5.1.4 Minimum-Variance Portfolio & the Constraints That Bind

Basic Prerequisites: 02 · The Efficient Frontier and Calculus & KKT Optimization.

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1. Intuition & Practical Objective

The minimum-variance (min-var) portfolio is the quietest one you can build - the point of lowest variance regardless of return. It is special for two reasons: it is the only point on the frontier whose weights do not depend on expected returns at all (they depend only on $\Sigma$), and it is the natural benchmark for every fancier claim (“beat min-var and you’ve added forecast value”). The practical objective of this page: compute it, see why it’s the least fragile object in the folder, and then confront the reality that real portfolios cannot short - turning the clean parabola into a constrained, corner-laden feasible set.

The deep lesson: constraints are what destroy the pretty closed forms. The unconstrained frontier is one smooth curve; add $w\ge 0$ and the feasible frontier becomes a piecewise curve whose pieces are the boundaries where some weights hit exactly $0$ - the corner solutions that Markowitz’s critical-line algorithm enumerated. Understanding which constraints bind (long-only, caps, turnover) is the gateway to real deployment.

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2. Mathematical Ground Truth & Derivations

The min-variance portfolio (Merton 1972, eq. 14). Minimize $\frac{1}{2}{w}^T\Sigma w$ subject only to $w^{T}1=1$:

$$\mathcal{L}=\frac{1}{2}{w}^T\Sigma w-\lambda (w^{T}1-1)\Rightarrow \Sigma w=\lambda 1\Rightarrow w_{\text{mv}}=\frac{{\Sigma }^{-1}1}{1^T{\Sigma }^{-1}1},{\mu }_{\text{mv}}=\frac{A}{C},{\sigma }_{\text{mv}}^2=\frac{1}{C}.$$

Crucially there is no $\mu$ left - the min-var portfolio depends only on covariance. This is why it is the workhorse robust portfolio: covariance estimates are far more stable than mean estimates (a fact Best–Grauer and Chopra–Ziemba exploit).

The constrained (long-only) frontier. Add non-negativity and a return floor:

$$\underset{w}{\min }\frac{1}{2}{w}^T\Sigma w\text{s.t.}{w}^{T}1=1,w\ge 0,w^T\mu \ge R^{\ast }.$$

The KKT conditions now include complementary slackness ${\nu }_i{w}_i=0$ (using ${\nu }_i$ for the per-asset multipliers to avoid colliding with the budget multiplier $\lambda$ above): whenever a weight wants to be negative, the optimum pins it at 0 and re-solves on the remaining assets. The efficient long-only set is therefore built from (a) the min-var portfolio (if all-majority-positive) through (b) successive “corner” portfolios where one asset exits, up to (c) the single-asset portfolio of the highest-return asset. At every corner the active set changes, and beyond the highest return only the best asset survives.

Why the min-var answer is “good enough” so often. Because $w_{\text{mv}}$ uses only $\Sigma$, it is (i) stable under mean noise, and (ii) frequently already positive for well-separated assets - so long-only doesn’t bind near it. The binding happens only as $R^{\ast }$ rises and the optimizer tries to short the low-return assets.

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3. Computational Implementation - min-var and its long-only frontier

Stdlib only. The unconstrained min-var is closed-form; the long-only frontier is a constrained convex quadratic solved by a dense grid on the simplex (exact enough for a 3-asset demo; production uses a QP/SOCP solver).

Read the table: from $R^{\ast }=0.08$ to $0.10$ the constraint never binds (min-var is already feasible), so weights are pinned at min-var. As $R^{\ast }$ rises past ~$0.11$, the optimizer must increase return and weights move along the efficient frontier; by $R^{\ast }=0.16$ it collapses to the corner solution $[0,0,1]$ - 100% in the highest-return asset (vol 0.50). The long-only frontier is exactly the concatenation of these segments: min-var → interior moves → all-in corner.

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4. Failure Modes & First-Principles Breakdowns

1. Corner solutions are the rule, not the exception. A $w\ge 0$ optimizer coupled with a high $R^{\ast }$ pushes weights to the simplex boundary; the naive habit of reading $[0,0,1]$ as “the model wants one asset” misses that it’s an artifact of the constraint plus the target, not a forecast.
2. Long-only ≠ no leverage. Even all-positive weights can still concentrate 100% in a single low-diversifying name - long-only is a sign constraint, not a diversification constraint. Adding caps ($w_i\le c$) is what actually compels spread.
3. The min-var portfolio is myopic: it ignores returns entirely, so a genuinely informative $\mu$ is wasted, and in crises all correlations rise, degrading its diversification (the correlation-breakdown failure of 01 · From Zero).
4. “More constraints” is not monotone good. Every additional constraint shrinks the feasible set, raising achievable variance at a given return and pushing weights into corners - the cost of realism is measurable, and sophisticated solvers (SOCP/QP with caps, turnover, impact) are how practitioners manage it (Transaction Costs).

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5. References

• Merton, Robert C.: An Analytic Derivation of the Efficient Portfolio Frontier, JFQA 7(4) (1972)
• Markowitz, Harry: Portfolio Selection, Journal of Finance 7(1) (1952) and Portfolio Selection (1959)
• Best & Grauer, On the Sensitivity of Mean–Variance-Efficient Portfolios…, RFS 4(2) (1991)
• Boyd & Vandenberghe, Convex Optimization

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6. Connected Graph Bridges

• Back: 03 · Tangency & CAPM · Index Hub
• Forward: 05 · Failure Modes · Transaction Costs & Turnover
• Sibling: Risk Parity & ERC · Covariance Shrinkage