# 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. – Sanjay Apr 18 at 11:08
• This sounds like homework. So, what did you try to solve this? Where is your specific problem? – g g Apr 22 at 21:40
• It's not homework. I have simulated the answer but am wondering if there is an analytical approximation for the distribution. – Fortranner Apr 23 at 17:36