This leads on from the previous blog entry on probabilistic models. The platform I am working on incorporates catastrophe models that produce Event Loss Tables or ELTs. These are tables containing Events, Ground Up Loss, Ground Up Loss standard deviation, loss frequency and various other loss perspectives. These loss tables are sorted in order of increasing ground up loss giving a cumulative loss distribution. A function that fits this curve is called a cumulative distribution function or cdf. Pricing a Cat XL treaty is done by integrating the CDF. This can be made in a number of ways:
The simplest way is by burning cost. This means just summing up the events taht trigger the treaty. This works very well in low layers because there are naturally more low cost events than high cost events. For high layers there may be only a few events and the accuracy is not sufficient. In these cases we can try to fit a points to a function such as Paratoo or log polynomial and then integrate the resulting function over the Cat XL layer. There are a number of problems with this approach, for example the numerical regression may give unreasonable coefficients that give unexpected results. The method that we use the most is to to sampling each loss assuming a distribution like log normal and the standard deviation and applying a burning cost.
Event loss tables can become quite large but are still manageable compared to Year Loss Tables. For example it is possible to make a loss estimation of a current event by making a query based on event or combination of events in the event loss table. It is not possible to combine different ELTs of different event sets
Some cat models incorporate some information over when an event occurred. This makes it possible to model clustering of events much more accurately. This has quite an impact on pricing. For example inuring treaties can be priced much more accurately. Imagine with an ELT you need to assume an average number of events in a year. There is a big difference between having an average of 2 events per year and a year loss table that says on the first year there are 0 events on the second year there are 4 events. Both have a mean of 2 events but in the first year the treaty is not touched and in the second year it can depend on the number of reinstatements.
Year loss tables are event based which makes them easy to combine with other YLTs.
Here are the steps needed to create a YLT from an ELT
1. Decide on how long the YLT is to be based
2. Determine an average number of events that happen in a year
3. Use a Poison distribution with a random sample to determine how many events happen in a particular year
(The poison distribution provides a sequence of integer numbers )
4. Sample from the ELT randomly these events
5. Based on the Mean Loss and STD Loss make a sample with a distribution such as a Beta or log normal to determine an actual loss using another random variable
The random variables are created from seeds. For a given seed the same sequence of random numbers are generated each time the application is run. Mathematically it turns out that this is an acceptable way of sampling and it means that the resulting YLTs are not needed to be stored. This is advantageous because in order to get events in the high layers a long simulation period is needed which means the storage requirements of a YLT is much bigger than an ELT.
Now let's drill into the details. Step 3 sounds very easy, make a sample random sample of a Poisson distribution. The thing is that Poisson is expressed in a formula like frequency = Function( Mean, Standard Deviation). For example in Excel it looks like POISSON(A3,$B$1,TRUE). The problem is that we need to make random samples of the number of events with a given mean and sd. To do this we need to use something called Inversion theory. There are some statistical packages that have this function built in. For example in r it looks like
pois <- rpois(100, lambda=3)
pois[1:10]
[1] 1 2 3 2 2 2 3 3 6 2
In C# you need to find a library to do this. I found 2 that look interesting open source projects that have these functions
http://code.google.com/p/pzmath/source/checkout
http://www.alglib.net/
