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

[Patrick Burns]

Adams, Zeno wrote:
>  << 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.
>
>  Just one remark: the arithmetic mean of log returns is only
> approximately equal to the geometric mean of net returns (see example
> below). I point this out because I have read this claim frequently and
> was puzzled when I didn't get exactly the same results.
>
>
>
> x <- numeric(100)
> x[1] <- 100
>
> set.seed(123)
> for (i in 2:100) {
> x[i] <- x[i-1] + rnorm(1,1,1) - 0.009*i*rnorm(1,1,1)^2
> }
>
> x1 <- c(x[-1],0)
> ret <- x1/x - 1
>
> #arithmetic mean of net returns:
> amean <- mean(ret[-100])*25000 ; amean
>
> #arithmetic mean of log returns:
> amean2 <- mean(diff(log(x)))*25000 ; amean2
>
> #geometric mean of net returns:
> gmean<- ((prod(1+ret[-100]))^(1/length(ret[-100]))-1)*25000; gmean
>
>   

'amean' is pretty much nonsense. 'amean2' and 'gmean'
should be equivalent except one is a log return and the
other is a simple return.  If you transform one return into
the other form, then you should get the same number.

r = log return
R = simple return

r = log(R + 1)
R = exp(r) - 1



Patrick Burns
patrick at burns-stat.com
+44 (0)20 8525 0696
http://www.burns-stat.com
(home of S Poetry and "A Guide for the Unwilling S User")

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