A time-under-water limit catches risks a depth limit misses when losses have thin tails. For regularly varying losses, the limits mostly flag the same extreme episodes. Yinhao Wu proves the distinction: in a Gaussian random walk with positive drift, the deepest drawdown eventually differs from the longest almost surely. Fixed lists of the deepest and longest episodes eventually have no members in common. Under regular variation, both lists contain the same episodes in the same order, with probability tending to one. The paper does not establish which side real ETF returns occupy.
Does the deepest loss mean the longest wait?
Wu studies completed drawdowns. Each episode begins at the last peak before a loss and finishes when the path first reaches or exceeds that peak. A tie counts as recovery. Depth H measures the fall from peak to trough; duration L measures the entire interval from peak to recovery. The clock includes both the descent and the climb back.
Cumulative log return follows a random walk with iid increments, mean µ > 0 and a positive probability of a negative step. Each recovery restarts the process, making successive (depth, duration) pairs iid. Among the first n completed episodes, Wu asks how the deepest episode's duration compares with the longest, where it ranks by duration, and how much the k deepest and k longest lists overlap.
The paper contains no data, simulations or performance figures. Closed-form examples supply its only numbers. All the SPY results below come from our own construction based on the idea.
Gaussian extremes pull the rankings apart
Start with losses that have a finite exponential moment. Their depth tail decays at rate α and their duration tail at rate β. During a deep episode, the path descends at a tilted speed v and recovers at drift µ. Reaching depth h and recovering therefore takes about (1/µ + 1/v)h periods.
Divide the deepest episode's duration by the longest duration. Wu proves that this ratio converges in probability to ρ = (β/α)(1/µ + 1/v), strictly between 0 and 1. The deepest episode's rank by duration grows like n^(1−ρ), apart from subpolynomial factors the theorem leaves unspecified. For every fixed k, the k deepest and k longest episodes eventually become disjoint almost surely.
For Gaussian increments, α = 2µ/σ², β = µ²/(2σ²) and v = µ. These give ρ = 1/2 regardless of drift or volatility. The deepest drawdown thus lasts asymptotically half as long as the longest. Its duration rank is n^(1/2), up to subpolynomial factors.
Wu explains the difference through exponential cost. A fall to depth h and the subsequent recovery take roughly 2h/µ periods. An episode with near-zero average motion can last about 4h/µ at the same exponential cost. The longest episode grinds sideways. The deepest descends at speed µ, then recovers at that speed.
The Gaussian value 1/2 does not carry over to every light-tailed model. With ±1 steps and an up-probability of 0.7, ρ ≈ 0.5144.
These figures concern extremes in an expanding collection of episodes. Maximum depth scales as log n/α; maximum duration scales as log n/β. They offer no recovery date for a drawdown already under way. And the ratio converges only in probability: almost surely, its limsup alone equals ρ.
When a single crash sets both rankings
For regularly varying losses, write P(X < −x) = x^(−γ)ℓ(x), with γ > 1. A single large negative increment determines an extreme episode's depth. It also creates the deficit that the drift must repay, so duration follows depth divided by µ.
Wu makes this precise across the full sample. The largest gap between H and µL, divided by the deepest depth, tends to zero almost surely. For each fixed k, the k deepest and k longest episodes have the same indices in the same order with probability tending to one. The duration ratio converges to one almost surely.
Growing lists retain the contrast. Let their sizes k_n increase without bound while k_n/n tends to zero. The overlap fraction tends to 0 with light tails and to 1 under regular variation. Both statements hold almost surely once k_n/log n grows without bound. Wu illustrates the heavy-tailed case with a shifted Pareto distribution: γ = 3/2 and c = 4 give µ = 1 and infinite loss variance.
For a trader, the distinction is direct. A duration limit largely repeats a depth limit at power-law extremes. With thin tails, it identifies different episodes, and the separation increases with the sample. The theorems cover finite exponential moments and regular variation. We did not find a treatment of intermediate tails, including lognormal or Weibull-type losses. The paper gives no verdict for an asset in that zone.
The limits of the result
Every result above assumes iid increments. That excludes volatility clustering, which could alter recovery times following a crash. In the heavy-tailed mechanism, recovery proceeds at the pre-crash drift µ. All the limits are asymptotic in episode count n or horizon N. We did not find rates for either the duration ratio or the rank. A rank of n^(1/2+o_P(1)) gives no guidance when n is ten or twenty.
Wu also extends the laws to a fixed window of N periods. Only completed episodes enter those rankings. If fewer than k episodes have finished, the missing statistics receive fixed values. The unrecovered drawdown at the window's end remains outside every theorem here, although it is usually the episode a risk committee cares about most. The requirement µ > 0 also excludes zero-drift assets.
What we ran on SPY
We applied the asset-agnostic idea to SPY. This is our construction, and it tests none of the theorems: SPY returns do not satisfy their iid assumption. Signals use daily adjusted closes; fills use adjusted opens.
The book stays in SPY until its closing drawdown reaches 10% or its time below the peak exceeds 63 sessions. Either event sends the entire position to cash at the next open. Re-entry waits for SPY to close at or above the frozen episode peak, then takes place at the following open. We charged $0.0040 per share, subject to a $1.00 minimum per order. Cash received zero interest.
From 2 January 2020 to 1 July 2024, our SPY construction earned 33.67% in total. Its Sharpe was 0.69 and its Sortino 0.62, with volatility of 10.91% and maximum drawdown of -18.75% (Calmar 0.36). That worst drawdown is nearly twice the 10% trigger. Next-open execution therefore failed to cap losses at the threshold. Wu supplies no performance figures for comparison with ours.
The result depends heavily on choices we made outside the theorem. The recovery latch keeps the book in cash until the previous peak returns, giving up the entire climb from the trough after every exit. In the Gaussian benchmark, that climb takes h/µ out of the deep episode's 2h/µ. Sharp crashes expose the other choice: the 10% trigger fires well before session 63. In those episodes, the OR rule behaves like a depth-only rule. The duration trigger adds exits during slow grinds, the episodes occupying the highest duration ranks in Wu's light-tailed case.
Our 10.91% volatility largely reflects time spent in cash. The four-and-a-half-year window from 2020 to mid-2024 contains too few completed episodes to speak to a result asymptotic in n. Our 33.67% return and -18.75% drawdown are products of the 10% and 63 sessions thresholds, the full-recovery latch and zero cash yield.
Should the book keep both clocks running?
The paper's use is diagnostic.
If the deepest episodes also lead the duration rankings in daily US ETF histories, that would be consistent with the heavy-tailed regime. A time-under-water clock would then add little to a depth stop. If the deepest episode repeatedly ranks much lower by duration, closer to the n^(1/2) picture, the duration limit is catching something the depth limit misses. That evidence would persuade us to keep both clocks running.
Our backtest stops at 2024-07-01, and everything after that date is deliberately left untouched so the same strategy can be checked out of sample later.