Brainstorms

Loss Ratio Models

by Stephen W. Philbrick

One of the first things I learned about insurance was the "law of large numbers." I learned that it justified the existence of insurance—explaining why an individual would transfer risk to a company that itself was risk averse—because the combination actually reduced the total risk. (Glenn Meyers stated it well in his 1982 paper when he said, "One of the main pillars of insurance theory has been the Law of Large Numbers.")

In general discussions about insurance companies, we tend not to get explicit about the mathematics of the law. We say risk tends to decrease as business volume increases. We might elaborate and explain the loss ratio of a larger business volume is expected to be "tighter" (all other things being equal). If we wanted to be more formal, we would say we expect the loss ratio's standard deviation to decrease for larger business volumes.

At the individual account level, we have analyzed the situation in great detail. Simon's Table M is effectively a distribution of loss ratios for individual risks. Hewitt explicitly modeled the loss ratios of individual risks. We have been less likely to model the loss ratio of an entire company.

With the advent of DFA, that is changing. One of the key components of a DFA model is the formal modeling of a company loss ratio.

For any specific company, we probably model the loss ratio based upon an explicit analysis. I wanted to create a more general model; a loss ratio model that would apply to any sized company. I decided to start with first principles.

Roughly speaking, we expect the loss ratio's standard deviation to drop with the square root of the size. If there were no such thing as parameter risk, we might expect (as Simon suggested in the Table M paper) the standard deviation would ultimately go to zero. However, we know parameter risk does exist (as Meyers and others have pointed out), so we might refine our statement to assume the standard deviation drops to some level representing industry parameter risk.

It seemed reasonable to me, but data can often be stubborn. With trepidation, I calculated standard deviations of accident year loss ratios for a number of companies (over a ten-year period) and plotted the results. To my pleasant surprise, the data for many business lines followed my expected pattern reasonably well. I was particularly interested in aggregating all casualty lines and property lines together. The casualty model worked acceptably. Setting a "floor" standard deviation at the industry aggregate level, and fitting the companies so the "excess" standard deviation dropped with the square root of the volume produced a reasonably fitting model.

The results for property were much different. The plot of standard deviations against company size looked more like a random pattern. Our working hypothesis is that catastrophe influence on property results overwhelms the reduction in standard deviation due to the law of large numbers.

One colleague noted the time frame selected (latest ten years) largely excludes the impact of asbestos and pollution claims. The use of a longer period might simply increase the "constant," that is, the overall level of the parameter risk, or it might totally change the results. As usual, if anyone else has examined this question, I would be interested to hear the results.