Portfolio construction: mean-variance, and why nobody uses it raw

SIG · Chapter 612 min readAsked at AQR, Two Sigma, Citadel, Point72

Assumes Risk models: covariance estimation, VaR and expected shortfall.

After this lesson you should be able to

  • Derive the mean-variance solution and say why it misbehaves.
  • Describe Black–Litterman and risk parity as responses to that.
  • Incorporate transaction costs into construction rather than after it.

Mean-variance optimisation is the correct answer to the wrong problem: it is optimal given the inputs, and the inputs are estimated so badly that the optimiser’s main achievement is amplifying their errors. Every practical method is a way of constraining it back toward sanity.

Equation 6.1

The solution

Maximise expected return per unit of variance and the weights come out proportional to inverse covariance times expected return.

w∗∝Σ−1μ,SRmax⁡2=μ⊤Σ−1μw^* \propto \Sigma^{-1}\mu, \qquad \mathrm{SR}^2_{\max} = \mu^\top\Sigma^{-1}\mu
Σ−1\Sigma^{-1}
The source of all the trouble: inverting a noisy estimate amplifies its worst directions.
μ\mu
Expected returns, which are estimated far worse than the covariance.
0.10.20.300.040.08Efficient frontierPortfolio volatilityExpected return
Figure 6.2 · The frontier, and why its top end is a fiction. The curve is flat near the minimum-variance point and steep at the right-hand end, which is the same thing as saying the weights there are extremely sensitive to the expected returns you fed in. Nobody trades the top corner, because it is mostly estimation error drawn confidently.

Why it produces absurd portfolios. Two errors compound. Expected returns need decades of data to estimate to any precision, so μ\mu is mostly noise. And Σ−1\Sigma^{-1} amplifies the smallest eigenvalues, which are exactly the noisiest directions. The optimiser then does what it was asked: it finds the combination that looks best given those numbers, which means loading enormously into a near-zero-variance direction that does not exist. The characteristic output is a portfolio with 400%400\% long and 300%300\% short positions in nearly identical assets — and it is not a bug, it is the honest answer to a question with garbage inputs.

MethodWhat it fixesHow
ConstraintsExtreme weightsPosition and leverage limits; long-only
ShrinkageA noisy covariancePull Σ\Sigma toward a structured target
Black–LittermanA noisy μ\muStart from equilibrium and tilt by stated views
Risk parityDependence on μ\mu at allEqualise risk contributions; ignore expected returns
ResamplingSensitivity to the inputsOptimise over many bootstrapped inputs and average
Table 6.3 · The practical responses. Constraints are crude and effective: a long-only constraint has been shown to work partly as an implicit form of shrinkage, because it prevents the optimiser acting on the differences it is least sure about.

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Portfolio construction: mean-variance, and why nobody uses it raw · QuantMax