1. Introduction
The formula for the workers compensation experience rating modification factor, or mod
is often presented asM=ZpAp+(1−Zp)Ep+ZeAe+(1−Ze)EeE.
The actuarial profession generally calls the terms
and in (1.1) primary credibility and excess credibility. Other notation, and background on experience rating, is given in Appendix A—Experience Rating Terminology and Formulas.The Actuarial Standards Board (ASB) (2020) defines credibility as
Credibility—A measure of the predictive value in a given application that the actuary attaches to a particular set of data (predictive is used here in the statistical sense and not in the sense of predicting the future).
The major goal of this article is to show that in the context of experience rating,
and are in better alignment with the ASB’s definition of credibility than are and Here is the insured’s average Dratio, i.e., the ratio of the insured’s expected primary losses to the insured’s total expected losses. None of the major papers on experience rating during the past 80 years or so explicitly state that and meet the standard definition for the credibility of and in common formulas for experience rating mods.Because the terms “primary credibility” for
and “excess credibility” for are so well established, we will continue to use those terms here. We use the terms effective primary credibility and effective excess credibility to refer to andIn NCCI’s Experience Rating Plan,
and are essentially functions of an insured’s expected number of claims. This is because and are functions of where is essentially the state average severity divided by 1,000, so is proportional to an insured’s expected number of claims (assuming all insureds in a state have the same expected average claim size).The mod formula (1.1) is, of course, equivalent to the formula
M=Ap+WAe+(1−W)Ee+BE+B
given in the NCCI experience rating manual. This can be verified easily using the relations
and as shown in Appendix A.Appendix A defines many of the terms commonly used in experience rating, such as split point, and terms specific to NCCI’s Experience Rating Plan, such as Experience Rating Adjustment (ERA) and Graduated Experience Rating Table (GERT). In addition, Appendix A defines notation used in this article and gives some wellknown formulas.
Experience rating in workers compensation is more than 100 years old, and the related literature is vast. The very first article in the first issue of the Proceedings of the Casualty Actuarial and Statistical Association (Rubinow 1914) mentions experience rating, and other articles in that volume discuss issues related to experience rating. The second volume of the Proceedings has two papers (Greene 1916; Woodward 1916) that give systematic discussions of features of experience rating. Since then, there have been many papers on experience rating. While it is not a goal of this article to summarize that literature, for those unfamiliar with the theory and history of experience rating, we suggest starting with Evans (2014, 2015) and Gillam (1992) and their reference or bibliography sections.
The next section presents an NCCI test of its Experience Rating Plan that is one of the main motivations for this article. Section 3 shows, both intuitively and mathematically, how the Dratio affects mods beyond establishing expected primary and excess losses. This leads to the concept of effective credibility. Subsequent sections use the concept of effective credibility to explore various aspects of current experience rating plans.
2. NCCI Mod Performance by State
In 2017, NCCI launched a comprehensive multiyear project to review the performance of its current plan. For this analysis, NCCI restated the split points for historical years to be in line with the current plan. For example, the restated split point used in the review for Rating Year 2014 is $15,000 for all NCCI states. Mod performance analyses discussed in this article are results of this project.
In the NCCI analysis, statelevel mod ranges—the differences between the 90th and 10th percentiles of actual mods by insured—show a strong correlation with state average severities. Figure 1 shows that the range shrinks as the state —essentially the state average severity divided by 1,000—gets larger. The mod range is for the lowestseverity state and for the highestseverity state. Other mod ranges were also examined, such as the difference between the 99th and the 1st percentiles and the 80th and 20th percentiles. All these ranges show similar patterns. Rating Years 2013 and 2015 have patterns similar to those observed for Rating Year 2014.
The strong correlation between the mod ranges and the state average severities is surprising, as we generally expect the mod range to be comparable across states. When NCCI reviewed mod performance by state, an opportunity for improvement became apparent.
NCCI performed quintile tests for each state and for certain groups of states; see Appendix B—Mod Performance Tests—for a description of these tests. In this analysis, the quintiles are defined by expected losses.
Quintile tests were done for the combined experience of the five highestseverity gross experience rating NCCI ERA states and for the combined experience of the five lowestseverity gross experience rating NCCI ERA states. Gross experience rating states are those that use groundup losses to calculate mods. As shown in Figure 2, for the lowestseverity states, the loss ratios, after application of the mod, decrease as the mod increases. Conversely, for the highestseverity states, the loss ratios, after application of the mod, increase as the mod increases. If mods were accurate, we would expect to see loss ratios after mod that are fairly flat across quintiles, with only random deviations from unity.
NCCI also performed the quintile test for each state. This analysis shows a progression in the loss ratios after application of the mod from a downward trend for lowseverity states, to a more random variation around unity for the mediumseverity states, to an upward trend for highseverity states. While there is some variation at the state level, most likely due to the low data volume in small states, this general pattern holds for most states.
Credibility comes to mind as a possible cause for these patterns. Perhaps in the lowseverity states, each insured’s experience was assigned too much credibility, and thus many insureds were assigned mods with credits or debits that were too big. Conversely, perhaps in the highseverity states, each insured’s experience was assigned too little credibility, and thus many insureds had mods with credits or debits that were too small.
We believe that this intuition is correct. However, as
and depend only on an insured’s expected number of claims, it would appear that severity has been removed from the mod calculation (1.1) or (1.2). How could it be that insureds in highseverity states have too little credibility assigned to their experience while insureds in lowseverity states have too much credibility assigned?3. Impact of the DRatio on the Mod
In this section, we show how the Dratio affects the mod for an insured. In particular, we show that the Dratio has an impact that goes beyond the determination of expected primary and excess losses. We begin with an intuitive explanation and then give a rigorous mathematical explanation. This discussion addresses the question raised at the end of the previous section.
In NCCI’s current plan, where one split point is applied to all states, average Dratios vary from state to state and are highly correlated with state average severities, as shown in Figure 3. Policies included in this figure are those subject to filings effective from October 1, 2018, to August 1, 2019, for intrastate insureds in NCCI ERA states. The split point for these policies was $17,000. The highestseverity ERA state had an average Dratio of 0.23, and the lowestseverity ERA state had an average Dratio of 0.47.
Consider two insureds in NCCI ERA states, one in a highseverity state and one in a lowseverity state, both with the same Figure 4 shows that is significantly less than for most insureds—the impact of the offsetting component is less than the impact of So overall, actual experience for an insured in a highseverity state has less impact on the mod than actual experience of an insured in a lowseverity state.
i.e., with similar expected claim counts. In (1.2), the values of and will be the same for these two insureds because these are both functions of Because both states have the same split point, a lower share of actual losses will be primary and a higher share will be excess for the insured in the highseverity state than for the insured in the lowseverity state. For the insured in the highseverity state, will have a smaller overall effect on the mod than for the insured in the lowseverity state. Conversely, will have a larger effect on the mod, but because is usually much less than 1.0—Now we give a mathematical approach. We write
so and view as an insured’s overall average Dratio across classes. We derive a formula for the mod as a weighted average of the insured’s actual primary loss ratio actual excess loss ratio and 1, using the respective weights and Starting from (1.1), we haveM=ZpApE+(1−Zp)EpE+ZeAeE+(1−Ze)EeE=ZpEpEApEp+(1−Zp)EpE+ZeEeEAeEe+(1−Ze)EeE=ZpDApEp+(1−Zp)D+Ze(1−D)AeEe+(1−Ze)(1−D).
Rearranging terms in the equation immediately above gives
M=DZpApEp+(1−D)ZeAeEe+D(1−Zp)+(1−D)(1−Ze).
Now
DZp+(1−D)Ze+D(1−Zp)+(1−D)(1−Ze)=DZp+D(1−Zp)+(1−D)Ze+(1−D)(1−Ze)=D(Zp+1−Zp)+(1−D)(Ze+1−Ze)=D+(1−D)=1,
so we can write (3.1) as
M=DZp⋅ApEp+(1−D)Ze⋅AeEe+(1−(DZp+(1−D)Ze))⋅1.
Equation (3.2) is a weighted average
of and using the respective weights and Clearly and for each weight.This shows that the weight given to the insured’s actual primary loss ratio in the mod is
and the weight given to the actual excess loss ratio is This is why we call effective primary credibility and effective excess credibility.Alternatively, (3.1) can be written as
M=D(ZpApEp+1−Zp)+(1−D)(ZeAeEe+1−Ze).
This can be interpreted as a weighted average of the credibilityweighted primary loss ratio (which we will call the primary mod) and the credibilityweighted excess loss ratio (excess mod) using weights of
and As the primary loss ratio gets a weight of in the primary mod and the primary mod gets a weight of in the overall mod, the primary loss ratio gets a weight of in the overall mod. Similarly, the excess loss ratio gets a weight of in the excess mod and in the overall mod.We now show how the concept of effective credibility answers the question asked earlier—how could it be that insureds in highseverity states have too little credibility assigned to their experience while insureds in lowseverity states have too much credibility assigned?
With a single countrywide split point, lowseverity states have much higher average Dratios than do highseverity states and thus much higher effective primary credibilities. Taking the experience shown in Figure 3, when one state’s average Dratio is 0.47 and another’s is 0.23, for insureds with similar and all else equal, an insured in the first state receives about twice the effective primary credibility as does an insured in the second state. Of course, higher effective primary credibility implies lower effective excess credibility because, all else equal, will be lower in lowseverity states. But because is less than for any insured, the magnitude of the difference in primary effective credibilities between the insureds in the two states is greater than the magnitude of the difference in excess effective credibilities. As such, the overall effective credibility for an insured with a given in a lowseverity state will be greater than the overall effective credibility for an insured with the same in a highseverity state.
NCCI increased the split point in 2013 because it saw a deterioration of mod performance over the years (Evans 2015). As losses increased with inflation each year, the pre2013 $5,000 fixed dollar split point resulted in lower average Dratios over time and thus lower effective credibilities. The increase of the split point corrected for this problem, and the indexation of the split point to a measure of claim severity going forward removed its sensitivity to inflation. The increase in the split point increased overall effective credibilities (the increase in effective primary credibility more than offset the decrease in effective excess credibility), and the indexation of the split point should prevent deterioration of effective credibilities.
While the increase was warranted and greatly improved plan performance on an allstatescombined basis, the optimized countrywide split point was not optimal for each individual state—it was too high for lowseverity states and too low for highseverity states. The effective credibility lens shows how the countrywide split point combined with similar
s and s for insureds in different states with similar expected claim counts results in greater weight being given to actual loss experience in lowseverity states than in highseverity states. Therefore, it is not surprising that the insureds in lowseverity states have too much credibility assigned to their experience while insureds in highseverity states have too little.Allowing the split point to vary by state could help bring effective credibilities more in line across states. The split point for each state could be selected so that there were comparable average Dratios across states. Having comparable average Dratios across states (and comparable
s and s for comparable insureds) would result in comparable effective credibilities across states.Note that we are not concluding that the effective credibility must be constant across states. There might be factors other than expected claim count that affect credibility. To the extent that loss distributions and other key factors vary by state, the optimal effective credibility may indeed vary by state. However, the magnitude of the variation in optimal effective credibilities across states is probably much smaller than the variation in effective credibilities under the current NCCI plan. The current variation in effective credibilities is largely driven by the use of a single split point for all states, which causes substantial variation in average Dratios across states.
4. Comparison of Fixed and Variable Split Point Experience Rating Plans
NCCI states and many independent bureau states currently (in 2020) use fixed split point experience rating plans. Under these plans, a single split point is used for all insureds in a state. That split point might vary over time.
Under one type of variable split plan, the split point varies by size of the insured—smaller insureds get lower split points, and larger insureds get higher split points. The mod formula is
M=Ap+EeE.
It is straightforward to show that this mod formula is the same as (1.1) using
and i.e., with 100% credibility assigned to primary losses and 0% credibility assigned to excess losses. Equivalently, (4.1) is (1.2) with California implemented this type of experience rating plan on January 1, 2017.The variable split plan mod can be written in terms of the average Dratio as follows. Starting with (4.1) we have
M=ApE+EeE=EpEApEp+EeE=DApEp+(1−D).
This looks exactly like the standard credibility weighted average formula
R=ZL+1−Z
with (Mahler and Dean 2001; LongleyCook 1962).
the actual primary loss ratio as and the complement of credibility applied toBecause the split point is lower for smaller insureds,
is smaller, so smaller insureds have less credibility applied to their primary loss ratio than do larger insureds.5. NCCI ERA and GERT Plans
Most NCCI states currently use the ERA plan, while some use the GERT plan. Under the ERA plan, only 30% of the medicalonly losses are included in the mod calculation, while 100% are included in the GERT plan. The main motivation for not using 100% of the medicalonly losses in ERA is to encourage insureds to report all of their claims, especially the small medicalonly claims.
In this section, we look at how effective primary and excess credibilities compare between NCCI ERA and GERT plans.
First we look at
and The calculation of and and so and uses eight credibility parameters, to (see Appendix A). For a given
is the same under GERT and ERA because to are the same under GERT and ERA.

is smaller under GERT than under ERA because and are larger in GERT than in ERA, while and are the same under GERT and ERA.
Now we turn to the Dratio. For a given state and split point, average Dratios are higher under GERT than under ERA because of the inclusion of the 100% medicalonly losses in GERT—most medicalonly losses are below the split point, and removing 70% of medicalonly losses in ERA removes a higher share of losses below the split point than above the split point. Depending on the state, the difference in average Dratios under ERA or GERT can range from 0.02 to 0.07 (based on data underlying filings with effective dates in 2014).
Overall, the impacts on effective credibility for GERT versus ERA are the following:

Effective primary credibility,
is larger under GERT than under ERA because is the same, and the average Dratio, is larger. 
Effective excess credibility,
is smaller under GERT than under ERA because is smaller, and is also smaller.
This means that a given insured would get different primary and excess effective credibilities if it were rated using GERT instead of ERA. Furthermore, because
is generally significantly bigger than for most insureds, this means an insured would receive a much higher effective credibility under GERT.To achieve the same effective primary credibility
and effective excess credibility regardless of whether a state uses ERA or GERT, the credibility parameters could be recalibrated so that
the
under GERT will be lower than under ERA to offset the higher and 
the
under GERT will be higher than under ERA to offset the lower
Alternatively, the split point could be adjusted so that the average Dratios are comparable regardless of whether the state uses ERA or GERT. This might be easier than adjusting the eight
parameters. In addition, adjusting the split point may provide a significant benefit when it comes to interstate experience rating.6. Interstate Rating
There are two types of experience modifications developed by NCCI: intrastate and interstate. An intrastate modification factor is issued when the insured has exposure in only one state that participates in the NCCI plan. An interstate modification is issued when the insured has exposures in two or more states that participate in the NCCI plan (NCCI 2019).
The current interstate rating mod uses weighted averages of statebased
s and s in mod formula (1.2) to determine a mod that applies across all states where an insured has exposure. The statebased s and s are lookup values from statespecific tables based on an insured’s total expected losses across all states, without adjustment for differences in average loss severity between states. This interstate rating mod formula can be used with states that use variable split points, as discussed previously, by using for these states. Appendix C—The Current NCCI Interstate Rating Mod—shows that this approach of weighting s and s is not consistent with the underlying principle that the credibilities and are based on an insured’s expected number of claims.Our view is that an interstate rating mod that recognizes effective credibility of losses by state would give a plan that is more consistent with commonly recognized credibility principles. In this section, we will demonstrate how an insured can be rated on an interstate basis under the following two scenarios:

Scenario 1: The effective credibilities are the same across states.

Scenario 2: The effective credibilities differ across states.
6.1. Scenario 1
One approach would be to have states adopt intrastate experience rating plans consistent among themselves so that, for example, two insureds that have similar operations and similar expected claim counts but that operate in different states would have similar Dratios,
s, and s. These two insureds would then have similar effective credibilities in both states. An interstate mod could be computed for an insured that operates in both of these states as though all of its operations were in either of the states. This example can be extended to any number of states.Having intrastate experience rating plans that are consistent between states might involve adjusting the
to align and between, for example, ERA and GERT states. Also, split points might have to vary between states so that average Dratios are similar from state to state.With the same or similar effective credibilities across all states,
and could be the aggregation of those components across all states, and the mod can be computed using (3.2).6.2. Scenario 2
Now we turn to an alternative approach to interstate rating that allows for differences in effective credibilities between states. In particular, this approach allows for interstate rating across current ERA, GERT, and variable split point plans.
The idea is to determine how much effective credibility each state would assign to a given insured’s experience and use a weighted average of those effective credibilities in the determination of an insured’s mod. In this proposal, we use the expected claim count for each state as the weight for that state’s effective credibility.
Consider an insured with exposures in several states. We index state values by
so for exposures and losses restricted to State is expected losses, is expected primary losses, is actual primary losses, is expected excess losses, and is actual excess losses. We will also use state Gvalues, Define by so is (1,000 times) the expected claim count for the insured for exposures in State Let be the sum of the across all states for the insured.Now we determine the effective primary and excess credibilities and the actual primary and excess loss ratios for the insured, based on weighted averages of these values across states.
For primary credibilities by state, we take

for NCCI ERA or GERT states and

for states with a variable split point plan.
For an ERA or GERT state, we take
Here, is based on state exposure and not countrywide exposure, unlike and which are based on countrywide exposure. For states with split points that apply to all insureds regardless of the size of the insured, such as ERA or GERT states, an insured’s average Dratio is not a function of the size of the insured.For a variable split point state, we first determine a split point for the state based on the countrywide expected losses. We then determine
for an insured using this split point, the associated Dratios, and the exposures in the state. Finally, For a variable split point state, is a function of the countrywide size of the insured because the split point is determined by the countrywide size of the insured. On the other hand, which is for all insureds, is not a function of the size of the insured in a variable split point state.For the weight to be applied to the primary loss ratio in the mod, we take
wp=∑iniDiZp,in.
This is the weighted average of the state effective primary credibilities, using expected claim counts by state as weights.
Similarly, for excess credibilities by state we take

for NCCI ERA or GERT states and

for states with a variable split point plan.
Note that for NCCI ERA states, the
will all be the same, and for NCCI GERT states, the will all be the same, but the for NCCI ERA states will be different from the for NCCI GERT states.For the weight to be applied to the excess loss ratio, we take
we=∑ini(1−Di)Ze,in.
Similar to the effective primary credibilities, this is the weighted average of the state effective excess credibilities, using expected claim counts by state as weights.
For the actual primary loss ratio we take
Rp=∑iEp,iAp,iEp,i∑iEp,i=∑iAp,i∑iEp,i,
which is the weighted average of the state primary loss ratios using expected primary losses by state as weights. This is, of course, the sum of the state actual primary losses divided by the sum of the state expected primary losses.
Similarly, for the actual excess loss ratio we take
Re=∑iAe,i∑iEe,i,
where
The mod is then
M=wpRp+weRe+1−wp−we=1+wp(Rp−1)+we(Re−1).
The mods that result from this proposal of weighting effective credibilities generally differ from the mods resulting from NCCI’s current procedure of weighting the
s and s (see Appendix C).7. ClaimFree Credits
When an insured has had no claims in the experience period used to compute its mod, its mod is
MCF=D(1−Zp)+(1−D)(1−Ze),
as is apparent from (3.1), (3.2), or (3.3). We call
the claimfree mod and the claimfree credit.In private correspondence, Saiying He points out that (7.1) can be written as
MCF=1−(DZp+(1−D)Ze).
Taking (4.2) with
so and comparing that to (7.2) shows that in NCCI’s Experience Rating Plan, the sum of the effective primary credibility and the effective excess credibility corresponds to the credibility in standard credibility formulas.Under the variable split point plan,
and so and the insured’s average Dratio is its claimfree credit. Because the split point varies by insured size, smaller insureds get smaller claimfree credits. This is reasonable as lossfree experience in one period is less indicative of a low loss ratio in future periods for a smaller insured than it is for a larger insured. So the Dratio not only plays a role in effective credibilities, but also serves as the claimfree credit.Under the NCCI plan, the Dratio does not vary by size of insured, but
and do vary by size of insured. Generally, as becomes smaller, the claimfree credit becomes smaller. As goes to zero, goes to 1, and the claimfree credit goes to zero. But for a given a higher Dratio gives a bigger claimfree credit. This is reasonable as a higher Dratio goes with a smaller severity, so a larger expected claim count for a given and so greater credibility for the insured’s experience.8. Conclusion
This article introduced the concepts of effective primary credibility and effective excess credibility. We discussed how effective credibilities align more closely with the ASB definition of credibility than do
and and specifically
how Dratios affect the weight assigned to an insured’s experience in experience rating,

why mod performance in NCCI’s Experience Rating Plan varies depending on state average severity,

the connection between NCCI’s Experience Rating Plan and a variable split point plan,

an approach to interstate rating based on effective primary and excess credibilities, and

the relation between claimfree credits and split points.
Of course, average claim size not only varies by state, but also varies between classes within a state. NCCI effectively recognizes the most significant severity differences between classes by grouping classes into hazard groups (Robertson 2009). While we studied mod performance by hazard group and by additional dimensions other than state, we have not included those results in this article for the sake of simplicity.
Acknowledgments
The authors thank Barry Lipton, Tom Daley, Donna Glenn, and Brett Foster of NCCI for their leadership and support. Saiying He, a member of the actuarial experience rating team at NCCI, performed a thorough review of all facts and results included in this article. We are grateful for Saiying’s significant contribution to this article. Leigh Halliwell and YenChieh Tseng of NCCI provided helpful reviews and comments that improved an early draft. We thank the Workers’ Compensation Insurance Rating Bureau of California for its supportive comments. We thank the anonymous reviewers for Variance, whose questions and suggestions led to improvements to our original submission. Last but not least, we thank our family members for their support as we wrote this article during the year when COVID19 began.