Investing Basics
A fall and its recovery are not the same size
Percentages are taken from different bases on the way down and the way back, so returning to the starting point always requires a larger rise than the fall that preceded it.
By Aditya Ramaswamy4 min read
The asymmetry, in one line of arithmetic
A balance that falls by half needs to double to get back. That sounds like a contradiction until the bases are made explicit: the fall was fifty per cent of the original amount, and the recovery has to be one hundred per cent of what is left. Same absolute sum, two different denominators, two very different percentages.
The general form is simple. The rise required to recover from a fall is the fall divided by one minus the fall. A 20% decline needs a 25% gain; a 33% decline needs a 50% gain; a 50% decline needs 100%; a 75% decline needs 300%. The relationship is not linear and it worsens sharply as the fall deepens.
This is arithmetic rather than a market phenomenon, and it applies to anything measured in percentages. It is also the single most useful thing to know about why deep losses are treated so differently from shallow ones.
Why an average return can mislead
The asymmetry has an immediate consequence for how returns are described. A holding that gains 50% and then loses 50% has not broken even — it is down 25%, because the loss applied to the larger balance the gain created. The two percentages average to zero and the outcome does not.
This is why the arithmetic average of annual returns overstates what an investor actually experienced, and why the compounded figure is the honest one. The gap between the two widens as returns become more volatile, which means a strategy with dramatic swings can report a respectable average while delivering a mediocre result.
It follows that volatility is not merely uncomfortable. Where a balance is being multiplied year after year, large swings reduce the compounded outcome relative to a steadier path with the same average, purely through the arithmetic of the bases.
A fall in value is not automatically a loss, and not automatically temporary
It is commonly said that a decline is only a loss when it is realised, and there is something to that: an investor who does not sell has not converted a lower valuation into a smaller amount of money. Holdings that fell and later recovered have done so many times, and selling into a decline removes the possibility of participating in whatever comes next.
But the phrase is misleading if it is taken as reassurance. A lower valuation is a real reduction in what the holding is worth today, and recovery is not a guarantee. Individual holdings can fall and never return. Broad markets have recovered from severe declines historically, though the time taken has varied enormously and past behaviour is not a commitment about the future.
The honest version sits between the two claims. Not selling preserves the possibility of recovery; it does not create an entitlement to one, and how long a recovery takes is outside anyone’s control.
When the fall arrives matters as much as its size
Two investors can experience an identical set of annual returns in a different order and finish in very different places, provided money is moving in or out along the way. This is usually called sequence risk, and it is one of the more underappreciated features of investing in real life rather than on paper.
The reason is that a percentage applies to whatever balance exists at the time. A poor stretch early on, while the balance is small, costs comparatively little in absolute terms and leaves the later contributions to work on a lower base. The same poor stretch arriving when the balance is at its largest removes a far greater sum.
The direction reverses for anyone withdrawing money. Selling during a decline to fund spending removes holdings at a depressed value and reduces the base that any recovery would apply to, which is why the timing of withdrawals relative to market conditions receives so much attention.
What follows from the arithmetic
The general implication is about matching money to horizons rather than about predicting anything. Money that might be needed at short notice sits badly in something that can fall, because a fall combined with a forced sale converts a paper decline into a settled one. That is the mechanism, and it is why buffers and horizons are discussed alongside returns rather than separately.
Nothing here says how much variability any particular person should accept. That depends on the horizon, on what the money is for, on other resources, on obligations and on how a household would actually behave during a severe decline — which is a much harder thing to predict about oneself than most people expect.
Investments can fall in value and may be worth less than the amount put in. The arithmetic of recovery is general and reliable; the decisions built on top of it are specific, and belong with a regulated adviser who can see the whole situation.
Common questions
Why do deep falls take so much longer to recover?
Because the required gain grows faster than the loss that caused it. A 20% fall needs a 25% rise, while a 60% fall needs 150%. At normal rates of growth the second takes many times longer than the first, even though the falls differ by a factor of three.
Does this mean volatile investments are worse?
It means volatility has an arithmetic cost as well as a psychological one, which is often left out of the comparison. Whether a more variable holding is worth it depends on whether the expected return compensates, and that is a judgement rather than a calculation.
Should I sell if my holdings are falling?
This article cannot answer that, and any source that answers it confidently without knowing your circumstances should be treated with suspicion. What the arithmetic shows is only that selling converts a valuation into an amount, and that timing relative to a decline changes the base any future growth applies to.
Aditya covers spending, saving, debt and the questions readers actually send in and thinks most subjects are more interesting once you know how they work.
