Geometric Average Returns

Kerry Back

Tesla

For each $100 of Tesla stock, shareholders experienced 100 → 50
and then 50 → 75.
They lost 25%, even though the average return was zero.
So, lose 50% and make 50% → lose 25%. Suppose you
- make 50% and then lose 50%?
- lose 50% and then make 100%?
- make 100% and then lose 50%?

Given returns $r_1, \cdots, r_n,$ the geometric average return is the number $r$ such that

\[(1+r)^{n}=(1+r_1)\cdots(1+r_{n})\]

So earning $r$ each period produces the same accumulation as the actual returns $r_1, \cdots, r_n.$ We solve for $r$ as

\[r=[(1+r_1)\cdots(1+r_n)]^{1/n}-1\]

The geometric average return is always less than the arithmetic average return. The difference is larger when returns are more volatile.

\[\sqrt{1.5 \times 0.5}-1=-0.134\]

\[\sqrt{2 \times 0.5}-1=0\]

Compounding continuously at rate $z$ $\Rightarrow$ $1 $\rightarrow e^z$ in a year.
Actual return is $r$ $\Rightarrow$ continuously compounded return is $\log (1+r)$
- because compounding continuously at rate $\log (1+r)$ means $1 $\rightarrow e^{\log (1+r)} = 1+r$.
Make 50% and lose 50% $\Rightarrow$ continuously compounded returns are $\log(1.5)$ and $\log(0.5)$ and average $<0$.
Make 100% and lose 50% $\Rightarrow$ continuously compounded returns are $\log(2)$ and $\log(0.5)$ and average $=0$.