The Recovery Tax

Measure it peak-to-trough, or you will understate every one

A drawdown is the fall from a **peak** to the lowest point that follows it — not the fall from where you started. The difference matters more than it sounds. An account that runs from $100,000 to $118,000 and then down to $88,000 has fallen 25.4% from its high and only 12% from the start, and it is the 25.4% that the household lived through and that the recovery has to undo. Measuring from the start flatters every account that had a good run before it fell, which is most of them. The maximum drawdown is the deepest such fall in the whole record, and it is the number to quote when comparing strategies because it is the worst experience the plan has to survive. Two related figures finish the picture: the **duration**, meaning how long the account stayed below its previous high, and the **recovery time**, how long the climb back took. A strategy that earns 12% a year while spending four years of every decade underwater is a different proposition from one that earns 9% with a two-month hole, and the difference will not appear in either average return. There is a reason professionals obsess over this rather than over returns. The drawdown is the point at which the plan is most likely to be abandoned — it is where the arithmetic of the account and the arithmetic of the household collide, and behavioural research on loss aversion says the collision is not symmetric. A loss feels roughly twice as large as an equivalent gain, which means the deep drawdown is precisely the moment when the decision to sell feels most rational and is most expensive. The path in the workbench — Peak, reached in Q2: $118,000 · Trough, in Q4: $88,000 · Maximum drawdown: 25.42% — not the 12% measured from the start ← · Gain needed to climb back: 34.09% Five quarters underwater and about 3.8 years of unbroken 8% gains to regain the old high. Nothing about the strategy changed in that time; the account was simply earning back what had been lost.

Why recovery is a division

The arithmetic is one line, and it is the reason this lesson exists. If an account falls to a fraction f of its peak, the gain needed to get back is 1 ÷ f − 1, which is loss ÷ (1 − loss). At a 10% fall that is 11.1%, close enough to the loss to be mistaken for symmetric. At 20% it is 25%. At a third it is 50%. At half it is 100%, and at three quarters it is 300%. The relationship is convex: each extra point of loss costs more than the last one to undo, without limit. Two practical conclusions follow. The first is that deep drawdowns are not simply painful versions of shallow ones — they are qualitatively different, because the recovery doubles your required return at the point where your capital is smallest. The second is that the depth of the hole is a function of position size rather than of market conditions. A strategy that falls 20% at one size falls 40% at double the size on the same price series, so the question "how much can this strategy lose?" is really the question "how big am I allowed to be?", and it is answered before the trade rather than after the fall. That is why drawdown control and position sizing are the same subject wearing two names. Setting the size sets the peak-to-trough fall; setting the fall you can live with sets the size. The next two lessons make the arithmetic explicit, and the rest of the subject is mostly the discipline of keeping the two connected when it feels bad to do so. loss ÷ (1 − loss) — −10%: +11.1% · −20%: +25.0% · −33%: +49.3% · −50%: +100% ← · −75%: +300% ← Halving size after a large drawdown is not cowardice and it is not a prediction: it is the arithmetic above, applied to the fact that a strategy in a deep hole needs a return it may not have.

Time underwater is the cost the number hides

Depth and duration are two different costs and only one of them fits in a headline. A maximum drawdown tells you how far below the peak the account went; it says nothing about how long it stayed there, and for a real investor the second number is often the more expensive one. There are three distinct time measures worth keeping: time to the trough, time from the trough back to the old peak (**time underwater**), and the fraction of all periods spent below a previous high. A strategy that loses 30% and recovers in eight months is a different experience from one that loses 30% and takes four years, even though the drawdown figure is identical. The length of a recovery is not a function of the depth alone, and this is where the arithmetic of the previous section needs an addition. Recovery depends on the **rate of return the strategy earns while recovering**: a 50% drawdown needs a 100% gain, and whether that arrives in two years or eight depends on the annual rate, which for an equity strategy is roughly stable. What makes long recoveries particularly damaging is that they are also the periods when money leaves: an investor who withdraws during the underwater phase converts a paper drawdown into a permanent shortfall, and an investor who stops contributing stops buying at the prices that make the recovery possible. The drawdown in dollars can therefore end up larger than the drawdown in the index, purely because of behaviour. Inflation adds a third dimension that rarely appears in the reported figure. A nominal recovery means the account is back to the old dollar value, but every year underwater is a year the purchasing power fell, so the real peak is a moving target and the real underwater period is longer than the nominal one. For a plan with a spending requirement — retirement withdrawals, an endowment payout — the relevant question is not whether the account regained its nominal high but whether the spending that came out during the drawdown damaged the terminal value. That is why drawdown analysis and withdrawal analysis are the same conversation, and why the deepest historical drawdowns are studied in real terms. The same −30%, three ways of being expensive — Depth: −30% peak to trough: what the headline reports · Duration: Time back to the old high — months or years, at a fixed rate of return · Real terms plus withdrawals: The nominal high is not the goal, and money taken out is never recovered ← Report all three when you evaluate a strategy: depth, months underwater, and the share of periods spent below a prior peak. A strategy can pass on depth and fail badly on duration, and duration is what makes people abandon a plan at the bottom.

How deep should you expect it to be?

The drawdown in a backtest is an observation. The number that belongs in a plan is an expectation, and there is a rough relationship worth carrying that turns volatility and horizon into a plausible worst case: expected maximum drawdown grows with volatility, and grows only slowly — roughly with the logarithm — as the holding period lengthens. A strategy’s worst drawdown over ten years is materially worse than its worst over one, but not ten times worse. The practical shape of that relationship is what matters. Halving the volatility roughly halves the expected maximum; doubling the horizon raises it by a fraction rather than by a multiple. It is an order-of-magnitude tool, not a precision instrument, and it is enough for its purpose, which is to stop a plan from being built on the best observed outcome. Three consequences follow. First, set the tolerance from the expected maximum rather than from the observed one, and then ask whether the account can actually sit through it — a thirty percent expectation is a plan to lose thirty percent, decided in advance. Second, treat any backtested drawdown as a lower bound, because the backtest’s window is a sample and there is no rule that the future resembles it. Third, since leverage scales the drawdown roughly proportionally, a tolerance places a ceiling on leverage regardless of how good the edge looks. Duration deserves the same treatment as depth and gets far less attention. A drawdown has a length, and the recovery math makes length the more expensive dimension: after a deep loss the required gain is a division rather than the same addition, so recovering from a large drawdown takes years of ordinary returns. A plan that states the maximum depth but not the maximum time underwater is a plan whose holder will be tested by something they did not prepare for. One refinement is easy to get wrong when the portfolio has more than one position. The portfolio drawdown is not the sum or the average of its parts’ drawdowns; it depends on how the parts move together, and correlations rise in a selloff, which means the portfolio’s worst case is closer to the simultaneous one than a calm-period correlation matrix suggests. Estimating the portfolio’s expected maximum from an average correlation is how a diversified book turns out to be a single position at the bottom. • Expected maximum drawdown grows with volatility and roughly with the horizon’s logarithm. • Plan from the expected maximum, not from the worst that has happened so far. • State the maximum time underwater as well as the depth; recovery is a division. • Portfolio drawdown depends on correlation at the worst moment, not on the average one. The single most useful sentence in a risk plan may be this one: “I expect to lose approximately this much at some point, and this is what I will be doing while it happens.” Without the second half, the first is a number rather than a plan.

Sources

• Peak-to-trough drawdown and the recovery ratioStandard drawdown arithmetic; Magdon-Ismail & Atiya on drawdowns in portfolio management
• Loss aversion: losses loom about twice as large as equivalent gainsKahneman & Tversky, prospect theory (1979)
• Why deep drawdowns predict abandonment rather than patienceBehavioural finance literature on the disposition effect and redemption timing
• Drawdown control as a sizing outputVan Tharp, "Trade Your Way to Financial Freedom"