From Expected Return to Position Size

Two forecasts can both say "expected return: 2%" and deserve wildly different position sizes. A 2% forecast on a stock with 15% annual volatility, estimated with high confidence, is a much bigger bet than a 2% forecast on a stock with 60% volatility, estimated on thin data. Converting a raw expected-return number into an actual trade size is a separate step from generating the forecast, and skipping it — sizing every position the same, or sizing purely by conviction — throws away most of what a good forecast is worth.

The mean-variance sizing rule

The standard starting point treats position size as proportional to expected return, scaled by risk aversion and by the inverse of variance:

$$w^* = \frac{\mu}{\lambda \, \sigma^2}$$

In words: the optimal position size grows with expected return $\mu$, shrinks as risk aversion $\lambda$ or volatility $\sigma$ rises, and — critically — shrinks with the square of volatility, so doubling an asset's riskiness cuts its ideal position size to a quarter, not a half.

Worked example

A quant is choosing between two stocks, both forecast to return 2% over the next month. Stock A has monthly volatility of 5%; Stock B has monthly volatility of 12%. With the same risk-aversion parameter $\lambda = 2$: Stock A's size is proportional to $0.02 / (2 \times 0.05^2) = 0.02/0.005 = 4$. Stock B's size is proportional to $0.02 / (2 \times 0.12^2) = 0.02/0.0288 \approx 0.69$. Even though the forecasts are identical, Stock A gets roughly six times the position size of Stock B, because its risk is so much lower per unit of expected return.

Confidence enters too

The formula above assumes the expected return $\mu$ is known with certainty, which it never is. In practice, $\mu$ is itself an estimate with a standard error, and a forecast built on thin data should be treated as smaller than its face value before it ever reaches the sizing formula — usually by shrinking $\mu$ toward zero first (see the related page on shrinkage intensity), then sizing off the shrunk number. Sizing off the raw, unshrunk forecast systematically overstates conviction and produces positions that are too large relative to what the evidence actually supports.

Think of the sizing problem the way this explorer treats a loss surface: too large a step (position) overshoots and adds risk without adding expected return; too small wastes an edge that was really there. The optimal size sits at a specific point set by the trade-off, not at either extreme.

What this means in practice

Desks that size positions by conviction ("I really like this one") rather than by the mean-variance-style calculation tend to concentrate risk in their highest-confidence, often highest-volatility ideas — exactly the ones where estimation error is largest. A disciplined sizing rule forces expected return, risk, and forecast confidence to all show up in the position size, not just the first of the three.