# Geometric and Arithmetic mean

You've just invested in an exciting start-up company. You reckon there is a 20% probability of quadrupling your money in a year's time, a 60% probability of a modest (10%) gain, and a 20% probability of a disaster (losing half your money).

The graphs illustrate these possibilities, in terms of five equally probable future outcomes.

The graphs show the same information, but one uses a linear axis for expected future value and the other uses a logarithmic axis.

Two different kinds of mean are visible: the arithmetic mean and the geometric mean.

The arithmetic mean (often simply called "the mean") is the expected value of your investment in a year's time.

Visually, the arithmetic mean is the average height of the bars on the left hand (linear) graph.

Click on either graph to change the position of the bars.

Both means are re-calculated.

#### Can you raise the geometric mean higher than the arithmetic mean?

You should find that the two numbers can be equal (when all the bars are the same height), but the geometric mean is never greater than the arithmetic mean.

The geometric mean can be...

The geometric mean can be thought of as the average result you would get from making this kind of investment again and again, over the long term.

Visually, the geometric mean is the average position of the bars on the right hand (log) graph.

#### Click on the bottom of the left-hand graph to bring one bar down to zero. What happens?

When one of a set of numbers is zero, the product of all of them is zero, so the geometric mean is zero.

What does this mean in the context of investment?

If you have a non-zero probability of going bust, and you reinvest with the same probabilities year after year, the long-term expected outcome is that you will lose all your money.