Two assets have variances 0.04 and 0.09 and covariance 0.055. Minimise portfolio variance subject to the weights summing to 1 and both weights being non-negative. Give the solution, and the KKT multiplier on the constraint that binds.

Two assets have variances 0.04 and 0.09 and covariance 0.055. Minimise portfolio variance subject to the weights summing to 1 and both weights being non-negative. Give the solution, and the KKT multiplier on the constraint that binds.

Approach: Solve the unconstrained problem first, check whether it violates the sign restriction, then write the KKT stationarity condition at the boundary point and solve for the multipliers.

w = (1, 0) with portfolio variance 0.04, and the KKT multiplier on w_2 >= 0 is 0.015. Ignoring the sign restriction, the unconstrained weight is (0.09 - 0.055)/(0.04 + 0.09 - 0.11) = 0.035/0.02 = 1.75, so w_2 = -0.75 and the non-negativity constraint is violated, which forces a corner solution. The KKT conditions for minimising 0.5 w^T Sigma w subject to w^T 1 = 1 and w >= 0 read Sigma w - lambda 1 - m = 0 with m >= 0 and complementary slackness m_i w_i = 0. At w = (1, 0) the gradient Sigma w is (0.04, 0.055). Since w_1 > 0, complementary slackness gives m_1 = 0 and therefore lambda = 0.04, and then m_2 = 0.055 - 0.04 = 0.015, which is non-negative, so the KKT conditions hold and the point is optimal for this convex problem. The multiplier 0.015 is the rate at which half the variance would fall per unit of shorting allowed, since that is the objective it was derived from, so the variance itself falls at 0.03 and the multiplier prices the binding constraint directly.

Follow-up: With three assets and a cap of 50% on each weight, how many constraints can bind at once, and how does the active set method exploit that?

Key concepts: kkt conditions, complementary slackness, binding constraint, corner solution.