Capacity Analysis for a Mid-Frequency Strategy
Your team has built a mid-frequency US-equity strategy. At $\
Hints
- Net Sharpe is gross Sharpe times $(1 - C(A)/\mu)$. Under $\sqrt{A}$ impact, net Sharpe is LINEAR in $\sqrt{A}$, not falling as /\sqrt{A}$.
- The crossover AUM is where impact has eaten exactly
/3$ of the gross per-trade return (since 1.0/1.5 = 2/3 ⇒ cost = 1/3).- Real capacity is constrained by name-level ADV concentration, not just aggregate AUM. Always ask for the concentration histogram.
Worked Solution
How to Think About It: Capacity analysis is one of the most important and most often handwaved analyses in mid-frequency systematic equities. The core question: as you scale AUM, what fraction of your alpha do you give back to market impact? Square-root-impact models give you a clean closed-form answer to leading order, but the assumptions matter — the answer is sensitive to alpha autocorrelation, universe liquidity distribution, and execution skill.
Key Insight: Under square-root impact, net Sharpe falls *linearly in* $\sqrt{\text{AUM}}$ — NOT as
/\sqrt{\text{AUM}}$. The crossover AUM is the point where impact cost has eaten exactly/3$ of the gross per-trade alpha (since gross Sharpe is 1.5 and the floor is 1.0).The Method:
Sketching the capacity curve: Let $\mu$ = expected per-trade gross alpha (in return units) and $C(A)$ = per-trade impact cost at AUM $A$. Under the square-root impact model, $C(A) = \kappa \sqrt{A / V}$ where $V$ is daily volume in the universe and $\kappa$ is an empirical constant calibrated from production fills. Gross Sharpe does NOT scale with AUM under the assumption stated. So: $ \text{Sharpe}_{\text{net}}(A) = \text{Sharpe}_{\text{gross}} \cdot \left(1 - \frac{C(A)}{\mu}\right) = 1.5 \cdot \left(1 - \frac{\kappa}{\mu} \sqrt{A/V}\right). $ The curve is linear in $\sqrt{A}$ within the model — flat near small $A$, falling steadily with $\sqrt{A}$, hitting zero when $C(A) = \mu$.
Solve for crossover: $\text{Sharpe}_{\text{net}}(A^*) = 1.0$ when $C(A^*)/\mu = 1 - 1.0/1.5 = 1/3$. Given $C(\
00M)/\mu = 0.05$ from the statement and the $\sqrt{A}$ scaling, $\sqrt{A^* / \00M} = (1/3)/0.05 = 6.67$, so $A^* = \00M \cdot 6.67^2 \approx \$4.4B$.The curve: flat near $A = \
00M$, visible degradation starting around $A = \B$, hits Sharpe 1.0 near $A \approx \$4.4B$, then asymptotes toward zero.(b) Most uncertain input: The $\kappa$ constant in the impact model — and specifically how well backtested impact estimates extrapolate to LARGER positions than ever ran in backtest. Impact at 10× scale is not a 10× extrapolation problem; it is a 'is the square-root model still right at this size?' problem. Realized impact often grows faster than $\sqrt{A}$ at the largest sizes because you exhaust the natural counterparty pool.
(c) Additional data: - *Realized impact estimates from live trading at the current AUM.* Backtested impact is only as good as the impact model used in backtest. - *Histogram of position concentration vs ADV.* If 30% of the portfolio is in names where you already trade > 5% of ADV, you are much closer to capacity than the aggregate AUM number suggests. - *Alpha-decay sensitivity to slower execution.* If you must spread fills over 2 days at higher AUM, what does that do to alpha capture? Often this halves the effective IC.
Answer: Net Sharpe $= 1.5 \cdot (1 - \kappa \sqrt{A/V}/\mu)$ — linear in $\sqrt{A}$ within the model. Crossover at $A^* \approx \$4.4B$ under the assumed 5% cost-of-$\mu$ at $\
00M$. The most uncertain input is the impact constant at sizes beyond live trading history. Ask for realized impact + concentration histogram + alpha decay under slower execution before committing.Intuition
Capacity is the single biggest reason mid-frequency strategies fail to scale at quant funds. A strategy that looks like a million-dollar opportunity at $50M is often a worse business at $5B than a lower-Sharpe but more scalable strategy. Capacity-aware research is therefore not a side activity — it is a first-class input to which alphas get production resources.
The square-root impact model is your starting point but not your endpoint. At very large sizes, impact tends to grow faster than $\sqrt{A}$ because you start exhausting the natural counterparty pool. Two Sigma's flagship strategies have publicly stated capacity considerations as a core constraint; capacity is often THE thing the senior MDs ask about in the final round.
- The crossover AUM is where impact has eaten exactly