Modern portfolio theory

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

Modern portfolio theory constructs portfolios by comparing expected return with variance risk and the effects of diversification.

The set of attainable return standard deviations and expected returns for budget-feasible risky investment portfolios. Negative weights are allowed when short selling is permitted. With positive definite covariance matrix and nonconstant means, the minimum-risk boundary is a hyperbola in standard-deviation/mean coordinates; its upper branch is the mean-variance efficient frontier. Adding a risk-free asset changes the budget constraint on risky holdings and produces a different attainable set.

For nondegenerate Gaussian terminal prices, a finite maximizer of a differentiable increasing concave non-affine utility, subject to a linear budget, lies in the span of $V^{-1}s$ and $V^{-1}\mu$. Gaussian integration by parts givesThe budget first-order condition makes this a multiple of $s$. For a nonzero portfolio, non-affineness and the full Gaussian support imply $\mathbb{E}{U}^{\prime \prime }<0$, so rearrangement gives the span assertion. Affine utility is a genuine exception: when means are proportional to prices, every budget-feasible portfolio is optimal, including portfolios outside that span.

With fixed budget $s^T\pi =x$, positive definite covariance $V$, and mean vector $\mu$ independent of $s$, set $A=s^T{V}^{-1}s$, $B=s^T{V}^{-1}\mu$, $C={\mu }^T{V}^{-1}\mu$ and $D=AC-B^2>0$. The minimum variance at mean $m$ is the displayed parabola. Its upper-mean branch $m\ge Bx/A$ is efficient: the lower branch has a competitor with equal variance and higher mean. If $\mu$ is proportional to $s$, only one mean is feasible and this two-dimensional parabola degenerates.

The efficient frontier is the set of portfolios for which no other feasible portfolio has at least as much expected return and no greater variance, with at least one strict improvement.

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 $V-b{b}^T/(b^T{V}^{-1}b)$.