D-Ratios and Credibility in Experience Rating

1. Introduction

The formula for the workers compensation experience rating modification factor, or mod $M$, is often presented as

$$\small M = \cfrac{Z_p A_p + (1 - Z_p) E_p + Z_e A_e + (1 - Z_e) E_e}{E}. \tag{1.1}$$

The actuarial profession generally calls the terms $Z_p$ and $Z_e$ 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, $D Z_p$ and $(1 - D) Z_e$ are in better alignment with the ASB’s definition of credibility than are $Z_p$ and $Z_e$. Here $D$ is the insured’s average D-ratio, i.e., the ratio $E_p/E$ 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 $D Z_p$ and $(1 - D) Z_e$ meet the standard definition for the credibility of $A_p/E_p$ and $A_e/E_e$ in common formulas for experience rating mods.

Because the terms “primary credibility” for $Z_p$ and “excess credibility” for $Z_e$ 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 $D Z_p$ and $(1 - D) Z_e$.

In NCCI’s Experience Rating Plan, $Z_p$ and $Z_e$ are essentially functions of an insured’s expected number of claims. This is because $Z_p$ and $Z_e$ are functions of $E/G$, where $G$ is essentially the state average severity divided by 1,000, so $E/G$ 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

$$M = \cfrac{A_p + W A_e + (1 - W) E_e + B}{E + B} \tag{1.2}$$

given in the NCCI experience rating manual. This can be verified easily using the relations $B = E (1 - Z_p)/Z_p$ and $W = Z_e/Z_p$, 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 well-known 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 D-ratio 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, state-level 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 $G$—essentially the state average severity divided by 1,000—gets larger. The mod range is $0.61$ for the lowest-severity state and $0.32$ for the highest-severity 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.

Figure 1.10th and 90th percentiles of mods by state

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 highest-severity gross experience rating NCCI ERA states and for the combined experience of the five lowest-severity gross experience rating NCCI ERA states. Gross experience rating states are those that use ground-up losses to calculate mods. As shown in Figure 2, for the lowest-severity states, the loss ratios, after application of the mod, decrease as the mod increases. Conversely, for the highest-severity 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.

Figure 2.Quintile tests for high and low G-value NCCI gross experience rating ERA states

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 low-severity states, to a more random variation around unity for the medium-severity states, to an upward trend for high-severity 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 low-severity 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 high-severity 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 $Z_p$ and $Z_e$ 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 high-severity states have too little credibility assigned to their experience while insureds in low-severity states have too much credibility assigned?

3. Impact of the D-Ratio on the Mod

In this section, we show how the D-ratio affects the mod for an insured. In particular, we show that the D-ratio 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 D-ratios 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 highest-severity ERA state had an average D-ratio of 0.23, and the lowest-severity ERA state had an average D-ratio of 0.47.

Figure 3.D-ratios by state

Consider two insureds in NCCI ERA states, one in a high-severity state and one in a low-severity state, both with the same $E/G$, i.e., with similar expected claim counts. In (1.2), the values of $W$ and $B/E$ will be the same for these two insureds because these are both functions of $E/G$. 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 high-severity state than for the insured in the low-severity state. For the insured in the high-severity state, $A_p$ will have a smaller overall effect on the mod than for the insured in the low-severity state. Conversely, $A_e$ will have a larger effect on the mod, but because $W$ ($= Z_e/Z_p$) is usually much less than 1.0—Figure 4 shows that $Z_e$ is significantly less than $Z_p$ for most insureds—the impact of the offsetting component $W \, A_e$ is less than the impact of $A_p$. So overall, actual experience for an insured in a high-severity state has less impact on the mod than actual experience of an insured in a low-severity state.

Figure 4.Primary and excess credibility by expected claim count

Now we give a mathematical approach. We write $D = E_p/E$ so $1 - D = E_e/E$ and view $D$ as an insured’s overall average D-ratio across classes. We derive a formula for the mod as a weighted average of the insured’s actual primary loss ratio $A_p/E_p$, actual excess loss ratio $A_e/E_e$, and 1, using the respective weights $D Z_p$, $(1 - D) Z_e$, and $1 - (D Z_p + (1 - D) Z_e)$. Starting from (1.1), we have

$$\small \begin{aligned} M &= Z_p \cfrac{A_p}{E} + (1 - Z_p) \cfrac{E_p}{E} + Z_e \cfrac{A_e}{E} + (1 - Z_e) \cfrac{E_e}{E} \nonumber \\ &= Z_p \cfrac{E_p}{E} \, \cfrac{A_p}{E_p} + (1 - Z_p) \cfrac{E_p}{E} + Z_e \cfrac{E_e}{E} \, \cfrac{A_e}{E_e} + (1 - Z_e) \cfrac{E_e}{E} \nonumber \\ &= Z_p D \cfrac{A_p}{E_p} + (1 - Z_p) D + Z_e (1 - D) \cfrac{A_e}{E_e} + (1 - Z_e) (1 - D). \nonumber \end{aligned}$$

Rearranging terms in the equation immediately above gives

$$\begin{align} M &= D \, Z_p \, \frac{A_p}{E_p} + (1 - D) \, Z_e \, \frac{A_e}{E_e} \\ &\quad + D \, (1 - Z_p) + (1 - D) \, (1 - Z_e). \end{align} \tag{3.1}$$

Now

$$\begin{aligned} D \, Z_p &+ (1 - D) Z_e + D \, (1 - Z_p) + (1 - D) \, (1 - Z_e) \\ &= D \, Z_p + D \, (1 - Z_p) + (1 - D) Z_e + (1 - D) \, (1 - Z_e) \\ &= D(Z_p + 1 - Z_p) + (1 - D) (Z_e + 1 - Z_e) \\ &= D + (1 - D) = 1,\end{aligned}$$

so we can write (3.1) as

$$\begin{align} M &= D \, Z_p \cdot \frac{A_p}{E_p} + (1 - D) \, Z_e \cdot \frac{A_e}{E_e} \\ &\quad + (1 - (D \, Z_p + (1 - D) Z_e)) \cdot 1. \end{align} \tag{3.2}$$

Equation (3.2) is a weighted average $w_1 Q_1 + w_2 Q_2 + w_3 Q_3$ of $Q_1 = A_p/E_p$, $Q_2 = A_e/E_e$, and $Q_3 = 1$ using the respective weights $w_1 = D \, Z_p$, $w_2 = (1 - D) \, Z_e$, and $w_3 = 1 - (w_1 + w_2)$. Clearly $w_1 + w_2 + w_3 = 1$ and $0 \le w_i \le 1$ for each weight.

This shows that the weight given to the insured’s actual primary loss ratio in the mod is $D Z_p$ and the weight given to the actual excess loss ratio is $(1 - D) Z_e$. This is why we call $D Z_p$ effective primary credibility and $(1 - D) Z_e$ effective excess credibility.

Alternatively, (3.1) can be written as

$$\begin{align} M &= D \left( Z_p \cfrac{A_p}{E_p} + 1 - Z_p \right) \\ &\quad + (1 - D) \left( Z_e \cfrac{A_e}{E_e} + 1 - Z_e \right)\!. \end{align} \tag{3.3}$$

This can be interpreted as a weighted average of the credibility-weighted primary loss ratio (which we will call the primary mod) and the credibility-weighted excess loss ratio (excess mod) using weights of $D$ and $1 - D$. As the primary loss ratio $A_p/E_p$ gets a weight of $Z_p$ in the primary mod and the primary mod gets a weight of $D$ in the overall mod, the primary loss ratio gets a weight of $D \, Z_p$ in the overall mod. Similarly, the excess loss ratio $A_e/E_e$ gets a weight of $Z_e$ in the excess mod and $(1 - D) Z_e$ 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 high-severity states have too little credibility assigned to their experience while insureds in low-severity states have too much credibility assigned?

With a single countrywide split point, low-severity states have much higher average D-ratios than do high-severity states and thus much higher effective primary credibilities. Taking the experience shown in Figure 3, when one state’s average D-ratio is 0.47 and another’s is 0.23, for insureds with similar $E/G$ 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, $1 - D$ will be lower in low-severity states. But because $Z_e$ is less than $Z_p$ 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 $E/G$ in a low-severity state will be greater than the overall effective credibility for an insured with the same $E/G$ in a high-severity 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 pre-2013 $5,000 fixed dollar split point resulted in lower average D-ratios 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 all-states-combined basis, the optimized countrywide split point was not optimal for each individual state—it was too high for low-severity states and too low for high-severity states. The effective credibility lens shows how the countrywide split point combined with similar $Z_p$s and $Z_e$s for insureds in different states with similar expected claim counts results in greater weight being given to actual loss experience in low-severity states than in high-severity states. Therefore, it is not surprising that the insureds in low-severity states have too much credibility assigned to their experience while insureds in high-severity 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 D-ratios across states. Having comparable average D-ratios across states (and comparable $Z_p$s and $Z_e$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 D-ratios 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

$$M = \cfrac{A_p + E_e}{E}. \tag{4.1}$$

It is straightforward to show that this mod formula is the same as (1.1) using $Z_p = 1$ and $Z_e = 0$, i.e., with 100% credibility assigned to primary losses and 0% credibility assigned to excess losses. Equivalently, (4.1) is (1.2) with $B = W = 0$. 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 D-ratio as follows. Starting with (4.1) we have

$$\begin{aligned} M &= \cfrac{A_p}{E} + \cfrac{E_e}{E} \\ &= \cfrac{E_p}{E} \, \cfrac{A_p}{E_p} + \cfrac{E_e}{E} \\ &= D \, \cfrac{A_p}{E_p} + (1 - D).\end{aligned}$$

This looks exactly like the standard credibility weighted average formula

$$R = Z \, L + 1 - Z \tag{4.2}$$

with $Z = D$, the actual primary loss ratio $A_p/E_p$ as $L$, and the complement of credibility applied to $1$ (Mahler and Dean 2001; Longley-Cook 1962).

Because the split point is lower for smaller insureds, $D$ 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 medical-only losses are included in the mod calculation, while 100% are included in the GERT plan. The main motivation for not using 100% of the medical-only losses in ERA is to encourage insureds to report all of their claims, especially the small medical-only claims.

In this section, we look at how effective primary and excess credibilities compare between NCCI ERA and GERT plans.

First we look at $Z_p$ and $Z_e$. The calculation of $Z_p$ and $Z_e$, and so $W$ and $B$, uses eight credibility parameters, $k_1$ to $k_8$ (see Appendix A). For a given $E/G$,

• $Z_p$ is the same under GERT and ERA because $k_1$ to $k_4$ are the same under GERT and ERA.

• $Z_e$ is smaller under GERT than under ERA because $k_5$ and $k_6$ are larger in GERT than in ERA, while $k_7$ and $k_8$ are the same under GERT and ERA.

Now we turn to the D-ratio. For a given state and split point, average D-ratios are higher under GERT than under ERA because of the inclusion of the 100% medical-only losses in GERT—most medical-only losses are below the split point, and removing 70% of medical-only 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 D-ratios 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, $D \, Z_p$, is larger under GERT than under ERA because $Z_p$ is the same, and the average D-ratio, $D$, is larger.

• Effective excess credibility, $(1 - D) \, Z_e$, is smaller under GERT than under ERA because $Z_e$ is smaller, and $1 - D$ 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 $Z_p$ is generally significantly bigger than $Z_e$ for most insureds, this means an insured would receive a much higher effective credibility under GERT.

To achieve the same effective primary credibility $D \, Z_p$ and effective excess credibility $(1 - D) Z_e$ regardless of whether a state uses ERA or GERT, the credibility parameters $k_i$ could be recalibrated so that

• the $Z_p$ under GERT will be lower than under ERA to offset the higher $D$ and

• the $Z_e$ under GERT will be higher than under ERA to offset the lower $1 - D$.

Alternatively, the split point could be adjusted so that the average D-ratios are comparable regardless of whether the state uses ERA or GERT. This might be easier than adjusting the eight $k_i$ 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 state-based $B$s and $W$s in mod formula (1.2) to determine a mod that applies across all states where an insured has exposure. The state-based $B$s and $W$s are lookup values from state-specific 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 $B = W = 0$ for these states. Appendix C—The Current NCCI Interstate Rating Mod—shows that this approach of weighting $B$s and $W$s is not consistent with the underlying principle that the credibilities $Z_p$ and $Z_e$ 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 D-ratios, $Z_p$s, and $Z_e$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 $k_i$ to align $Z_p$ and $Z_e$ between, for example, ERA and GERT states. Also, split points might have to vary between states so that average D-ratios are similar from state to state.

With the same or similar effective credibilities across all states, $A_p$, $E_p$, $A_e$, and $E_e$ 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 $i$, so for exposures and losses restricted to State $i$, $E_i$ is expected losses, $E_{p, i}$ is expected primary losses, $A_{p, i}$ is actual primary losses, $E_{e, i}$ is expected excess losses, and $A_{e, i}$ is actual excess losses. We will also use state G-values, $G_i$. Define $n_i$ by $n_i = E_i/G_i$ so $n_i$ is (1,000 times) the expected claim count for the insured for exposures in State $i$. Let $n$ be the sum of the $n_i$ 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

• $Z_{p, i} = \cfrac{k_3 n + k_4}{(k_1 + k_3) n + k_2 + k_4}$ for NCCI ERA or GERT states $i$, and

• $Z_{p, i} = 1$ for states $i$ with a variable split point plan.

For an ERA or GERT state, we take $D_i = E_{p, i}/E_i$. Here, $D_i$ is based on state exposure and not countrywide exposure, unlike $Z_{p, i}$ and $Z_{e, i}$, 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 D-ratio 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 $E_{p, i}$ for an insured using this split point, the associated D-ratios, and the exposures in the state. Finally, $D_i = E_{p, i}/E_i$. For a variable split point state, $D_i$ 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, $Z_{p, i}$, which is $1$ 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

$$w_p = \cfrac{\sum_i n_i D_i Z_{p, i}}{n}.$$

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

• $Z_{e, i} = \cfrac{k_7 n + k_8}{(k_5 + k_7) n + k_6 + k_8}$ for NCCI ERA or GERT states $i$, and

• $Z_{e, i} = 0$ for states $i$ with a variable split point plan.

Note that for NCCI ERA states, the $Z_{e, i}$ will all be the same, and for NCCI GERT states, the $Z_{e, i}$ will all be the same, but the $Z_{e, i}$ for NCCI ERA states will be different from the $Z_{e, i}$ for NCCI GERT states.

For the weight to be applied to the excess loss ratio, we take

$$w_e = \cfrac{\sum_i n_i (1 - D_i) Z_{e, i}}{n}.$$

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

$$R_p = \cfrac{\sum_i E_{p, i} \, \cfrac{A_{p, i}}{E_{p, i}}}{\sum_i E_{p, i}} = \cfrac{\sum_i A_{p, i}}{\sum_i E_{p, i}},$$

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

$$R_e = \cfrac{\sum_i A_{e, i}}{\sum_i E_{e, i}},$$

where $E_{e, i} = E_i - E_{p, i}$.

The mod is then

$$\begin{gathered} M = w_p R_p + w_e R_e + 1 - w_p - w_e \\ = 1 + w_p (R_p - 1) + w_e (R_e - 1). \end{gathered}$$

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 $B$s and $W$s (see Appendix C).

7. Claim-Free Credits

When an insured has had no claims in the experience period used to compute its mod, its mod is

$$M_{CF} = D (1 - Z_p) + (1 - D) (1 - Z_e), \tag{7.1}$$

as is apparent from (3.1), (3.2), or (3.3). We call $M_{CF}$ the claim-free mod and $1 - M_{CF}$ the claim-free credit.

In private correspondence, Saiying He points out that (7.1) can be written as

$$M_{CF} = 1 - (D Z_p + (1 - D) Z_e). \tag{7.2}$$

Taking (4.2) with $L = 0$, so $R = 1 - Z$, 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 $D Z_p + (1 - D) Z_e$ corresponds to the credibility $Z$ in standard credibility formulas.

Under the variable split point plan, $Z_p = 1$ and $Z_e = 0$ so $M_{CF} = 1 - D$, and the insured’s average D-ratio is its claim-free credit. Because the split point varies by insured size, smaller insureds get smaller claim-free credits. This is reasonable as loss-free 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 D-ratio not only plays a role in effective credibilities, but also serves as the claim-free credit.

Under the NCCI plan, the D-ratio does not vary by size of insured, but $Z_p$ and $Z_e$ do vary by size of insured. Generally, as $E$ becomes smaller, the claim-free credit becomes smaller. As $E$ goes to zero, $M_{CF}$ goes to 1, and the claim-free credit goes to zero. But for a given $E > 0$, a higher D-ratio gives a bigger claim-free credit. This is reasonable as a higher D-ratio goes with a smaller severity, so a larger expected claim count for a given $E$, 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 $Z_p$ and $Z_e$ and specifically

• how D-ratios 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 claim-free 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 Yen-Chieh 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 COVID-19 began.

Appendices

Appendix A—Experience Rating Terminology and Formulas

We give a high-level summary of NCCI’s current (2020) Experience Rating Plan. For full rules, consult NCCI’s “Experience Rating Plan Manual for Workers Compensation and Employers Liability Insurance—2003 Edition.”

Experience rating modification factors, mods, are determined at the insured level and are used in determining the insured’s workers compensation premium. NCCI’s Experience Rating Plan is defined at the state level, with rules that allow for experience rating modification factors for insureds that have operations in more than one state. There are essentially two state variations in the NCCI plan—ERA, which is used in most NCCI states, and GERT, which is used in two NCCI states.

NCCI’s Experience Rating Plan at the state level, whether ERA or GERT, is based in part on the following:

• $SACC$ is the state average cost per case, based on unlimited claim amounts, except that employer’s liability claims are capped at basic limits. This average is based on three years of undeveloped Statistical Plan reported claims.

• $SRP$ is the state reference point and is 250 times $SACC$, rounded to the nearest $5,000.

• $SAL$ is the state accident limit and is 10% of $SRP$.

• $G$ is $SRP$ divided by $250,000, rounded to the nearest $0.05$. Apart from rounding rules, $G$ is the state average cost per case divided by $1,000.

An insured’s mod is generally based on its reported losses for the three-year period ending one year prior to the insured’s rating effective date, which is the earliest effective date of the insured’s policies that use the mod.

A rating year is a one-year period of rating effective dates. For example, policies included in Rating Year 2014 are all policies for insureds with rating effective dates during the period January 1, 2014, to December 31, 2014.

A filing season is a period of effective dates for NCCI workers compensation rate filings. For example, the 2019–2020 filing season refers to filings effective from October 1, 2019, to September 30, 2020.

Each state that uses the NCCI plan has a split point used to determine primary and excess losses. The split point is a dollar amount, and for the 2018–2019 filing season is $17,000 in all states. Changes to state split points are usually effective on dates that rate filings are effective.

Ratable losses are losses that are capped at the $SAL$ and exclude certain catastrophe and occupational disease claims. Primary losses are ratable losses capped at the split point. Excess losses are ratable losses in the layer between the split point and the $SAL$. In ERA states, primary and excess medical-only losses are reduced by 70%. In GERT states, medical-only losses are not subject to this adjustment. There are additional rules for capping losses from multiclaim occurrences. For Rating Year 2014, across all states where NCCI provides ratemaking services, about 89% of losses and 97% of claims were within the SAL.

Some additional notation that will be useful is as follows:

$$\begin{aligned} A_p &= \ \text{actual primary losses} \\ A_e &= \ \text{actual excess losses} \\ E &= \ \text{expected total losses limited to the SAL} \\ E_p &= \ \text{expected primary losses} \\ E_e &= \ \text{expected excess losses} \\ Z_p &= \ \text{credibility of primary losses} \\ Z_e &= \ \text{credibility of excess losses} \\ B &= \ \text{ballast} \\ W &= \ \text{weight} \\\end{aligned}$$

Eight constants, $k_1$, …, $k_8$, are used to determine $B$ and $W$. The constants for the current plan are given in Table 1.

The ballast $B$ is a dollar-denominated value given by

$$B = E \, \cfrac{k_1 \cfrac{E}{G} + k_2}{k_3 \cfrac{E}{G} + k_4},$$

subject to some rounding rules and a minimum value of 2,500 times $G$.

The NCCI plan defines $C$ as

$$C = E \, \cfrac{k_5 \cfrac{E}{G} + k_6}{k_7 \cfrac{E}{G} + k_8},$$

subject to a minimum of 60,000 times $G$.

Note that $B/E$ and $C/E$ are functions of $E/G$.

Then $W$ is given as $W = (E + B)/(E + C)$, subject to some rounding rules. As $W = (1 + B/E)/(1 + C/E)$, $W$ is a function of $E/G$.

The mod, $M$, for an insured is given by

$$M = \cfrac{A_p + W A_e + (1 - W) E_e + B}{E + B},$$

which is (1.2).

The credibilities $Z_p$ and $Z_e$ are defined as $Z_p = E/(E + B)$ and $Z_e = E/(E + C)$. These credibilities can be written in terms of the $k_i$ as

$$Z_p = \cfrac{k_3 \, \cfrac{E}{G} + k_4}{(k_1 + k_3) \, \cfrac{E}{G} + k_2 + k_4} \tag{9.1}$$

and

$$Z_e = \cfrac{k_7 \, \cfrac{E}{G} + k_8}{(k_5 + k_7) \, \cfrac{E}{G} + k_6 + k_8}. \tag{9.2}$$

Equations (9.1) and (9.2) show that $Z_p$ and $Z_e$ are functions of $E/G$.

Some relationships among $B$, $W$, $Z_p$, and $Z_e$ are given by

$$\begin{aligned} Z_p &= \cfrac{E}{E + B}, \quad Z_e = W \, Z_p = \cfrac{W \, E}{E + B}, \\ B &= E \left( \cfrac{1 - Z_p}{Z_p} \right), \quad W = \cfrac{Z_e}{Z_p}.\end{aligned}$$

For current NCCI ERA states,

$$Z_p = \frac{E/G + 700}{1.1 \, E/G + 3270}, \quad Z_e = \frac{E/G + 5100}{1.375 \, E/G + 155100},$$

when $E/G$ is large enough that the minimum $B$ or $C$ does not apply.

The mod can also be written in terms of $Z_p$ and $Z_e$ instead of $B$ and $W$ as

$$M = \cfrac{Z_p A_p + (1 - Z_p) E_p + Z_e A_e + (1 - Z_e) E_e}{E},$$

which is (1.1), or as

$$M = 1 + Z_p \frac{A_p - E_p}{E} + Z_e \frac{A_e - E_e}{E}.$$

The equivalence of formulas (1.1) and (1.2) for the mod is established using formal algebra, as we now show. We start with (1.2), substitute for $B$ and $W$ in terms of $Z_p$ and $Z_e$, and rearrange terms, as follows:

$$\small\begin{aligned} M &= \cfrac{A_p + W A_e + (1 - W) E_e + B}{E + B} \\ &= \cfrac{A_p + \cfrac{Z_e}{Z_p} A_e + \left(1 - \cfrac{Z_e}{Z_p} \right) E_e + E \left( \cfrac{1 - Z_p}{Z_p} \right)}{E + E \left( \cfrac{1 - Z_p}{Z_p} \right)} \\ &= \cfrac{\cfrac{1}{Z_p}\left(\phantom{\cfrac{}{}}Z_p A_p + Z_e A_e + (Z_p - Z_e) E_e + (1 - Z_p) E \right)}{\cfrac{1}{Z_p}\left(\phantom{\cfrac{}{}}Z_p E + E(1 - Z_p) \right)} \\ &= \cfrac{Z_p A_p + Z_e A_e + (Z_p - Z_e) E_e + (1 - Z_p) E}{E} \\ &= \cfrac{Z_p A_p + (1 - Z_p) E_p + Z_e A_e + (Z_p - Z_e) E_e + (1 - Z_p) E_e}{E} \\ &= \cfrac{Z_p A_p + (1 - Z_p) E_p + Z_e A_e + (1 - Z_e) E_e}{E} \, . \end{aligned}$$

Appendix B—Mod Performance Tests

Tests known as quintile tests are commonly used to test the performance of experience rating plans. Such tests take a retrospective look at the performance of the experience rating plan on some set of insureds.

To describe the tests, we first need to define the terms experience period and rating period. The experience period is the three-year period (generally) during which the losses used in determining an insured’s mod occur. The rating period is the one-year period (again, generally) during which a policy is in effect whose premium includes an adjustment by the mod.

In the quintile test, the set of insureds is segmented into five subsets based on their mods. In principle, insureds are put into order by their mods; then the 20% (or so) with the lowest mods are in the first subset, the 20% with the lowest mods of those remaining are in the second subset, and so on until the 20% with the highest mods are in the fifth subset. Quintiles might be based on a count of the number of insureds or by expected losses. That is, the first quintile might be the 20% of insureds with the lowest mods or might be the lowest-mod insureds that account for 20% of the total expected losses across all insureds in the test.

When the quintiles are based on counting insureds, the quintile that includes insureds that have mods of 1.00 tends to comprise smaller insureds. Mods for small insureds cannot deviate from 1.00 to the degree that mods for larger insureds can. Small insureds have small credibility, limiting the magnitude of debit or credit mods. Also, small insureds are subject to lower caps on debit mods than are larger insureds. Use of expected losses as the basis for the quintiles avoids having a quintile that consists mainly of small insureds.

The test then has two parts. The first part looks at loss ratios during the rating period without consideration of the mod. More specifically, for each quintile, a loss ratio is computed where

• the numerator is the sum of the actual losses reported during the rating period for the insureds in the quintile, and

• the denominator is the sum of the expected losses during the rating period for the insureds in the quintile, before the application of the mod.

For ease of interpretation, these loss ratios are normalized so that the total loss ratio across all quintiles is 1. That is, the actual losses and expected losses are rescaled to equal one another, both before and after the application of the mod.

In this first part, the test looks at whether, and to what degree, the loss ratios increase as we go from the first quintile, with the lowest mods, to the fifth quintile, with the highest mods. The difference between the loss ratio for the fifth quintile and the loss ratio for the first quintile is called the predictive power of the plan. A larger predictive power is considered to be better because it shows greater loss ratio differentiation. Essentially the test is looking at how well the plan did in terms of identifying insureds that would have had low loss ratios without the mod and identifying insureds that would have had high loss ratios without the mod.

The second part of the test looks at loss ratios during the rating period after application of the mod. For each quintile, a loss ratio is computed as above, except that the denominator is the sum of modified expected losses across insureds, i.e., the sum of expected losses times the mod for each insured. Again, these loss ratios are normalized so that the loss ratio across all quintiles is 1.

In this second part, the test looks at the degree to which quintile loss ratios differ from 1. This is called the accuracy of the plan. Generally speaking, small and random deviations from 1.00 for the loss ratios indicate more accurate mods than larger or more systematic deviations. Essentially this part of the test looks at how well the plan did at adjusting insured expected losses so that each quintile of insureds has the same loss ratio after application of the mod.

Appendix C—The Current NCCI Interstate Rating Mod

In this appendix, we show that for the current (2020) NCCI plan, $Z_p$ and $Z_e$ for a multistate insured differ from what $Z_p$ and $Z_e$ would be if these were based on the insured’s expected claim count.

Assume one insured has exposures in ERA State 1 and ERA State 2. The expected losses from States 1 and 2 are $E_1$ and $E_2$, the expected claim counts are $n_1$ and $n_2$, and the average severities (divided by 1,000) for the two states are $G_1$ and $G_2$. The insured’s total expected loss is $E = E_1 + E_2$.

The total expected claim count times 1,000 for the insured is $n = n_1 + n_2 = E_1/G_1 + E_2/G_2$.

Consistent with (9.1), the $Z_p$ calculated using $n$ is

$$\begin{gathered} Z_p = \cfrac{k_3 \, n + k_4}{(k_1 + k_3) \, n + k_2 + k_4} \\ = \cfrac{k_3 \, \left( \cfrac{E_1}{G_1} + \cfrac{E_2}{G_2} \right) + k_4}{(k_1 + k_3) \, \left( \cfrac{E_1}{G_1} + \cfrac{E_2}{G_2} \right) + k_2 + k_4}. \end{gathered}\tag{9.3}$$

NCCI does not use (9.3) to compute interstate mods but rather weights state-level $B$s and $W$s. To determine the NCCI interstate rating mod, we compute $B_1$ and $B_2$ for the two states as

$$B_1 = E \, \cfrac{k_1 \cfrac{E}{G_1} + k_2}{k_3 \cfrac{E}{G_1} + k_4}, \ B_2 = E \, \cfrac{k_1 \cfrac{E}{G_2} + k_2}{k_3 \cfrac{E}{G_2} + k_4}.$$

The $B$ for the interstate mod is the weighted average of $B_1$ and $B_2$, so

$$B = \cfrac{E_1 B_1 + E_2 B_2}{E}.$$

The $Z_p$ implicit in the interstate mod is then

$$\small \begin{gathered} Z_p = \cfrac{E}{E + \cfrac{E_1 B_1 + E_2 B_2}{E}} \\ = \cfrac{E}{E + \cfrac{E_1}{E} \, E \, \cfrac{k_1 \cfrac{E}{G_1} + k_2}{k_3 \cfrac{E}{G_1} + k_4} + \cfrac{E_2}{E} \, E \, \cfrac{k_1 \cfrac{E}{G_2} + k_2}{k_3 \cfrac{E}{G_2} + k_4}} \\ = \cfrac{1}{1 + \cfrac{E_1}{E} \, \cfrac{k_1 \cfrac{E}{G_1} + k_2}{k_3 \cfrac{E}{G_1} + k_4} + \cfrac{E_2}{E} \, \cfrac{k_1 \cfrac{E}{G_2} + k_2}{k_3 \cfrac{E}{G_2} + k_4}} \end{gathered}\tag{9.4}$$

$$\tiny=\cfrac{\left( k_3 \cfrac{E}{G_1} + k_4 \right) \, \left( k_3 \cfrac{E}{G_2} + k_4 \right)}{\left( k_3 \cfrac{E}{G_1} + k_4 \right) \left( k_3 \cfrac{E}{G_2} + k_4 \right) + \left( k_3 \cfrac{E}{G_2} + k_4 \right) \left( k_1 \cfrac{E_1}{G_1} + \cfrac{E_1}{E} k_2 \right) + \left( k_3 \cfrac{E}{G_1} + k_4 \right) \left( k_1 \cfrac{E_2}{G_2} + \cfrac{E_2}{E} k_2 \right)}.$$

To see that the $Z_p$s resulting from Equations (9.3) and (9.4) are in fact different, it suffices to plug in some values for the several variables. The biggest differences occur when $E_1 = E_2$ and $G_1$ is substantially different from $G_2$.

As an example, consider a hypothetical insured with operations in two states. We assume $E_1 = E_2 = 20000$, $G_1 = 5$, and $G_2 = 20$. Then $E = E_1 + E_2 = 40000$, $n_1 = 20000/5 = 4000$, $n_2 = 20000/20 = 1000$, and $n = n_1 + n_2 = 5000$. Using (9.3) gives

$$Z_p = \cfrac{5000 + 700}{1.1 \cdot 5000 + 2570 + 700} = 0.650.$$

For the NCCI interstate rating calculation, we have

$$B_1 = 40000 \, \cfrac{0.1 \cdot (40000/5) + 2570}{40000/5 + 700} = 15494,$$

and

$$B_2 = 40000 \, \cfrac{0.1 \cdot (40000/20) + 2570}{40000/20 + 700} = 41037,$$

and weighting these using $E_1$ and $E_2$ gives $B = 28266$. Then $Z_p = E/(E + B) = 0.586$. As such, the interstate $Z_p$ calculated according to NCCI’s Experience Rating Plan in this hypothetical example is about 10% less than the $Z_p$ calculated using the sum of the insured’s expected claim counts by state, per (9.3).

Similarly, the interstate $Z_e$ calculated according to NCCI’s Experience Rating Plan generally differs from the $Z_e$ calculated using expected claim counts by state.