# Risk and geometric expected value

In probability and statistics, one often talks about the expected value of a random variable, but very rarely about the geometric expected value. However, in finance, the geometric expected value is extremely important.

What is geometric expected value ?

If you have positive a random variable A with n outcomes A_1,…,A_n with equal probability 1/n, then the geometric expected value of A is:

GE(A) = (A_1….A_n)^{1/n}

In general:

GE(A) = exp ( E ( ln A) )

where E is the expected value, which we will call also the arithmetic expected value

Why is it important ?

The geometric expected value (or geometric mean) gives you the correct measure of the average performance of a portfolio. For example, if a portfolio grows 100% in 1 year, and loses 50% the next year, then what’s its average performance during the two year period ? The arithmetic mean would give you (100%-50%)/2 = 25%, which is totally wrong. After the 2-year period, the total performance is zero, so the average performance is zero, and that’s what the geometric mean value gives you: The geometric mean of 50% (= 100%-50%) and 200% (= 100% + 100%)  is 100%.

Inequalities

GE (A) < E(A)

(The sign < here means “less or equal”)

The equality holds if and only if A is a constant

The difference E(A) – GE(A) > 0 is a measure of loss of performance or risk due to uncertainty or volatility of A.

So we have the following principle: risk reduces performance

If A and B are two positive random variables then we always have:

GE(A+B) > GE(A) + GE(B)

The equality holds if and only if A and B are proportional, i.e. there is a constant c such that B = cA.

This inequality is the basis of the diversification principle:

If A and B have the same geometric expected performance, then by taking (A+B)/2 instead of A or B alone, you will get a better geometric expected performance.

Even if B has a lower geometric expected performance than A, then changing a bit of A for a bit of B may still increase your geometric expected performance

In other words, diversification increases performance by reducing risk

### 1 comment to Risk and geometric expected value

• Chung

Thank you for these significant posts. You should lead us more and more to details of the interesting financial math. Will read carefully before raising some questions for clarification, especially nature of some fomulas.