Math: break-even year-over-year

Self-tutoring about math: the tutor mentions perhaps an oft-considered question.

Suppose an investment loses x% this year. What percent does the resulting sum need to gain next year in order to break even?

Imagine an investment of initial value $250. If, for instance, the investment loses 12% this year, it’s worth 250(1-12/100)=220 at year end. The % growth needed, the following year, to return to $250 is (250/220 – 1)x100% = 13.64%.

In general, if an investment loses x% this year, its result will need to grow by (1/(1-x/100) – 1)x100% next year to break even. So, by the example above, an investment that loses 12% this year will need to gain
(1/(1-12/100) -1)x100% = 13.64% next year to break even.

Source:

Killip, Brian T. Mathematics for Business: The CGA Reference Handbook. Toronto: Harcourt Brace and Company, 1993.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

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