Actuaries occasionally encounter inconsistencies in mortality experience that cannot be explained by typical causes of fluctuation (i.e., underwriting, market conditions, intrinsic randomness). For example, the actual-to-expected (A/E) ratio for a particular issue year may be unusually high or low compared to those in surrounding years. In this case, the actuary may wish to lessen the effect of that subgroup while not completely ignoring its impact on overall experience. This article describes a technique that uses credibility theory to achieve this result.
Credibility Theory
Limited Fluctuation Credibility Theory (LFCT) is a method for assigning full credibility to an experience study based on claim count. Full credibility means it is appropriate to base a mortality assumption solely on a study’s experience and to ignore any industry data. In addition, LFCT also determines partial credibility, where the experience study is weighted against industry experience.
Recalculating Subgroup Experience
Before deciding whether to limit the impact of anomalous subgroups, the actuary should calculate credibility-adjusted experience. This is done by weighting each subgroup’s A/E ratio with a corresponding fully credible value. In calculating credibility-adjusted experience, assume the study’s overall A/E ratio is fully credible and can stand on its own. Therefore, the claim count of the overall study becomes the threshold for full credibility.
The credibility factor for each subgroup is then defined as the square root of the quotient of its claim count over the study claim count. These credibility factors are then used to adjust each subgroup’s A/E ratio by weighting them against the overall study.
This produces a set of credibility-adjusted ratios, which are renormalized in the final step to conserve the actual total claim amount, thus preserving the overall study A/E ratio. Once this process is complete, any particular subgroup’s experience may be discarded without entirely losing its credibility-weighted impact on total experience.
An Example
The experience for issue years 2001-02 is very high compared to later years (see Figure 1). Further review uncovers no rational explanation for the anomaly, and the actuary would like to disregard the ’01-’02 experience in determining mortality assumptions.
| Issue Year Subgroup | Actual Claim Count | Actual
Amount | Expected Amount | A/E Ratio
by Amt |
| 2001-02 | 133 | 49,734,937 | 27,105,783 | 183.5% |
| 2003+ | 401 | 90,735,072 | 136,750,640 | 66.4% |
| Total | 534 | 140,470,009 | 163,856,423 | 85.7% |
Figure 1 - Claims in 2001-02 are much higher than expected.
We assume here that the overall study A/E ratio of 85.7 percent is fully credible and 534 claims represents the count threshold for this determination. If we had no direct knowledge of the subgroup A/E ratios, we might expect that 85.7 percent would be the experience in each subgroup. Therefore, we will credibility-weight the actual A/E ratio for each subgroup withthe overall study’s 85.7 percent ratio.
Figure 2 shows the results of these calculations. The credibility factor for each subgroup is the square root of [subgroup claim count / total study claim count].
| Issue Year Subgroup | A/E Ratio by Amt | Credibility Factor | Overall Study A/E Ratio | Adjusted A/E Ratio |
| 2001-02 | 183.5% | 0.499 | 85.7% | 134.5% |
| 2003+ | 66.4% | 0.867 | 85.7% | 68.9% |
| Total | 85.7% | 1.000 | | 79.8% |
Figure 2 - Credibility factors adjust the A/Es based on sample size.
The credibility adjustment to the subgroup A/E ratios has decreased the overall study experience to 79.8 percent, which we want to renormalize to preserve the 85.7 percent actual experience.
In Figure 3, we ratio up the adjusted actual amounts by [85.7 / 79.8] and calculate final A/E ratios. Now the actuary can discard the 2001-02 experience and use the credibilityadjusted 2003+ experience of 74.1 percent in determining the current mortality assumption.
| Issue Yr
Subgroup | Actual Amount | Expected Amount | Adjusted Actual Amt | Normalized Actual Amt | Final A/E Ratios |
| 2001-02 | 49,734,937 | 27,105,783 | 36,461,189 | 39,177,129 | 144.5% |
| 2003+ | 90,735,072 | 136,750,640 | 94,270,788 | 101,292,880 | 74.1% |
| Total | 140,470,009 | 163,856,423 | 130,731,977 | 140,470,009 | 85.7% |
Figure 3 - Once the A/Es are renormalized, the actuary can disregard the 2001-02 results as outliers.
Conclusion
Credible experience studies usually provide a good basis for future mortality expectations. However, data anomalies and results of extremely unlikely events can sometimes creep into the picture. With a robust analysis system, these outliers can be identified by comparing them to surrounding data. Then, by applying the process detailed in this article, actuaries can remove these elements and provide a clearer view upon which to base their mortality assumptions.
