Why a loss is harder to recover than it was to make

The same distance, measured twice 

Two figures are involved and only one of them stays put. The fall is a percentage of the figure the account started at. The recovery is a percentage of the smaller figure the account arrived at. The money amount in between is identical in both directions, because it is the same gap being crossed twice, but the number it is divided by is not the same on the way back, and a percentage is nothing more than a division. The recovery is the same money measured against a smaller base, so it is a larger percentage. Always, and by an amount that has nothing to do with anything happening in a market.

The formula falls straight out of that observation. The gain required to return an account to its starting figure is the loss divided by what remains after the loss. The numerator is the gap. The denominator is not the figure the account started at but the one it is currently standing on. Every property described in the rest of this lesson is a property of that denominator, and none of it is a property of trading.

The lower half of the block is the half most often left out, and leaving it out misstates what the asymmetry actually is. It is not a property of losses. It is a property of the base. After a rise, the fall that hands the gain back is smaller than the rise that produced it, by exactly the same mechanism and for exactly the same reason. Percentages are asymmetric in both directions because the second one is always measured against whatever the first one left behind.

Why the requirement grows faster than the loss 

Reading down the falls in the block above, the requirement does not keep pace with the loss. It pulls away from it, and it pulls away faster the further down the column the reading goes. The reason sits entirely in the denominator. Each additional percentage point of loss subtracts from the amount remaining, so the divisor is shrinking at the same time as the numerator is growing. Two effects push the same quotient in the same direction at once, and a quotient with a growing top and a shrinking bottom does not grow in a straight line.

The consequence at the far end is worth stating plainly, because it is where the arithmetic stops behaving like ordinary percentages. As the remaining fraction approaches nothing, the divisor approaches nothing, and a fixed gap divided by a vanishing base grows without limit. There is no depth at which the requirement levels off. An account reduced to nothing has no percentage recovery at all, not a very large one, because there is no base left to take a percentage of.

The same denominator explains a second result, and it is the one that catches readers out most often. Sequential percentage changes compound rather than add rather than add, so a fall and a rise of the same percentage do not cancel. The rise is applied to the reduced figure, so it returns less money than the fall removed, and the account lands short of where it began. The order the two are applied in makes no difference whatsoever, which the block below shows by running it both ways.

What position size changes 

So far the fall has simply been stated. In an account it is produced, and it is produced by two quantities rather than one. A price move of a given percentage produces a money result proportional to the contract value of the position it moved against. That money result is then divided by the account's equity to become the percentage fall the arithmetic above operates on. The price move is one input. The size of the position is the other, and it is the one that decides which part of the curve an account ends up standing on.

This is where the mechanic built earlier in the module does its work. Profit and loss are calculated on the full contract value while only a percentage of that value is posted as margin, so an adverse move is measured against the whole contract, a running loss can exhaust the margin posted entirely, and losses are not limited to the amount deposited. A favourable move is measured on exactly the same basis and to exactly the same degree. The percentage fall an account records is therefore not the percentage the instrument moved. It is that move scaled by how large the position was relative to the account.

Two things separate the two positions, and they are the same fact seen twice. The smaller position's recovery requirement sits barely above its own fall, while the larger position's has already begun to pull away from it. The smaller position also leaves more free margin standing behind it, so the same adverse move carries the account less far toward the level at which positions are closed on the platform's initiative. Size is the single quantity behind both, which is why the arithmetic of recovery and the arithmetic of close out are not two separate subjects.

The argument this arithmetic is used to make 

Position sizing is the practice of deriving contract size from a stated fraction of account equity rather than choosing it by feel, and the argument trading literature builds on the arithmetic above is narrower than it is usually reported to be. It is not that a size limit produces a profit, and it is not that a size limit prevents a loss. Neither claim follows from anything on this page. The bounded version is that a limit keeps the percentage falls an account can record in the region where the recovery requirement is close to the fall itself, rather than the region where it climbs away from it. That is a statement about arithmetic and about nothing else.

Traditions differ over the fraction, over whether the fraction is measured against equity or balance, and over whether it is fixed or recalculated after each result. No fraction removes the risk of loss, no fraction guarantees survival of a sequence of losses, and this page puts forward no figure for anybody. A lesson knows nothing about the circumstances of the person reading it, which is the reason a specific limit is absent here rather than an oversight.

One further term belongs in the requirement, and every block above excluded it. Recovery is measured gross in this lesson. Positions opened in the course of one carry the cost of opening and closing them, and a position held past the daily cut off carries a financing adjustment for as long as it stays open. Those amounts are debited from the same equity the percentage is measured against, so the gain required in practice is the gross arithmetic requirement plus whatever the positions taken cost to run. Costs move the requirement in one direction only. They never make it smaller.

Where practitioners disagree 

The first disagreement is about the unit itself, and it has a concrete consequence rather than a philosophical one. One convention measures a fall as a percentage of equity, on the grounds that it is the only unit that compares an account to itself across time and across sizes. Another objects that a percentage of equity is unstable in any account that takes deposits and withdrawals, because a deposit reduces the measured fall without a single position having improved, and a withdrawal deepens it without a single position having worsened. Both conventions are in live use, they can report materially different figures for the identical trading, and which one a given report uses is stated in that report rather than deducible from the numbers in it.

The second concerns what, if anything, follows from the asymmetry about size after a fall. One tradition holds that because the requirement climbs with depth, reducing size after a fall keeps an account away from the steep region. The counter is arithmetic rather than temperamental: a smaller position also returns less from any favourable move, so reducing size lengthens the road back at the same time as it flattens the risk of extending it. The two effects work against each other, and which one dominates depends on the sequence of moves that actually follows, which no arithmetic on this page contains. The disagreement therefore persists rather than resolving, and this page takes no position on it.