7 Ergodicity Deficit Across Tail Classes
When the Peters Formula Breaks Down
Status: Planned — research literature surveyed, writing not yet begun
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 formula, however, assumes Gaussian dynamics. It is undefined for \(\alpha \leq 2\) (where variance is infinite) and has not been calibrated to an empirically estimated tail exponent \(\hat{\alpha}\) across any equity cross-section.
7.2 The Open Question
How does the ergodicity deficit change as a function of \(\alpha\) in the range \((2, 4)\)? Is there a regime transition at \(\alpha = 2\) where the standard formula breaks down? Can the deficit be estimated directly from Hill estimator output?
Our 715-stock cross-section with measured \(\hat{\alpha}\) per ticker provides the instrument to answer these questions empirically for the first time.
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 \(\alpha \approx 3\)
- Gap: no paper bridges Peters + Gabaix. Novelty: HIGH.