When is the deepest drawdown also the longest?
Yinhao Wu
Abstract
Does the deepest drawdown take the longest to recover? For random walks with iid increments of positive finite mean, we find that the deepest and longest episodes are eventually distinct almost surely under light-tailed losses, but coincide with probability tending to one under heavy-tailed losses. More precisely, when losses have a finite exponential moment, the deepest episode's duration, relative to the longest among $n$ episodes, converges in probability to an explicit $ρ\in(0,1)$, while its descending duration rank is $n^{1-ρ+o_{\mathbb{P}}(1)}$. These limits hold for every fixed number of leading depths; in the Gaussian model, $ρ=1/2$, independently of drift and variance. The separation reflects the different exponential costs of reaching a large depth and sustaining a long episode. For regularly varying losses, a common large loss governs both depth and duration: the duration ratio tends to one almost surely, and for every fixed $k$, the $k$ deepest and $k$ longest episodes have the same indices in the same order with probability tending to one. For any $k_n\to\infty$ with $k_n=o(n)$, the proportion of common episodes among the $k_n$ deepest and $k_n$ longest converges in probability to zero in the light-tailed case and to one in the heavy-tailed case. These ranking laws also hold for completed drawdowns within a deterministic observation horizon.
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