Return

We have to be able to talk about the relative performance of different products and investments. The word to describe that is Return.

‘Return’ on an investment is equivalent to ‘interest’ on savings. It is is expressed the same way - as a percentage. A one-year return is the total gain on the investment in the year expressed as a percentage of the opening value. The ‘total gain’ is the gain in value of the investment plus the cash returned to the investor (e.g. dividends in the case of a share).

If you have savings in a risk investment (e.g. property or shares) you should be hoping for a long-term return a few percentage points above what you are getting for risk-free cash. So with interest rates at 4% you might be hoping for 6% per annum for example.

But investments such as shares are highly volatile. Gains or losses on a portfolio in one year of 25% are not unusual. With much larger results for individual shares.

This is when an unexpected mathematical property comes into play.

Percentage Games

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 the results of the two years are swapped - lose 50% and then make 100%. {£1,000 minus 50% goes to £500. £500 plus 100% goes back to £1,000}.

General Rule

This is a simple example to illustrate a much more surprising 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 (40-30+35+30-35=40, divide by 5 gives 8%). But the compounded annual return is only 7% (trust us).

This may seem trivial (and breaks our ‘no maths’ rule) but we have shown you that small percentage differences make a big value difference in lifetime saving. You must prepare for the world of hype.

Words do bad Maths

Here are some examples of misrepresentation:

• "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}

Beware the innocent interpretations of journalists. A fund manager may tell them: "we lost 50% last year but we are already up 60% in 2024". 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.

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. It is also because geometric averages come in lower.