Geometric Mean vs. Arithmetic Mean

CFA Level I
#1

Hello,

Can anyone explain the intuition behind why geometric mean is less than arithmetic mean, and why the different between geometric and arithmetic mean increases with variability in returns?

Thanks!

#2

If I were you, I’d put a bunch of (positive) random numbers into an Excel spreadsheet, and compute the arithmetic and geometric means for 2, 3, 4, 5, . . . such numbers. You should be able to see it very quickly.

Here’s a quick proof of it for two numbers, a, b ≥ 0:

(a − b)² ≥ 0

a_² − 2_ab + _b_² ≥ 0

_a_² + b_² ≥ 2_ab

Let a = √_c_, and b = √_d_. Then, _a_² = c, _b_² = d, and,

c + d ≥ 2√(cd)

(c + d) / 2 ≥ √(cd)

Arithmetic average of c & d ≥ Geometric average of c & d

#3

“Let c = √a, and d = √b. Then, a² = c, b² = d,”

Did you mix up the notation? Shouldn’t c2 = a and d2 = b?

#4

Yes.

Fixed.

Thanks.

#5

I am still confused.

Is there a real life example (with stock returns perhaps) that could explain this reasoning?

#6

Let c = 4, d = 5.

Then the arithmetic mean is (4 + 5) / 2 = 4.5; the geometric mean is √(4×5) = √20 = 4.47.

4.47 < 4.5.