A strategy does not own its backtest. It owns only the return that remains after the market has charged for implementation.
The distance between a research result and a realized portfolio is not administrative. It is mathematical. A signal may be statistically valid, economically intuitive, and robust across samples, yet fail because the act of trading consumes the edge faster than capital can harvest it. This is where many strategies die. Not in the regression. Not in the simulation. In the market's cost of absorbing size.
Capacity is often discussed as though it were a marketing number: a fund can manage a certain amount before performance deteriorates. That language is too imprecise for serious systematic research. Capacity is a property of an interaction between alpha, turnover, liquidity, volatility, concentration, execution, and the shape of market impact. It is not a single number attached to a strategy. It is an equilibrium between expected value and friction.
Atamus Capital does not publish proprietary strategy rules, signals, feature definitions, datasets, data transformations, model architectures, candidate-generation methods, training procedures, parameters, investment universes, portfolio construction methods, execution processes, trade-level information, position-level information, or performance results. This note examines implementation and capacity through closed-form calculations, deterministic numerical experiments, and controlled Monte Carlo experiments under fully specified probability laws. No Atamus Capital strategy return, backtest, live result, internal threshold, research pipeline, holding period, implementation assumption, model-development workflow, investment universe, portfolio construction method, execution venue, order-routing logic, transaction-cost estimate, or capacity estimate is disclosed. The models, parameters, horizons, and simulations below are public illustrations for studying implementable alpha. They are not Atamus Capital internal settings.
Implementation is where statistical edge becomes an economic claim. We study the capacity problem through implementation shortfall, concave market impact, and the square-root impact law. If the average per-dollar implementation concession of trading notional \(q\) in a liquidity pool of value \(V\) is modeled as \(I(q)=\eta\sigma\sqrt{q/V}\), then total impact cost scales as \(q^{3/2}\). When deployed capital is represented by a normalized scale variable \(a\), gross edge after linear costs is \(B\), and square-root impact contributes \(K\sqrt a\) to average cost, expected net return becomes \(r_{net}(a)=B-K\sqrt a\). Total expected net value is \(\Pi(a)=aB-Ka^{3/2}\). The marginal-capacity point, where an additional unit of capital contributes zero expected value, is \(a_\star=(2B/3K)^2\). The average break-even point, where net expected return reaches zero, is \(a_0=(B/K)^2\). Thus the profit-maximizing economic capacity is only \(4/9\) of the average break-even scale under the square-root model. Using fully disclosed model-based calculations, we show how impact curvature, liquidity concentration, and parameter uncertainty transform capacity from a slogan into a distribution. The conclusion is operational: edge that cannot survive implementation is not edge at institutional scale.
1. The object being measured
Let a research model imply a desired trade. The paper portfolio assumes that the position is acquired at the decision price. The implemented portfolio is acquired through real orders, real liquidity, real delay, and real price response. Implementation shortfall is the difference between those two worlds.
For a buy order of intended quantity \(N>0\), let \(P_0\) denote the decision price and let executed child quantities \(\Delta N_1,\ldots,\Delta N_m\) fill at prices \(P_1,\ldots,P_m\). If the order is fully executed, a simplified dollar shortfall is
where \(F\) collects explicit commissions, fees, taxes, and other directly charged costs. The corresponding decision notional is \(Q=NP_0\), and \(C_{IS}/Q\) is the normalized shortfall in return units. For a sell order, signs reverse. With partial execution, an opportunity-cost term must be added for the unfilled quantity. With multi-asset portfolios, the expression becomes vector-valued and must include cross-impact, financing, borrow, settlement, operational constraints, and residual risk.
The high-level decomposition is
This expression is not a trading recipe. It is an accounting identity for the sources of implementation loss. The central point is that implementation is not a single haircut applied after research is complete. It is a stochastic transformation of the strategy itself.
View data
| Participation | Impact | Spread | Fees | Delay | Total shortfall |
|---|---|---|---|---|---|
| 1% | 15.0 bps | 2.0 bps | 0.5 bps | 3.0 bps | 20.5 bps |
| 5% | 33.5 bps | 2.0 bps | 0.5 bps | 3.0 bps | 39.0 bps |
| 10% | 47.4 bps | 2.0 bps | 0.5 bps | 3.0 bps | 52.9 bps |
| 20% | 67.1 bps | 2.0 bps | 0.5 bps | 3.0 bps | 72.6 bps |
| 25% | 75.0 bps | 2.0 bps | 0.5 bps | 3.0 bps | 80.5 bps |
2. Square-root impact and convex dollar cost
A widely used stylized model for large metaorder impact is the square-root form. In this note, \(q>0\) denotes traded notional and \(I(q)\) denotes the average per-dollar implementation concession in return units:
Here \(V\) is a reference liquidity value over the execution horizon, \(\sigma\) is volatility over that horizon, and \(\eta>0\) is an impact multiplier. If a source defines square-root impact as peak price displacement rather than average execution concession, the average-execution conversion is absorbed into \(\eta\). The model says that the per-dollar concession grows with participation \(q/V\), but concavely. Doubling the order does not double the per-dollar impact.
Total dollar impact is different. It is
The per-dollar cost is concave in size. The total dollars paid to implement the order are convex in size. This distinction is where capacity begins.
For \(q>0\), the marginal dollar impact cost is
The marginal cost of the next dollar is \(3/2\) times the average impact cost. A strategy can look viable on average while the next unit of capital is already unattractive.
More generally, suppose per-dollar impact follows
Then total impact cost is proportional to \(q^{1+\delta}\). The square-root case is \(\delta=1/2\), giving exponent \(3/2\). A linear per-dollar impact model has \(\delta=1\), giving total cost exponent \(2\). The exponent matters because it determines how quickly implementation consumes edge.
View data
| Participation | Average impact | Total cost of reference liquidity |
|---|---|---|
| 1% | 15.0 bps | 0.15 bps |
| 5% | 33.5 bps | 1.68 bps |
| 10% | 47.4 bps | 4.74 bps |
| 20% | 67.1 bps | 13.42 bps |
| 25% | 75.0 bps | 18.75 bps |
3. From order impact to strategy capacity
Let \(a\geq0\) denote normalized deployed capital. It is a scale variable, not a dollar estimate. In a production environment, \(a\) would be tied to a specific instrument universe, turnover profile, execution horizon, liquidity model, and portfolio construction process. None of those internal quantities is disclosed here.
Assume that, before nonlinear impact, the strategy has annual expected edge \(B>0\) after linear costs:
Under a square-root impact model, suppose average nonlinear cost per dollar of capital is
where \(K>0\) is an aggregate implementation coefficient. Then annual expected net return is
Total expected net value, in normalized dollars of expected return per unit base scale, is
Two capacity notions follow immediately.
The average break-even scale solves
Therefore
At \(a_0\), the strategy's expected return after nonlinear impact is zero. Scaling beyond this point is expected to destroy value in average-return terms.
The marginal economic capacity solves
Since
we obtain
Thus
Under the square-root model, the scale that maximizes expected net dollars is less than half the scale where average net expected return reaches zero.
This distinction is not semantic. At \(a_0\), average expected return is zero, but marginal expected value is already negative:
A manager who treats average break-even capacity as the usable capacity has already crossed the economic optimum.
View data
| Scale | Average net return | Total expected value | Marginal value |
|---|---|---|---|
| 0.00 | 7.00% | 0.0000 | 7.00% |
| 1.00 | 4.50% | 0.0450 | 3.25% |
| 3.4844 | 2.33% | 0.0813 | 0.00% |
| 7.8400 | 0.00% | 0.0000 | -3.50% |
| 10.00 | -0.91% | -0.0906 | -4.86% |
Proposition 1. General impact exponent capacity
Let
Then the average break-even scale is
and the marginal economic capacity is
Therefore
Proof. Average break-even requires \(\Pi(a)/a=B-Ka^\delta=0\), which gives \(a_0=(B/K)^{1/\delta}\). Marginal capacity requires \(\Pi'(a)=B-(1+\delta)Ka^\delta=0\), which gives \(a_\star=[B/((1+\delta)K)]^{1/\delta}\). Dividing the two expressions gives the ratio. \(\square\)
For \(\delta=1/2\), the ratio is \((3/2)^{-2}=4/9\). For \(\delta=1\), the ratio is \(1/2\). The precise number depends on the impact exponent. The principle does not: marginal capacity arrives before average break-even capacity.
4. A disclosed numerical illustration
The central illustration uses
These are model-based values chosen for arithmetic clarity. They are not Atamus Capital estimates, targets, thresholds, or strategy parameters.
The average net expected return is
The marginal economic capacity is
The average break-even scale is
At the marginal-capacity point, the average expected net return is still positive:
But the next unit of capital has zero expected marginal contribution. The strategy is not dead at \(a_\star\). It is economically full.
The maximum expected net value is
With the displayed values,
in normalized annual expected value units.
View data
| Input | Value |
|---|---|
| Edge after linear costs \(B\) | 7.00% |
| Impact coefficient \(K\) | 2.50% |
| Marginal economic capacity \(a_\star\) | 3.4844 |
| Average break-even scale \(a_0\) | 7.8400 |
5. Why diversification of liquidity matters
Capacity is not only about total capital. It is about how trade demand is distributed across liquidity.
Suppose the strategy trades across \(n\) liquidity buckets with normalized turnover-demand weights \(u_1,\ldots,u_n\), where
Under identical volatility and liquidity assumptions for illustration, the aggregate square-root impact coefficient is proportional to
The function \(x\mapsto x^{3/2}\) is convex. Concentration raises \(H_{3/2}\). Since capacity is proportional to \(K^{-2}\) in the square-root model, concentration can reduce capacity materially even when total capital and total turnover are unchanged.
A useful effective-bucket count is
For equal weights across \(n\) identical buckets, \(u_i=1/n\), so
and
Concentration reduces the effective number of liquidity buckets. It is possible for a strategy to look diversified by number of instruments while being concentrated by implementation burden.
The stylized Figure 5 example compares four normalized weight profiles across eight buckets. The high-concentration case has an effective bucket count of 3.7335 rather than 8.0000. Its impact coefficient is 1.4638 times the equal-weight case, implying a capacity multiple of only 0.4667 relative to equal deployment under the identical-bucket assumptions.
View data
| Scenario | \(H_{3/2}\) | Effective buckets | Impact multiplier | Capacity multiple |
|---|---|---|---|---|
| Equal across 8 liquidity buckets | 0.3536 | 8.0000 | 1.0000 | 1.0000 |
| Balanced concentration | 0.3651 | 7.5008 | 1.0327 | 0.9376 |
| Moderate concentration | 0.4110 | 5.9213 | 1.1624 | 0.7402 |
| High concentration | 0.5175 | 3.7335 | 1.4638 | 0.4667 |
6. Sensitivity is squared
The square-root capacity formulas create a severe sensitivity:
Taking differentials,
A 10 percent reduction in edge after linear costs reduces break-even capacity by approximately 19 percent:
A 10 percent increase in the impact coefficient reduces break-even capacity by approximately 17.36 percent:
This is why capacity estimates should not be treated as point values. They are nonlinear functions of quantities that are themselves estimated with uncertainty.
Figure 6 makes this point with a disclosed controlled experiment under stated assumptions. We draw
with rejection resampling when \(B\leq0\), and
Across 100,000 replications, the marginal-capacity quantiles are
| Quantile | Marginal capacity |
|---|---|
| 5% | 1.0497 |
| 25% | 2.1391 |
| 50% | 3.3814 |
| 75% | 5.2334 |
| 95% | 9.6287 |
The average break-even capacity quantiles are
| Quantile | Break-even scale |
|---|---|
| 5% | 2.3619 |
| 25% | 4.8130 |
| 50% | 7.6083 |
| 75% | 11.7751 |
| 95% | 21.6645 |
The experiment is model-based and controlled under stated assumptions. Its role is not to estimate capacity for any strategy. Its role is to show that when capacity is a nonlinear function of estimated edge and estimated cost, uncertainty in either input becomes amplified.
View data
| Quantile | Marginal capacity | Break-even scale |
|---|---|---|
| 5% | 1.0497 | 2.3619 |
| 25% | 2.1391 | 4.8130 |
| 50% | 3.3814 | 7.6083 |
| 75% | 5.2334 | 11.7751 |
| 95% | 9.6287 | 21.6645 |
7. What capacity is not
Capacity is not assets under management. A strategy can have large assets and small implementation burden if turnover is low, positions are highly liquid, and rebalancing is slow. Another strategy can have small assets and severe implementation burden if it is concentrated, high turnover, and liquidity demanding.
Capacity is not average daily volume. Liquidity depends on volatility, order-flow imbalance, spread, depth, resilience, cross-impact, urgency, information leakage, and the ability to source liquidity without revealing intent.
Capacity is not a single ex-ante haircut. A constant cost deduction cannot capture the nonlinear effect of scale. If average impact grows as \(\sqrt a\), then a fixed cost deduction is mathematically inconsistent with the very mechanism it is meant to represent.
Capacity is not a promise. It is a conditional statement under assumptions. Those assumptions include alpha persistence, turnover, liquidity, volatility, execution horizon, participation constraints, crowding, opportunity cost, and operational reliability.
8. The allocator-relevant question
The allocator-relevant question is not only whether a model has historical edge. It is whether the edge is harvestable at the proposed scale.
A research claim of the form
is incomplete. It must become
and for economic scaling it must satisfy
for the intended capital level. The first inequality asks whether average expected return remains positive. The second asks whether additional capital is still expected to add value. These are different tests.
Under the square-root model,
so average viability requires
while marginal value creation requires
Because \(a_\star=4a_0/9\), the stricter economic condition arrives first.
9. Why this belongs in research, not operations
Implementation is often assigned to operations after the model is built. That division is dangerous. If implementation costs are nonlinear, then the research object itself changes with scale. The model at small capital and the model at large capital are not economically identical.
Atamus treats implementation awareness as part of systematic research. A signal that cannot be implemented is not a delayed success. It is an incomplete claim. A portfolio that survives a backtest but fails after impact has not merely paid too much to trade. It has revealed that the paper strategy was not the investable strategy.
The correct order is therefore not:
It is:
This is not conservatism for its own sake. It is arithmetic.
10. Conclusion
The market does not pay gross alpha. It pays what remains after the strategy interacts with liquidity.
The square-root impact model provides a simple but powerful lesson. If per-dollar impact cost grows as \(\sqrt a\), then average net expected return declines as \(B-K\sqrt a\), total expected value is \(Ba-Ka^{3/2}\), and the scale that maximizes expected dollars is
The scale at which average edge disappears is
They are not the same. The former is the economic capacity. The latter is the average break-even boundary. Confusing them means confusing a full strategy with a dying one.
Capacity is not a marketing number. It is a nonlinear consequence of liquidity, volatility, turnover, concentration, execution, and uncertainty. That is why implementation belongs inside the research process, not after it.
Edge does not die because a formula says it must. It dies when a strategy asks the market to absorb more information, urgency, and size than the edge can pay for.
References
- Andre F. Perold, "The Implementation Shortfall: Paper vs. Reality", Journal of Portfolio Management 14(3), 4-9, 1988.
- Robert Almgren and Neil Chriss, "Optimal Execution of Portfolio Transactions", Journal of Risk 3(2), 5-39, 2001.
- Robert Almgren, Chee Thum, Emmanuel Hauptmann, and Hong Li, "Direct Estimation of Equity Market Impact", Risk 18(7), 58-62, 2005.
- Bence Toth, Yves Lemperiere, Cyril Deremble, Joachim de Lataillade, Julien Kockelkoren, and Jean-Philippe Bouchaud, "Anomalous Price Impact and the Critical Nature of Liquidity in Financial Markets", Physical Review X 1, 021006, 2011.
- Frederic Bucci, Michael Benzaquen, Fabrizio Lillo, and Jean-Philippe Bouchaud, "Crossover from Linear to Square-Root Market Impact", Physical Review Letters 122, 108302, 2019.
- Albert S. Kyle and Anna A. Obizhaeva, "The Market Impact Puzzle", working paper, 2018.
- Andrea Frazzini, Ronen Israel, and Tobias J. Moskowitz, "Trading Costs of Asset Pricing Anomalies", working paper, first circulated 2012.
Disclaimer
Research notes published by Atamus Capital are provided for general informational and research purposes only. They do not constitute investment advice, trading advice, a recommendation, an offer to sell, or a solicitation to buy any security, fund interest, account, or investment product.
This note does not disclose Atamus Capital's proprietary strategies, signals, feature definitions, datasets, data transformations, model architectures, candidate-generation methods, training procedures, parameters, portfolio construction methods, execution processes, investment universe, research thresholds, model-development workflow, or investment decisions.