Capacity: Why a Strategy That Works at One Size May Not Work at Ten Times That Size

A trading strategy is not a fixed object. It is a function of its own size. The backtest that produced it was run on prices the strategy did not move, because on paper it traded nothing. In practice every order it sends moves the price it is trying to capture, by an amount that grows with the order, and the strategy run at ten times the capital is not the strategy that was tested at one times the capital with the numbers scaled up. It trades against a market that pushes back harder, keeps a smaller share of each forecast, and at some size keeps nothing at all. That size is its capacity, and it is one of the central constraints in systematic investing.

This piece works through the arithmetic. We use the square-root impact model, a stylized but well-documented description of how prices respond to trading, and follow a hypothetical strategy as the capital behind it grows. The exercise yields a few results that we find clarifying: the share of alpha consumed by impact grows with the square root of capital; the size that maximizes net dollars is a fixed fraction of the size at which net return reaches zero; and capacity depends on the speed of a strategy so steeply that a tenfold change in holding period can change capacity by a factor of a hundred. None of the numbers below describe any actual portfolio. The relationships between them are the point.

The Square-Root Law, Applied to Capital

The model has one substantive ingredient. The price impact of trading, measured from the price before an order begins to the price at which it is completed, grows in proportion to volatility and to the square root of the order's share of the volume that normally trades.^[1] In symbols, the impact per dollar traded is Y · σ · √ρ, where σ is daily volatility, ρ is the participation rate, and Y is a constant that empirical work places somewhere near one. Throughout this piece we use Y = 0.7 and σ = 2% a day, add a half-spread of one basis point on every dollar traded, and let the strategy trade a universe whose combined daily volume is $10 billion. All of these values are illustrative; the arithmetic that follows does not depend on them.

Capital enters through the participation rate. A strategy with capital C that turns its book over every h trading days trades roughly 2C dollars per turn, a sale and a purchase, spread over h days, so its daily participation is 2C ÷ (hV), where V is the volume of the universe. Doubling the capital doubles the participation and, through the square root, raises the impact on each dollar traded by about 41%. Because the strategy also trades twice as many dollars, its total impact cost grows with capital to the power three halves. That exponent, one and a half rather than one, is the whole story of capacity.

Figure 1 draws the consequence for three hypothetical strategies that are identical on paper. Each would earn a gross 6% a year at negligible size, before any trading cost. They differ only in speed: the first turns over every five trading days, the second every twenty-one, the third every hundred and twenty-six. The horizontal axis is capital on a logarithmic scale; the vertical axis is the net expected return after the spread and impact.

The three curves begin near the same level and end in very different places. The fast strategy pays the spread a hundred times a year and gives up about a percentage point to it before impact enters; in the model it reaches zero net return at a size on the order of $30 million. The monthly strategy crosses zero at a scale about a hundred times larger. The slow strategy is still positive at the right edge of the chart, at which point the model has stopped being trustworthy for a different reason: its participation rate has risen past anything the square-root law was estimated on, and the binding constraint is no longer impact but the plain fact that the universe does not trade enough for the strategy to be larger. The same paper return, three very different sizes at which it survives.

Where the Alpha Goes

A useful way to read the same arithmetic is as a division of the gross alpha into three parts: the share paid to the spread, the share consumed by impact, and the share that reaches the investor. The spread's share is fixed by turnover and does not depend on size. Impact's share grows with the square root of capital, so each tenfold increase in capital multiplies it by about 3.2. Figure 2 shows the division for the monthly strategy of Figure 1 at six sizes, from $10 million to $3 billion.

The pattern is not linear, and that is what makes capacity hard to see from inside. At $10 million, impact takes a few percent of the gross alpha; at $100 million, on the order of a sixth; at $1 billion, more than half; at $3 billion, essentially all of it. A manager who watched net returns decline gently from the first size to the third might reasonably conclude that the strategy had weakened. It has not. It has been scaled into the steep part of a square-root curve, and the next doubling costs more than the last one did.

Two numbers fall out of the model and are worth carrying around. Write the net return as a − b√C, where a is the paper alpha net of the spread and b collects the impact terms. Net return reaches zero at a break-even size of (a ÷ b)². Net dollars, which is what a strategy is ultimately judged on, are C · (a − b√C), and that quantity is maximized at four ninths of the break-even size, where the net return has fallen to a third of a and impact takes the other two thirds.^[2] Beyond that point more capital produces fewer dollars, not just a lower percentage, and the strategy is not merely diluted by its size but harmed by it. Figure 3 collects the arithmetic.

The four ninths deserves emphasis because it is so much smaller than intuition suggests. Break-even is the size at which a strategy earns nothing; the size at which it earns the most is less than half of that. A strategy run at its break-even size is not fully used but overused by a factor of more than two, and the difference goes to the market as impact. Definitions matter for the same reason. A “capacity” quoted without saying whether it means the break-even, the dollar optimum, or the size at which returns fall below some hurdle can differ by a factor of three from the same number quoted under a different definition.

Capacity and the Clock

Figure 1 held the paper alpha constant and varied speed. A different and in some ways more natural question is what happens to capacity when the quality of a forecast is held constant and the horizon over which it is harvested is varied. We take the forecast's information coefficient, its correlation with the return over the holding period, as fixed, so that the expected return of a position grows with the square root of the holding period, as the dispersion of returns itself does.^[3] Holding longer then helps twice: the forecast has longer to pay, and the book turns over less often, so each dollar of capital is traded fewer times and at lower participation. Figure 4 plots the resulting break-even capacity against holding period on logarithmic axes, for a base forecast and one twice as strong, alongside a ceiling that stands for a limit on the share of daily volume the strategy is permitted to be.

Two features stand out. The first is the slope. In the base case, capacity grows roughly with the square of the holding period, so a strategy that holds for a month has on the order of a hundred times the capacity of one that holds for a couple of days, and the stronger forecast's line sits roughly four times higher, because capacity grows with the square of the edge net of the spread. The exact exponent depends on how the forecast is assumed to scale with horizon,^[3] but the direction is robust: the clock is the largest single lever a strategy's designer has. The second feature is the ceiling. Past a certain horizon the impact-based estimate exceeds what a sensible participation limit would allow, and capacity becomes a question of how much the universe trades rather than how much trading costs. For slow strategies that is the constraint that actually binds, and it is why breadth, the number of instruments a strategy can trade with the same forecast, matters as much as speed.

The Levers

If capacity is a property of a strategy rather than a fact about a market, it can be designed for. Figure 5 lists the levers that appear in the model, the parameter each one moves, how capacity scales with it, and what each one costs. The last column is the important one. Every lever is a trade, and none of the trades is free.

Slower signals are the strongest lever and the most expensive in a way the model does not show. A strategy that holds for a quarter makes a third as many independent decisions as one that holds for a month, its diversification across time is correspondingly weaker, and the period over which it can be told apart from noise is correspondingly longer. More instruments scale capacity linearly, which is less than the clock offers but more reliable, provided the forecast actually carries over to the added markets and the added markets are not the old ones under different names.^[4] Better execution moves both the spread and the impact coefficient, and it is the lever a firm controls most directly, but patience has a price of its own: an order worked slowly to reduce impact is filled later, and the forecast it was acting on has decayed in the meantime.

Stronger forecasts enter squared, which makes research the highest-leverage activity in the model and also the least controllable. Smoothing, trading only part of the way toward the target when the forecast changes, reduces turnover for a modest sacrifice of the forecast, and in the model it behaves like a partial move along the clock. In practice the levers are used together, and the design question is not which one to pull but where the combination places a strategy relative to the capital it is expected to carry.

How We Think About It

Three habits follow from taking the arithmetic seriously. The first is to estimate capacity before a strategy is allocated capital, from the same model that costs its trades, and to treat the estimate as a property of the strategy in the same way that its expected return and its risk are. A strategy without a capacity estimate is not fully specified. The second is to measure impact live, order by order, against what the model predicted, because the coefficient Y and the volume V are not fixed: they move with the regime, and a strategy that was inside its capacity in a calm market can be outside it in a stressed one without any change in its size. The third is to size strategies toward the dollar optimum rather than the break-even, and to be honest that the two differ by more than a factor of two.

The deeper point is that capacity is where a strategy meets the market's finite ability to absorb it, and the market does not care how good the forecast was. A forecast that is correct but cannot be traded at the size at which it is held is not alpha; it is a description of what someone smaller could have earned. We would rather know that number early, and design around it, than discover it from the returns.