How do actuaries calculate the premium of catastrophic insurance given the shortage of data?

Since catastrophes seldom happen, there aren't enough data points for meaningful statistical analysis. So, how do actuaries go about doing their calculation for the premium of these kinds of insurance given the lack of data points?

My response is from a US-based, non-life, reinsurance actuary's perspective. Clearly you are correct in that most of our estimates are simply that—estimates. It is not like pricing private passenger auto, or motor as our colleagues across the pond would say. You have millions of drivers, hundreds of thousands in each state, so the Central Limit Theorem is your friend and classification/regression models work like a charm. For the low-frequency/high severity cases it is different.

Perhaps the most important point to keep in mind is that when dealing with the situation you describe, everyone agrees that the actuary's job is not to find a great model with high predictive power. As you say, that is impossible at times. Rather, our job is to reduce the principal's uncertainty. And by principal, I mean the person for whom we are working: a client, a policyholder, our boss. Our work is aimed at putting reasonable bounds around the uncertainty so better financial decisions can be made. For example, technically, an insurer can be on the hook to pay the maximum policy limit for every policy they wrote in a given year. The probability of that happening is so remote that it is likely the sun will collapse into a brown dwarf first. By providing sound estimates and ranges—even though we know our point estimate has a 100% probability of being wrong (it's a continuous distribution after all)—we supply the principal with the knowledge to better estimate the capital needed to support the risk. All the techniques we do use require a measure of judgement, which is why—especially in reinsurance—there is a tendency for actuaries and other risk analysts to be "top-heavy", meaning they lean to experienced, credentialed practicioners who have decades of experience from which to draw. That is probably the most important element: drawing conclusions from similar situations.

Specifically speaking of natural catastrophes, there has been a lot of work done over the past decade in refining wind, quake, flood, wildfire, NBC terror attack, and non-NBC terror attack models around the world. We will often work with meterologists, seismologists, hydrologists, and others on refining these models which can then be used for prediction.

For those events for which there aren't even nascent models, here are some techniques which may be used:

• Pooling with similar-enough risks

  • While certain catastrophes like the Beirut warehouse harbor explosion of a few years ago are unique, they share similarities with other such events. Maybe it wasnt a nitrate-based explosion but a gas explosion. In this particular example's case, while the frequency may be different the severity would be similar. We can pool similar enough risks together and make deductions based on a larger pool. Sacrificing homogeneity for credibility as it were, which is the constant balancing act of the actuary.

• Curve Fitting and Extrapolation

  • As @Alex said in his comment, there are statistical methods to extrapolate from known data. Fitting a distribution to the largest-seen similar risks and using that distribution's tail to predict severity of the unseen risks is a standard method. EVT is another, but it still requires a pool of data from which to fit and basically splines a GPD onto the tail of another distribution below the threshold.

• Scenario-based Testing or Simulation

  • Sometimes there is no way to generate a credible parametric distribution for use in modeling. However, we can judgementally suggest events such as the probable maximum loss (PML) or the maximum forseeable loss (MFL) which are usually estimates of how bad things can get at various points on the distribution. So we can build models that include particular events 10% of the time or 1% of the time and fold those in to the more analytcially tractable elements of the client's exposure to loss to get a better representation of just how bad things are when things go bad than the principal would have had without these judgemental suggestions

• Qualitative Suggestions

  • Sometimes we cannot even do the above. Then our job is not necessarily to model the events, but to provide insight on how best to mitigate such events should they occur. Helping a principal understand their own risk appetite and risk tolerance is a key element of how we provide value.