Mean-Variance Across Strategies and Why It Fails

Handing a chef ingredient measurements accurate to the gram, mean-variance optimization can tell you the exact ratio for a perfect sauce. Hand the same recipe-optimizer measurements guessed to within 30% of the true amount, and it will still spit out an exact-looking ratio — six decimal places of confidence built on numbers that were barely more than a shrug. That is what happens when Markowitz's mean-variance framework, built for allocating across asset classes with decades of clean return history, gets pointed instead at a portfolio of trading strategies, each with a few years of noisy, often-correlated live track record. The math doesn't know the difference between a precise input and a guess — it optimizes just as confidently either way.

The same formula, a much worse input

Mean-variance optimization picks portfolio weights $w$ to maximize expected return for a given level of risk, solving

$w^* = \Sigma^{-1} \mu$

(up to a scaling constant), where $\mu$ is the vector of expected returns for each strategy and $\Sigma$ is the covariance matrix of strategy returns. In words: the optimal weight on each strategy depends on its expected return, adjusted by how it moves with every other strategy in the book — strategies that are correlated with others get down-weighted relative to what their standalone return alone would suggest, because they're not adding much diversification.

The formula is identical to allocating across stocks or asset classes. What's different, and what breaks it, is the quality of $\mu$ and $\Sigma$ when the "assets" are strategies: three years of monthly returns is 36 data points, autocorrelated because strategies share underlying risk factors, and $\mu$ itself is often still being revised as the strategy is only lightly live-tested (see What to Watch in a Strategy's First Week).

Worked example 1. Two strategies each have an estimated Sharpe ratio of 1.0 over a 3-year backtest, with monthly return standard deviation of 4%. The standard error on a 3-year (36-month) mean-return estimate is $4\%/\sqrt{36} \approx 0.67\%$ per month, or roughly 2.3% annualized — meaning the true annual expected return behind an estimated 8% (a Sharpe-1 result at that vol) could plausibly sit anywhere from roughly 3% to 13% and still be consistent with the data. Feed two such noisy estimates into $w^* = \Sigma^{-1}\mu$ and the optimizer will confidently overweight whichever one's estimate happened to land higher — even if the "true" expected returns are identical and the difference is pure estimation noise. Michaud (1989) called this the optimizer acting as an "estimation-error maximizer": it doesn't just fail to correct for noisy inputs, it actively concentrates the portfolio in whichever inputs are most overstated by chance.

Drag the correlation on this two-asset frontier up toward 1.0 — that's closer to what a book of trading strategies often looks like in practice, since many "different" strategies share exposure to the same underlying risk factors (momentum, value, market beta) even when they look unrelated on the surface. At high correlation the frontier collapses and the optimizer's precise-looking weights become extremely sensitive to tiny changes in the input estimates.

Correlation instability makes it worse

Worked example 2. Suppose two strategies — one a value-tilted equity book, one a trend-following futures book — show an estimated correlation of $\rho = 0.1$ over their overlapping 3-year live history, suggesting they're excellent diversifiers for each other, and the optimizer allocates 60% and 40% respectively based partly on that low correlation. But that 3-year window happened to exclude any sharp equity drawdown; historically, value-style and trend-following strategies have shown correlation spiking toward $\rho = 0.5$–0.7 specifically during equity crashes, when trend-followers' short equity positions and value's long cyclical tilt can both get hit by the same liquidity event, or diverge sharply depending on the crash's character. An allocation built on the calm-period correlation of 0.1 understates the risk the combined book actually carries in exactly the scenario — a crisis — where correlation risk matters most. This is the strategy-level version of the well-known equity finding that correlations rise in crises (see Correlation Breakdown in Crises); it's just as true, and just as underappreciated, one level up at the strategy-allocation stage.

What this means in practice

Practitioners rarely run naive mean-variance across a live strategy book without modification. Common fixes include shrinking $\mu$ toward a common mean or toward zero (treating small differences in estimated Sharpe ratios as mostly noise), shrinking $\Sigma$ toward a structured target like a single-factor model (see Shrinkage), capping any single strategy's weight regardless of what the optimizer wants, and stress-testing the allocation against a crisis-period correlation matrix rather than only the calm-period one actually observed. Risk parity — sizing by risk contribution rather than optimizing expected return at all — is a common alternative precisely because it sidesteps the need to estimate $\mu$, the noisiest and least reliable input in the whole formula.