Good morning Julien, Thanks for having taken the time to answer. Quoting julien cuisinier <j_cuisinier at hotmail.com>:
Benoit, I may miss something also, but for me these are two separate issues 1. what is the daily return (Geometric Vs arithmetic)
For me a daily return is 1. Net Return = V(i)/V(i-1) - 1 2. Log return = log(V(i)/V(i-1)) Geometric or arithmetic are not a sample daily return they are average return calculated in two different ways. What is nice with log return, is that the arithmetic mean of the daily log return is equal the log return of the geometric mean of the net return.
2. How do I annualize daily returns (geometric Vs arithmetic) if I put random prices in a spreadsheet over 252 days (1 year), then compute arithmetic & geometric daily returns >>
You also need to take the log of the daily ratios in the case of a log return before computing arithmetic or geometric average.
I use both vectors to compute annualized average daily return using both technique (arithmetic & geometric) >> the closest to the real return (as (Price(252)/Price(1)-1, so what an investor would actually get over a year) I get is by taking geometric annualization of the log returns...geometric annualization of arithmetic returns still yields close approximation but arithmetic annualization got it off the chart...
Arithmetic with net return is useless as you said. But arithmetic with logr is usefull because: exp(arithmetic average of daily log return) - 1 = geometric average of net returns Therefore arithmetic with the logr give the same as of geometric with net return. What is new for me is that quants also use geometric annualization with logr if I understand well. I thought that quants were only using arithmetic annualization with logr and geometric with net returns. If you have good urls on the motivations on applying geometric on logr, please do not hesitate to share them. I am interested. If you have a simple R code that shows your computations of annualization of logr and show geometric on logr is a better aggregation, please do not hesitate to post it. I think it would help me to be sure I understood everything you wanted to explain. Thanks for your answer.