8 Minimum Track Record Length Under Fat Tails
How Long Must Performance History Be Before Skill Is Declared?
Status: Planned — research literature surveyed, writing not yet begun
8.1 Motivation
Bailey and López de Prado (2012) derive the Minimum Track Record Length (MinTRL) — the number of observations needed to reject, at a given confidence level, the null hypothesis that a manager’s Sharpe ratio is zero. The derivation assumes finite fourth moment (\(\kappa > 4\)).
Kao et al. (2025) proved in 2025 that the Sharpe ratio converges at rate \(n^{1-2/\kappa}\) when \(\kappa \in (2, 4)\) — slower than the \(\sqrt{T}\) Gaussian rate — toward a stable distribution with tail index \(\kappa/2\).
The practical table that follows from these results does not exist: “At tail exponent \(\hat{\alpha}\) and significance level \(p\), the minimum track record is \(N\) years.”
8.2 The Claim
At \(\hat{\alpha} = 2.69\) (the median in our equity sample), the minimum track record required to declare skill at 95% confidence is substantially longer than Gaussian theory implies. The precise factor is derivable analytically from the Kao et al. (2025) convergence rate and verifiable against our cross-sectional \(\hat{\alpha}\) distribution.
The implication is philosophical as much as practical: for a large fraction of the equity universe, the sample sizes available to any practitioner are insufficient to distinguish skill from luck at conventional significance levels — not because the practitioner is unlucky, but because the distribution does not permit it.
8.3 Key prior work
- Bailey and López de Prado (2012) — Probabilistic Sharpe Ratio and MinTRL
- Bailey and López de Prado (2014) — Deflated Sharpe Ratio
- Kao et al. (2025) — convergence rate under fat tails (critical new anchor)
- Vlasiuk (2025) — Lévy-stable scaling of performance metrics
- Gap: no explicit \(N(\hat{\alpha}, \text{significance})\) table. Novelty: HIGH.