Sizing Down When the Forecast Is Uncertain

A signal built on ten years of stable data and a brand-new signal tested for three months can both produce the exact same expected-return number for tomorrow. They should not get the same position size. The first forecast has been through many market conditions and has a well-estimated track record; the second is, in an honest sense, still mostly a guess dressed up as a number. Sizing down when a forecast is uncertain — separately from sizing based on the forecast's expected return and the position's volatility — is what keeps a portfolio from getting hurt by its newest, least-proven ideas.

Building uncertainty into the sizing rule

The standard mean-variance sizing rule scales position size inversely with the outcome's variance. Adding forecast uncertainty means also accounting for the variance of the estimate itself:

$$w^* = \frac{\mu}{\lambda \, (\sigma^2 + \sigma_\mu^2)}$$

In words: the position size still grows with expected return $\mu$ and shrinks with the asset's own volatility $\sigma^2$, but now there's a second term, $\sigma_\mu^2$, the variance of the forecast estimate itself — a forecast built on little data or an unstable relationship has a large $\sigma_\mu^2$, and that inflates the effective denominator, shrinking the position even though $\mu$ and $\sigma$ haven't changed.

Worked example

Two signals both forecast a 1.5% expected return on similar-volatility stocks ($\sigma^2 = 0.0009$ in both cases), with risk aversion $\lambda = 2$. Signal A is a well-established value factor with an estimated forecast standard error, from years of history, of $\sigma_\mu = 0.3\%$, so $\sigma_\mu^2 = 0.000009$ — small relative to $\sigma^2$. Its position size is proportional to $0.015 / (2 \times (0.0009 + 0.000009)) \approx 0.015/0.001818 \approx 8.25$. Signal B is a newly built alternative-data signal with only six months of history, giving a much wider forecast standard error of $\sigma_\mu = 1.2\%$, so $\sigma_\mu^2 = 0.000144$. Its position size is proportional to $0.015/(2 \times (0.0009+0.000144)) \approx 0.015/0.002088 \approx 7.18$ — smaller, purely because the forecast itself is less trustworthy, despite an identical expected return and identical asset volatility.

The explorer above shows how sample means from a small sample scatter more widely around the truth than means from a large sample — a new signal with little history is exactly the small-sample case, and that wider scatter is what $\sigma_\mu^2$ is measuring.

What this means in practice

In practice, $\sigma_\mu^2$ is usually approximated by the standard error of whatever regression or estimation procedure produced the forecast, or by out-of-sample track-record volatility for signals that have been running live. New signals almost always warrant explicit position-size caps or discounts until enough live history accumulates to estimate $\sigma_\mu$ with any confidence, rather than being sized at full conviction from day one.