Hello
when calculating the Geometric Mean for exam questions,how to know when to select either of these formulas ?
finally, when I should use (n) for the period and I should use (n-1)
for example in this question 25 in textbook (CFA® Program Curriculum 2023 • LEVEL 1 • VOLUME 1) - page 157
The second one is used for returns, the first for everything else.
2nd formula is the correct formula for return. You will end up using the 2nd formula in level 3 too when you calculate capture ratios etc.
First formula is an easy formula to calculate. In fact, i have seen books using arithmetic average to calculate mean of returns.
It is a question of convenience vs. accuracy.
Not really.
It’s a matter of knowing what you’re doing. When compounding returns, you have to know that you take the product of the growth factors (1 + r_i), not the product of the returns (r_i).
When we say “the geometric mean return” what we really mean is “the return associated with the geometric mean of the growth factors”. It’s a lot more than a mere matter of convenience.
@S2000magician - I think people should be aware of the errors and the limitations being introduced by approximation. If return numbers are close to each other, arithmetic mean and geometric mean, calculate by any of the formulas, given by OP, will give average return numbers very close to each other. And, that is why some books have taken liberty to choose convenience over accuracy.
However, this formula, given by OP and shown below, to calculate geometric mean, has some serious issues -
For example - Try to calculate the geometric mean on the following set of returns using the formula shown above -
10%, 20%, 30%, 40%, 50%, -20%
(note the last return is minus 20%)
@Firas1978 will have a very crystal clear idea if he calculates the geometric average return on this set of return-data by the two formulas he mentioned.
I agree.
But using \bar X_G = \sqrt[n]{X_1X_2\cdots X_n} when calculating the geometric mean return isn’t a problem of approximation; it’s a problem of using an inappropriate formula.
That’s my point: when calculating geometric mean returns, you calculate the geometric mean of the growth factors (and then subtract 1), not the geometric mean of the returns themselves.
It’s not a matter of accuracy versus convenience. It’s a matter of accuracy versus absurdity.