The Drawdown and the Comeback

The convexity, and what it does to a plan

The arithmetic of recovery is a division. If the equity falls by a fraction D, the gain needed to restore the old high is D ÷ (1 − D): a 10% fall needs 11.1%, a 20% fall needs 25%, a 33% fall needs 50%, a 50% fall needs 100%, and an 80% fall needs 400%. The curve is convex, which means the cost of the hole grows faster than the hole, and it means the deepest drawdowns are not scaled-up versions of the shallow ones — they are a different category of problem, because the required return is outside anything the market reliably supplies. The consequence for a policy is that the drawdown has to be bounded before it happens, and the bound is a risk decision rather than a return decision. A trader who decides in advance that a 15% drawdown halves the unit has chosen a maximum cost for being wrong, and the choice is available at the beginning and not in the middle: by the time the account is down 40%, the size that would produce the recovery is also the size that would make ruin possible, and that is the moment when the decision is hardest to make well. The risk-side lessons on ruin and the Kelly curve make the same point from the survival direction; this lesson is the same result stated as the depth of a hole. There is a second-order effect that is easy to miss and expensive to ignore. The recovery time depends not only on the depth but on the expectancy of the remaining trades: at +0.2R a trade and a 1% unit, a 6% drawdown is about thirty trades of good work, while the 7% day from the tilt lesson is six weeks. A drawdown is therefore experienced as a duration, not as a number, and the second thing it changes is the trader’s willingness to sit through that duration without altering the process. Most of the damage in a long drawdown is done by the changes made inside it. What a fall costs to undo — Falls 10%: Needs 11.1% · Falls 20%: Needs 25% ← · Falls 33%: Needs 50% · Falls 50%: Needs 100% ← · Falls 80%: Needs 400% Every row is D ÷ (1 − D). The curve steepens without limit, which is why a stop at the account level — a maximum drawdown rather than a stop on a position — is the only protection that scales with the depth of the problem.

What the drawdown does to the decision-maker

A drawdown does not merely reduce the balance, it changes the person making the next decision, and the change has a specific and well-documented shape. Experimentally, a prior loss makes people more risk-averse — the position they take afterwards is smaller, not larger — with one exception that matters enormously here: if a single bet offers the chance to break even, risk-seeking rises. Thaler and Johnson called it the break-even effect, and it is the mechanism behind the trade that turns a manageable drawdown into an account-ending one. The trader who has been cautious for weeks becomes aggressive precisely at the moment when an additional loss would be unrecoverable, because that is the moment the next trade is framed as the one that restores the high. The professional version of the same effect runs through incentives rather than feelings. A manager who is behind a benchmark, or whose fund is approaching its high-water mark, has a payoff structure that rewards more risk and it is usually the mandated response to the career consequences of closing the gap. The institution is doing what the individual does, for reasons that are entirely rational from inside its own incentives, and the aggregate result is the same: risk-seeking at the bottom and the possibility of ruin that a constant risk budget would have ruled out. The third change is to activity. A drawdown makes the plan feel slower than the situation, so the response is to trade more, widen stops, or move into something that moves faster — all of which raise the risk, which is the variable that just produced the problem. The honest countermeasure is not an attitude: it is a written drawdown rule naming the trigger (a percentage from the equity high), the action (halve the unit, or pause, or reduce the number of open positions), and the condition for restoring it (two planned trades followed, or a month of the process). Written in advance, it converts the worst moment in an account’s life into a pre-agreed procedure, which is the same move the whole mastery rung makes. • Set the maximum drawdown in advance: at what fall does the unit halve, and at what point does trading pause? • Name the restore condition as well, so the rule has an exit as well as an entry. • Compute the recovery in trades — drawdown in R divided by expectancy per trade — so the duration is known rather than discovered. • Beware the break-even bet: it is framed as the recovery and it is the position that carries the ruin risk. • Keep the process constant inside the drawdown; most of the damage is done by the changes made under pressure. De-risking in a drawdown lengthens the recovery and reduces the chance of ruin, and those are genuinely in conflict. The trade-off is a policy decision to be made when equity is at its high, with numbers, rather than resolved by whichever feeling is strongest in the middle of a losing month.

The drawdown changes the right size, and not in the direction you want

The instinct after a drawdown is to size up, and it has a name and an experimental basis. The **break-even effect** describes the finding that people who have just lost become more willing to take risk with what remains — not because the odds improved but because the remaining decision is now framed as a choice between recovering the loss and accepting it. The **house-money effect** is its mirror image: people who have just won also take more risk, because the gain feels like money they did not have. Both effects push position size up after an unusually large change in the account, in either direction, and both are reversals of what the arithmetic asks for. What the arithmetic asks for depends on how the size is expressed. If risk is a fixed fraction of the account, the position automatically shrinks in dollars after a loss, which is the right behaviour delivered for free. If risk is a fixed dollar amount, or a share count, the position does not shrink, and after a large drawdown it becomes a larger fraction of what is left — the same trade carrying more of the account than it did at the peak. The systematic version of the correction is a **drawdown-scaled size**: risk a fraction of the *current* account value rather than of the peak, or scale the fraction itself down as the drawdown deepens and restore it as the account recovers. Neither is elegant, and both reduce the rate at which the account can come back. That is the point: the state in which you need the most leverage to recover quickly is the state in which you can least afford to be wrong. There is a second-order effect that makes the recovery slower than the simple arithmetic suggests. Volatility costs money in a compounding account — a sequence of returns has a lower compound rate than its arithmetic average, and the gap grows with the volatility — so an account that has just experienced a large loss is typically experiencing higher volatility at the same time. The required gain is computed from the loss alone, but the achievable rate of return is lower in the aftermath, which is why recoveries take longer than the division suggests and why the pressure to make up time builds exactly when the conditions are worst. Writing the rule before the drawdown is the only defence that does not require a decision in that state. A workable drawdown rule has two parts: the risk fraction scales with the current account rather than the peak, and a written threshold at which the size is deliberately cut further. Both are anti-martingale, which is to say both are the opposite of intuition.

The recovery with cash flows: why the arithmetic changes when you are adding or withdrawing

The division in this lesson — a fall of D needs a gain of D ÷ (1 − D) — describes an account that is left alone. Almost no household account is left alone, and cash flows change the arithmetic in both directions. On the accumulation side, contributions are a tailwind in a drawdown: an account that falls 30% and receives regular contributions recovers its value in less time than the percentage arithmetic suggests, because the new money is buying at the lower prices and the denominator being restored is growing. That is the mechanism behind the claim that a market fall is good for a young saver, and it is true precisely and only while the contributions are large relative to the balance — for a large balance with small contributions, the effect is negligible, and the simple division is the right approximation. The distinction is worth making concrete: on a $20,000 account with $600 a month going in, a 30% fall is largely a buying opportunity; on a $2 million account with the same contribution, it is arithmetic. On the withdrawal side the asymmetry is brutal, and it is the mechanism the personal-finance sequence-of-returns discussion exists for. A withdrawal made during a drawdown is sold at the low point, and unlike a contribution it can never be bought back at that price. The recovery division therefore understates the damage: an account withdrawing four percent a year from a portfolio that falls 30% in year one has taken that year’s spending out of a shrunken base, which locks in the loss and raises the required return on the remaining balance. Two accounts with identical portfolios and identical average returns can finish decades apart depending on the order in which the returns arrived, and the account with the withdrawals is the one that is most exposed to the order. The practical consequence is that the required-return arithmetic has to be computed on the **balance net of the withdrawal schedule**, not on the balance itself: a 20% fall with a 4% withdrawal in the same year needs materially more than 25% to restore the high-water mark, and the exact figure is a small table rather than a formula. Both cases share the same prescription, which is the one this lesson already gives in a different key: decide the cash-flow rule before the drawdown, not inside it. For an accumulator that means pre-committing to keep contributing on schedule, because the moment to buy is the moment it feels worst. For a withdrawer it means holding a **cash buffer** sized to the withdrawal schedule — commonly a year or two of spending — so the portfolio is not being sold during the decline, and writing the rule that refills the buffer from gains rather than from a judgement about the market. Neither is a return forecast; both are the same statement this lesson makes about the recovery, translated into cash: the depth of the hole and the duration of the recovery are both facts about the whole account, and the flows are part of the account rather than something external to it. A 30% fall, three account types — Left alone: Needs 42.9% to restore the high — the D ÷ (1 − D) case · Contributing monthly: Recovers sooner: the new money buys at the low, if the balance is small relative to the flow ← · Withdrawing 4% a year: Worse than the division implies: the sale locks in the loss on a smaller base ← · The common prescription: Accumulators keep contributing; holders keep a cash buffer of a year or two of spending The link to the personal-finance withdrawal-rate lesson is exact: this is the same convexity seen from the spending side, and the buffer is the same instrument that lesson builds. A drawdown is not a portfolio event with the person standing outside it; the flows are part of the position, and the rule about them is a risk rule rather than a cash-management detail.

Sources

• Gambling with house money and trying to break evenThaler & Johnson (1990), Management Science
• Prospect theory and risk-seeking in the domain of lossesKahneman & Tversky (1979), Econometrica
• Optimal investment strategies for controlling drawdownsGrossman & Zhou (1993), Mathematical Finance