# Geometric mean vs arithmetic mean

Fund
Arithmetic Mean (%)
Geometric Mean (%)

SLASX
2.64
−0.65

PRFDX
4.31
1.59

“The difference between the geometric mean returns of the two funds (2.24%) is greater than the difference between the arithmetic mean returns of the two funds (1.67%). How should the analyst interpret these results?”

Taken directly from reading 8 - the paragraph after describes that geometric is compound growth.

I do understand that the larger the difference between the SLASX Arithmetic and geometric means the greater the variability.

Not clear about the significance of “greater variability between: arithmetic mean of two different samples vs geometric mean of two different samples”.

Thanks

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Arithmetic mean is not time weighted and Geo mean is.

For Arithmetic mean return earned in year 1 and year 2 have same weights where as in geometric mean return earned in year 1 would have lager weights than year 2 because we have held that asset longer than year 2.

I think it’s a bad question.

The difference between the arithmetic and geometric mean returns for a single sample is based on the standard deviation of the returns for that sample - for a large enough sample (And from what I recall), the relationship is

Geometric mean = Arithmetric mean - 1/2 Std deviation of returns.

But the question is asking about the difference between arithmetic and geometric mean returns and how it changes between two samples.  Think of this difference as the “spread” for SLASX it’s 2.64- (-0.065) = 3.29%, while for PRFDX it’s 4.31-1.59 =  2.62%.

Since the spread is lower for PRFDX, it must have a lower variability of returns (that affects the second term on the right hand side of the equation above).

But the reason I think it’s a bad question is that first, you have to make some assumptions (a large enough sample), and also know the relationship.  And since it’s been a while since I looked at the L1 curriculum, I don;t know if they explicitly state that relationship and the assumptions necessary for it to hold.

You keep using that word.  I do not think it means what you think it means.

Geometric mean equals to arithmetic mean minus half variance. The larger the difference between them the larger the dispersion is - riskier. (given the only difference between them is half variance).

KaanKaya wrote:
Geometric mean equals to arithmetic mean minus half variance. The larger the difference between them the larger the dispersion is - riskier. (given the only difference between them is half variance).

First, it’s standard deviation, not variance.

Second, they’re not equal; it’s only an approximation.

Simplify the complicated side; don't complify the simplicated side.

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S2000 - I believe the relationship holds “in the limit” (but I could be misremembering there).  But even if so, for all practical purposes, you are correct.

You keep using that word.  I do not think it means what you think it means.