# Percentage Losses, Percentage Gains

by on Aug 8, 2013 · 3 comments

As a followup to my post about predicting portfolio growth, I wanted to share a few more thoughts on how misleading percentages can be.

When thinking about the effect of percentage gains and losses on the value of your investment portfolio, it’s important to keep in mind that they don’t work in perfect opposition.

What I mean by this is that it takes (for example) a 100% gain to offset a 50% loss, or vice versa. Yes, this is a simple mathematical truism, but it’s something that many people don’t fully appreciate. And this sort of misunderstanding can be costly.

Here’s how it worksâ€¦

Let’s say you start with \$100. If you then lose 50%, you’ll have \$50. You’ll now have to double your money (100% gain) to get back to where you started.

Of course, this applies to larger or smaller percentages, as well. If you have \$100 and lose 10%, you’ll be down to \$90. If you then gain 10%, you’ll be back up to \$99, short of where you started.

And yes, it’s the same regardless of order. Starting with \$100, if you gain 10% you’re at \$110. Now if you lose 10% it falls to \$99.

Why does this matter? Mainly because it’s very easy to be misled by percentages. If you’re given a choice between investments with the following sequences of annual returns, which would you choose?

`Scenario 1: +50%, -50%, +50%`
`Scenario 2: +10%, +5%, +8%`

Well, if you look at a simple average of the returns over years, it’s a no brainer — Scenario 1 has an average return of 16.7%/year whereas Scenario 2 has an average return of 7.7%/year.

But if you run the numbers, you’ll see that \$100 in Scenario 1 would grow to \$112.50 whereas \$100 in Scenario 2 would grow to \$124.74 over the three years in question.

So things are not always what they seem. The compound annual growth rate (CAGR; see here for details) of Scenario 1 is 4.0%/year whereas the CAGR of Scenario 2 is 7.6%/year. Things look rather different now, don’t they?

The difference lies in the consistency of the returns (i.e., volatility). The more consistent the returns, the closer the CAGR will be to the arithmetic average. The more variable the returns, the more these two values will diverge.

#### Hat tip to a reader named Chris for suggesting the topic.

1 Chris Peplinski August 8, 2013 at 10:41 am

EXCELLENT explanation! Thanks.

2 Kurt @ Money Counselor August 8, 2013 at 5:31 pm

Isn’t mathematics a wonder!

3 Martin August 13, 2013 at 12:16 am

@Kurt, it is, but sometimes perfectly boring. Nevertheless, I like using it and playing with it.

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