11 Against Optimization
Why Fat Tails Make the Optimal Strategy Epistemically Unavailable
Status: Planned — target journal: Erkenntnis or Synthese
11.1 Motivation
Portfolio optimization — maximize expected utility subject to distributional constraints — requires that the distribution of returns be estimable from available data. Under Gaussian assumptions, moment estimators converge at rate \(\sqrt{T}\) and the optimization problem is well-conditioned for realistic sample sizes.
Under fat-tailed distributions, this assumption fails systematically. Taleb (2025) (Ch. 3–4) proves that sample estimators of mean and variance converge far slower than \(\sqrt{T}\) under power-law tails, making standard statistical inference unreliable at realistic sample sizes. Michaud (1989) showed that even under Gaussian assumptions, sample estimation error dominates the optimization, producing portfolios that are optimized for measurement noise rather than true parameters.
Under fat tails with \(\hat{\alpha} \in (2, 3)\), the Michaud problem is dramatically worse.
11.2 The Claim
The claim is epistemological: optimization is not merely imprecise under fat tails — it is epistemically unavailable. The data required to close the estimation gap exceeds any realistic investment horizon.
This can be stated precisely. Chapter 7 establishes the minimum track record required to detect a given information ratio at significance level \(p\) under tail exponent \(\hat{\alpha}\). The analogous result for optimization holds: the minimum sample size to identify the optimal portfolio within a given tolerance is a function of \(\hat{\alpha}\), and at \(\hat{\alpha} = 2.69\), that minimum exceeds the observable sample by a derivable factor.
The rational response to this result is robustness — the barbell strategy that Taleb advocates — rather than optimization. This chapter provides the mathematical justification for that preference as a theorem, not an intuition.
11.3 Connection to the Precautionary Principle
The Taleb et al. (2014) precautionary principle argues that for systems with fat-tailed harm distributions, expected-value frameworks fail. Combined with the unavailability of optimization established here, PP implies a specific portfolio structure: maximize resilience to estimation error (robustness) rather than expected utility of an estimated distribution (optimization). The two arguments — PP and estimation unavailability — converge on the same structural recommendation from different directions.
11.4 Key prior work
- Taleb (2025) — statistical consequences of fat tails; moment estimation failure
- Michaud (1989) — estimation error dominates optimization under Gaussian assumptions
- Taleb et al. (2014) — precautionary principle under systemic fat-tail risk
- Peters (2019) — ergodicity and the failure of ensemble optimization (Chapter 6 connection)
- Gap: mathematical proof that optimization is epistemically unavailable (not merely imprecise) at empirically calibrated \(\hat{\alpha}\). Novelty: MEDIUM-HIGH.