Module 3, Part 1 of 2

An average is not a promise

Every figure in this course so far has assumed a steady rate of return. Real markets move up and down instead, and that changes the answers by more than you would expect. Two things follow from it. Almost everyone gets one of them wrong in one direction and the other wrong in the opposite direction.

Time
About 45 minutes if you attempt the questions
Level
High school
You need first
Module 2, both parts.
By the end of Part 1
You can say why a quoted average overstates what you will earn, and when the order of returns matters.
Part 1 Averages, and the order things happen in. Part 2 Why cash is not the safe option.

First, three from Module 2

Answer from memory, and do not look back until afterwards. Nothing below is gated on getting these right.

Guess before you read on

You put $10,000 into a fund

In its first year it gains 50%. In its second year it loses 50%. You add nothing and take nothing out. What is it worth at the end of the second year?

The average and the outcome are two different numbers

Add 50% and subtract 50% and the average is zero. Do it to actual money and you have lost a quarter of it. Nothing went wrong and nobody took a fee. The gain was a percentage of $10,000 and the loss was a percentage of $15,000, so the loss was worth more dollars.

This generalizes. Whenever returns move around, the rate that actually produced your ending balance sits below the average of those returns. They are only equal when every year is identical, which no market has ever managed.

So there are two honest numbers, and they are not the same:

  • The average return is what you get by adding the yearly returns and dividing. It is the number in the advertising.
  • The compound rate is the steady rate that would have left you with the same money. It is the number that describes what happened to you.
The gap between them is built into the arithmetic. Nobody was unlucky. It will not average out if you wait longer, and it does not depend on which years happened to be the good ones. One thing controls the size of that gap: how much the returns move around.

Watch the gap open

Both lines below have the same average return. The dashed line earns it steadily every year. The solid line earns it in a real-looking pattern of good and bad years. Raise the volatility and watch what a constant average is worth.

Market 1042. The same number always produces the same run, so you can go back to a result that surprised you instead of losing it, and a class can all look at the same market.

Set the swing to 0 and the two lines merge: with no volatility the average is the outcome. Every step up from there separates them.

What the volatility costs you
—

Steady, at the average The same average, unevenly
Average return
—
What you actually earned
—
Ending balance
—
What this model can and cannot show you. Returns are drawn from a simple bell curve, then nudged so their average lands exactly on the one you set. That nudge is on purpose. It guarantees the two lines share an average, so the only thing separating them is how much the returns move around. Left alone, a real thirty year run would drift off its own average and muddy the comparison. Two warnings about the model itself: real markets throw up extreme years more often than a bell curve does, so the worst runs here are gentler than the worst runs history has actually produced; and every year here is treated as unrelated to the year before it, which keeps the arithmetic simple and is not how markets behave. The steady comparison line ends at —.

Does the order matter?

Here is a question with a surprising answer. Two people get the same set of returns over the same years, but in a different order. Does one of them end up better off?

The answer depends entirely on whether money is moving. Try all three settings below before reading further.

Market 7, starting from $200,000.

Start on the first setting and look at the two ending balances. Then switch to taking money out and look again. The returns never changed.

Difference between the two orders
—

As it happened The same years, reversed
As it happened
—
Reversed
—

Before moving on

You now have two results: a quoted average overstates what you earn, and the order of returns matters enormously to some people and not at all to others. Which of the two changes more about how you would invest, and why?

Seven problems

Estimate first, then check. An estimate within 25% counts.

Save or hand in your answers

This writes out your guess, every problem you attempted with the answer beside it, and anything you typed.

Everything here is read out of this browser and written into a file on your own machine. Nothing is uploaded, because there is no server to upload it to. Clearing removes Module 3's answers only.

End of Part 1

Part 2 takes on the word "safe". Cash does not move around, which makes it feel like the option with no risk in it, and Module 2 already showed what inflation does to money that is not growing. It also covers what mixing different holdings together actually buys you, and why the most expensive mistakes in investing are made by people reacting to a number on a screen.

Start Part 2