Brainstorms

New and Improved Bornhuetter-Ferguson

by Stephen W. Philbrick

I received an E-mail informing me that the deadline for this column was rapidly approaching as I was heading for the Bermuda airport. I was leaving for a four-day, four-actuary, four-hundred-mile bicycle trip through the Carolinas; not the best opportunity for writing. Luckily, I was awaiting my flight at the airport with Ted Dew, and had some time before takeoff. I told him (and remind readers) that my goal is to provide a forum to other actuaries with ideas to share.

He hesitated at first, but then told me about a refinement to a reserving method that he is working on. Ted likes the Bornhuetter-Ferguson (BF) method, but he recognizes that the determination of the Initial Expected Loss Ratio (IELR) is the weak link in the method. Good price monitoring tools can improve the reasonability of the IELR's, but, in practice, the selection of the IELR's is often heavily subjective.

Even when the IELR is based upon solid information, the standard application of the method uses the same IELR even after substantial emergence occurs. Some people advocate a second application of the method (that is, use the implied loss ratio from the first application as an IELR for a second application). However, this seems terribly ad hoc.

Ideally, we want to calculate the best possible estimate of the expected (not the actual) loss ratio for the year. (Our ultimate goal is an estimate of the actual loss ratio for the year, but the BF method requires a good estimate of the expected loss ratio as an input.) Before the year starts, a formula incorporating price changes, exposure changes and other trends should represent the best possible information. However, as the year progresses, the actual year's experience should provide information relevant to the expected loss ratio for that year.

The emerged experience to date is not a perfect estimator of the expected loss ratio for two reasons:

1. The true reporting pattern is not known with certainty.
2. Inherent randomness of the loss process means that the actual loss ratio will depart from the expected loss ratio.

Consequently, it seems reasonable that the best estimate of the IELR might be some weighting of the loss ratio implied by the experience to date and the original estimate calculated before the year started. We can express this as a formula by letting:

• IELR represent Initial Expected Loss Ratio
• SULR represent "Standard" Ultimate Loss Ratio
• n represent the year associated with the estimate
• Z represent the weighting factor.

Then we can express the calculation as:

IELR_n = Z x SULR_n-1 + (1-Z) IELR_n-1

It would be interesting to calculate the theoretical weights representing the credibility of these two values. (Or perhaps someone can point out that this problem has already been solved and provide a reference.)

Short of a theoretically rigorous solution, Ted Dew suggests that the reporting pattern itself has some desirable properties consistent with what we would expect for the set of credibility factors. Specifically, most reporting patterns tend to increase with age, as we would expect the credibility associated with maturing actual experience. Long-tail lines have lower expected emergence than short-tail lines, implying a lower credibility at early emergence dates, again consistent with intuition.

One potential flaw is that I would argue that the credibility associated with actual experience should never reach 100 percent, but this is unlikely to be a practical problem, as the BF and other methods tend to converge before the reporting pattern reaches 100 percent.

Does anyone have any thoughts on whether this approach is reasonable (or already discussed in the literature) or whether it can be improved? Perhaps the square or the square root of the reporting pattern makes more sense, or perhaps there is some other easily calculable amount that can be used as a weighting factor.

After submitting this column, I started reading the papers in the Fall 1998 Forum. One of the papers by Paul Struzzieri and Paul Hussian specifically discusses the selection of weights to be used in a BF method. (You can find this paper in the 1998 Fall Forum.) They also reference a forthcoming paper by Spencer Gluck in PCAS Volume LXXXIV that will address the question for the BF as well as the Stanard-Bühlmann (Cape Cod) method.