I think there are 2 points worth discussing:
1. the geometric average of discrete returns is close to but not exactly equal to the arithmetic average of continuous (or log) returns
2. the different assumptions concerning return behavior when annualizing

Concerning the second point:
set.seed(200)
P <- seq(100,120, length.out = 253) + rnorm(253,0,1) ; P

#Anualized geometric average returns from discrete daily returns:
P1 <- c(NA,P[-length(P)])
R.disc <- P/P1-1 ; R.disc  #discrete returns
R.disc.avg <- prod(1+R.disc, na.rm = TRUE)^(1/252) -1 ; R.disc.avg #geometric average of daily returns
R.disc.avg.ann <- (1+R.disc.avg)^252 -1 ; R.disc.avg.ann # annualized 

If you annualize in this way then you assume that the returns from each day are reinvested and thus compound over time.


#Annualized arithmetic average returns from continuous daily returns
R.cont <- c(NA,diff(log(P))) ;R.cont #continuous returns
R.cont.avg <- mean(R.cont, na.rm = TRUE) ; R.cont.avg
R.cont.avg.ann <- mean(R.cont, na.rm = TRUE)*252 ; R.cont.avg.ann 
# annualized

With this way to annualize you do not assume any reinvestments


Concerning the first point:
Note that R.disc.avg is not the same as R.cont.avg

However, the connection between the two is 

exp(R.cont.avg) -1 == R.disc.avg

hope this helps
(I also hope that I’m correct and Patrick wont tell me I got it all wrong (again))

Zeno


-----Original Message-----
From: r-sig-finance-bounces@stat.math.ethz.ch on behalf of julien cuisinier
Sent: Thu 7/29/2010 11:16 AM
To: r-sig-finance@stat.math.ethz.ch
Subject: [R-SIG-Finance] Annulized returns from daily? Back to the roots
 

Hi List,

 

 

 

This is a very simple question and I am feeling bit silly to ask, but googling around does not seem to allow me to confirm what I thought...

 

 

What I thought correct is (assuming 252 trading days in a calendar year):

 

arithmetic daily returns (Pt/Pt-1) & Geometric annualization (1+ Arithmetic Average Daily Ret) ^ 252 -1

OR 

Geometric daily returns (Log(Pt/Pt-1)) & Arithmetic Annnualization (Arithmetic Average Daily Ret)*252

 

AND Both calculation should give the same final annualized return. 

 

 

But a simple test in XL proves me wrong...

 

 

What I do I do wrong? 

 

Many thanks in advance - any feedback appreciated - & apologies for the simplicity of the question

 

 

 

Best regards,

Julien

 

 

 
 		 	   		  
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