Geometric mean return

CFA Level I Quant
#1

The average return for Portfolio A over the past twelve months is 3%, with a standard deviation of 4%. The average return for Portfolio B over this same period is also 3%, but with a standard deviation of 6%. The geometric mean return of Portfolio A is 2.85%. The geometric mean return of Portfolio B is:

  1. less than 2.85%.
  2. equal to 2.85%.
  3. greater than 2.85%.

Solution

A is correct. The more disperse a distribution, the greater the difference between the arithmetic mean and the geometric mean.

This answer is confusing to me. Can anyone pls explain why it’s A? I thought according to the answer, it’s supposed to be C?

#2

The explanation says that for a given (arithmetic) mean return, as the dispersion (standard deviation) of returns increases, the geometric mean return decreases.

Portfolio B’s dispersion of returns is larger than portfolio A’s dispersion of returns, so portfolio B’s geometric mean return should be less than portfolio A’s geometric mean return.

1 Like
#3

Thanks a lot magician!

#4

My pleasure.

#5

Hi there,

Appreciate if you can further elaborate on the rationale of

Dispersion increase → GM decreases
@ same AM

#6

Take a simple example:

r_1 = 5\%, r_2 = 5\%
AM = 5\%, GM = 5\%
r_1 = 4\%, r_2 = 6\%
AM = 5\%, GM = 4.9952\%