Insurance Premium Modelling – Introducing Multiple Insured Parties

Last month, we took a look at the way an insurer would have to operate if they only insured a single party. As we discovered, it would be a hard business to run and not a very profitable one if an insurer only dealt with a single incidence of risk. They’d need to keep reserves equivalent to the maximum possible pay out under the policy and the premiums would be unlikely to support this kind of ring fencing of cash.

Fortunately, that’s not how insurance works – insurance is all about combining similar risks from many parties in order to share some of that risk and for insurers to be able to charge reasonable premiums and still make a profit.

The Loss Formula for Multiple Parties

Our simple loss formula needs an update at this point; it’s going to look something like this:

(Where X is the loss and N is the number of losses in the year.)

The trouble with this formula is that it is clear that the loss is going to vary from year to year; some years will be better for our insurer in that they will beat the odds and their insured parties will have fewer accidents than the average and some years will be worse when they also beat the odds but their insured parties have more accidents than the average.

There are two things we need to know for this; the first is the distribution of loss and the second is the frequency of loss for the risk.

This is where the maths gets a bit more complicated.

The Distribution of Loss

This will normally be calculated using a Pareto distribution. Where the α depends on the specific risk. For our car insurer they’d have to calculate this distribution based on the chances of theft, fire, mechanical failure, etc. and then combine the results.

The Frequency of Loss

The frequency of loss can be calculated in many ways but one of the simplest ways to do so is with a Poisson distribution. This only works well in the event that losses are unrelated and relatively rare so it might not be suitable for all types of premium calculation.

These two formulae can then be used to work out the total expected losses in multiple years; to some degree of certainty. You can then treat the expected loss in any given year as an average of the possible losses across multiple years.

However, there’s still more complexity to add to this model and that’s what we’ll be looking at in the next part of this series next month.