Mean-Variance Position Sizing

Mean-Variance Optimization, introduced by Harry Markowitz (1952), is the mathematical foundation of modern portfolio theory. Applied as a dynamic single-asset strategy, it determines the theoretically optimal position size at each point in time given estimates of expected return (mu) and variance (sigma²). Unlike binary long/short signals, it produces a continuously variable position that shrinks as uncertainty increases and grows as the risk-adjusted return improves.

The Merton (1969) intertemporal consumption model shows that the optimal risky asset allocation for a CRRA investor is exactly w* = mu / (sigma² × risk_aversion). This is the formula we implement. With risk_aversion = 1, the position is the unconstrained Kelly fraction — the growth-optimal bet size. Higher risk aversion shrinks the position, acting as a conservative multiplier on the Kelly fraction.

The key advantage over a binary signal: position sizing is proportional to the signal strength. When mu is large relative to sigma², the strategy takes a large long position. When mu and sigma² are both large (high-momentum but also high-volatility), the position is moderated by the risk. This volatility-targeting property means the strategy naturally de-risks during volatile periods — unlike fixed-size strategies that maintain full exposure through market turbulence.

Key assumptions: (1) The rolling estimates of mu and sigma² are unbiased predictors of future return and variance. In practice, expected return estimation is notoriously noisy — the signal-to-noise ratio in mu estimates is very low at daily to weekly horizons. (2) Returns are approximately normally distributed — fat tails mean the Gaussian variance understates true risk. (3) The position is continuously rebalanced — in reality, transaction costs create a "no-trade zone" around the optimal position.

The strategy's practical failure mode is estimation error in mu: rolling sample means of returns are extremely noisy. A 60-day rolling mean of daily returns has a standard error of approximately sigma / sqrt(60), which is typically larger than the true expected daily return. This noise creates excessive position turnover and whipsaw losses. Practitioners often use shrinkage estimators (e.g., James-Stein) or factor-based expected return models to reduce estimation error.