The index fund arrives after the trade has faded
The opportunity modeled here has mostly been competed away. The authors put the evidence up front. Greenwood and Sammon, cited in the introduction, find that abnormal returns around S&P 500 announcements fell from 4.94% in the 1990s to 1.02% in 2010-2020. Around implementation, they dropped from 3.55% to 0.26%. Pegoraro et al. study 2010-2024 additions from outside the index family and report implementation-day turnover near 20% of shares outstanding, alongside an average abnormal return of -0.09%. Huge volume, no displacement. Activity instead moved ahead of the announcement: abnormal returns over the prior 100 trading days rose from 9.60% to 18.68%.
Fiechtner and Blanchet derive the equilibrium behind those flows. The paper reports no measurement on real data at all, and says so.
The model has n opportunists trading d assets over [0, T]. Membership becomes public at T_ann. By T_impl, the index fund must finish its rebalance and hold the new index portfolio. Before the announcement, every trader has a private probability vector over m possible constituent sets. Each knows its own vector and the population average, then receives no further information before the reveal.
The indexer never enters as a player. It trades a fixed quantity, calculated self-financing at a fixed reference price, uniformly over [T_TWAP, T_impl] as a stand-in for the closing auction. Traders incur a quadratic execution cost on contemporaneous flow. Their orders also move a transient impact state, which decays at rate R in an Obizhaeva-Wang framework extended to a matrix. Opportunists earn money by selling into indexer demand at prices they helped displace.
For the numerical illustration, d = 5 hand-picked names and m = 10 scenarios cover every two-asset subset. The run spans T = 22 trading days, with T_ann = 10, T_TWAP = 19, T_impl = 20 and n = 20. In each of the two incumbents, the indexer holds seven days of ADV, representing USD 2.112 billion of assets. Lambda and Gamma are taken from one table in Cartea and Jaimungal. The authors choose resilience by hand: rho = 1.386294 per day, equivalent to a half-life of half a trading day.
What the theorem delivers
A unilateral deviation in a closed-loop game changes the inventories and prices entering every rival's feedback function. Their realized trading therefore adjusts. Much of the execution-game literature avoids this problem through open-loop controls, where rivals' rate processes remain fixed. The authors instead construct a subgame-perfect equilibrium.
Their two-sided bounds on the Riccati solution H_n hold uniformly over n at least 2. The solution cannot blow up, and the proof needs no short-horizon or small-impact side condition. The state reduces to (X^i minus Xbar, Xbar, I). Computation then requires one 3d by 3d Riccati ODE and a fixed set of 3d linear ODEs, regardless of the trader count. Each trader uses its own belief vector and the population average, never the complete profile.
One convention carries the existence result. Terminal inventory is marked at the fundamental price S_0(T), rather than the impacted midprice. The authors state directly that their a priori argument fails with midprice marking. They also say parameter values exist for which the altered Riccati equation blows up in finite time, and they do not construct that case. With the terminal penalty fixed at q_T/lambda = 13,000 per day, ending positions are near zero, so the convention has little numerical effect. It has everything to do with the existence claim.
Does competition help the forced buyer?
Skepticism belongs here. In the illustrations, indexer implementation shortfall falls as n rises, with the biggest gains at small n. Mean opportunist wealth and total opportunist wealth both decline with n. The savings combine adverse displacement from positioning before T_TWAP with offsetting trades and lower execution costs during the window.
The paper varies trading windows, execution costs, impact strength and resilience. Only resilience gives a non-monotonic response. Savings peak at an intermediate rho because rapid decay removes the indexer's own impact, whereas slow decay preserves the opportunists' earlier displacement through implementation. At low rho under this Lambda, the net mid-price effect is negative and opportunists leave the indexer worse off. With larger Lambda, the authors write, execution-cost savings can exceed the adverse net price effect, allowing opportunists to help even when resilience is low. They do not show that case.
Then come the levels. Figure 10 varies the two timing gaps T_impl minus T_ann and T minus T_impl. At n = 20, its contours run from USD 640 to 960 million relative to the no-opportunist benchmark under the same parameters. We then did our own arithmetic, which the paper does not report. Pricing the average-belief indexer order from their Table 3 at the initial prices in their Table 1 gives roughly USD 799 million of buys and USD 799 million of sells, or about USD 1.6 billion gross. Savings of 640 to 960 million on a 1.6 billion gross order equal 40 to 60 percent of the traded notional. Measured implementation-day abnormal returns on real additions average -0.09% over the period cited earlier.
The authors never present those levels as empirical magnitudes. Their abstract and conclusion make a directional claim: competition and impact decay govern whether anticipatory trading increases or reduces the indexer's cost. Our objection concerns reading Figure 10 as a magnitude, rather than anything the authors claim. They write that they "illustrate the model without seeking a complete empirical fit to real-world index rebalances as the goal here is to showcase the qualitative properties of the model." On those terms, readers can retain the signs and shapes of the comparative statics. The contour levels establish internal consistency for the parameterization and say nothing about real index funds.
Disagreement appears in volume
The paper's cleanest economic separation lies in the state decomposition. Average belief alone drives the common state (Xbar, I). A trader's own belief minus the average drives its deviation from average inventory. Someone who expects a stock to enter the index may still short it when its expected indexer purchase falls below the crowd's estimate.
Mean-preserving dispersion leaves aggregate signed flow and the full impact path unchanged, while increasing gross opportunist volume. At fixed dispersion, incremental gross volume per trader rises with n and reaches a plotted RMS of about 0.12. The model charges that extra turnover solely through its quadratic costs: the Lambda execution term and the inventory penalties Q and Q_T. There is no spread or borrow cost, even though aggregate equilibrium inventory crosses through zero during the TWAP window at n = 20 and is covered after implementation.
Post-announcement inventory deviations match across all ten scenarios. Positions created by pre-announcement disagreement freeze at T_ann, after which the reveal changes only the common state.
Conditions for an empirical claim
The mean-field version is the implementable policy because it removes the need to observe live average inventory. The paper acknowledges that average inventory is not reported in practice. The authors sketch a Lee-Ready proxy based on signed flow, then identify three defects. Resting limit orders inherit the sign of the counterparty's aggressive side. Aggressive orders from unrelated participants enter the measure. Auctions and off-exchange prints lack an identifiable liquidity-taking side.
This convenience has a price. The approximate-Nash property applies only from the prescribed zero initial state and is not subgame perfect. Rate convergence is clean: eps_n = 1/n plus the difference between finite and limiting average belief. For i.i.d. types, this produces O_p(n^-1/2) paths and an O_p(n^-1) Nash gap. The constant C is declared independent of n but never quantified. At n = 30, the smallest population shown in the mean-field figures, the rate therefore says nothing about the level.
We could not test any of this. Deploying or checking the mechanism requires a point-in-time calendar of announced and implemented constituent changes, together with the indexer's target quantities and schedule. Testing the impact mechanism also requires closing-auction order flow. We have OHLCV bars.
An estimate of Gamma from reconstitution tape and of R from anything at all would change my view, especially if paired with measured indexer shortfall. The current inputs are one borrowed table and a hand-set half-life. If real resilience lies near the peak of the paper's savings curve, its conditional conclusion becomes a number worth disputing: competition and impact decay determine whether anticipatory trading raises or lowers the indexer's cost.
Until then, the result is proved on every finite horizon under their Assumption 2.8. Its parameters were chosen by the authors to display qualitative behaviour, and those same parameters determine whether opportunists help or hurt. We made a related argument about auction-window effects that are real and too small to trade in an earlier note. Here the illustrated effect is large, while the paper declines to estimate its size anywhere else.