# Monte Carlo simulation of low probability events with impact

I wonder if I'm lacking the terminology to phrase this question: I would like to model, in Excel, the risk of a fairly rare event, E, happening over time, where time is divided into chunks called periods, each with n trials. So the event has a low probability (p < 0.1%) of occurring but a high variable impact, hence the client's interest.

If the event impact were constant I would use the inverse binomial function to model the occurance of events in each period as BINOM.INV(n, p, RAND()). A possible approach is then to layer the impact distribution (assume Normal) on top of this: NORM.INV( RAND(), $\bar{x}$, $s$ ). By repeating this many times I can build up a monte-carlo simulation model (into which I need to feed other parameters).

It feels contrived and unrobust, probably inevitably: I don't have much data on how often E occurs, or its impact impact, so clearly I can't be confident about when these events will occur in the future.

Here's some data: In a sample of 7000 "trials", the event happened 5 times with impacts 6.3, 6.3, 6.7, 6.8 and 7.6. There are 100 trials per period.

My specific questions, which may not be the right ones, are:

1. Given the need to model this event, how appropriate is this combined simulation model?
2. What better ways are there to model these kinds of events - that have both a probability of occurring and an variable impact?

• This is a very common insurance problem, combining modelling of the rare event with something like a gamma distribution for the impact - suggest you look in the actuarial literature. – Peter Ellis Jul 17 '13 at 8:49
• you probably want Poisson rather than Binomial for number of events in a period (unless there really is a maximum on the number of times an event can occur). – TooTone Jul 17 '13 at 10:23
• Thanks @TooTone for the relevant reminder of the Poisson. A quick sense check made it seem like binomial/poisson would give pretty similar results in this study, perhaps due to large n and small p. – Jon Jul 18 '13 at 7:06