Lies, Damned Lies and Percentages

The Mystery of Percentages

Here's a simple warm-up question: you invest $100 in a stock which then rises 20% followed by a drop of 20%. How much is it now worth?

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It's OK, I will give the answer but it really is worth having a stab at your first intuitive response!

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When I used to give a similar question to my students many would smugly blurt out the wrong answer, whereas others were more wary and distrusted such a simple question, rightly suspecting there was a trap lurking in there.

There isn't actually a trap as such, but a realisation that percentages do not work like natural numbers. Percentages are not additive - a 10% rise followed by another 10% rise does not equal 20%. Percentages are multiplicative - this is the reason for the counter-intuitive power of compounding interest in a deposit account, as well as the devastating effect on wealth of long-term inflation. It is this constant battle between these two percentages that can result in voracious greed as everyone wants to beat the market, to make a real positive return on investments.

I cannot hold off answering the initial teaser. A 20% profit would make the initial $100 rise to $120 but then the 20% loss is 20% of the new $120 (not the original $100), which is $24 so that the final total is just $96 - a loss of 4%. We would also get the same result if we had a 20% drop first followed by a rise.

We can now also answer the question in the previous paragraph. A 10% increase will take the total to $110 with a further 10% on the new total bringing the compound interest to $21 and the grand total to $121.

I write about this not to share a lesson in elementary percentages but because in the current situation of volatile stock markets and interest rates a true grasp of percentages will help you ignore some of the more damaging advice meted out by financial advisers and websites.

If we take a working definition of a bear market as a drop of at least 20% from its peak - and therefore the opposite bull market as the mirror image rise of 20% from its bottom - then we can now see that even when we get the bulls waving their flags the market will still be below what it was before the current bear market. And as the percentage swings get larger so does the difference between the false arithmetic adding of percentages and the true multiplication. The US markets are now below 50% from their peaks and such a drop followed by a subsequent 50% rise will leave the markets still losing 25% of their value. A 50% drop requires a 100% rise to get back to par.

Notice here that a 50% rise would be considered a very bullish rally and yet we would still be in bear market territory based on the previous high. Bulls and bears are not absolute positions but are relative descriptions based on the trend from the last turning point. Statistics may lie but so can percentages in the hands of financial pundits desperate for investors.