Boor, Joseph. 2022. “Rebalancing the Off-Balance Factor with the Complement of Credibility.” Variance 15 (1).
• Table 1. Allocation of an Overall Change to a Few Subsets with Sample Loss Ratio Data
• Table 2. Ratemaking Scenario with Off-Balance Factor Using Sample Class Ratemaking Data
• Table 3. Complement of Credibility Off-Balance Allocations for Limited Fluctuation Credibility with the Few Subsets/Classes in Table 1
• Table 4. Complement of Credibility Off-Balance Allocations for Limited Fluctuation Credibility on Class Data from Table 2
• Table 5. Off-Balance Factor Computations for Best Estimate Credibility (Method 2)
• Table 6. Test Correction Factor Computations for Best Estimate Credibility (Method 2) Under Capping - Iteration 1
• Table 7. Test Correction Factor Computations for Best Estimate Credibility (Method 2) Under Capping - Iteration 2 (Final)

## Abstract

The ratemaking algorithms used to calculate class factors, territory factors, allocations of rate changes to coverages and other types of rating values often require either an off-balance factor (off-balance correction), or, when capping is involved, a test correction factor. The current approach is to multiply a common off-balance or test correction factor by all the post-credibility rates, so that the weighted average of the final rates matches the overall rate indication. The present correction algorithm appears to have disparate impacts on some classes, though. For example, when correction factors are applied uniformly to the post-credibility rates, rates for already fully credible classes may be artificially raised or lowered.

However, mathematical equity does not specifically require that the current correction algorithm be used. Given that flexibility, it is important to use the most effective correction process possible. This paper considers an alternative process: applying the correction factor to the complement of credibility instead. It presents two ratemaking scenarios. In the limited fluctuation credibility scenario applying the off-balance correction to the complement of credibility appears to result in rates that are more reasonable than those the present process creates. Rates of fully credible classes are not altered, and the off-balance is split among the lower credibility classes that generate it. Further, this process does not sacrifice or mitigate the reduction in volatility that credibility provides. This process also creates the most plausible rates possible given the circumstances. Further, the mathematics of optimization suggest that the complement of credibility should also be used to distribute the off-balance in best estimate ratemaking. If one begins with best estimate credibility rates for each class, the paper shows that the final off-balance adjusted class rates have the minimum expected squared error in predicting the loss costs. Thus, that simple change in the ratemaking formula appears to be helpful in a wide variety of situations.

Accepted: June 16, 2020 EDT

# Appendices

## A. The Best Estimate $$T_i$$’s Should be Proportional to the Complement of Credibility Too

The section will show that the off-balance basis $$(1-Z_i)M$$ from limited fluctuation credibility is als