Arithmetic versus geometric average. (a) Let a 1 , a 2 > 0 a 1 , a

Question:

Arithmetic versus geometric average.

(a) Let $a_{1}, a_{2} > 0$ and show that

$\frac{a_{1} + a_{2}}{2} \geq \sqrt{a_{1} a_{2}}$

(b) Use Jensen's inequality applied to the log function to show that the arithmetic average dominates the geometric average

$\frac{1}{N} \sum_{i = 1}^{N} a_{i} \geq {(a_{1} \times \dots \times a_{N})}^{1 / N}, a_{i} > 0$

(c) Let $r_{n}$ be the periodic return of an asset $r_{n} = A_{n} / A_{n - 1} - 1$ . If the asset value cannot go negative, then $1 + r_{n} \geq 0$ . What can you say about the $N$ period arithmetic average, $r_{A}$ , defined via

$1 + r_{A} = \frac{1}{N} \sum_{i = 1}^{N} (1 + r_{i})$

versus the compounded average, $r_{G}$ , defined via

${(1 + r_{G})}^{N} = (1 + r_{1}) \times \dots \times (1 + r_{N})$

of the periodic returns?