# Percentage Games

*Lots of tricks can be played with the clever use of percentages. It makes for snappy journalism. But don't confuse it with analysis.*

## Example

If you make 100% one year and lose 50% the next you are back where you started. {£1,000 plus 100% goes to £2,000. £2,000 less 50% goes back to 1,000}. The arithmetical average of your 2 years of returns is 50% (or 25% per year). Your actual return is zero.

It's the same if you do it the other way round - lose 50% and then make 100%. {£1,000 minus 50% goes to £500. £500 plus 100% goes back to 1,000}

## General Rule

This simple example extends to a more general truth: the arithmetical average of a series of annual returns is *always* greater than the true annual return. For example, five annual returns of +40%, -30%, +35%, +30%, -35% arithmetically average 8% per year, but the compound annual return is only 2%.

This is particularly useful for misrepresenting the performance of highly volatile investments:

- "The XYZ Fund lost 50% last year, but this year it's already up 70%". {This means it is down 15% over two years - 100 goes to 50 goes to 85.}
- "The ABC Fund lost 50% in it's first year but in the next 5 years it averaged 20% a year and never lost money". {It went -50, 0,0,100,0,0 - giving a zero return}
- "The PQR Fund has averaged 20% a year for five years and only lost money once". {It went +180, -80, 0,0,0 - it has actually lost money cumulatively}

## Words do bad Maths

Beware the innocent interpretations of journalists. A fund manager may tell them: "we lost 50% last year but we are already up 60% in 2003". The journalist may write this as "XYZ lost 50% last year but has already more than made it back". But we know that is wrong now, don't we!

Advertising copywriters sometimes make the same mistake.

## Performance Tables

You will sometimes see ranking tables of funds or fund managers expressed as their average annual outperformance of a market index. With rare exceptions these will be arithmetic averages. This is not just because geometric averages are harder to calculate (although not difficult). It is also because they come in lower, and it would become more obvious that funds tend to underperform the market.