Hi everyone, I have a question regarding calculating the shortfall probability in the context of a retirement plan and would greatly appreciate any useful advice on how to solve it.

Imagine you have a retirement plan with a financial institution. At t=0, you deposit $10,000 with the instituion. At the start of every month, you will withdraw 5% of your account balance for own consumption. The remaining funds is carried over to the next period with an interest rate of 2%. On top of that, the first $6,000 of your account balance will receive an additional 1% as bonus. Since the amount you withdrew every month (x) is a percentage of the account balance brought forward from the previous month, it varies from month to month. Hence, you are interested to know what is the probability that the amount withdrawn will be less than your benchmark value of $400 (this is know as the shortfall probability i.e. P(x<$400)). Given these assumptions, what is the shortfall probability?

In my opinion, it is not possible to solve this question analytically and obtain a closed form solution because the evolution of the account balance depends on the interest rate earned and we cannot tell in advance what is the probability that it will earn "(2% + 1%) on the full amount" or "2% on full amount + 1% on the first $6,000". Since we cannot solve for the account balance and amount withdrawn analytically, we also cannot solve for its shortfall probability analytically. As such, I believe we have to rely on a Monte-carlo simulation to run say 10,000 independent paths and see what's the probability of the shortfall. Nevertheless, this is just my opinion and I may be mistaken. Therefore, any advice from anyone with ideas will be greatly appreciated.

Thank you!