If an investment averages 0% a year, did you break even?

Up 50% one year, down 50% the next. The two returns average out to 0%, which sounds like breaking even. But each percentage is taken from a different starting balance, so they don't cancel out.

The average return vs. what's actually in the account

The average
0% / yr
+50% and −50%, added up and divided by two
What $100 became
$75
A loss of a quarter of the money in two years
The real yearly rate
−13% / yr
The steady return that would also turn $100 into $75 in two years

$100 invested, year by year

The same 50% takes away more dollars than it added, because it is 50% of a bigger balance.

Line chart: $100 grows by $50 to $150 after a 50% gain, then drops by $75 to $75 after a 50% loss, ending $25 below the starting $100, shown as a dashed line. $0 $50 $100 $150 STARTING $100 +$50 50% of $100 −$75 50% of $150 $100 $150 $75 Start After year 1 up 50% After year 2 down 50%

A percentage is always a percentage of something, and after a gain or a loss that something has changed. The 50% gain was 50% of $100. The 50% loss was 50% of $150. So a 50% loss needs a 100% gain to recover, not another 50%, and the "average return" can say 0% while a quarter of the money is gone.

The bigger the swings, the bigger the gap. After the S&P 500 fell about 37% in 2008, it needed a gain of about 59% just to get back to even. The same math applies to anything that grows or shrinks by percentages: prices, pay, populations, a store's sales. Before trusting an average of percentages, ask what actually happened to the total.

If prices rise 10% one year and fall 10% the next, are they back where they started?

How big a gain does it take to recover from a loss?
Loss$100 becomesGain needed to get back to $100
−10%$90+11%
−20%$80+25%
−25%$75+33%
−37%$63+59%
−50%$50+100%
−75%$25+300%
−90%$10+900%
Two different kinds of "average return"

The simple average adds the yearly percentages and divides: (+50% + −50%) ÷ 2 = 0%. The compound average asks what one steady rate, repeated every year, would land on the same ending balance: $100 × 1.5 × 0.5 = $75, and the steady rate that also gets there is about −13.4% a year (√0.75 − 1). The compound average is never higher than the simple one, and the more the returns swing, the further apart they get. Only the compound average tells you what happened to the money.

Related exhibits

Sources: Calculations shown on this page; S&P 500 2008 total return of about −37%, S&P Dow Jones Indices historical index returns.