Note on Average Return Rates

Introduction

I came cross this Quora post, which is about arithmetic mean and geometric mean in the context of investment return. Based on my reading, I want to discuss three average return rates defined as: For n-years' return rates, $r_1,r_2,\dots ,r_n$, where $r_i>-1$,

$$r_a=\frac{\sum _{i=1}^n{r}_i}{n},$$

and

$$r_g=\sqrt[n]{\prod _{i=1}^n(1+r_i)}-1,$$

and

$$r_d=\frac{\prod _{i=1}^n(1+r_i)-1}{n}.$$

Note that $r_a$ is an arithmetic mean, $r_g$ is a geometric mean. In $r_d$, the sub-script ’d' stands for “direct”; $r_d$ comes from my intuition.

In the next section, we will have a number of inequalities.

Inequalities

Lemma (Bernoulli’s inequality): If $h>-1$, then

$$(1+h{)}^n\ge 1+nh,$$

where $n$ is a positive integer.

Inequality 1:

$$r_g\le r_a.$$

Proof:

$$r_g\le \frac{\sum _{i=1}^n(1+r_i)}{n}-1=r_a.$$

Inequality 2:

$$r_g\le r_d.$$

Proof:

$$\begin{array}{rl}{r}_d& =\frac{(1+r_g{)}^n-1}{n}\\ & \ge \frac{1+n{r}_g-1}{n}\text{[using Bernoulli's ineqality]}\\ & =r_g\end{array}$$

Remark: If all $r_i>0$, then obviously $r_d>r_a$. In the case that some of $r_i$s are positive and the others are negative, the sign of $r_d-r_a$ depends on the values of $r_i$s.

Inequality 3:

$$r_d\le \frac{(1+r_a{)}^n-1}{n}.$$

Proof:

$$\begin{array}{rl}{r}_d& \le \frac{(\sum _{i=1}^n(1+r_i)/n{)}^n-1}{n}\\ & =\frac{(1+r_a{)}^n-1}{n}.\end{array}$$

Inequality 4:

$$r_a\le \frac{(1+r_a{)}^n-1}{n}.$$

Proof:

$$\frac{(1+r_a{)}^n-1}{n}\ge \frac{1+n{r}_a-1}{n}=r_a.$$