In the last post, we looked at a simple model of an endowment, and showed how various interest and withdrawal rates lead to different balances in the account at the end of 50 years. In this post, we'll look at Monte Carlo Simulation as a method for predicting the likely end state of the account when we are uncertain what the average interest rate and withdrawal rate will be.

We'll start by specifying probability distributions for these two variables. We'll choose to represent our interest rate as a normal distribution with mean 0.05 and standard deviation 0.01, and our withdrawal rate as a normal distribution with mean 100 and standard deviation 10.
In our Monte Carlo calculation, we draw a value from each of the input distributions, and use them as parameters for the model (red and blue). Then we simulate the model over its 50 year time span, and measure the result at year 50 (purple).

In the animation that follows, we build up histograms for the input and output parameters by grouping both input parameters and outputs into bins. We can compare the input distribution with its histogram to get a sense for how well our calculation has recreated that distribution by taking samples from it.