Given a sample rt, t = 1, 2, . . . , T, of simple returns or total returns over one period, say, a year. The arithmetic mean a is defined as a := 1 T

T

∑

t=1 rt.

The geometric mean g is an average return that takes into account the compounding nature of growth. Mathematically, we write T

∏

t=1

(1 + rt) =: (1 + g)T.

Though rt may be different for each t, the forward value of a dollar invested from t = 0 grows at an average rate of g per period, resulting in (1 + g)T at time t = T. The geometric mean is obtained as g =

T

∏

t=1

(1 + rt)

1 T

− 1.

(a) Show that when |rt| ≈ 0, t = 1, 2, . . . , T, the geometric mean is approximately equal to the arithmetic mean:

g ≈ a.

(b) Show that the geometric mean return is always equal or smaller than the arithmetic return. Namely, g ≤ a.

(c) Let Pt, t = 0, 1, 2, . . . , T, be the prices of an asset. Show that g =  PT P0 1 T − 1. (2.5)