Why Losses Require Disproportionately Larger Gains

The denominator changes, and that is the whole effect

Percentages feel symmetrical. Down ten, up ten, back where you started. They are not symmetrical, and the reason is not subtle once it is named: the two percentages are percentages of different amounts.

The loss is measured against the value you had before it. Whatever survives is a smaller amount, and the gain that would restore the original value is measured against that smaller amount. A bigger job, expressed as a percentage of a smaller base, produces a bigger percentage. Nothing else is going on.

Write it out with a starting value of 100 and a loss fraction d:

value remaining = 100 × (1 - d)
amount missing  = 100 × d
gain required   = amount missing / value remaining
                = d / (1 - d)

At d = 0.1 the required gain is 0.1 / 0.9, or 11.11%. At d = 0.2 it is 25%. The numerator grows while the denominator shrinks, which is why the two numbers separate faster than intuition suggests.

Fifty per cent down, one hundred per cent up

The single most useful case to hold in your head is the halving.

Starting reference:      100
A 50% loss removes:       50
Value remaining:          50
To return to 100, the 50 must gain 50
50 gained on a base of 50 is a 100% gain

There is no trick in it. Half has to become a whole, and half becoming a whole is a doubling. The 100% figure is not a claim about how hard the doubling is or how long it takes. It is the size of the hurdle stated in the same units the loss was stated in, which is the only way to compare the two honestly.

The same reading works in reverse and is worth knowing, because it is where the arithmetic bites hardest. A position that is down 80% has 20% of the reference left. Turning 20 into 100 is a 400% gain. The loss went up by a factor of 1.6 between 50% and 80%; the gain required went up by a factor of four.

The curve, and where it stops having an answer

Plotted against the size of the loss, the required gain is a curve that starts almost flat and then turns upward sharply.

Two things are worth taking from the table rather than from the shape alone.

The first is that the early part is genuinely mild. A 10% fall needing an 11.11% rise is close enough to symmetrical that treating it as symmetrical costs almost nothing. The asymmetry is not a fact about small declines. It is a fact about large ones.

The second is what happens at the end. At a 100% loss the value remaining is zero, and there is no percentage that turns zero into anything. The honest statement is that no finite gain exists, not that the required gain is very large. A calculator that prints an enormous number there is telling you something false in a format that looks precise. New money can rebuild a balance from zero, but new money is not a percentage recovery of what was lost, which is a distinction the rest of this article keeps returning to.

Getting back to the old balance

Once a return assumption is added, the identity turns into a duration. If the remaining value compounds at a constant effective annual rate r:

value remaining × (1 + r) ^ t = reference
t = ln(1 / (1 - d)) / ln(1 + r)

At a 50% loss and an assumed 8% a year, t is about nine years. That figure is worth sitting with, because it is where the abstract percentage becomes a period of somebody’s life. It is also worth being precise about what it is not. It is not a forecast, it is not a probability, and it does not become more likely because it has been calculated.

Two boundary cases matter more than they might appear to.

If the assumed return is zero or negative and there was a loss, the old balance is not reached at all under a constant-rate model. The right output there is that sentence, not a very long duration and not a negative one.

If the loss was total, no positive rate helps either, for the reason above. Zero compounds to zero.

Getting back to the original plan

Here is the part that is usually left out, and it is the reason this article exists rather than a table of percentages.

Before the loss, the money was on a path. Call the return that defined that path p. The loss did not pause that path; it removed you from it. While the remaining value is climbing back toward the old number, the counterfactual path that never had the drawdown is still compounding at p, and it is moving away from the old number too.

So there are two separate finish lines:

• the old balance, a fixed horizontal target equal to the value before the loss;
• the original plan, a rising target equal to where the uninterrupted path would be at the same date.

Catching the first is not catching the second. With no contributions, the second has a closed form of its own:

t = ln(1 / (1 - d)) / ln((1 + r) / (1 + p))

Take the same 50% loss, a recovery return of 8% and a plan return of 6%. The old balance is reached after about nine years. The original plan is caught after about thirty-seven years. Roughly twenty-eight years separate two events that are both commonly described as “recovered”.

The denominator explains why. The first duration is driven by r alone. The second is driven by the gap between r and p, which is two percentage points here, and a two-point gap closes a hole slowly. Narrow the gap and the second duration lengthens sharply. Close it entirely and something stricter happens: if r is not above p, and both paths receive the same contributions, the plan is never caught under constant rates. Not caught late. Not caught.

That is not a pessimistic reading. It is what the algebra says, and a model that produced a number there would be inventing one.

New money rebuilds a balance. It does not rebuild performance.

Add contributions and a third confusion appears, and it is the one most likely to be flattering.

Suppose a position fell 40% and the holder keeps paying money in. The balance climbs, passes the old value, and the account shows a number higher than the one before the loss. It is tempting to read that as the investment having recovered. It is usually not.

The balance is made of two things: what the existing capital did, and what was added. Only the first is investment performance. The second is money that already belonged to the holder and would have existed under any scenario, including the one where they had sold and put it in a jar.

The test that separates them is simple and worth doing explicitly: run the same return assumption with no contributions and see whether that path reaches the reference at all. If it does not, the balance came back because money was added, and saying so is not pedantry. It is the difference between a strategy that worked and a strategy that was subsidised.

There is a stricter version of the same point for the plan comparison. If new contributions are added only to the recovery path and not to the uninterrupted one, a catch-up will appear that is made entirely of timing rather than performance. The comparison is only meaningful when both paths receive the same amounts on the same dates, which is why the UBWHY model gives them one shared schedule rather than two matching ones.

A constant return is a scenario, not a market path

Every duration above assumes the same return every year, applied smoothly. Real paths are not like that, and the difference is not cosmetic.

The order in which returns arrive changes outcomes whenever money moves in or out of a position. Two sequences with identical average returns can produce different ending balances once contributions or withdrawals are involved, because each contribution meets a different part of the sequence. Constant-rate arithmetic cannot see that at all: it has one rate and no sequence.

This is a real limitation and it is worth stating plainly rather than in a footnote. A constant-return model answers a narrow question well: under this assumption, how long? It does not answer what will happen, and it cannot be made to by choosing a more careful rate. The rate is an input you supplied, and the output inherits everything that is uncertain about it.

There is no rate in the calculator that UBWHY suggests, no historical average prefilled, and no “typical” recovery period offered. A number the tool put there would be an assumption the reader never made, wearing the authority of a default.

Why “I need to make it back” is not an investment rule

Everything above is arithmetic. This section is the boundary around it, and it matters more than any of the formulas.

A previous value is a fact about the past. It is not evidence that an asset will return to it. Prices reached in the past were reached under conditions that may no longer hold, and the market has no memory of what anybody paid. The reference value in a drawdown calculation is there because you selected it, not because it has been established as correct, fair or reachable.

The amount already lost is not a reason to keep holding. That money is gone whichever option is chosen next, which means it is identical across every option and cannot distinguish between them. Letting it decide is the sunk-cost error in its purest form, and it is especially easy to fall into after a loss, because the recovery target feels like a debt the asset owes.

It does not. The asset is not repaying anything. It will do whatever it does from today, and so will every alternative.

There is a second version of the same trap that is harder to spot: the belief that a loss should be recovered in the same product that produced it. Nothing in the arithmetic supports that. The hurdle is a property of the amount of money that remains, not of where it is held. Money moved somewhere else faces exactly the same required gain, and faces it with whatever prospects that alternative actually has.

UBWHY has no view on whether you should hold, sell, add money, average down or switch, and neither the calculator nor this article will develop one. That decision needs facts about the specific product, the specific alternatives and the specific person, none of which are inputs to a recovery formula.

What is actually worth comparing from today

If the recovery target is not the decision, something has to be, and the useful reframing is short.

The money currently in the position is an amount you hold right now. The question is which available use of that amount offers the best expected outcome for the risk it carries, judged from today, with the loss treated as history rather than as a target.

That reframing changes what is worth investigating:

• the mechanism of the product, and whether the thing that caused the decline is still present;
• the costs, because recurring costs raise the required return before any recovery is possible;
• the alternatives, priced on the same money over the same period, with the same contributions;
• the liquidity, because an exit that is slow, partial or dependent on a counterparty is not really an option until it has been tested;
• the concentration, because a position that has already fallen a long way is often a larger share of someone’s thinking than of their portfolio.

The required gain is still useful in that frame. It tells you the size of the hurdle any option has to clear, which is a genuine input to a comparison. It is just not the answer to it.

Where this arithmetic shows up in real products

Drawdown arithmetic is not only a stock-market topic. It becomes decisive wherever three conditions overlap: capital that can fall a long way, an exit that is not immediate, and an income stream that is easier to see than the capital loss underneath it.

• Assets that depreciate while paying out. Hardware, equipment and other wasting assets can produce income every day while their resale value falls faster than the income accumulates. The payout is visible and frequent; the capital decline is neither.
• Positions with a limited or peer-to-peer exit. Where a sale needs a counterparty rather than an order book, the recovery question and the exit question are the same question, and neither has a guaranteed answer.
• Holdings with a lock or a notice period. A lock removes the option to act on a reassessment, so a drawdown during one is a drawdown you carry to its end.
• Leveraged or borrowed positions. Leverage moves the arithmetic into a different regime entirely, because a large enough drawdown can end the position before any recovery is possible. That is outside what this model covers.
• Concentrated single holdings. The identity is the same, but a large drawdown in a single asset removes the diversification that would otherwise make the required gain a question about a portfolio rather than about one decision.

In every case the arithmetic does the same limited job. It converts a loss into the hurdle that follows it, so the hurdle can be compared with what the product plausibly offers instead of being estimated by feel.

The formulas, the assumptions and the limits

The whole model, in one place.

gain required            q = d / (1 - d)
old balance, returns only  t = ln(1 / (1 - d)) / ln(1 + r)
original plan, returns only t = ln(1 / (1 - d)) / ln((1 + r) / (1 + p))
monthly factors            (1 + r) ^ (1/12) and (1 + p) ^ (1/12)
target-period return       r = (1 / (1 - d)) ^ (1/T) - 1

What it assumes:

• the drawdown happens once, at the start, measured against a reference value you chose;
• returns are effective annual returns and stay constant for the whole period;
• contributions arrive on schedule and are identical on every compared path;
• nothing is rounded until it is displayed.

What it does not include: probability, volatility, the order in which returns arrive, inflation, tax, fees, spreads, leverage, liquidation, withdrawals, or any judgement about whether the reference value was reasonable. A nominal return to the old number is not a return to the same purchasing power, and this model does not claim otherwise.

Where an answer does not exist, the model says so rather than producing a large number: no finite gain at a total loss, no recovery through returns alone at a non-positive rate, no plan catch-up when the recovery return does not exceed the planned one, and nothing found within a hundred years of searching. That last one is a limit on the calculation, not a forecast horizon.

Run it on your own figures

The Drawdown Recovery Calculator applies exactly the model above to numbers you enter: your loss, your reference value, and, if you want a duration, your own return assumption.

It shows the required gain first, keeps the old balance and the original plan as separate results, names the cumulative new money whenever contributions are part of the scenario, and states non-reachable cases as sentences rather than as figures. It runs entirely in your browser: UBWHY does not receive the values you enter, stores none of them, and puts none of them in a link.