[R-SIG-Finance] Returns used to compute the alpha and the beta

[Benoit.Schmid at unige.ch]

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.