Minimise w^T Sigma w subject to w^T 1 = 1 and w^T mu = m, with Sigma positive definite. Derive the closed form solution and the multipliers, and say what the optimal variance looks like as a function of m.

Minimise w^T Sigma w subject to w^T 1 = 1 and w^T mu = m, with Sigma positive definite. Derive the closed form solution and the multipliers, and say what the optimal variance looks like as a function of m.

Approach: Form the Lagrangian with two multipliers, set the gradient of the quadratic form to zero, then solve the two by two linear system the constraints impose on the multipliers.

w = Sigma^{-1}(lambda * 1 + gamma * mu) with the two lagrange multipliers fixed by the constraints, and the optimal variance is the parabola (A m^2 - 2 B m + C)/(A C - B^2) in m. Form the Lagrangian L = w^T Sigma w - 2 lambda (w^T 1 - 1) - 2 gamma (w^T mu - m). The gradient of the quadratic form gives 2 Sigma w - 2 lambda 1 - 2 gamma mu = 0, so w = Sigma^{-1}(lambda 1 + gamma mu), a closed form solution once the multipliers are known. Define A = 1^T Sigma^{-1} 1, B = 1^T Sigma^{-1} mu and C = mu^T Sigma^{-1} mu. Substituting into the two constraints gives A lambda + B gamma = 1 and B lambda + C gamma = m, so lambda = (C - B m)/(A C - B^2) and gamma = (A m - B)/(A C - B^2). The minimum variance is then w^T Sigma w = lambda + gamma m = (C - 2 B m + A m^2)/(A C - B^2). This is the efficient frontier: variance is quadratic in the target return, its vertex sits at m = B/A with variance 1/A, and A C - B^2 > 0 by Cauchy-Schwarz applied in the Sigma^{-1} inner product whenever mu is not a multiple of 1.

Follow-up: Every optimal portfolio is a combination of two fixed portfolios. Name them and prove the two fund separation result from the formula above?

Key concepts: lagrange multipliers, quadratic form, efficient frontier, closed form solution.