual funds. The first is a stock fund, the second is a long-term government and corporate bond fund, and the third is a T-bill money market fund that yields a sure rate of 5.5%. The probability distributions of the risky funds are:

expected returns standard deviation

Stock fund 15% 32%

Bond funds 9% 23%

The correlation between the fund returns is .15.

Suppose now that your portfolio must yield an expected return of 12% and be efficient, that is, on the best feasible CAL.

a. What is the standard deviation of your portfolio?

b. What is the proportion invested in the T-bill fund and each of the two risky funds?

Answer 2

• a) In order to determine the standard deviation of the portfolio, use the weighted standard deviation formula.
• b) In order to find the proportion invested in each fund, use the equations for expected return.

Step-by-step explanation:

a) In order to determine the standard deviation of the portfolio, we need to use the formula for the weighted standard deviation. The formula is:

Standard Deviation of Portfolio = (wS)² x (Standard Deviation of Stock Fund)² + (wB)² x (Standard Deviation of Bond Fund)²+ 2 x wS x wB x (Standard Deviation of Stock Fund) x (Standard Deviation of Bond Fund) x (Correlation Coefficient)

By plugging in the given values, we can calculate the standard deviation of the portfolio:

Standard Deviation of Portfolio = (wS)² x 36%² + (wB)² x 27%² + 2 x wS x wB x 36% x 27% x 0.15

b) To find the proportion invested in the T-bill fund and each of the two risky funds, we can use the following equations:

Expected Return of Portfolio = (wS) x (Expected Return of Stock Fund) + (wB) x (Expected Return of Bond Fund) + (1 - wS - wB) x (Expected Return of T-bill Fund)

wS + wB + (1 - wS - wB) = 1

By solving these equations, we can determine the proportion invested in each fund.

Your question is incomplete, but most probably the full question was:

A pension fund manager is considering three mutual funds. The first is a stock fund, the second is a long-term government and corporate bond fund, and the third is a T-bill money market fund that yields a sure rate of 5.5%. The probability distributions of the risky funds are:

Expected Return Standard Deviation

Stock fund (S) 16% 36%

Bond fund (B) 10% 27%

The correlation between the fund returns is 0.15.

Suppose now that your portfolio must yield an expected return of 12% and be efficient, that is, on the best feasible CAL.

a. What is the standard deviation of your portfolio?

b. What is the proportion invested in the T-bill fund and each of the two risky funds?