Been reading some notes that say when calculating returns, using the geometric methodology (1+returns, performing a division between the two returns and - 1 from the result) for computing returns is proven to be mathematically sound over a period of time as compared to arithmetic return computation.
Can someone explain why is that so?
In addition, for a single day, the arithemtic return should be the same as the geometric return?
Many thanks for clearing these doubts.
//Added the following clarifications
I am currently reading the following paper by morningstar (http://corporate.morningstar.com/us/documents/MethodologyDocuments/MethodologyPapers/EquityPerformanceAttributionMeth.pdf?bcsi_scan_aef1ec8c787364dd=JzQUZczAUPY5RaKpjawEbIdLMXNPAAAAw/PsCg==&bcsi_scan_filename=EquityPerformanceAttributionMeth.pdf)
If you refer to page 11/54, there is a section that talks about the difference between using arithmetic versus geometric methdology. Based on understanding, does it means that geometric return takes into account of the effect of compounding (due to taking into account the current accumulated capital "1+"r) that makes it mathematically sound over a period of time?
Suppose you are deposit in a bank whose interest rate changes every year, and the interest is paid as a lump sum at the end of each year. Suppose that the interest rates over the first $n$ years are $r_1,r_2,\ldots,r_n$ respectively. If $P$ is the initial deposit, then after the $n$ years have passed, the total amount of money in the bank will be:
$$P(1+r_1)(1+r_2)\cdots(1+r_n)$$
We would like to find an average interest rate over this time. Whatever the average $r$ is, we would expect it to satisfy the property, that if the interest rate had been $r$ all along, the total amount of money in the bank would be the same. This means that we would expect
\begin{align*} P(1+r)^n &= P(1+r_1)(1+r_2)\cdots(1+r_n) \\ 1+r &= \sqrt[n]{(1+r_1)(1+r_2)\cdots(1+r_n)}. \end{align*}
As you can see, this makes the geometric mean more appropriate, in the manner you described. In general, it makes sense in a very rough sort of way that some kind of geometric mean would be more appropriate than the arithmetic mean in this context: after all, rates/returns are multiplicative in nature, it makes sense that a mean which multiplies the rates instead of adding them will be more accurate.