Geometric Mean vs. Arithmetic Mean

[keep_running]

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!

 

[S2000magician]

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
a² − 2ab + b² ≥ 0
a² + b² ≥ 2ab
Let a = √c, and b = √d. Then, a² = c, b² = d, and,
c + d ≥ 2√(cd)
(c + d) / 2 ≥ √(cd)
Arithmetic average of c & d ≥ Geometric average of c & d

 

[keep_running]

“Let c = √a, and d = √b. Then, a² = c, b² = d,”
Did you mix up the notation? Shouldn’t c2 = a and d2 = b?

 

[S2000magician]

keep_running wrote:”Let c = √a, and d = √b. Then, a² = c, b² = d,”
Did you mix up the notation? Shouldn’t c2 = a and d2 = b?

Yes.
Fixed.
Thanks.

 

[keep_running]

I am still confused.
Is there a real life example (with stock returns perhaps) that could explain this reasoning?

 

[S2000magician]

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.