Leverage, Liquidation, and the Asymmetry of Losses

Deriving the liquidation price

Take an isolated long. You post equity E and the venue lets you control a notional position N = L × E, where L is the leverage. If the price falls by a fraction d, the position loses N × d, so your remaining equity is E − N·d. The venue closes you out when that remaining equity falls to the maintenance requirement, which is a rate m applied to the current value of the position, N × (1 − d).

Set them equal: E − N·d = m·N·(1 − d). Substitute E = N ÷ L and divide through by N, and every reference to the size of the position disappears: 1 ÷ L − d = m − m·d. Collect the d terms and you get d = (1 ÷ L − m) ÷ (1 − m). The liquidation price for a long entered at P is therefore P × (1 − d), and for a short the same argument with the sign reversed gives d = (1 ÷ L + m) ÷ (1 + m). Note what is not in the formula: your account size, the asset, the direction of the market. Only the leverage and the maintenance rate.

The common shortcut is to say a position at L× dies on a 1 ÷ L move — 5 percent at 20×, 10 percent at 10×. That is the answer when m is zero, and it is always too generous. With a 0.5 percent maintenance rate, an entry at 100 liquidates a 5× long at 80.402 rather than 80, a 10× long at 90.452 rather than 90, a 20× long at 95.477 rather than 95 and a 100× long at 99.497 rather than 99. On the short side, where the position grows as it moves against you, the distances are wider: 5.473 percent at 20× rather than 4.523. Every one of these figures assumes a clean fill with no fees and no gap through the liquidation level, and all three of those assumptions fail in exactly the conditions that produce liquidations.

At 20×, the fatal move fits inside a normal day

Put numbers on the leverage levels people actually use. At a 0.5 percent maintenance rate, a 2× long survives a 49.75 percent fall, a 5× long 19.60 percent, a 10× long 9.55 percent, a 20× long 4.52 percent, a 50× long 1.51 percent and a 100× long 0.50 percent. Venues do not hold the maintenance rate constant across leverage tiers, and higher leverage usually carries a higher rate: at 2.5 percent the 20× figure falls to 2.56 percent, and at 5 percent a 20× position is not offerable at all, because the maintenance requirement would already equal the whole deposit. That is why the table above marks the high-leverage cells at a 2.5 percent rate as impossible rather than as small numbers — an exchange has to lower m as it raises the maximum L, and the specific schedule differs by venue and by instrument and must be read there.

Whether 2.56 or 4.52 percent is a lot depends on the asset, and for a volatile one it is not. Recall the arithmetic from the companion article on volatility: an asset with a 60 percent annualised volatility has a daily standard deviation of 3.141 percent. That is a one-standard-deviation day. On such an asset the 2.56 percent liquidation distance at 20× is less than a single ordinary day's typical movement, and the 4.52 percent distance is about one and a half of them — a magnitude that occurs regularly, on no news, several times a month. This is the sentence worth keeping: at high leverage the market does not need to be wrong about your thesis to close your position, it only needs to be noisy.

The recovery asymmetry, which applies without leverage too

Lose a fraction L of your capital and you are left with (1 − L) of it. To get back to where you started, that remainder must be multiplied by 1 ÷ (1 − L), so the gain required is g = 1 ÷ (1 − L) − 1. The function is convex and it accelerates hard. Losing 5 percent needs 5.26 percent back. Losing 10 percent needs 11.11. Losing 20 percent needs 25.00. Losing 25 percent needs 33.33, losing a third needs 50.00, losing half needs 100.00. Losing 60 percent needs 150 percent; 75 percent needs 300; 90 percent needs 900; 95 percent needs 1,900; 99 percent needs 9,900 percent, a hundredfold, to stand where you already stood.

This has nothing to do with leverage or with any particular market. It is a property of percentages, and it is why a portfolio that alternates gains and losses of equal percentage size drifts downward — the same fact the companion article on volatility calls volatility drag. The asymmetry also explains why avoiding large losses matters more than capturing large gains: a single 50 percent loss cancels a doubling, whatever order they arrive in.

Leverage multiplies both the loss and the recovery requirement

Combine the two. A position at k× turns a market move of x into an equity move of kx. A 2 percent adverse move costs 4 percent of equity at 2×, 10 percent at 5×, 20 percent at 10×, 40 percent at 20× and 50 percent at 25×. Feed each of those into the recovery formula and the required equity gain is 4.17, 11.11, 25.00, 66.67 and 100.00 percent respectively. Now translate back into market movement, for a position that is re-margined to keep the same leverage on the reduced equity: the market has to move 2.08 percent at 2×, 2.22 at 5×, 2.50 at 10× and 3.33 at 20× to undo a 2 percent move. At 20×, the market must travel 1.67 times as far back up as it fell down.

One distinction matters here and is often blurred. If you open a position of fixed notional and never touch it, a price that falls and then returns exactly to your entry leaves you flat: the asymmetry is in the equity percentage, not in the price. The extra distance appears when leverage is maintained — when you are re-margined, when you top up to hold the same multiple, or when you hold a product that rebalances to a constant leverage each day. In the maintained case the position shrinks with the equity, so the same market recovery buys back less than it cost. Constant-leverage products are the everyday instance of this, and it is the reason regulators require them to warn that they are not designed to be held over long periods.

Repeated leveraged bets lose even with a positive edge

This is the result that surprises people, and it is the same geometric-mean argument the companion article on volatility develops. Build a market that is genuinely favourable: each period it moves +2 percent with probability 0.52 and −2 percent with probability 0.48. The expected move is 0.52 × 2 − 0.48 × 2 = +0.08 percent per period. That is a real edge, of the kind almost nobody has, and it is positive at every leverage: hold the position at k× and the expected arithmetic return per period is 0.08k percent, which grows without limit as k rises.

The compound growth rate is a different quantity. Because outcomes multiply, what matters is the expected logarithm: g(k) = 0.52·ln(1 + 0.02k) + 0.48·ln(1 − 0.02k). At 1× that is +0.0600 percent per period. At 2× it is +0.0800 percent, the maximum, which lands exactly on the Kelly fraction (p − q) ÷ b = 0.04 ÷ 0.02 = 2. At 3× it falls back to +0.0600. It crosses zero at k = 3.9957 — essentially 4× — and goes negative from there: −0.1012 percent per period at 5×, −1.2302 at 10×, −7.0231 at 20× and −12.1869 at 25×. Above 50× a single adverse period is total ruin, since 0.02 × 50 = 1.

Over 250 periods the exact binomial probabilities make the picture concrete. Unleveraged, this favourable market leaves you below your starting capital 32.87 percent of the time. At 2× that rises slightly to 37.56 percent, still with the best compound outcome. At 5× it is 57.50 percent, and 9.19 percent of runs are down 90 percent or more. At 10× you finish below where you started 82.88 percent of the time and are down 90 percent or worse in 57.50 percent of runs. At 20× the figures are 99.54 and 99.06 percent. The expected return has been positive throughout. What increasing leverage did was convert a favourable game into an unfavourable one, without changing the game.

Funding, fees, and the honest conclusion

Everything above assumed the position is free to hold. It is not. Perpetual futures charge a periodic funding payment between longs and shorts, and that mechanism — how the rate is set, which side pays, and why it exists — is derived in the companion article on funding rates rather than repeated here. What matters for this article is how leverage scales it: funding is charged on the notional, and your notional is k times your equity, so the cost as a percentage of your own capital is k times the headline rate. At a stated rate of 0.01 percent per eight hours, which is 0.03 percent a day on notional, a 1× position pays 0.03 percent of equity a day and a 20× position pays 0.60 percent — 16.52 percent over thirty days. That rate is an illustration and not a quotation; real funding varies continuously, can be negative, and must be read from the venue.

Add trading fees, which are also charged on notional and therefore also multiplied by k, and add slippage on entry and exit, and the drag becomes a third mechanism working in the same direction as the two above. None of it changes the shape of the conclusion, and all of it makes the conclusion stronger.

The conclusion is not encouraging and there is no honest way to make it so. European regulators require firms offering leveraged retail products to publish the proportion of retail accounts that lose money, and the analysis behind those product-intervention measures reported figures in the region of three-quarters to nearly nine-tenths across providers. That is not a claim about any particular firm or instrument, and the current disclosure for any product you are looking at is on that provider's own page — but the direction is consistent with everything derived above. This article exists to make the arithmetic legible, not to suggest that understanding it converts a losing structure into a winning one. Nothing here is advice, and the safest thing any of these formulas can tell you is how much you would need to be right about.

Frequently asked questions

Is the liquidation price simply my entry minus 1 ÷ leverage?

No, and the shortcut is always optimistic. The correct distance is (1 ÷ L − m) ÷ (1 − m), where m is the maintenance margin rate, and the shortcut is the special case m = 0. At a 0.5 percent maintenance rate the true figures are 19.60 percent at 5× rather than 20, 9.55 at 10× rather than 10, and 4.52 at 20× rather than 5. At a 2.5 percent rate the 20× distance is only 2.56 percent. Fees deducted at entry and any funding already charged reduce your equity before the price moves at all, which pulls the level closer still, and in fast markets the fill can be worse than the trigger level. Treat any published liquidation price as the best case rather than the boundary.

Why is the liquidation distance different for a short?

Because a short position grows as it moves against you. When the price rises, the notional you owe rises with it, so the maintenance requirement — which is a percentage of the current notional — rises too, while your equity falls. Running the same algebra with the signs reversed gives d = (1 ÷ L + m) ÷ (1 + m) instead of (1 ÷ L − m) ÷ (1 − m). At 20× with a 0.5 percent maintenance rate a long survives 4.523 percent against it and a short survives 5.473 percent, so the short has slightly more room in percentage terms. That is not an advantage: the long's worst case is bounded, since the price cannot fall below zero, whereas the short's loss has no ceiling, which is the more important asymmetry between the two sides.

If my expected return is positive, why does more leverage make things worse?

Because the expected return and the rate at which money actually compounds are two different quantities, and only the second determines what you end up with. Leverage multiplies the expected return by k and the variance by k squared, and the compound growth rate is roughly the expected return minus half the variance — so the penalty grows faster than the reward. On the worked example, a market with a genuine +0.08 percent edge per period compounds at +0.0600 percent unleveraged, peaks at +0.0800 at 2×, crosses zero at about 4× and falls to −1.2302 percent per period at 10×. Over 250 periods that 10× position finishes below its starting capital 82.88 percent of the time and down 90 percent or more in 57.50 percent of runs, all while the expected return stays positive. This is the same volatility-drag mechanism as in the companion article, applied to a position whose volatility you chose.

Does a stop-loss protect me from liquidation?

It can move the exit closer than the liquidation level, which is worth something, but it does not remove the arithmetic. A stop is an instruction to close at market once a trigger is touched, so the price you actually receive depends on what is available at that moment; in a fast move it can be materially worse than the trigger, and if the market gaps through both levels the stop and the liquidation happen at essentially the same place. The deeper problem is that at high leverage the stop has almost nowhere to go: at 20× the liquidation sits 2.56 to 4.52 percent away depending on the maintenance rate, so any stop must sit inside that, which on a volatile asset is well within one day's ordinary movement. Where to place a stop relative to an asset's own volatility is worked through in the companion article on stop placement.

Is there a leverage level that is mathematically safe?

No level is safe, and the arithmetic only identifies which levels are self-defeating. On the worked example the compound growth rate is maximised at 2×, which coincides with the Kelly fraction (p − q) ÷ b = 2, and goes negative above about 4×. But that number depends entirely on knowing p and b — the true probability and size of your edge — and nobody knows those; they are estimated from a sample, and an overestimated edge produces an overleveraged position with a negative compound return that still looks positive on paper. Practitioners who use this framework routinely apply a fraction of it for that reason. The honest summary is that the arithmetic can tell you a level above which you certainly lose over time, and cannot tell you a level at which you are safe. Nothing in this article is a recommendation to use any leverage at all.