Minimum Variance Portfolio: Asset Allocation with Correlation

To find the portfolio of minimum variance, we need to calculate the weights of assets A and C that minimize the variance of the portfolio.

Let's assume the weight of asset A is 'x' and the weight of asset C is '(1-x)'.

The expected return of the portfolio (Rp) is given by: Rp = x * Ra + (1-x) * Rc

The variance of the portfolio (σp^2) is given by: σp^2 = x^2 * σa^2 + (1-x)^2 * σc^2 + 2 * x * (1-x) * ρ * σa * σc

Where: Ra = return of asset A = 4% Rc = return of asset C = 2.75% σa = standard deviation of asset A = 6% σc = standard deviation of asset C = 4.5% ρ = correlation coefficient between assets A and C = -0.4

We want to minimize the variance of the portfolio, so we need to find the value of 'x' that minimizes σp^2.

Let's differentiate σp^2 with respect to 'x' and set it equal to zero: d(σp^2)/dx = 2x * σa^2 - 2(1-x) * σc^2 + 2(1-2x) * ρ * σa * σc = 0

Simplifying the equation: 2x * σa^2 - 2(1-x) * σc^2 + 2(1-2x) * ρ * σa * σc = 0 2x * σa^2 - 2σc^2 + 2σc^2x + 2(1-2x) * ρ * σa * σc = 0 2x * σa^2 - 2σc^2 + 2σc^2x + 2σa * σc - 4x * ρ * σa * σc = 0 2x * (σa^2 + σc^2 - 2ρ * σa * σc) + 2σa * σc - 2σc^2 = 0 x * (σa^2 + σc^2 - 2ρ * σa * σc) + σa * σc - σc^2 = 0 x * (36 + 20.25 + 4.8) + 27 - 20.25 = 0 x * 61.05 + 6.75 = 20.25 x * 61.05 = 13.5 x = 13.5 / 61.05 x = 0.221

Therefore, the weight of asset A in the minimum variance portfolio is approximately 0.221 and the weight of asset C is (1-0.221) = 0.779.