# Distribution of portfolio values with constant spending rate

If your portfolio is invested in an asset that follows a geometric Brownian motion, and you withdraw a constant dollar amount at the beginning of each year, is there an approximate analytical distribution for the portfolio value after N years and for the expected lifetime of the portfolio before depletion?

For example, if you start with 1,000,000 dollars in the stock market, which has annualized geometric return of 8% and volatility of 15%, and you withdraw 50,000 dollars at the beginning of each year, what is the distribution of portfolio values after 10 years?

• This is an excellent question, indeed. I don't have the answer for it myself. If you look at the Black-Scholes model with dividends then the assumption is that dividends are handed out on a continuously basis. The fact that financial mathematicians have made that assumption to construct the BS model tells me that the analytical distribution you are looking for does not exist. Apr 18 '19 at 11:08
• This sounds like homework. So, what did you try to solve this? Where is your specific problem?
– g g
Apr 22 '19 at 21:40
• It's not homework. I have simulated the answer but am wondering if there is an analytical approximation for the distribution. Apr 23 '19 at 17:36