10.3 Quadratic programming and portfolios

A quadratic programme has a quadratic objective \(\frac 12x^{\mathsf T}Qx + c^{\mathsf T}x\) and linear constraints; it is convex when \(Q\) is positive semidefinite. With equality constraints \(Ax = b\) only, the KKT conditions are a linear system, \[ \begin {pmatrix} Q & A^{\mathsf T} \\ A & 0 \end {pmatrix}\begin {pmatrix} x \\ \mu \end {pmatrix} = \begin {pmatrix} -c \\ b \end {pmatrix} , \] and with inequalities, active-set methods guess which constraints are tight, solve this system, and adjust the guess, much as the simplex method adjusts its basis.

The best-known application is Markowitz’s portfolio selection (1952). With expected returns \(\mu \) and covariance matrix \(\Sigma \) of \(n\) assets, a portfolio with weights \(w\) has expected return \(\mu ^{\mathsf T}w\) and variance \(w^{\mathsf T}\Sigma w\). An investor minimises risk for a required return: \[ \text {minimise } w^{\mathsf T}\Sigma w \quad \text {subject to}\quad \mu ^{\mathsf T}w \geq r, \quad \textstyle \sum w_j = 1, \quad w \geq 0 . \] Solving for every \(r\) traces the efficient frontier, the best attainable trade-off between return and risk (Figure 10). Diversification shows in the numbers: the minimum-variance portfolio of three assets has standard deviation \(0.170\), below the \(0.2\) of the least risky asset alone, because the assets are imperfectly correlated.

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