What is the formula for the variance of a portfolio of two assets? And how would you calculate the variance of a portfolio composed of 40% asset A with an annual standard deviation of 15% and 60% asset B with an annual standard deviation of 25%, given that the correlation between the two assets is 0.4?

The formula for the variance of a portfolio of two assets is given by:

σ_p² = w_A²σ_A² + w_B²σ_B² + 2w_Aw_Bσ_Aσ_Bρ_AB

where - σ_p² is the variance of the portfolio - w_A and w_B are the weights of asset A and asset B in the portfolio, respectively - σ_A and σ_B are the standard deviations of asset A and asset B, respectively - ρ_AB is the correlation coefficient between the returns of asset A and asset B

Given that the portfolio is composed of 40% asset A (with an annual standard deviation of 15%) and 60% asset B (with an annual standard deviation of 25%), and the correlation between the two assets is 0.4, we can plug these values into the formula to find the variance of the portfolio:

σ_p² = (0.4²)(0.15²) + (0.6²)(0.25²) + 2(0.4)(0.6)(0.15)(0.25)(0.4)

Calculating the values:

σ_p² = 0.16 × 0.0225 + 0.36 × 0.0625 + 2 × 0.4 × 0.6 × 0.15 × 0.25 × 0.4

σ_p² = 0.0036 + 0.0135 + 0.036

σ_p² = 0.0531

So the variance of the portfolio is 0.0531. If you want to find the standard deviation of the portfolio, take the square root of the variance:

$$\sigma_p = \sqrt{0.0531} \approx 0.2302$$

The annual standard deviation of the portfolio is approximately 23.02%.