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.

Portfolio optimization chooses asset allocations to optimize an objective under specified constraints. In a one-period mean-variance optimization model, the return vector $R$ has mean $\mu$ and covariance matrix $\Sigma$. Portfolio weights $w$ give expected return ${\mu }^{T}w$ and return variance $w^T\Sigma w$. A standard problem isThe constraints describe a fully invested portfolio, a target expected return and no short selling. Different assumptions produce different optimization problems. Harry Markowitz made the dependence on covariance and the tradeoff between mean and variance central to portfolio selection.^[ref]

For two uncorrelated assets with expected returns $4\%$ and $8\%$, each with standard deviation $20\%$, a weight $w$ in the first gives variance $0.04(w^2+(1-w{)}^2)$. Its unconstrained minimum is at $w=1/2$, yielding mean $6\%$ and standard deviation $\sqrt{0.02}\simeq 14.14\%$. A target of at least $7\%$ instead requires $w\le 1/4$, making the constrained minimum $w=1/4$. The same covariance matrix therefore gives a different solution when the objective's feasible set changes.

Estimated means and covariances are model inputs, rather than known future outcomes. Sampling uncertainty, transaction costs and additional allocation restrictions can materially change the resulting weights.

The Black-Litterman model combines equilibrium-implied expected returns with additional views whose uncertainty is stated explicitly. Fischer Black and Robert Litterman introduced it to make portfolio optimization less dependent on unstable unconstrained return estimates.^[ref]

In a standard Gaussian formulation, let the unknown return-mean vector have prior $\mu \sim N(\pi ,C)$, often $C=\tau \Sigma$ for asset-return covariance matrix $\Sigma$. Views take the form $q=P\mu +\epsilon$, with independent $\epsilon \sim N(0,\Omega )$. The Gaussian conditional expectation formula gives posterior meanThe matrix $P$ describes absolute or relative views, while $\Omega$ describes their uncertainty.

For one mean with prior $4\%$, prior variance $0.0004$, and a view of $8\%$ with equal variance, the posterior mean is $6\%$. Increasing the view variance pulls the result toward $4\%$; making it very small pulls toward $8\%$. The uncertainty $C$ about a mean is distinct from the covariance matrix $\Sigma$ of fluctuating returns, and must not be substituted for it without specifying the portfolio model.

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.

Let terminal asset prices $S$ have mean $\mu$ and positive definite covariance matrix $V$, with a Gaussian distribution. Consider a finite optimizer of $\mathbb{E}[U({\pi }^{T}S)]$ subject to the linear budget $s^T\pi =w$. Assume $U\in C^2(\mathbb{R})$ is increasing, concave and non-affine, and its derivatives satisfy the integrability needed for differentiation and Gaussian integration by parts. Then every optimizer lies in the linear span of $V^{-1}s$ and $V^{-1}\mu$.

The first-order condition and Gaussian identity giveFor $\pi \ne 0$, the terminal wealth has full Gaussian support. Continuity and non-affine concavity make $U^{\prime \prime }<0$ on some interval, so $\mathbb{E}{U}^{\prime \prime }({\pi }^{T}S)<0$. Rearranging proves the two-fund representation. The zero portfolio belongs to the same span automatically.

The regularity conditions justify the displayed derivative proof. Affine utility is an exception to its conclusion: if means are proportional to initial prices, every budget-feasible portfolio can be optimal, including portfolios outside this 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 nonzero mean vector $b$ and positive definite covariance matrix $V$. With unrestricted holdings, the minimum-variance portfolio achieving target excess expected wealth $c$ isA Lagrange multiplier or the Cauchy-Schwarz inequality in the $V$ inner product proves this: every component orthogonal to $V^{-1}b$ adds variance without changing $b^T\theta =c$. The minimum variance is $c^2/(b^T{V}^{-1}b)$.

For constant absolute risk aversion utility with coefficient $\gamma >0$ and Gaussian excess payoffs, expected-utility maximization gives $\theta ={\gamma }^{-1}{V}^{-1}b$. This coincides with the target-mean portfolio when $c>0$ and $\gamma =1/\lambda$. A negative target has $\lambda <0$ and cannot correspond to a positive risk-aversion parameter. If $b=0$, only target $c=0$ is feasible, and the zero portfolio uniquely minimizes variance.

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