The problem provides the expected returns and risk (standard deviation) for two funds, A and B. It also provides the correlation coefficient between the returns of the two funds. The problem asks for: (1) the portfolio's expected return if 55% is invested in fund B; (2) the portfolio's standard deviation; (3) the weight of fund A in the minimum variance portfolio; and (4) the weight of fund B in the minimum variance portfolio.

1. Problem Description

The problem provides the expected returns and risk (standard deviation) for two funds, A and B. It also provides the correlation coefficient between the returns of the two funds. The problem asks for: (1) the portfolio's expected return if 55% is invested in fund B; (2) the portfolio's standard deviation; (3) the weight of fund A in the minimum variance portfolio; and (4) the weight of fund B in the minimum variance portfolio.

2. Solution Steps

(1) Expected rate of return

Given:

Expected return of Fund A, $E(R_A) = 13.6\% = 0.136$

Expected return of Fund B, $E(R_B) = 7.9\% = 0.079$

Weight of Fund B, $w_B = 55\% = 0.55$

Weight of Fund A, $w_A = 1 - w_B = 1 - 0.55 = 0.45$

The expected return of the portfolio is given by:

$E(R_P) = w_A * E(R_A) + w_B * E(R_B)$

$E(R_P) = 0.45 * 0.136 + 0.55 * 0.079$

$E(R_P) = 0.0612 + 0.04345 = 0.10465$

$E(R_P) = 10.465\%$

(2) Standard deviation of the portfolio

Given:

Standard deviation of Fund A, $\sigma_A = 2.2\% = 0.022$

Standard deviation of Fund B, $\sigma_B = 1.1\% = 0.011$

Correlation coefficient between A and B, $\rho_{AB} = 0.34$

The portfolio variance is given by:

$\sigma_P^2 = w_A^2 * \sigma_A^2 + w_B^2 * \sigma_B^2 + 2 * w_A * w_B * \rho_{AB} * \sigma_A * \sigma_B$

$\sigma_P^2 = (0.45)^2 * (0.022)^2 + (0.55)^2 * (0.011)^2 + 2 * 0.45 * 0.55 * 0.34 * 0.022 * 0.011$

$\sigma_P^2 = 0.2025 * 0.000484 + 0.3025 * 0.000121 + 0.00082218$

$\sigma_P^2 = 0.00009801 + 0.0000366025 + 0.0000091509$

$\sigma_P^2 = 0.0001437634$

The standard deviation of the portfolio is:

$\sigma_P = \sqrt{\sigma_P^2} = \sqrt{0.0001437634} = 0.0119901376$

$\sigma_P = 1.199\%$

(3) Weight of fund A in the minimum variance portfolio

$w_A = \frac{\sigma_B^2 - \sigma_A \sigma_B \rho_{AB}}{\sigma_A^2 + \sigma_B^2 - 2\sigma_A \sigma_B \rho_{AB}}$

$w_A = \frac{(0.011)^2 - (0.022)(0.011)(0.34)}{(0.022)^2 + (0.011)^2 - 2(0.022)(0.011)(0.34)}$

$w_A = \frac{0.000121 - 0.00008228}{0.000484 + 0.000121 - 0.00016536}$

$w_A = \frac{0.00003872}{0.00043964} = 0.08807152$

$w_A = 8.807\%$

(4) Weight of fund B in the minimum variance portfolio

$w_B = 1 - w_A = 1 - 0.08807152 = 0.91192848$

$w_B = 91.193\%$

3. Final Answer

1. Portfolio's expected rate of return: 10.465%

2. Portfolio's standard deviation: 1.199%

3. Weight of fund A in the minimum variance portfolio: 8.807%

4. Weight of fund B in the minimum variance portfolio: 91.193%