It’s been called the cruel math of investing, and sometimes used to justify a more conservative approach. Whenever your account loses a percentage of its value, it has to go up by a greater percentage to put you back where you started. For example, a $40,000 account would drop to $30,000 in a 25% loss; getting it back to $40,000 requires growth of 33%. If your account drops by 50% you need to grow it by 100% to recover. Yet there’s a reason this math isn’t cruel at all, and shouldn’t affect your judgment about how much risk to take with your investments.
Unfortunately for those of you who hate mathematics, the reason involves two of the things math haters hate the most: statistical distributions and logarithms. Stick with me, though, I promise this won’t hurt a bit.
Normal distribution
Normal distribution
In statistics we frequently encounter a normal distribution, which is represented by what is often described as a bell-shaped curve. Items appearing in the middle of the distribution have the greatest probability of occurring. As you move farther away from the middle, the probabilities get smaller and smaller. We encounter many more people of roughly average height than people who are extremely tall or short, for example. The normal distribution curve is symmetrical, with each side being a mirror image of the other.
Investments
We haven’t found a reliable way to predict how investments will perform. The best we can do for for an individual investment, or a portfolio of investments, is to estimate the expected return (roughly speaking, the average rate of growth), and the standard deviation, which measures the degree to which the investment tends to wander away from its expected return. A low-risk investment such as short-term bonds would have a small expected return with a tall, skinny-looking bell curve because the results seldom vary much from what we anticipate. Stocks have a higher expected return but a flatter looking curve indicating it’s easier for them to produce results that are far better — or far worse — than average.
If we did a statistical analysis of the actual results produce by an investment such as stocks the resulting graph would not resemble a bell curve. Investment returns don’t exhibit a normal distribution. The probability of a result that’s 25% above the expected return is greater than the probability of a result that’s 25% below the expected return. The resulting curve isn’t symmetrical.
Logarithms
To get a graph resembling a bell curve, we have to convert differences from the expected return into logarithms. Remember logs? Don’t worry, hardly anyone does. Here’s the key point: logarithms are positive for numbers bigger than 1 and negative for numbers smaller than 1. When you turn a fraction upside down, you get exactly the same log multiplied by negative one. The log for 4/3 is a positive number (because 4/3 is bigger than 1), and the log for 3/4 is the negative of that same number.
Here’s the payoff. We get a symmetrical curve when we convert investment returns to logarithms. That means the probability of getting an outcome 33% higher than expected (4/3 of the expected result) is the same as the probability of getting an outcome 25% lower than expected (3/4 of the expected result). We saw earlier that if your investments lose 25% they have to gain 33% for you to recover, making it seem as if losses are difficult to recover. Yet a 33% gain occurs with the same frequency as a 25% loss.
All this is not to say we should disregard investment risk. Managing risk is one of your most important tasks as an investor. Low risk is associated with slow rates of growth, however, and your investments have to be bold enough to reach your goals. If you’re overly risk-averse, it may help to understand that higher-percentage rewards occur with the same frequency as smaller-percentage losses.
Tags: investment risk
