For Business

Trading Math

How to Turn Monthly Into Yearly Returns

A monthly return becomes a yearly one by multiplying twelve factors together, not by adding twelve percentages, and a yearly return becomes a monthly one by taking a root, not by dividing by twelve. This paper shows you the twelve steps, the two formulas, the same arithmetic on a real ten-day list in both directions, and the one calculator key both sums need.

Difficulty
Intermediate
Reading time
5 min
Pages
7
Reviewed by
Saman Rahbar

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Twelve steps, each on the last

Two per cent a month for twelve months is not twenty-four per cent a year. Each month's two per cent is taken on the balance at the start of that month, which already holds the earlier months' gains, so the steps grow. Twelve steps of two per cent land at 26.8 per cent, because the twelve factors of 1.02 multiply together to 1.268. The gap between 24 and 26.8 is small at two per cent a month and large at larger rates; at ten per cent a month it is the difference between 120 and 214.

The two formulas

Up from a month to a year: one plus the monthly rate, raised to the twelfth power, minus one. Down from a year to a month: one plus the yearly rate, raised to the power of one twelfth, minus one. Both are on a card, because they are the two directions of the same fact: periods join by multiplying their factors, so they split by taking roots. Never times twelve, never divided by twelve — those are the sums for something that does not compound, and a balance does.

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