4 The Investable Habitat

What this establishes. The population a factor is measured on is not a preliminary to the research — it is an identifying assumption of every cross-sectional claim the research makes. This chapter argues that the admission conventions inherited from the academic literature encode a capacity constraint that does not bind on a sole operator, states the resulting habitat claim as a falsifiable hypothesis, and reports the completed experiment that refuted the programme’s original position on it.

What it contributes. A capacity-conditional definition of the population a small-capital operator should both study and trade, together with a pre-registered test of whether edge survives there once survivorship, cost, and thin cross-sections are honestly charged. It also reports a negative result: the programme’s founding hypothesis — that removing the smallest names would raise measured factor quality — was tested and not supported.

Descriptive detail about the populations themselves — how each is constructed, what it covers, and where it is unreliable — lives in The Study Population. This chapter is the argument; that appendix is the reference.

4.1 1. The universe is an identifying assumption

A cross-sectional factor premium is a statement about a population. Change the population and the statement changes, sometimes in sign. This is not a caveat to be noted and set aside; it is the joint hypothesis under which every information coefficient in this programme is computed.

The literature is unambiguous that population choice does most of the work in the anomaly literature. Hou, Xue & Zhang (2020) replicated 452 published anomalies and found that the majority fail to clear conventional significance once microcapitalisation stocks are de-weighted — not because the original results were fabricated, but because microcaps are roughly three per cent of aggregate market capitalisation while being a majority of listed names, so an equal-weighted cross-section is dominated by firms that contribute almost nothing to the investable opportunity set. Fama & French (2008) reach the same conclusion by decomposition: several prominent anomalies are concentrated in, or confined to, the smallest size group.

We take the implication seriously in both directions. If a premium exists only among the smallest names, a claim about “the cross-section of equity returns” is overstated. But the converse deserves equal weight and is rarely stated: if a premium exists only among the smallest names, that is a claim about where the premium lives, not evidence that it is absent.

Which of those two readings is correct depends entirely on whether the smallest names are investable — and that is not a property of the market. It is a property of the investor.

4.2 2. The gap: an institutional convention, not a law

The conventions that govern admission in published factor research — capitalisation breakpoints taken from the largest exchange, value weighting, price and liquidity screens — exist for a defensible reason. A finding that cannot be implemented at the scale of the institutions who will act on it is not useful to them, and value weighting is the honest way to report what a capacity-constrained investor could have earned.

These conventions therefore encode an assumption about the reader: that they deploy enough capital for market impact to bind. Berk & Green (2004) formalise why that assumption is load-bearing, showing that diseconomies of scale in active management are sufficient on their own to compete away apparent skill as assets grow. Under that model the disappearance of a premium at scale is not evidence the premium was illusory; it is evidence the strategy has a capacity, and that capacity has been exceeded.

The gap this chapter addresses is that the conventions are applied as though they were properties of the market rather than of the investor’s size. For an operator whose capital is small enough that participation rates stay negligible, the constraint that motivates the convention does not bind — and the convention, applied unreflectively, excludes from study precisely the part of the market that operator can most plausibly reach.

We are careful about what this does and does not license. It does not license ignoring the microcap critique: the measurement pathologies that make small names treacherous are real, and §7 treats them as the central threat rather than as a footnote. It licenses one narrower claim — that the population boundary is a free parameter of the research design, that its correct value depends on deployable capital, and that inheriting it unexamined from a literature written for a different investor is a modelling choice we should make deliberately rather than by default.

4.3 3. The contribution: a capacity-conditional habitat

We therefore propose that this programme’s population be defined by reachability at the operator’s actual capital, and that the same population be used for both measurement and deployment.

The claim has a shape worth stating plainly, because it is easy to mistake for special pleading. We are not claiming an edge that others cannot see. We are claiming that a class of edges is visible to everyone and actionable only below a capital threshold — and that establishing where that threshold sits, per liquidity band, is a measurable quantity rather than a rhetorical position.

The programme’s operating frame makes this the natural boundary. A single operator, running a daily decision cycle, deploying capital that is small relative to the daily turnover of even modest names, faces the exact inversion of the institutional constraint: what excludes an institution from the smallest names — the impossibility of accumulating a position without moving the price — is not a binding constraint at that scale. The costs that do bind are spread and fixed charges, which are bounded and estimable.

4.4 4. The completed experiment: narrowing did not raise measured quality

The programme’s founding position was the opposite of the one argued above. It held that factor research was producing insufficient cross-sectional edge, and that the suspected cause was an unfiltered, microcap-inclusive population. The remedy proposed was to narrow.

That position was tested. We constructed nine candidate population definitions varying along four axes — capitalisation tier by exchange-referenced percentile, liquidity by trailing dollar-volume rank, volatility relative to the per-date median, and a sector-neutral scoring variant — and evaluated factor information ratios across them.

The narrowing did not deliver the expected lift. The evaluation returned no information-ratio improvement sufficient to justify moving off the incumbent definition, and the incumbent was retained.

We report this as a result rather than as a false start, because it discriminates between two explanations that had been indistinguishable. If thin, noisy, small names were diluting a real premium, removing them should have raised measured quality. It did not. That is evidence against dilution as the explanation for weak measured edge, and it redirects attention to the alternatives — that the premia are genuinely weak in this sample, that the measurement is insufficiently corrected, or that the edge is concentrated in exactly the names the narrowing removed.

Two limitations bound how far this result travels, and both are material. First, the comparison was made on information ratio gross of transaction costs; a population comparison that does not price execution cannot separate a real edge from an uncollectable one, and this one did not. Second, the provenance of the resulting baseline is not settled: a subsequent promotion of a small-capitalisation variant was recorded, but carries an internal flag indicating the promotion was forced, a second indicating measured coverage was implausible, and a third marking the re-evaluation as not ready. We state the population this programme currently measures on in The Study Population as the running configuration reports it, and we flag the promotion’s status as an open provenance question rather than presenting it as a settled decision.

4.5 5. The hypothesis

H1. For an operator deploying capital small enough that market impact is negligible relative to spread, the executable factor-premium frontier extends into a small-capitalisation, low-liquidity habitat that is not investable at institutional scale. Factor evidence measured on that habitat — net of realistic transaction costs and corrected for delisting bias — remains significantly positive.

The hypothesis is pre-registered here, with its falsifiers, its variables, and its decision rules fixed before the measurement is run.

Null. Net-of-cost, survivorship-corrected factor information ratios in the widened habitat are drawn from the same distribution as those in the incumbent population.

Falsifiers. Any one of the following refutes H1 as stated:

F1 — no gain from reach. After the survivorship haircut and net-of-cost adjustment, information ratios in the widened band are statistically indistinguishable from the incumbent band. The reach buys nothing.
F2 — rediscovered artifact. The widened band’s advantage does not survive exchange-referenced breakpoints and value weighting. We have reproduced the microcap artifact Hou, Xue & Zhang (2020) document, not found a habitat.
F3 — unreachable in practice. The capital at which the band’s net premium reaches zero (§6) falls below the operator’s plausible capital range. Reachable in principle, not in practice.

The widening under test. The candidate definition lowers the point-in-time price floor from five to two units of currency, lowers the trailing dollar-volume floor from one million to two hundred and fifty thousand, and introduces a point-in-time capitalisation floor of fifty million. Measured membership on a recent date moves from 1,468 to approximately 1,899 instruments, a rise of about 29 per cent. Neither definition applies a capitalisation floor today; the incumbent’s is disabled.

Status. The widened definition is specified but not in force, and no factor evidence has yet been measured on it. Every statement about it in this chapter is a prediction, not an observation, and is worded accordingly.

4.6 6. Method: solving for the capacity frontier

Rather than assume a capital figure and ask whether the habitat is reachable, we solve for the capital at which it stops being reachable. The output is a frontier rather than a verdict, and it remains valid as capital changes.

For a liquidity band \(B\) and factor \(f\), let \(\pi_B\) be the annualised gross premium to the long-short spread portfolio of \(f\) within \(B\), and \(\tau_B\) the annualised two-sided turnover the strategy requires. Let \(s_B\) be the median effective half-spread among names in \(B\), estimated from daily high-low ranges by the Corwin & Schultz (2012) estimator. For capital \(K\) spread across \(N_B\) names of median daily dollar volume \(V_B\), the participation rate per rebalance is

\[q(K) = \frac{K\,\tau_B}{N_B\,V_B}\]

and we take impact to follow the square-root form that Almgren, Thum, Hauptmann & Li (2005) estimate directly from institutional order data, giving a net premium

\[\pi_B^{\text{net}}(K) = \pi_B - \tau_B\left[s_B + \lambda\sqrt{q(K)}\right]\]

The band capacity is then

\[K_B^{*} = \sup\{\,K > 0 : \pi_B^{\text{net}}(K) > 0\,\}\]

and the habitat is the set of bands whose capacity exceeds the operator’s capital with margin. H1 predicts that \(K_B^{*}\) for the widened band is comfortably above a sole operator’s capital while being far below the capital of any institution that would find the premium worth pursuing.

Variables fixed in advance. Bands are defined by trailing dollar-volume percentile of the admitted population, computed per date, in five strata. \(\pi_B\) is the annualised mean of the daily decile-spread return within band, at the factor’s declared horizon. \(\tau_B\) is measured from the realised rebalancing schedule, not assumed. \(s_B\) is the per-date median across band members, winsorised at the first and ninety-ninth percentiles. \(N_B\) is the median per-date membership count.

Honest limitations of this estimator. The spread term is measured; the impact term is not. The impact coefficient \(\lambda\) is the weakest element of the specification — we have no order-level data from which to estimate it, and will therefore report the frontier as a family of curves across a range of \(\lambda\) rather than as a single line, stating the range over which the qualitative conclusion holds. Where the conclusion depends on \(\lambda\), we will say so rather than pick a value. A frontier computed from the spread term alone is a strict upper bound on capacity and will be labelled as such.

4.7 7. What must be corrected before the number is believable

The habitat H1 identifies as reachable is also the habitat in which every measurement pathology is at its maximum. This is the central threat to the hypothesis, and treating it as a caveat rather than as the main event would be the chapter’s most likely failure.

Delisting bias. Shumway (1997) shows that returns to delisted firms are systematically missing from the standard research database, and Shumway & Warther (1999) show the omission is largest precisely among small firms on the exchange where most small firms trade — large enough that they attribute a substantial part of the measured size effect to the bias itself. Brown, Goetzmann, Ibbotson & Ross (1992) show more generally that survivorship inflates measured performance in a way that is easily mistaken for skill. A habitat argument that is blind to this reproduces the size effect and mislabels it as an edge. The programme’s correction is a delisting-return imputation applied to instruments that leave the population, and a haircut subtracted from reported information ratios rather than noted alongside them; the haircut measured to date is of the order of three hundredths of an information ratio at a quarterly horizon.

Transaction costs. Novy-Marx & Velikov (2016) show that a large fraction of documented anomalies do not survive realistic costs, and that the survival rate is worst in exactly the low-liquidity names where gross premia look strongest. Chen & Velikov (2023) reach the same place with a wider net-of-cost treatment. Frazzini, Israel & Moskowitz (2015) provide the counterweight worth stating for balance: measured from live institutional order data, realised costs are materially lower than the academic estimates most anomaly-mortality results rely on, which cuts in favour of the habitat’s viability rather than against it. We take from this pair that the cost estimate is the pivotal input, not a nuisance parameter, and that its uncertainty must be carried through to the conclusion.

Venue and currency contamination. Until recently the admitted population included instruments listed on eight non-US venues, measured at 2,468 listings — 20.4 per cent of the admitted population and 33 per cent of all stored factor values — scored on a US market-day calendar in non-US currency units, one of which is denominated in minor units and so inflated by a factor of roughly one hundred. That inflation propagated into the very dollar-volume floor intended to exclude illiquid names. Factor evidence computed before this was corrected rests on a contaminated cross-section, which bears directly on §4’s negative result and is why we report that result’s limitations rather than only its verdict.

Thin cross-sections. The admitted population grows from roughly 1,550 instruments in the mid-1990s to roughly 6,900 by the mid-2020s. Early-period cross-sectional statistics rest on a materially smaller sample, and a result that appears to strengthen over time may be reporting only the growth of the denominator. Any period-by-period claim in this programme must be read against that.

4.8 8. Evidence is habitat-specific

The argument so far concerns one population. In practice this programme measures factor evidence on a single research population while each strategy deploys into its own, narrower one — so the information ratio a factor is promoted on is not the information ratio it is traded at.

H2. Factor evidence does not transfer across populations. Information coefficients measured on the research population do not predict deployed information coefficients on the narrower population a strategy actually trades.

Falsifier. Per-population factor rankings agree — the rank correlation of factor quality between the research population and each deployment tier is indistinguishable from unity. Transfer holds, and habitat-qualified validation is unnecessary.

If H2 holds, two things follow that the programme does not currently do. Factor evidence must be recorded per population rather than once globally, so that a factor’s validated status is qualified by the habitat it was validated in. And a strategy must be prevented from deploying on a population its factor was never validated on — a coherence condition that is presently unenforced.

This is stated as a hypothesis rather than a finding because it has not been measured. The per-tier evidence required to test it does not exist yet. We record it here, with its falsifier fixed, so that the test when run is confirmatory rather than exploratory.

4.9 9. Threats to validity

The habitat and the measurement are in tension. This is the threat that most endangers H1, and it is structural rather than incidental: the band that is most reachable is the band where survivorship, cost, and sampling error are all largest. A positive result is therefore only as good as the corrections in §7, and we should expect a positive result to be fragile to those corrections rather than robust to them.

The negative result of §4 was measured gross. It cannot distinguish a genuinely absent edge from an edge that exists gross and dies net — which is the same distinction H1 turns on. The two results are not independent, and neither is decisive alone.

Selection on liquidity is not a clean cut. The coverage requirement embedded in the dollar-volume floor excludes names that trade rarely, and rare trading correlates with the characteristics several factors are constructed from. The floor is therefore not a neutral threshold but a filter partially correlated with the signal.

Post-publication decay. McLean & Pontiff (2016) show that anomaly returns decline substantially after publication, and any habitat claim built on documented factors inherits that decay. A premium surviving in the small-cap band may be surviving because arbitrage capital has not yet found it — which is a reason to expect the edge to be temporary rather than structural, and should temper any capacity conclusion.

The impact coefficient is unestimated. §6 is explicit that \(\lambda\) is not measured from our own data. A conclusion that depends on its value is not established.

4.10 10. What would change our mind

Stated before the measurement, so that they bind.

We would abandon H1 if the widened band’s net-of-cost, survivorship-corrected information ratio fails to exceed the incumbent band’s at conventional significance after multiple-comparison correction across the bands tested.
We would treat the result as an artifact if the advantage does not survive exchange-referenced breakpoints and value weighting.
We would treat the habitat as unreachable if the capacity frontier places \(K_B^{*}\) below the operator’s plausible capital range under any defensible impact coefficient.
We would abandon H2 if per-population factor rankings prove indistinguishable, and would then drop habitat-qualified validation from the programme rather than build it.
We would revisit §4’s negative result if re-running the population comparison net of cost, on a cross-section corrected for the venue contamination of §7, reverses its verdict — which is the single most informative measurement available to this chapter and has not yet been made.

Next in the argument: the factor evidence measured on this population, in Factor Catalogue; the populations themselves, in The Study Population.