The Complement of Credibility

This Research Summary was first published in the IIB Bulletin, 2015, Vol. 1, Iss. 3, pp8-9 

https://iib.gov.in/IRDA/Articles/IIB%20Bulletin%20Q3%202014-15.pdf

Co-Author: Vishnuvardhan Pallreddy, IIB

Joseph A. Boor, FCAS, Ph.D. has been a working casualty actuary since 1979 and a Fellow of the Casualty Actuarial Society since 1988. Over a long and varied career he has had roles as diverse as regulator, Chief Actuary, consultant, and regional actuary. Currently, he works as an actuary for the Office of Insurance Regulation of the State of Florida in Tallahassee, Florida. He is the author of several published articles, including theoretical contributions to the theory of credibility and the optimum weightings of years of data and single-topic ratemaking papers on the complement of credibility in pricing and tail factors in loss reserving. He has a Bachelor’s of Arts degree in Mathematics from Southern Illinois University in Carbondale, a Master of Science degree in Mathematics from Florida State University, and a Doctor of Mathematics degree from Florida State University.

Complements have a special role in ratemaking exercises where the data is sparse or has high deviation from the mean over the years and hence has low credibility. “The Complement of Credibility” paper by Joseph A. Boor (1996) provides a good overview of the qualities and effectiveness of a good credibility complement and explains different credibility components commonly used. The paper also looks at the practical aspects of selecting a complement.

Overview

According to the paper, “The complement of the credibility deserves at least as much actuarial attention as the base statistic (historical loss data)”. Special attention should be given to its unbiasedness and accuracy. In some cases, interdependence must be avoided. Ease of computation and implementation must be reasonable. Explainability of statistics used must be considered, too.

Fundamental Principles

The paper explains a few issues that an Actuary must consider before selecting the complement of credibility:

Practical Issues: Complement should be readily available. The best possible statistic to use is the next year’s loss costs which are unknown. It has to be chosen from the available statistics. Ease of computation would also be a factor to consider as it involves time, costs and also increased chances of error.

Competitive Market Issues: The rates that are eventually produced will be subject to market competition. If rates are too high or too low, the outcome will not be desirable. So, the rate should be neither too high nor too low over a large number of loss cost estimates (unbiasedness) and the rate should have as low an error variance as possible (accuracy).

Regulatory Issues: Regulators typically require that the rates are not inadequate, not excessive, and not unfairly discriminatory. This implies that rates should be as unbiased as possible. It could also be implied that rates should be as accurate as possible, as highly inaccurate rates pose a greater risk of insolvency through random inadequacies.

Statistical Issues: For greater accuracy, error variance should be lower. If complement of credibility has low error variance in its own right and relatively independent of base statistic (which receives the credibility), the resulting rate will be more accurate.

When both the base statistic and complement are unbiased, the predictions are generally best when there is actually a negative correlation between the two errors (that is, they offset) but this rarely occurs in practice. So, a complement of credibility is best when it is statistically independent of the base statistic.

Based on the above four issue that must be considered by an Actuary when selecting a complement, Boor summarizes the desirable qualities that a complement of credibility should have:

1.       Accuracy as predictor of next year’s mean loss costs

2.       Unbiasedness as a predictor of next year’s mean subject expected losses

3.       Independence from the base statistic

4.       Availability of data

5.       Ease of computation

6.       Explainable relationship to the subject loss costs

Commonly used Components

Boor goes on to compare different types of complements used by Actuaries for First Dollar Ratemaking and Excess Ratemaking. Few often used methods for First Dollar ratemaking discussed in the paper are:

·         Using loss costs of a larger group including the class-Bayesian Credibility

·         Using loss costs of a larger related class

·         Harwayne’s method

·         Trending present rates

·         Applying the rate change from a larger group to present rates; and

·         Using competitors’ rates

For Excess Ratemaking, the four methods discussed in the paper are:

·         Increased limits factors

·         Derivation from a lower limits analysis

·         Analysis reflecting the policy limits sold by the insurer; and

·         Fitted curves

The use of the complement of credibility may differ from case to case, depending on the “availability of data” and reasonability of effort. For example, in the case of excess (or large) losses, fitted curves uses data available with the Insurers and is generally unbiased, but is complex to compute and may be difficult to communicate as well.

For pure premium ratemaking, using competitors’’ rates may be easy to use, especially for new companies or companies with low experience, but may suffer from inter-company difference in portfolio mix and may be harder to obtain.

“In Harwayne’s method, actuaries use countrywide (excepting the base state being reviewed) class data to supplement the loss cost data for each class, but they adjust countrywide data to remove overall lost cost differences between states (or provinces)”.

Many such practical insights on the above listed methods are provided in the paper along with the model itself and examples. As the Indian market moves towards an era of ratemaking, his paper is a valuable guide to the choice of a complement of credibility. The paper can be read at the following link: https://www.casact.org/pubs/proceed/proceed96/96001.pdf

Reference:

Boor, Joseph. A., “The Complement of Credibility" (Proceedings of the Casualty Actuarial Society,  Vol. LXXXIII, Part 1, No. 158, 1996, 32p; https://www.casact.org/pubs/proceed/proceed96/96001.pdf