3 Not All Lottery Stocks Are Equal
Source-of-Tail as a Missing Dimension in the Fat-Tail Taxonomy
SSRN: Pending review (submitted 2026-06-29) Status: Under editorial review
3.1 Abstract
The Taleb (2025) taxonomy classifies equities by tail exponent: THIN, SUBEXP, FAT, and SUPER_FAT (α ≤ 2). Among SUPER_FAT tickers (n = 78 in our 715-equity study universe), we identify a 13.7 percentage-point spread in payoff-weighted expected value between two economically distinct subpopulations that the taxonomy treats as equivalent: FAT_STRUCTURAL stocks (n = 70), whose extreme returns arise from a continuous power-law tail, achieve +7.74% expected value per signal with skill score +0.423; FAT_BINARY stocks (n = 8), whose extreme returns arise from the resolution of discrete binary events (FDA decisions, clinical trial readouts), achieve −5.96% with skill score −0.922. Binary win rates are approximately 49% for both groups — the Hill estimator generates the same classification for populations that are empirically indistinguishable by direction but have opposite economic outcomes.
Source-of-tail classification is not an extension of the existing taxonomy. It is a prerequisite for determining whether the Hill estimator’s output is meaningful at all.
3.2 1. The Binary Event Problem
Not all extreme equity returns have the same origin. For most companies, the largest return realizations are draws from a continuous distribution — the structural right tail of the return process, driven by compounding of good outcomes. For a specific and identifiable class of companies, the largest returns are instead the resolution of discrete binary events of large magnitude: a Phase 3 clinical trial succeeds or fails; an FDA decision is approved or rejected.
These two generating mechanisms produce statistically similar tail exponents when measured by the Hill estimator, but they are economically and probabilistically distinct.
3.3 2. Taxonomy Extension: FAT_STRUCTURAL and FAT_BINARY
We define the mixture process:
\[X = (1 - B) \cdot Z_S + B \cdot J\]
where \(B \sim \text{Bernoulli}(p)\), \(Z_S\) is drawn from a continuous power-law distribution with tail exponent \(\alpha_S\), and \(J\) is a random variable representing the binary event payoff.
FAT_STRUCTURAL: The top-\(k\) order statistics are dominated by draws from \(Z_S\). The Hill estimator is consistent for \(\alpha_S\).
FAT_BINARY: The top-\(k\) order statistics contain a non-negligible fraction of \(J\) realizations (see Chapter 3 for the formal inconsistency argument). The Hill estimator converges to a quantity depending on \((\alpha_S, p, J^+)\) that is not separately identified from order statistics alone.
3.4 3. Empirical Results
| Subgroup | n | EV per signal | Skill score | Binary win rate |
|---|---|---|---|---|
| FAT_STRUCTURAL | 70 | +7.74% | +0.423 | ~49% |
| FAT_BINARY | 8 | −5.96% | −0.922 | ~49% |
| Spread | 13.7pp | 1.345 | ~0 |
The binary win rates are statistically indistinguishable between groups. The Hill estimator cannot see the difference. The economic outcomes are opposite.
The FAT_BINARY result (n = 8) provides directional evidence only at this sample size. No magnitude claim should be made without confidence intervals. The structural result (n = 70) is on firmer statistical ground.
3.5 4. Sector Composition
The FAT_STRUCTURAL subgroup is dominated by Financials (25%), Energy (13%), Utilities (11%), and Real Estate (10%) — rate-sensitive, value-oriented names with left-tail risk and positive drift. The FAT_BINARY subgroup is dominated by Healthcare (biopharmaceutical names facing binary regulatory events).
This sector decomposition is the mechanism: the taxonomy conflates two economically opposite types of stock under a shared statistical label.
3.6 5. Implications
The consequence is not merely taxonomic. Risk models that apply uniform tail-class multipliers to all SUPER_FAT stocks are mis-specified. The appropriate multiplier for FAT_STRUCTURAL and FAT_BINARY stocks differs in sign, not just magnitude.
The formal mathematical argument for why the Hill estimator cannot distinguish these cases is developed in Chapter 3.
→ Chapter 3 provides the mathematical foundation: why the Hill estimator is inconsistent for the structural tail exponent when the return process contains a binary-event component.