Capacity-Adjusted Return Curves

A research memo says a strategy runs at "18% net return, Sharpe 2.0." That sentence is missing a word: at what size? An allocator who reads it as a fixed property of the strategy — like a Sharpe ratio is often treated — will size the mandate however the budget allows, discover the return degrades as the book grows, and conclude the desk oversold the strategy. The desk didn't lie. It just reported one point on a curve as if it were the whole curve.

The curve, not the point

Every dollar deployed pushes on the price a little, and that push scales roughly with the square root of how much of the day's volume you're consuming (see The Square-Root Impact Law). Gross edge per trade is roughly fixed in basis points; cost per trade grows with size. A capacity-adjusted return curve plots net return, or net Sharpe, against book size and shows exactly where the two lines cross — instead of reporting the return at one arbitrarily chosen size, usually the smallest one, because that's what the initial backtest happened to use.

Worked example

A statistical arbitrage book: gross edge of 25 bps per round trip at negligible size, 80 round trips a year, 10% annualised volatility. At zero size that's 20% gross return, Sharpe 2.0. Market impact model: 60 bps times the square root of participation rate $p$, paid twice per round trip.

$$\text{net bps per trade} = 25 - 2 \times 60\sqrt{p}.$$

Book size	Participation	Impact (both ways)	Net per trade	Net annual	Net Sharpe
$2m	0.2%	5.4 bps	19.6 bps	15.7%	1.57
$20m	2.0%	17.0 bps	8.0 bps	6.4%	0.64
$30m	3.0%	20.8 bps	4.2 bps	3.4%	0.34
$42m	4.2%	24.6 bps	0.4 bps	~0%	~0.04
$60m	6.0%	29.4 bps	−4.4 bps	−3.5%	−0.35

The memo's "Sharpe 2.0" figure only exists near zero. By $42 million the edge is gone; a mandate sized above that pays the market to trade. Dollar profit, by contrast, is book size times net return per dollar, and that product peaks well before break-even — maximising $W \times (25 - 120\sqrt{p})$ lands near $16 million, earning roughly 12.6% net, about $2.0 million a year at a Sharpe near 1.1. The size that makes the most money is neither the size in the pitch nor the size where the Sharpe finally hits zero.

Build the curve by re-running the backtest at each size with impact costs applied, not by scaling a single zero-size P&L series — position caps, partial fills, and liquidity screens all bind differently as size grows, and only a full re-simulation captures that. See Capacity-Constrained Backtesting for the mechanics.