Why Average Returns Mislead: Arithmetic vs Geometric Mean

0:00Your statement: +10% a year, and less money ▶

Not long ago you moved some savings into a fund, and you have not looked since. Tonight you finally open the statement. The page is short: a line for the deposit, and then a line for every year. Say your deposit was a thousand dollars, made once, and say the first year was wonderful. The fund went up sixty percent. Then came the year you would rather forget. Say the fund then went down forty percent. So how did you do, on average? You add them up the obvious way: plus sixty, minus forty, that leaves twenty. Spread over two years, that is ten percent a year. At ten percent a year, year after year, your thousand dollars should have become twelve hundred and ten. Then you look at the balance, and it says nine hundred and sixty dollars. That is less than the thousand you put in. Your average says you gained ten percent a year. Your account says you lost money. So which of the two tells you what your money actually did? Your sum skipped one thing: each percent landed on a different amount of money.

1:03Level 1: no swing ▶

Let's build up to your two years, one level of swing at a time. Every level keeps the same average, ten percent a year, so only the swing changes. Level one has no swing at all: plus ten, and then plus ten again. The first year adds a hundred dollars, and the second adds a hundred and ten, because its ten percent is taken of more money. You end at twelve hundred and ten dollars, exactly what the average promised. When every year is the same, the average and the account agree. Call that twelve hundred and ten the mark, and keep it in view: every level is measured against it.

1:37Level 2: a small swing ▶

Level two swings a little: plus twenty percent, and then nothing at all. The average is still ten percent a year. Your thousand becomes twelve hundred, and then simply stays there. That is ten dollars short of the mark. A small swing gives a small miss, and almost nobody would notice it.

1:55Level 3: a bigger swing ▶

Level three swings harder: plus thirty-five, and then minus fifteen. Same average again, ten percent a year. Make your guess now: how far below the mark does this one land? The thousand climbs well above the mark, to thirteen hundred and fifty. Then fifteen percent of that comes off. You end sixty-two dollars and fifty cents short of the mark. The swing grew, and the miss grew much faster.

2:20Level 4: up 60%, down 40% ▶

Level four is your fund: plus sixty, and then minus forty. The first year turns your thousand into sixteen hundred, a gain of six hundred dollars. The second year takes forty percent of sixteen hundred, and that is six hundred and forty. The loss is bigger than the gain, because it was taken from a bigger pile. That leaves your nine hundred and sixty dollars: two hundred and fifty below the mark. Swapping the two years changes nothing: with the bad year first, you land on the very same balance. So it is the account that tells you what your money did, and the average that missed.

2:54A return multiplies: the geometric mean ▶

Here is what the averaging missed. A return does not add to your money, it multiplies it. Plus sixty percent means times one point six, and minus forty means times zero point six. Two years means both, one after the other, and together they make zero point nine six. Adding the percents gave you plus twenty, but multiplying the factors gives you minus four percent in total. Now ask the question your average was meant to answer: what one steady rate lands on the same balance? That rate, applied twice, has to make zero point nine six. So one plus the rate is the square root of zero point nine six, about zero point nine eight. That is a loss of about two percent a year, and two years at that exact rate land on your balance. That steady rate is called the geometric mean, and it is the rate your money actually moved at. For more years you take the matching root: a square root for two years, a cube root for three, and so on. So you multiply all the yearly factors, take the root for the number of years, and subtract one. The sum you did in your head is the arithmetic mean: add the returns, and divide by the number of years. One question is left: why does a wider swing cost so much more?

4:08The picture: perimeter vs area ▶

There is a picture behind both names, and it answers that question. Draw your two years as a rectangle: one side one point six, the other zero point six. Its area is your money: zero point nine six for every dollar you put in. Adding the returns is adding the sides: one point six plus zero point six is one point one plus one point one. So one plus the arithmetic mean keeps the perimeter: a square of one point one. That square has an area of one point two one: the mark. One plus the geometric mean keeps the area instead: the side of the square that holds exactly your money, about zero point nine eight. Now cut off the piece of the rectangle that sticks out past the square, turn it, and lay it on top. One corner stays empty, a half by a half, an area of a quarter: there are your two hundred and fifty dollars. Each of your years sits fifty points from the average, a half, and that half, squared, is what goes missing. On level two each year sits ten points away, a tenth, and a tenth squared is a hundredth: ten dollars. On level three it is twenty-five points, a quarter, and a quarter squared is a sixteenth: sixty-two fifty. From level two to level three, the swing grew two and a half times and the miss six and a quarter: the swing gets squared. And a number squared is never negative, so your money never beats the mark; it only reaches it when every year is the same. That is why calm years barely miss the mark, and wild years miss it by a lot.

5:36The check: does the average survive? ▶

So does the average survive as a quick guess of how fast your money grew? Hold it against the exact rate, one level at a time. On level one it matches exactly. On level two it says ten, against nine and a half. On level three the gap widens, and on level four it says ten against minus two. So how far apart are the two averages on an actual market, over a whole lifetime?

6:0298 years of the S&P 500 ▶

Take ninety-eight years of the S and P five hundred, with every dividend put back in. These are real yearly returns, collected by Aswath Damodaran at New York University. Add all ninety-eight and divide, and the average year was plus eleven point nine percent. Multiply all ninety-eight factors and take the root, and the money grew ten point zero percent a year. The two sound close, like the kind of gap you would round away. Now say you put a hundred dollars in at the start of nineteen twenty-eight, once, and never touch it. By the end of twenty twenty-five, it has become about one point one six million dollars. Grow the same hundred dollars at the average year instead, and you get about five point eight six million. The average describes five times the money that the hundred dollars actually became. These are past returns, before fees, taxes and inflation, and they are not a forecast.

6:56Where you meet each average ▶

So which of these two does a fund show you? A US mutual fund prospectus shows a bar for each year, and next to the bars, the average annual total return. That average is not the simple one: the S E C prescribes the formula, and it compounds. The simple average turns up when someone averages yearly returns by hand, and that someone is often you. It is not wrong, it just answers a different question: what did one year give you, on average? For things that do not build on each other, like the lengths of a few planks, it is exactly the right average. Which average to use for guessing the future is a harder question, and it belongs to another video. But your money builds on itself, so the question of what it did has one answer.

7:39Back to your statement ▶

Now look again at the statement you opened at the start. Your average year really was plus ten percent, and your money really did shrink by about two percent a year. Both numbers are true, and only one of them is what your money did. A return multiplies, a swing gets squared, and a simple average never sees it, and that's the math behind the money. Next time: one more number from those ninety-eight years gets you from eleven point nine to ten, within a few hundredths of a point. Before you go: how far below the mark did you guess level three would land? I'm the MoneyMathician, and if you subscribe, the next one will find you. Until then: do the math!