A log returns distribution with median zero

Hi,

I have a time series of monthly log returns, to which I subtracted the median, so that they have zero median. I did this because I want the price distribution (kernel) for each month to have median X. So that for the first month P = X*exp(ln), such that the median of P is X. And for the second P'=X*exp(ln)*exp(ln'). I do this for twelve months.

The problem is that although the time series of returns have median zero, when I fit ksdensity and run 500.000 iterations, the median is not zero and so after 12 months P'''''''''''' is very different from X. Anyone knows what can I do?

Thanks,

[Michael Haderlein]

Is this some kind of homework and you are in the same class with the user who asked this?

Anyway, what did you normalize? If median(ln) is zero, then median of P=X*exp(ln) is something but not X. Please give more precise details here.

[Alexandra]

Hi Michael,

No, I am not related with the other question. The thing is, I want to simulate a random distribution of some price P. I use log returns ln(p1/P0). But I want the distribution of P do have median X. So I start from X*exp(ln(p1/p0)) with many returns from historical data. P does not have median X. I tried to make sure historical returns have median 0, but the random return simulated do not have median zero. Was I clear? many thanks,