Geometric mean application in finance ratio question

[Vital]

Homework Statement

Hello.
There is a financial metric called time weighted rate of return, which is computed using the following formula:
1) if we compute daily returns, or other returns within a year:

r _tw = (1+r₁) x (1+r₂) x...x (1+r _{nth year}),
where r _tw is the time weighted rate of return
r_n are period returns; for example, if we compute daily returns, then there will be 365 (1+r) returns multiplied on each other

2) if we have returns for a few years, then the formula is

r _tw = [(1+r₁) x (1+r₂) x...x (1+r _{nth year})] ^(1/n) - 1

Homework Equations

For example:
We are given quarterly rates of return, hence the time weighted rate of return will be computed in the following way:

(1+r1)(1+r2)(1+r3)(1+r4)−1=(1.20)(1.05)(1.12)(0.90)−1=0.27or27%

But if we have the same returns but they are not for each quarter within one year, but each return is a yearly return, hence we have returns for 4 years, then we use the geometric mean:[(1+r1)(1+r2)(1+r3)(1+r4)]^(1/n)−1=[(1.20)(1.05)(1.12)(0.90)]^1/4−1=6.16%

The Attempt at a Solution

My question:
Please, help me to understand why if we compute returns within 1 year period we do not take the n-th root of the product, but when we compute the return for several years we do take the n-th root. What is the math behind it?

I will be grateful for your explanations.
Thank you!

 

[BvU]

What is the math behind it

You determine the annual rate. So you multiply part-of-year rates until you have a full year, or you take the nth root to reduce n years to a single year.

You can extend taking powers to partial exponents, so that you can use one and the same formula to calculate back to one year (in case the rate is constant):

(1+r)^1/n,​

for example: if n = 1/4 you get (1+r_1/4)⁴ with r_1/4 the rate per quarter
and if n = 4 you get (1+r₄)^1/4 with r₄ the full rate over the four years

 

[Vital]

You determine the annual rate. So you multiply part-of-year rates until you have a full year, or you take the nth root to reduce n years to a single year.

You can extend taking powers to partial exponents, so that you can use one and the same formula to calculate back to one year (in case the rate is constant):

(1+r)^1/n,​

for example: if n = 1/4 you get (1+r_1/4)⁴ with r_1/4 the rate per quarter
and if n = 4 you get (1+r₄)^1/4 with r₄ the full rate over the four years

Thank you very much. It is clear now, and I am happy that now I understand how it works, though it seems that I should have understood that from the very beginning )))