The US capital distribution curve can keep its shape without small firms outgrowing large ones. With the coefficients Baker and Smyth estimate, firm death does the work.
The mechanism earns the paper an implementer's afternoon. Firms follow geometric Brownian motion, with each capitalization governed by birth and death intensities that depend on current cross-sectional rank. As the number of firms grows, the empirical CDF of log capitalizations obeys a reaction-diffusion equation. The zeroth-order term integrates birth intensity minus death intensity, then adjusts for dilution as entrants push incumbents down the ranks.
On CRSP, the fitted reaction term is bistable. It is negative on ranks (0, 0.63) and positive above them, with an interior zero at 0.63 and a bootstrap band of [0.61, 0.65]. A bistable reaction term admits a unique traveling wave. The long-run capital distribution therefore keeps a fixed shape while moving up the log-capitalization axis at constant speed.
A disclosure belongs up front. We could not calibrate the paper's rank-dependent birth and death intensities because we lack the required CRSP delisting-reason event data. Our own US equity coverage begins around 2010 and screens universes annually by capitalization instead of using point-in-time CRSP membership. The traveling wave, the two timescales and the growth identities come from the calibrated model. A backtest of a weighting rule cannot validate them.
The authors use the Center for Research in Security Prices monthly file from December 1925 to December 2024. They aggregate common shares on NYSE, AMEX and NASDAQ to the firm level by permco. The resulting file contains 26,100 firms and 3,751,795 firm-months, with 3,155 listed firms in an average month. Estimation mainly covers 1975 to 2024, following the AMEX and NASDAQ coverage expansions.
After cleaning, the sample has 19,567 entrances and 16,253 exits. Aggregate rates are 5.78e-3 and 4.98e-3 per firm-month, leaving the average firm with roughly a 6.0% annual chance of exit. Those exits cluster near the bottom. Among events fitted on 1975-2024, the lowest 5% of ranks account for 31% of all exits. Once merger targets are excluded, that share rises to 53% of the remaining exits.
The money claim comes from portfolio growth along the wave. For any p below the critical index alpha, which is 1.11 on 1975-2024, the capitalization growth rate of a p-diversity-weighted portfolio is an explicit integral against the profile. Equal weighting sets p=0; the market portfolio sets p=1.
Equal weighting earns 17.5% a year from excess growth in the variance curve. It surrenders 8.1% through the sub-front drift of its overweighted ranks and another 6.0% through exits, leaving +3.4% a year over the front speed. The market portfolio comes in at -0.5%. Their frictionless 1975-2024 spread is 3.9% a year.
That diminished edge supplies the authors' headline. Their abstract gives it in five words: turnover reclaims most of what rebalancing gains. The calibration also "places the market just inside the boundary of the diverse phase", while the conclusion says the most recent fifteen years lie at that boundary. The authors treat both findings as model outputs. On those terms the treatment is fair, since the margin is estimated rather than assumed.
The estimate creates a harder problem for the formula. In a 2010-2024 refit, alpha falls to 0.98. The mean-field derivation at p=1 requires p < alpha. During the latest regime, then, the machinery producing the 3.9% and +1.0% top-set results lies outside its stated range of validity.
How reproducible is the calibration?
The estimation recipe is explicit. It starts with One hundred equal rank bins and fits Poisson regressions whose log-intensities are natural cubic splines, with knots at quantiles of observed event ranks. Each regression uses Eight basis functions, selected by AIC from a four-to-eight search. An offset of log(N/J) leaves the coefficients carrying shape alone, after which the estimates are rescaled to aggregate rates.
Volatility is estimated from monthly log returns in fifty rank bins and smoothed by spline. It declines from 0.98 annualized among the smallest firms to 0.31 among the largest. The authors solve the wave by implicit Euler, using a 0.1-month step and a 2,000-point grid initialized with the January 1975 empirical CDF.
Exits are identified through delisting codes. Codes 300-399 and successor-flagged rows are removed as reorganizations, eliminating 8,577 spurious events. About 27% of the apparent exits at the endpoints of raw CRSP series are therefore unrelated to economic exit.
The Monte Carlo check is clean. At 5, 10 and 20 years, the n=10,000 particle system matches the partial differential equation to sup-norm 0.010, 0.013 and 0.011.
Judgment enters later. The quantile density g is estimated with a Gaussian kernel over the last twenty-four cross-sections, each centered on its median. Because g forms the denominator of the turnover term f~/g, the bandwidth chosen by an implementer feeds into the recovered drift, the tail exponent and every portfolio integral.
The tail calculation uses sigma^2(1), defined in the paper as the observed value at rank 0.98, then holds the volatility curve constant beyond that point. This truncation matters. Alpha is 1.11 in the model and 1.06 in the data. Its quoted band, [1.08, 1.14], bootstraps the reaction term while keeping volatility and drift at point estimates. Every rule satisfying 0 <= p <= 1 requires alpha above p. Keeping the market portfolio at p=1 is precisely why alpha must exceed one: "Every rule with 0≤p≤1 is therefore subcritical."
Another judgment concerns the drift level. The speed identity determines only c minus B(1), leaving the front speed to be supplied separately. Under constant coefficients, the wave gives c = 0.100 a year, compared with measured typical-firm growth of 0.059 and a turnover correction of +0.0412. The empirical median-log-cap trend is 0.085, and the authors use it to position the survivor drift curve.
The measured volatility curve is applied across ranks [0.02, 0.98] and continued constantly outside them. Both the 17.5% excess-growth term and the -8.1% mean relative drift are calculated on this trimmed grid. Within it, volatility approaches 0.98 and recovered drift reaches -0.26 a year at rank 0.12.
The authors anticipate the in-sample objection. Intensities, volatility and drift are fitted on 1975-2024, and the wave is assessed over that same period. They explicitly acknowledge that testing the shape with the recovered drift would be circular. The relevant prediction is therefore the zero-calibration wave, computed from measured volatility and any constant drift, with no fitted shape parameter.
Against the CRSP cross-section, that wave records 0.25 root-mean-square relative market-weight error for the top 100 firms. The constant-volatility baseline gives 0.89. Results of 0.11 and 0.09 come from the recovered-drift version, which the paper declines to use in its growth accounting.
Across all ten decades, the zero-calibration wave converges. Top-100 errors range from 0.15 to 0.60, with a median of 0.29. Giving every decade its own recovered drift lowers the median to 0.22, yet also creates the table's worst observation: 0.81 in the 1980s, described by the paper as one clear miss. The paper states that each decade panel is refitted separately. I did not find a forward test where coefficients estimated in one window predict a later window's curve. That is the test I would most want.
The no-turnover experiment makes the strongest case. Once entry and exit are set to zero, neither solver settles. Profile drift rises to 0.0368 and 0.0694, compared with 0.0008 to 0.0042 when turnover is present. At 100 years, interquartile width expands to 7.9 and 14.0 log units, versus 3.8 in the data.
The recovered drift also violates the chord condition of Jourdain and Reygner at every rank, by as much as 0.031 a year. A drift-stabilized model needs the chord condition for an equilibrium to exist. Under these measured coefficients, there is no equilibrium.
Six months below, fifty-three years above
The reaction term's endpoint slopes imply two relaxation half-lives: about 6 months at rank zero and about 53 years at rank one. Bootstrap bands span 5.5-6.4 months and 46-61 years. A full-century refit produces 7 months and 53 years.
Taken literally, lower-rank disturbances disappear within a year, while concentration at the leading edge can persist for a generation. This interpretation carries much of the paper's explanation for 2015-2024. Over that period, total capitalization increased 10.0% a year, median capitalization contracted 1.2%, and the shape term supplied 9.5%.
The authors give two qualifications. Exponential relaxation is proved under constant coefficients; for rank-dependent coefficients it remains an expectation. They also describe the two-timescale division as a heuristic interpretation of the two spectral branches.
Persistence is measured directly as well. Centered quantiles decorrelate with a half-life near 7 months at the s=0.05 quantile and 37 years at s=0.99, against the calibrated 53 years. The measurement overlaps the half-century used to fit the intensities. The paper calls it an independent test. It also observes that a century contains barely two half-lives of the slow mode.
The 2010-2024 refit gives the sharper warning. Its top slope weakens to -0.56% a year, the top half-life lengthens to 124 years, and alpha drops to 0.98. At that supercritical boundary, the mean-field portfolio formulas cease to apply.
Windows extending into the 2000s lose bistability altogether. Exit rates were 7.4% a year against entry rates of 4.1%, leaving the reaction term non-positive at essentially every rank. Seventeen of nineteen ten-year windows retain the bistable pattern; both exceptions include the early 2000s. Across decades, entry rates range from 1.8% to 9.5%, while exit rates range from 0.3% to 7.4%. By the paper's own figures, a calibrated wave belongs to its era.
Turnover consumes the equal-weight premium
The 3.9% spread measures capitalization-growth differences and excludes dividends. The paper identifies three gaps between that quantity and an investor's total return. Dividends remain outside the model, while share issuance enters drift rather than shareholder return. Exit losses are charged in full, although an equally weighted position recovers 88 cents on the dollar on average and 76 cents after delistings for cause. Transaction costs are absent even though full-universe strategies rebalance among the smallest, most expensive ranks.
Recovery works in the opposite direction from the other adjustments, as the paper acknowledges. Most of the charged loss returns to the investor despite leaving the listed market.
The implementable result is smaller, and it is the figure I would retain. Limiting equal weighting to a top set creates leakage, the diffusive capital flow across the membership boundary when firms enter or leave. Under the 85%-coverage rule, excess growth is 6.2% a year and leakage consumes 5.2%, producing +1.0% over the front. A top-500 equal-weight strategy also reaches +1.0%. Relative to the market portfolio at -0.5%, the authors' own numbers imply a 1.5 point frictionless gap before costs.
Membership under the coverage rule changed markedly. It averaged 679 firms in the late 1970s, reached 907 in 1996 and fell to 407 in 2020-24.
Mergers make up 42% of true exits and are treated as pure deaths. The model destroys the target's capital and omits the acquirer's upward jump. The paper tests and repairs the first issue. After merger targets are removed, the bottom-heavy pattern remains: the bottom five percent of ranks contains 53% of remaining exits, compared with 31% of all exits.
The death intensity therefore keeps its shape after mergers are excluded. The missing acquirer jump remains, and the authors flag it for future work. The median merger target leaves at rank 0.53, exactly where a mid-rank overweight would fall.
Our run used a much narrower book
We tested the implementable side: long-only monthly p-diversity weighting on a p grid from 0 to 1, subject to a 10% position cap. Targets were formed at the month-end close and filled at the following session's real close. We charged commissions of $0.004 per share, with a $1 minimum per order.
The intended universe was an annual top 500. The realized run instead traded 20 mega-cap names from January 2020 through July 2024: AAPL, AMZN, AVGO, BRK-B, COST, GOOGL, HD, JNJ, JPM, LLY and ten others.
Our own backtest produced 66.6% total return, 12.05% CAGR, a Sharpe of 0.63 and 13,487 trades over that 4.5-year window. These figures describe one pooled path across the six p values tested, namely 0, 0.25, 0.5, 0.75, 0.9 and 1. They cannot be assigned to any single weighting rule, and the run selected no p.
For comparison, the paper reports +3.4% a year over the front at p=0 and -0.5% at p=1. Its 3.9% spread covers the full CRSP universe during 1975-2024. Those figures are frictionless capitalization-growth differentials relative to wave speed, with dividends excluded. Our result is one absolute, dividend-inclusive total-return path after commissions on twenty surviving firms. The objects differ. Our 12.05% has no front-speed benchmark, so it cannot be netted down to their +3.4%.
Most of the discrepancy comes from setup, and we cannot close it. The paper's 17.5% excess-growth contribution integrates variance across every rank, where volatility rises from 0.31 at the top to 0.98 at the bottom. A universe of twenty mega-caps offers little of that dispersion. The source of the equal-weight premium is largely missing from our book.
The risk statistics make the mismatch plain: 23.46% annualized volatility, 1.05 beta to SPY and a -37.10% maximum drawdown. Our book behaved like a levered-market exposure with a rebalancing rule attached.
Our period is also especially unfavorable to the claim. In the paper's decade table, the 2020s have a +13.3% annual shape term and the market portfolio runs +14.0% over the front. It is the single decade in which cap weighting should outperform equal weighting. In the other direction, twenty surviving mega-caps bear effectively none of the paper's 6.0% exit drag, flattering any low-p path we ran.
Pooling the p paths also means we never measured the cross-p spread, which is the paper's actual deliverable. The visible results cannot fully explain the difference. Our run measures our implementation, and it represents one automated pass rather than a verdict on the authors' work.
The test still missing
The turnover-stabilization result is the paper's strongest contribution. It rests on quantities that do not rely on the recovered drift: exits concentrated within the bottom 5% of ranks, and a no-turnover system whose profile never settles.
The portfolio result is narrower than the 3.9% headline implies. Its tradable version earns +1.0% over the front before costs. The diversity condition supporting the formula is alpha = 1.11 in-sample and 0.98 for 2010-2024.
The nearest sensitivity checks are the full-century refit, with a crossing at 0.61 and half-lives of 7 months and 53 years, and the decade-by-decade refits. Both remain in-sample by construction, as the authors state. I would next fit the intensities and volatility curve on 1975-1999, then use them without refitting to predict the 2015-2024 capital distribution curve and its Pareto exponent.
Our backtest stops at 2024-07-01, and everything after that date is deliberately left untouched so the same strategy can be checked out of sample later.