## Meaningful Ranges for Reserves

by Victoria GrossackActuaries have discussed much in recent years about what a best estimate should look like, but given less thought, perhaps, to what it means to have a meaningful range. Part of the problem is that we first need to ask—meaningful to whom? Auditors, regulators, management, mathematicians and even the general public have very different ideas about how to interpret a range. This article discusses different possible understandings of ranges and gives advice on how to speak about them.

Every actuary reading this article knows that actuarial science is an inexact science, attempting as it does to predict the future. Although a point must be selected in order to create the balance sheet and the income statement, it is unlikely that the final amount paid will be the number booked. So, in order to give some understanding to the variability, people often refer to reasonable ranges. Even now, though, we may find that we are using the same term with widely different meanings. What is a reasonable range? Here are some perspectives, all of which may change according to the speaker or the listener.

1. All possible outcomes perspectiveA reasonable range goes from the lowest

possibleoutcome and the highestpossibleoutcome.This definition may seem naïve, but there is logic to it and it may be the definition most accepted by the general population. A range would be

unreasonableif it did not contain the final result, so for a range to be reasonable it must go from the least possible to the highest possible.

2. All probable outcomes perspectiveA reasonable range goes from the lowest probable outcome to the highest probable outcome and the point chosen should be the most likely.

Here we may run into trouble into the definition of probable, as well as most likely, with many articles already published on the subject.

3. Mathematical view perspectiveGiven some underlying assumptions on the underlying probability distribution, a point can be chosen as the mean, and a range can be created as a confidence interval around the mean.

This definition looks good until one starts poking too hard at the underlying assumptions. How accurate is the selected probability model? How certain is that tail factor? Are the residuals to the fitted distribution behaving normally? Also, who decides what level of accuracy is important for the confidence interval? With this interpretation of the meaningful range, a wider range implies more uncertainty.

4. Management view perspectiveA reasonable range means that we can book anywhere we like within it and still be reasonable, with the proviso that the next estimate provided by the actuarial department will give us the same flexibility, and the reserves should never show adverse development in the future.

5. Auditor view perspectiveA reasonable range is the difference we will allow between our estimate and the number booked by the client. It may be +/- 5%, with some allowance for the uncertainty of a line of business.

Notice that perspectives 1, 2 and 3 are concerned with calculation and measurement, whereas perspectives 4 and 5 are concerned with actions that can be taken as a result of the numbers. And the perspective may change as the phrase "meaningful range" or "reasonable range" is used by different people. In fact, the perspective may change as these phrases are used by the same person,

without the speaker even realizing it.So, what is the reserving actuary to do?

I have no one-size-fits-all solution, except that the reserving actuary should take care when using the word "range." The actuary should check—in advance, if possible—what the primary user believes about ranges, and if there is a company policy on how ranges are applied. If not, the actuary should make recommendations on how the range can be used while delivering them. The actuary may especially want to emphasize how the range should not be used.

Ranges are in demand these days, especially with the regulatory changes in the U.S. (Sarbanes-Oxley) and Europe (International Financial Reporting Standards). Nevertheless, despite the demands for ranges, those requesting them often ignore them in the end. Dealing with ranges increases the level of complexity considerably, and many people are either math-phobic or they have more important things to work on besides understanding the various interpretations of ranges.

There may never be total consensus on the definition and interpretation of "meaningful range." And that's as it should be, as a meaningful range should reflect what is meaningful for the decision maker given that decision maker's needs and capabilities. However, a meaningful range should also consider the users of that range and the degree to which they may rely on the range being reasonable.