I'm trying to estimate the distribution of future portfolio values based on the distribution of a portfolio's returns.
First, to define some variables:
- Rt = simple return for period t
- rt = ln(1+Rt)
- Vt = portfolio value at period t
If we assume rt is a normally distributed random value and an initial investment of $1, ln(Vt) = r1 + r2 + ... + rt. Since ln(Vt) is a sum of normally distributed random variables, ln(Vt) is also normal. Its mean is t * mean(rt), and its variance is t * var(rt).
MY PROBLEM: I need to get the distribution of portfolio values based on a starting value that is not 1 dollar and cash inflows/outflows each period thereafter. I can get the mean by adding ln(starting_value). However, I am not sure how to modify the variance from the scenario above to adjust for the fact that my scale is no longer based on a starting value of $1. If I just take the cumulative sum of the variance of the returns like I did in the scenario above, the portfolio's variance each period is far too small.
Any help would be greatly appreciated. I can provide a spreadsheet with a fairly simple example calculation if that helps clarify what I'm trying to do.