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.

What do a bull and bear market actually mean?

If you haven't heard the terms bull market and bear market being used then you obviously have never seen or read the financial press. But what do these words really mean?

Astonishingly for such common words they don't actually have any well-defined meanings. A quick search yields a lot of useless dictionary definitions but nothing quantitative. Yes, a bull market is one that is in an upward trend, with a bear market being the opposite. But that seems to be it; purely qualitative definitions. We must be able to do better than this!

According to The Vanguard Group, "While there’s no agreed-upon definition of a bear market, one generally accepted measure is a price decline of 20% or more over at least a two-month period." Indeed that 20% drop from its peak was the definition used on most financial news media as the current markets dropped precipitously. However, Ultra Financial Systems defines it "as a market that loses 15% as measured by a stock market index such as the Standard and Poor's 500 Index (S&P 500) which consists of the stock prices of 500 U.S. corporations." And About.com has this to say on the topic: "A common rule of thumb states that if the market is down 10%, it's a Market Correction, and if it's down 20%, it's a Bear Market. While common, it's not very useful. There is a better way." The site then goes on to list some 12 factors that will help you determine what's going on in the stock market.

So, it seems as if a bear market can be defined by many different parameters, the only thing in common is that prices are heading down. But percentages can give an illusory feeling of what a market is really doing. For example, a 20% drop followed by a 20% rise in no way takes us back to the original peak but rather at 96%, so with another 4% to make up to reach parity. As the percentage swings get bigger the
difference between such peak-to-peak values grows alarmingly - a 50% drop in a stock price followed by a 50% rally will bring the price up to just 75% of the original peak. Indeed a 50% drop in price then requires a 100% increase from this low just to break even. What this means is that our bear or bull markets have to be taken within the context of where we are measuring from.

A bull market that follows a bear market could still be showing a loss compared to that market's previous high. If we take a 20% drop from a high as the start of a bear market then, by symmetry, we should take a 20% rise from its low as the start of a bull market. But as we have shown above, this seems more a matter of semantics than hard mathematics. If a $100 stock drops 50% to a low of $50 and then rallies 20% to $60 then that $60 mark signals the start of a bull market from its low, but as we can see, we are still showing a $40 loss.

So the first lesson is that a bull or bear market is a description that is relative to the last high or last low of the market in question. A new bull market will take a long time to reach a new high compared to the 2007 level. But I think from an investing perspective we can improve on the above definitions. The easiest and best long term indicator is the 200-day moving average on a daily price chart. I have discussed this before but here wish to show how it too can define the current bear market, and it can also be used to define the next bull market.

Any important indicator, whether it is the 200DMA or the 20% market swing level, will show increased volatility at that level as the market participants struggle to figure out if the level will hold or be breeched. These indicators are therefore not precise numbers but have a fuzzy edge to them. As the Vanguard definition warns, their 20% level needs to be broken for "at least a two month period". For an investor, waiting 2 months for confirmation is a long time to be sitting there losing money. The advantage of the 200DMA is that the investor can slowly start to take profits and limit exposure to a bear market.

Taking last Friday's closing prices the S&P 500 was at 756.55 (S&P 500 chart). It's recent low was last Monday, touching 666.79. In just five days this is an increase of 89.76 points, equivalent to 13.5%. A 20% rise would take the S&P 500 to 800. Could we call that the start of a new bull market? In contrast, the 200DMA stands at about 1045, still about 300 points away from the index level and equivalent to a 57% rise from Monday's low. Historically, the 200DMA has been far better for the investor than a random 20% level. When the S&P took a dive last January and all the talk was of a bear market the 200DMA had already been broken during both November and December. The final proof that the market needed oxygen was in May 2009 when the 200DMA proved to be too high to scale and we are now at half of that level - that's half of the May 2009 peak, not the October 2007 all-time high.

The ideas behind bull and bear markets are that they should have long term secular meanings and not be short term indicators to sucker people into buying stocks. They are indicators of trends and not absolute levels. Indeed, the financial press have spectacularly failed to protect investors and report the gravity of this bear market. I expect the corporate news media to start effervescing as we reach the 800 level. Switch them off and just follow the numbers.