Returns used to compute the alpha and the beta
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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