7 Ergodicity Deficit Across Tail Classes
When the Peters Formula Breaks Down
Status: Technical revalidation — prior working-paper derivation held
Literature survey complete (see project repository). Writing begins after Chapters 1–5 are finalized.
7.1 Motivation
Peters (2019) establishes that expected utility maximization — the foundation of standard portfolio theory — implicitly assumes ergodicity: that the time-average growth rate of a compounding process equals its ensemble average. Under geometric Brownian motion, this assumption fails, and the ergodicity deficit equals \(\sigma^2/2\).
The Peters expression is exact within its geometric-Brownian setup. Extending it requires more than substituting an estimated tail exponent. Infinite-variance stable laws (\(0<\alpha<2\)), finite-variance regularly varying models with infinite fourth moment, and bounded return models must be analyzed separately. In particular, Bessel-corrected sample variance remains unbiased whenever variance exists; an infinite fourth moment produces instability and problematic squared-error risk, not automatic upward bias.
7.2 The Open Question
How do bias, dispersion, and coverage of growth estimates change across valid return models? When is \(\mathbb{E}[\log(1+r)]\) defined, and how sensitive is it to the lower support near \(r=-1\)? Can a tail estimate identify the quantities needed for time-average growth, or are body shape, skewness, dependence, and truncation also required?
The historical 715-stock cross-section can motivate model families, but its current survivorship-conditioned calibration cannot answer these questions. A point-in-time universe, complete return models, and prospective outcome gates are required before estimating an empirical ergodicity deficit across tail strata.
7.3 Key prior work
- Peters (2019) — ergodicity economics framework
- Peters (2011) — optimal leverage from non-ergodicity
- Gabaix et al. (2003) — empirical equity tail exponent \(\alpha \approx 3\)
- Research gap: connect time-average growth, complete heavy-tail models, and prospective equity outcomes without treating a tail exponent as sufficient. Priority and novelty claims remain on hold until the literature search and empirical calibration are independently complete.