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

[Adams, Zeno]

 << 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




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