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, $r_1,\ r_2,\ \ldots,\ r_n$, 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{align} r_d & = \frac{(1 + r_g)^n - 1}{n}\\ & \ge \frac{1 + nr_g - 1}{n} \ \ \ \hbox{[using Bernoulli's ineqality]}\\ & = r_g \end{align}

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{align} r_d & \le \frac{(\sum_{i=1}^n (1+r_i) / n)^n - 1}{n}\\ & = \frac{(1 + r_a)^n - 1}{n}. \end{align}

Inequality 4:

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

Proof:

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