Magazine · · 2 min read
Why a 50 per cent loss needs a 100 per cent gain
Losses and gains are not symmetric: lose half and you have to double your money to get back to even. The formula behind it, a table — and what follows from it.
The asymmetry in one formula
From 100 to 50 is minus 50 per cent. From 50 back to 100 is plus 100 per cent, because the gain is computed on the smaller amount. In general, after a loss d you need a gain of d ÷ (1 − d) to get back to where you started.
Gain needed = loss ÷ (1 − loss).
For small losses the difference is barely noticeable: after a 10 per cent loss you need 11.1 per cent, after 20 per cent you need 25. Then it grows quickly: 30 per cent needs 42.9, 50 needs 100, 70 needs 233.3 and 90 needs 900 per cent.
What that means in years
A percentage is abstract; time is not. StrikeAndYield therefore converts it into years — how long recovery would take at 7 per cent a year: about 1.6 years after a 10 per cent loss, 3.3 after 20 per cent, 10.2 after 50 per cent and 17.8 years after 70 per cent.
At 7 per cent a year, a value that halved needs just over ten years to come back.
The curve is convex: twice the loss needs more than twice the recovery. Beyond about 60 per cent it stops being a recovery problem and becomes, in effect, a different portfolio.
The average is not the return
The same arithmetic explains why an average can mislead. Plus 50 per cent followed by minus 50 per cent averages zero — yet you are down 25 per cent, because 100 becomes 150 and then 75.
+50% and −50% average zero and really come to −25%.
The gap between the arithmetic mean and the geometric mean actually achieved grows with the square of volatility. It is not a fee or a cost but arithmetic — and it applies to everything that swings strongly.
What follows — and what does not
The calculation itself is not an opinion but mathematics. It explains why finance puts so much weight on limiting large losses: an avoided collapse is worth more than an extra percentage point of expected return.
Mathematics, not a forecast: large losses cost more than they look.
This article is for information only and is neither investment advice nor a recommendation. StrikeAndYield explains financial products and their mechanics; its full page on the arithmetic of drawdowns has the table, a calculator and related effects such as the order of returns.
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