 Central Limit Theorem approximation of the total impact of a number of risks | Vose Software

# Central Limit Theorem approximation of the total impact of a number of risks

An example of a Monte Carlo simulation risk analysis model for the Insurance and reinsurance industry

Technical difficulty: 4

Techniques used: Monte Carlo simulation in Excel

# Model description

Imagine that you are an insurance company with several different policies. For each policy, you have the number of policy holders, the expected number of accidents per policy per year, the mean and standard deviation of the cost of each accident, and each policy has its own deductible and limit. It is a simple, though perhaps laborious, exercise to model the total payout associated with one policy and to sum the aggregate payout using simulation.

Now imagine that you feel there is likely to be some correlation between these aggregate payouts: perhaps historic data has shown this to be the case. Using standard simulation methods, we would simulate the individual payouts - we cannot then correlate the aggregate payout distributions. In order to generate a correlated sum, we would need to run a second simulation – which is laborious and clumsy.

However, we can include a correlation if we use FFT methods to construct the aggregate loss distributions. The model shown below shows the aggregate loss distribution of five different policies being correlated together via a Clayton(10) copula. Note that the equations used in Cells C21:C25 use one minus the Clayton copula values which will make the large aggregate claim values correlate more tightly than at the low end. This example model is simulating the loss distribution for a number of policies where the aggregate loss distribution for policies are correlated in some fashion.

Other examples of adding correlation to aggregate calculations can be found in this topic.