Distributions, Standard Deviation, and Fat Tails

Why this matters

Two investments can have the same average return and be nothing alike. One delivers close to its average every year. The other swings between large gains and large losses. Arithmetic and Geometric Averages showed that the swings cost money. This lesson gives you the tools to describe them, and shows where the most common tool stops working.

A distribution

A distribution is a count of how often each outcome occurred. Here are 98 years of returns on the S&P 500 including dividends, sorted into ranges, calculated for this lesson from Aswath Damodaran's yearly data.¹

The table says more than any single number. Shares rose in 72 years and fell in 26. The most common result was a gain of 10% to 20%. A year close to the long-run average wasn't especially typical: more years were above 20% or below zero than fell between 0% and 20%. Fees, taxes, and inflation are left out, and the past doesn't predict the future.

The middle: average and median

The arithmetic average of those 98 returns is 11.9%. The median, the middle value when they're lined up in order, is 15.2%.¹ The two differ because the distribution is lopsided. A handful of very bad years, including a loss of 43.8% in 1931, drag the average down without moving the median. More than half the years, 54 of 98, beat the average.

When the average and the median are far apart, a few extreme values are doing the work, and it's worth finding out which.

The spread: standard deviation

Standard deviation measures how far results typically fall from their average. A spreadsheet calculates it with one function, but doing it by hand once shows what it means.

Return	Distance from the 8% average	Distance squared
10%	+2	4
−5%	−13	169
20%	+12	144
15%	+7	49
0%	−8	64
Total		430

Find the average, 8%. Measure each return's distance from it. Square the distances, so that negatives don't cancel positives, and add them: 430. Divide by one fewer than the number of returns, which is the convention for a sample: 430 ÷ 4 = 107.5. Take the square root: 10.4%. The returns typically sit about 10 percentage points from their average.

Because the distances are squared, a result far from the average counts much more than several results close to it. That makes standard deviation sensitive to the big moves, which is what you'd want from a measure of risk. It also treats a surprise gain exactly like a surprise loss, which isn't how an investor feels about them.

The rest of this lesson is for members

Still to come in this lesson: The bell curve, Fat tails, What follows for an investor, the key takeaways, and 2 quiz questions with explanations.

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