The Power of Analogies
By Stephen W. Philbrick

A well-constructed analogy can be a powerful explanatory tool. Years ago, I stumbled on to the idea of using a target shooting analogy to help explain credibility concepts. Even if a person has never fired an arrow at a target, the concept is clear and understandable and helps provide insights into the more difficult concept of credibility. Generally, the use of an analogy is one-directional—a well-understood, simplified concept is used to provide insights into a less understood issue. I recently ran into a situation where the power of the analogy could be used in both directions. John Buchanan recently shared with me a presentation he made at a CARe meeting on a hybrid method for rating reinsurance layers (A Hybrid Experience / Exposure Method). The details of the approach itself may only be of interest to those in reinsurance, but I found aspects of it applicable to reserving. The core of the idea is an organized blending of two approaches to rating, the exposure method and the experience method. John remarked that his approach reminded him of the Bornhuetter-Ferguson method used in loss reserving.

Instantly, the power of the analogy helped me follow the approach. I could look for one element that corresponded to the expected loss ratio component of a BF, and another corresponding to the link ratio component. The BF and Hybrid techniques both shift reliance from the responsive method (LDF/experience) to the stable method (ELR/exposure) as time or layers advance, respectively. The BF approach has an implicit weighting of the two indications, as others have noted, while the weighting is more explicit in John's example.

So far, this is simply the normal use of an analogy. The weighting of exposure and experience rating in reinsurance is time-tested, and the analogy simply helped me see the elements more clearly. However, John's method went far deeper than simply weighting two indications. He noted that the frequency of large claims is usually more stable than the total dollars, so he incorporated frequency aspects into the weighting procedure. So I decided to turn the analogy around, to see if it could be used in the other direction. If incorporating frequency into the reinsurance rating method had value, is there a place for incorporating frequency into the BF method?

The use of frequency-severity in reserving, or even in a BF technique, isn't new. The CAS Foundations textbook discusses using a frequency-severity approach to select the initial loss ratios. However, applying the BF technique to claims counts themselves is less common. While I hadn't seen this approach used, it has been done before. Phone calls to Mike Angelina and John Buchanan verified that both had used these approaches in the past. I suspect others have done so routinely, even though it hasn't been written about extensively.

Despite the fact that the specific example did not identify a new reserving approach, the concept of exploring analogies in both directions is valuable. In some cases, the analogy itself will break down, but in many circumstances, exploring whether the aspects of one approach can be reproduced in the other approach may provide insights in both directions.

For example, John Buchanan emphasized that the key aspect of presenting his approach wasn't the weighting of the two indications, but the need to reconcile the two indications. When two different estimates of the same value differ, one should confirm that the differences could be attributable to noise. If not, then some critical assumptions underlying one (or both) of the two approaches need to be questioned. A weighting approach is an improvement over a single method only if it is averaging out random variations. A weighting approach can mask underlying problems if it deters the analyst from investigating the flawed assumptions associated with one or more of the methods. (As an aside, Isaac Mashitz notes that one can get false comfort when two methods produce similar answers. He has seen situations where both the exposure and experience method produce misleading answers, and cautions that checking the underlying assumptions needs to occur even when the methods produce similar results.)

Running half a dozen methods and weighting the results may appear satisfying, but a good analyst recognizes that the underlying assumptions of each method should be examined, whether for reserving or pricing. If the critical parameters cannot be corrected, it may well be better to drop the method from the average, rather than include it. John Buchanan also pointed me to Sholom Feldblum's excellent paper in the 2003 Proceedings (The Stanard-Bühlmann Reserving Procedure: A Practitioner's Guide). This paper provides an excellent discussion of the Stanard-Bühlmann reserving method. For those not familiar with the method, he explains it relative to the more familiar Bornhuetter-Ferguson method, explicitly relying on the power of the analogy to explain a new concept in terms of a familiar one. Interestingly, he specifically discusses applying the Stanard-Bühlmann to claim counts. The next generation of reinsurance pricing tools may analogously evolve the way that the BF and its derivatives evolved.