Mean-variance optimization

Mean-variance optimization balances expected linear return against a positive-definite quadratic variance penalty.

Without a risk-free intercept or constraints, every undominated linear portfolio lies on the nonnegative ray through $V^{-1}b$.

Let risky excess payoff have mean $b$ and positive-definite covariance $V$. The minimum-variance portfolio with target excess expected wealth $c$ isFor constant absolute risk aversion utility with coefficient $\gamma$, the optimal Gaussian portfolio is $\theta ={\gamma }^{-1}{V}^{-1}b$. The two choices coincide when $\gamma =1/\lambda$.

Let $\xi \sim N(b,\Sigma )$, where $b\ne 0$ and $\Sigma$ is positive definite. For any increasing concave objective of $m+{\theta }^T\xi$, every unique optimal portfolio has the formIndeed, the component orthogonal to ${\Sigma }^{-1}b$ in the $\Sigma$ inner product contributes independent mean-zero Gaussian risk without changing the mean. Removing it cannot reduce expected concave utility. A negative coefficient is dominated by the corresponding positive coefficient, which has the same variance and a larger mean.

A portfolio dominates another when it has no smaller mean and no larger variance, with at least one strict improvement.

Projection onto the mean-variance portfolio $V^{-1}b$ leaves residual covariance $V-b{b}^T/(b^T{V}^{-1}b)$.