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

[Benoit Schmid]

Good morning,

Thanks for having replied to my previous questions.

julien cuisinier wrote:
> 1. Log returns as a matter of precision, it assume continuous 
> compounding...which is what your security price does, so it is a better 
> estimate of your true return ==> geometric aggregation since it is the 
> correct one (I believe) that factor for compounding >> do it manually in 
> excel, you will see that this formula will revert the correct final 
> return while the "multiply by number of periods" will only revert an 
> approximation

I have tested it on a spreadsheet.

Aggregation of 252 days of log return is the sum of log return:
log(V(252)/V(1)) = log(V(252)/V(251)*V(251)/V(250)*...*V(2)/V(1))
                 = log(V(252)/V(251)) + log(V(251)/V(250)) + ... + log(V(2)/V(1))
Therefore for log return, we take the sum to aggregate.
If you suppose that the daily alpha is constant and is in log return term, 
then alpha(252 days) = 252 * alpha(1 day)

Aggregation of 252 days of net return (non log) is a geometric aggregation:
V(252)/V(1) - 1 = ((V(252)/V(251) - 1 + 1)*(V(251)/V(250) - 1 + 1)*...*(V(1)/V(0)) - 1
                = ((R(252) + 1)*(R(251) + 1)*...*(R(1) + 1)) - 1
If you suppose that the daily alpha is constant and is in net return term, 
then alpha(252 days) = ((alpha + 1)^252) - 1

This is why I say that aggregation (that is used for annualization) is a sum 
in the case of log return and is a geometric aggregation in the case of a net return.

Therefore I do not understand why you are using ((alpha+1)^252)-1 instead of 252*alpha 
when you work with log return.
For me ((alpha+1)^252)-1 is the correct value for net (non log) return.

What am I missing?

Thanks in advance for your answer.