Portfolio Depletion Risk Under Constant Annual Withdrawals

Summary

The document poses a quantitative finance problem: how portfolio values evolve when the underlying investment follows geometric Brownian motion and a fixed dollar amount is withdrawn at the beginning of each year. It asks whether an approximate analytical distribution is available for portfolio value after a chosen horizon, as well as for the time until the portfolio is depleted.

An illustrative setup gives an initial investment, an annualized geometric return and volatility, a fixed withdrawal, and a ten-year valuation horizon. However, the document contains only the question and no proposed derivation, formula, simulation, or answer. It therefore frames the modeling challenge but provides no evidence about the resulting distribution. Any solution would need to account for the interaction between stochastic returns, withdrawal timing, and the possibility of depletion; the prompt does not specify additional assumptions such as treatment of partial final withdrawals.

Key ideas

• The problem models investment returns as geometric Brownian motion with fixed annual dollar withdrawals.
• Withdrawals occur at the beginning of each year, affecting the subsequent invested balance.
• The requested outputs are the portfolio value distribution at a fixed horizon and lifetime until depletion.
• The document states an example setup but provides no solution or analytical approximation.

Tags

# Distribution of portfolio values with constant spending rate


# 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?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.