Estimating Future Value: Arithmetic or Geometric

Why do we use the arithmetic average to estimate expected return (assuming we believe in the historical sample as a good estimator)

We should compound at the arithmetic average if we wish to estimate an investment’s expected value (return). We should compound at the geometric average when we wish to estimate the likelihood that an investment will exceed or fall below a target value.

Consider an investment that has an equal chance of increasing by 25% and decreasing by 5%. There is an equal chance that a dollar invested will grow to $1.25 or decline to $0.94 after one period. The expected value after one period is $1.10 which is equal to (1+arithmetic average of the two possible returns). Subsequently, there are four equally likely outcomes at the end of two periods. See the following diagram for the possible paths of $1.00 investment.

Binomial Tree of Investment Paths

The expected value at the end of the second period is equal to $1.21 which is the probability-weighted outcome.This also corresponds precisely to the quantity (1+arithmetic average of 10%)². If we calculate the geometric average for this example and compound it forward for two periods, we arrive at a terminal value of $1.1875, which does not equal to the expected value.

The expected value assumes that there is an equal chance of experiencing any of the possible paths. A path of high returns raises the expected value over multiple periods more than a path of equal-magnitude low returns lowers it. This disproportionate effect is the result of compounding.