Corro, Dan, and Yen-Chieh Tseng. 2021. “NCCI’s 2014 Excess Loss Factors.” Variance 14 (1).
• Figure 1.1. Major components of the 2014 methodology and of the annual update of ELFs
• Table 2.1. Data used in analysis
• Table 6.1. Countrywide ALAE percentages by claim group
• Table B.1. Maximum additional duration to close after 10 years
• Table C.1. Credibility k value by claim group
• Exhibit D.1. Sample calculation of expected claim counts by claim group and hazard group for an NCCI state
• Exhibit D.2. Sample calculation of expected severity by claim group and hazard group for an NCCI state
• Exhibit E.1. Sample calculation of expected claim counts by hazard group for the PT claim group for an NCCI state
• Exhibit E.2. Sample calculation of expected severities by hazard group for the PT claim group for an NCCI state
• Exhibit G.1. Sample calculation of expected severities including ALAE by claim group and hazard group for an NCCI state
• Table H.1. Probability of multiclaim occurrences containing different numbers of claims
• Table H.2. Losses and claim counts by type of occurrence
• Table H.3. Losses and claim counts by type of occurrence and claim group
• Table H.4. Severity relativities by type of occurrence and claim group
• Table H.5. Per-claim excess ratio to per-occurrence excess ratio conversion
• Exhibit I.1. Sample calculation of per-occurrence excess ratios not including ALAE by loss limit for hazard group A for an NCCI state

Abstract

An excess loss factor is a measure of expected loss that is in excess of a given per-occurrence limit. The National Council on Compensation Insurance (NCCI) uses excess loss factors in its retrospective rating plan as well as in aggregate and class ratemaking.

NCCI computes annual updates of excess loss factors by state and hazard group for certain limits ranging from $10K to$10M. These annual updates are filed with regulators in 37 NCCI states.

Periodically, NCCI reviews the methodology behind these annual updates. Such a review concluded in 2014 and made significant changes to the methodology used. This paper describes the new methodology and highlights some improvements over prior NCCI methodologies.

Accepted: March 13, 2019 EDT

Appendices

Appendix A.1. Mathematics of Excess Ratios

Given a claim severity distribution with the cumulative distribution function (CDF) $$F(x)$$, we define the following standard terms:

• mean $$\equiv \mu_{F} = \int_{0}^{\infty}{{xf}(x){dx}}$$

• variance $$\equiv \ \sigma_{F}^{2} = \int_{0}^{\infty}{\left( x - \mu_{F} \right)^{2}f(x){dx}}$$

• standard deviation $$\equiv \sigma_{F} = \sqrt{\sigma_{F}^{2}}$$

• survival function $$\equiv S_{F}(x) = 1 - F(x) = \ \int_{x}^{\infty}{f(y){dy}}$$

• excess ratio function $$\equiv R_{F}(x)$$$$=\frac{\int_{x}^{\infty}{(y - x)f(y){dy}}}{\int_{0}^{\infty}{{yf}(y){dy}}}$$$$=\frac{\int_{x}^{\infty}{S_{F}(y){dy}}}{\mu_{F}}$$

• mean residual lifetime $$\equiv {MRL}_{F}(x)$$$$=\frac{\int_{x}^{\infty}{(y - x)f(y){dy}}}{\int_{x}^{\infty}{f(y){dy}}}$$$$= \ \frac{\mu_{F}R_{F}(x)}{S_{F}(x)}$$.

When $$\mu_{F} = 1$$, we have the equation:

\begin{align} \int_{0}^{\infty}{R_{F}(x)dx = \ \frac{\left( 1 + {{CV}_{F}}^{2} \right)}{2}}\text{ where }\\ CV_{F} =\text{ Coefficient of Variation}\ = \frac{\sigma_{F}}{\mu_{F}}. \end{align}

We use EVT to select a generalized Pareto distribution tail (select the splice point $$a$$ and parameters $$m$$ and $$b$$—see the following section) that applies to all states. This procedure was done for each claim group via the following steps:

Appendix A.2. Useful Formulas

Lognormal

In what follows, $$\Phi(z) = \frac{1}{\sqrt{2\pi}}\int_{- \infty}^{z}e^{- \frac{t^{2}}{2}}{dt}$$ denotes the CDF of the standard normal distribution. Therefore, the CDF of the lognormal distribution with parameters $$\mu$$ and $$\sigma$$ is given by:

$F(r) = \Phi(z)\text{ where }z = \frac{lnr - \mu}{\sigma}$

its mean by:

$\overline{r} = e^{\mu + \frac{\sigma^{2}}{2}}$

and its excess ratio function by:

$R_{F}(r) = 1 - \Phi(z - \sigma) - r\frac{1 - F(r)}{\overline{r}}\$

and its MRL function by:

${MRL}_{F}(r) = \frac{\overline{r}\left( 1 - \Phi(z - \sigma) \right)}{1 - \Phi(z)} - r{\ .}$

Generalized Pareto Distribution (GPD)

For the parameterization used here, which is not standard,[6] the CDF of the GPD is given by:

$G(b,m;x) = \left\{ \begin{matrix} 1 - \left( \frac{b}{mx + b} \right)^{\frac{m + 1}{m}} & m \neq 0 \\ 1 - e^{- \frac{x}{b}} & m = 0. \\ \end{matrix} \right.$

Its mean is just the $$b$$ parameter. Its excess ratio function is:

$R_{G}(x) = \left\{ \begin{matrix} \left( \frac{b}{mx + b} \right)^{\frac{1}{m}} & m \neq 0 \\ e^{- \frac{x}{b}} & m = 0 \\ \end{matrix} \right.\$

and its mean residual lifetime function is linear:

${MRL}_{G}(x) = mx + b.$

The case $$m > 0$$ is the usual Pareto distribution.

Splicing

Fix a splice point a, and suppose we have a claim severity distribution with CDF F that we want to modify to a new distribution with CDF $$\widetilde{F}$$ so that for losses greater than a, the new distribution $$\widetilde{F}$$ follows a second distribution with CDF $$G$$. The case of interest is when $$F$$ is a mixture of lognormal distributions and $$G$$ is a GPD. It is natural to specify this “spliced distribution” in terms of its survival function:

$S_{\widetilde{F}}(x) = \left\{ \begin{matrix} S_{F}(x) & x \leq a \\ S_{F}(a)S_{G}(x - a) & x \geq a. \\ \end{matrix} \right.$

Now suppose we have the equation:

${MRL}_{F}(a) = \mu_{G}.$

Then $$\mu_{\widetilde{F}} = \mu_{F}$$, and one can readily determine the excess ratio function of the spliced distribution $$\widetilde{F}$$ from those of $$F$$ and $$G$$:

$R_{\widetilde{F}}(x) = \left\{ \begin{matrix} R_{F}(x) & x \leq a \\ R_{F}(a)R_{G}(x - a) & x \geq a. \\ \end{matrix} \right.\$

When $$G$$ is GPD with parameters $$m$$ and $$b = {MRL}_{F}(a)$$ we have:

$R_{\widetilde{F}}(x) = \left\{ \begin{matrix} R_{F}(x) & x \leq a \\ R_{F}(a)\left( \frac{b}{m(x - a) + b} \right)^{\frac{1}{m}} & x \geq a. \\ \end{matrix} \right.$

Appendix B.1. Parameters of the Lognormal LDF Dispersion Models: Overview

The kernel density distribution that the individual claim development and dispersion (D&D) model initially assigns to the open claim of size $$x$$ (reopened claims are treated the same as open) and report $$t$$ is a lognormal distribution with standard parameters:

$\ln(x) + \mu(t;x)\text{ and }\sigma^{2}(t)$

• Ratemaking LDFs by state, claim grouping, and report are converted to “open-only” LDF factors

• Those open-only factors are used to modify the first parameter by an additive constant by state, claim grouping, and report (flat factor in entry ratio space)

• That adjustment assures that the model has the same expected loss at ultimate as implied by the ratemaking LDFs, by state, claim grouping, and report

Appendix B.2. Parameters of the Lognormal LDF Dispersion Models: Details

As noted in Section 3, the D&D model adjusts open claims to an ultimate basis in two steps—first to a 10th report and then from a 10th report to ultimate:

• Step 1 (through 10th report)—D&D varies by size of claim based on regression

• Linear regression considers individual claim development from report $$t$$ to report 10 and relates it with the claim amount at report $$t$$

• A linear regression model is determined using open claims for each of the 20 report-claim grouping combinations as follows:

• four reports $$t$$, for $$t = 6,7,8,9;$$ and

• five claim groupings

• Step 2 (beyond 10th report)—D&D does not vary by size of claim[7]

• Consider the pattern of variability of observed logarithms of annual LDFs from reports $$t$$ to $$t + 1$$, for $$t = \ 4,\ 5,\ 6,\ 7,\ 8,\ 9$$

• Use exponential regression, tempered with judgment, and project those patterns from report $$t > 9$$ to claim closure

The D&D steps 1 and 2 are not correlated. Sizes of loss and per-claim development beyond 10th report are not related.

Appendix B.3. Parameters of the Lognormal LDF Dispersion Models: Step 1 Further Details

For claims open at report t and within a claim grouping, a linear regression estimates the log per-claim LDF as a function of a transformed per-claim loss amount at report t.

• The regression equation is used to estimate the mean $$\mu_{1}(t;x)$$ of the $$t$$th to 10th log LDF; this estimate varies with the size of claim $$x$$.

• Transforming the size of claim metric improves the fit, especially for the largest claims.

• In place of entry ratio $$x$$ as explanatory variable, the revised model uses a compressed size of claim metric $$\gamma(x)$$:

$\gamma(x) = \ln x\text{ for }x \geq 1;\ \gamma(x) = \ x - 1\text{ for }x \leq 1.$

• The variance of the distribution of the residual gives an estimate of the variance $$\sigma_{1}^{2}(t)$$ of the $$t$$th to 10th log LDF; this estimate does not vary with the size of claim $$x$$.

• The proportion $$\rho(t)$$ of claims open at report $$t$$ that remain open at 10th is calculated from the claim data used in the regression.

Appendix B.4. Parameters of the Lognormal LDF Dispersion Models: Step 2 Further Details

The model assumes that beyond 10th report,

• the mean of the distribution of log annual LDFs is 0 for all claims and all years and

• the annual claim closure rate is constant for each claim grouping.

The variance $$\sigma_{2}^{2}$$ of the second step is estimated for each claim grouping using

• a constant annual claim closure rate,

• projected variances of the distributions of annual log per-claim LDFs,

• exponential decay model as appropriate (becoming linear with a log vertical scale),

• a judgmentally assigned asymptote as the long-term estimate of the variance of the log annual LDF, and

• formulas for the decay model.

Let $$y(t) =$$ empirical variance of annual per-claim LDF from report $$t$$ to $$t + 1$$. The formula to project variance is

$y(t) = a + c \cdot e^{{bt}}$

in which $$a$$ is an assumed asymptotic long-term variance[8] and $$b$$ and $$c$$ are constants to be estimated. The linear regression model with coefficient vector $$= \beta$$,

$\ln\left( y(t) - a \right) = \ln c + bt = \beta_{0}\ + \beta_{1}t + \epsilon(t),$

is used to yield the estimates $$c = e^{\beta_{0}}$$ and $$b = \beta_{1}$$. The formula for the variance of 10th-ultimate log LDF is as follows, letting $$1–s$$ be the constant annual claim closure rate and $$N$$ be the maximum duration to closure after report $$t = 10$$:

\begin{align} \sigma_{2}^{2} \approx &\left( \frac{a}{1 - s} \right)\left( 1 - \frac{Ns^{N}(1 - s)}{1 - s^{N}} \right) \\ &+ \left( \frac{ce^{10b}}{1 - e^{b}} \right)\left( 1 - \left( \frac{e^{b}(1 - s)}{1 - s^{N}} \right)\left( \frac{1 - \left( se^{b} \right)^{N}}{1 - se^{b}} \right) \right). \end{align}

The maximum additional duration to closure $$(N)$$ after 10 years by claim group is given in Table B.1.

Table B.1.Maximum additional duration to close after 10 years
Claim Group N (years)
Fatal 25
PT 30
Likely to develop PP/TT 20
Not likely to develop PP/TT 15
Medical only 10

In summary, to each open claim of size $$x$$ at latest report $$t$$, the D&D model assigns a mean $$\mu(t;x)$$ and variance $$\sigma^{2}(t)$$ to the log per-claim LDF distribution; with just two uncorrelated component steps:

$\mu(t;x)\ = \ \mu_{1}(t;x)\ \text{and}\ \sigma^{2}(t)\ = \ \sigma_{1}^{2}(t)\ + \rho(t)\sigma_{2}^{2}$

where:

• $$\mu_{1}(t;x) =$$ linear estimate for the mean of the $$t$$th to 10th log LDF

• $$\sigma_{1}^{2}(t) =$$ estimated variance of the $$t$$th to 10th log LDF

• $$\rho(t) =$$ proportion of claims still open at 10th report

• $$\sigma_{2}^{2} =$$ estimated variance of the 10th to ultimate log LDF.

The values $$\mu(t;x)$$ and $$\sigma_{2}(t)$$ are the standard parameters for the lognormal density model for the LDFs of an open claim of size $$x$$ at latest report $$t$$ (prior to balancing with ratemaking LDFs by state, claim group, and report).

Appendix C.1. Parameters of Countrywide Excess Ratio Curve as a Lognormal Mixture and Pareto Tail

This section describes the ingredients that go into the parametric form for expressing the excess ratio $$R(r)$$ as a function of entry ratio $$r$$. The severity distribution is a mixture of two lognormal distributions with parameters $$\mu_{1},\mu_{2}$$ and $$\sigma_{1},\ \sigma_{2}$$, respectively.[9] The weight assigned the first component is denoted $$\omega_{1}$$ and the weight assigned the second component is denoted $$\omega_{2} = (1 - \ \omega_{1})$$. A Pareto tail distribution is spliced onto the lognormal mixture at an entry ratio denoted by $$a$$ and termed the splice point (the splicing preserves an entry ratio mean of 1). The parameters used for the Pareto distribution are denoted $$b$$ and $$m$$, chosen to exploit the characterization of the Pareto distribution as one having a linear mean residual lifetime function of slope $$m$$ and intercept $$b$$, which is also the mean of the distribution.

The CDF of the lognormal components are:

$F_{i}(r) = \Phi(z_{i})\text{ where }{\ z}_{i} = \frac{\ln r - \mu_{i}}{\sigma_{i}},\text{ where }i = 1,\ 2.$

Their means are:

$\overline{r_{i}} = e^{\mu_{i} + \frac{{\sigma_{i}}^{2}}{2}}$

and in particular:

$1 = \omega_{1}\overline{r_{i}} + \left( 1 - \omega_{1} \right)\overline{r_{2}}$

as we are working with entry ratios. The excess ratio functions of the lognormal components are:

$R_{i}(r) = 1 - \Phi\left( z_{i} - \sigma_{i} \right) - r\frac{1 - F_{i}(r)}{\overline{r_{i}}}{\ .}$

The CDF of the lognormal mixture portion is:

$F(r) = \omega_{1}F_{1}(r) + \left( 1 - \omega_{1} \right)F_{2}(r),\ r \leq a$

and the excess ratio function for the lognormal mixture portion is loss-weighted average:

$R(r) = \omega_{1}\overline{r_{1}}R_{1}(r) + \left( 1 - \omega_{1}\overline{r_{1}} \right)R_{2}(r),\ r \leq a.$

The probability of surviving to the splice point is:

$S = 1 - F(a)$

and since the mean residual lifetime at the splice point must preserve a mean of 1, we have the following equation that can be used to find the value $$b$$ to assign to the b parameter:

$R(a) = bS.$

The CDF of the Pareto tail portion is:

$F(r) = 1 - S\left( \frac{b}{m(r - a) + b} \right)^{\frac{m + 1}{m}},r \geq a.$

Finally, from the formula for the excess ratio of a Pareto distribution, we have the formula for the Pareto tail portion as:

$R(r) = S\left( \frac{b}{m(r - a) + b} \right)^{\frac{1}{m}},r \geq a.$

Appendix C.2. Deriving the State Excess Ratio Curves from the Countrywide Curves

The process starts with a credibility-weighted relativity $$r$$ of the standard deviation of state log losses to that of the countrywide. That is,

\begin{align} r = &Z\left( \frac{{σ\, for\, logged\, losses\, for\, claim\, group\, in\, state}}{{σ\, for\, logged\, losses\, for\, claim\, group\, countrywide}} \right) \\ &+ (1 - Z). \end{align}

Here, the credibility weight is determined as

$Z = \frac{N}{N + k},$

where k varies by claim group, as shown in Table C.1, and N is the expected number of such claims in the claim group for the state.

Table C.1.Credibility k value by claim group
Claim Group k
Fatal 60
PT 33
Likely to develop PP/TT 73
Not likely to develop PP/TT 129
Medical only 373

For the state curve, we use the same $$w_{i}$$ as for the countrywide. We replace each of the $$\mu_{i}$$ and $$\sigma_{i}$$ with $$r\mu_{i}$$ and $$r\sigma_{i}$$, respectively. This method multiplies the standard deviation of the log losses by a factor of $$r$$. We then replace the new $$\mu_{i}$$ with $$\mu_{i} + c$$, where $$c$$ is the constant that produces a mean of 1 for the state curve. Note that adding this constant does not change the standard deviation of the logarithmic entry ratio of the state curve. This yields the lognormal mixture for the state curve.

Now we determine the parameters for the Pareto distribution. We keep the splice point $$a$$ and the slope $$m$$ the same as for the countrywide and determine the $$b$$ parameter for the state curve. The parameter value b can be determined by matching the mean residual lifetimes of the lognormal mixture and the GPD tail. To this end, the mean residual lifetime of a lognormal component at $$a$$ is:

${MRL}_{i}(a) = \frac{e^{\left( \mu_{i} + \frac{{\sigma_{i}}^{2}}{2} \right)}\left( 1 - \Phi\left( \frac{\ln(a) - \mu_{i}}{\sigma_{i}} - \sigma_{i} \right) \right)}{1 - \Phi\left( \frac{\ln(a) - \mu_{i}}{\sigma_{i}} \right)} - a{\ .}$

The mean residual lifetime of the lognormal mixture at splice point $$a$$ is, therefore, the sum weighted by the frequency of claims surviving to the splice point:

${MRL}(a) = \frac{\sum_{i = 1}^{2}{w_{i}\left( 1 - \Phi\left( \frac{\ln(a) - \mu_{i}}{\sigma_{i}} \right) \right){MRL}_{i}(a)}}{\sum_{i = 1}^{2}{w_{i}\left( 1 - \Phi\left( \frac{\ln(a) - \mu_{i}}{\sigma_{i}} \right) \right)}}{\ .}$

Since the overall mean of the GPD is the $$b$$ parameter, setting $$b = MRL\ (a)$$ will produce a distribution function $$F(r)$$, defined as in the previous section, whose mean residual lifetime at splice point $$a$$ is also $$b$$ and whose overall mean is therefore also 1. This last $$b$$ is the state-specific $$b$$ parameter. This procedure now gives the complete state curve.

We compared the empirical claim experience before and after applying D&D and found that the state relativity of the standard deviation of the logarithmic loss to countrywide was very similar. Accordingly, interim updates to the relativities are based on empirical experience.

Appendix D.1. Claim Count Model

Claim counts are assumed to follow a negative binomial distribution.

The model can be written as:

$\log\left( \mu_{{ghr}} \right) = \delta_{{shr}} + \gamma_{g} + \xi_{s} + \eta_{{hg}} + \psi_{{sg}} + \omega_{{sh}} + \rho_{r}$

where:

• $$\mathbf{\mu_{ghr}}\ =$$ expected number of claims in claim group $$g$$, state $$s$$, hazard group $$h$$, and policy period $$r$$

• $$\mathbf{\delta_{shr}}\ =$$ log of payroll in state $$s$$, hazard group $$h$$, and policy period $$r$$

• $$\mathbf{\gamma_{g}}\ =$$ factor for claim group $$g$$

• $$\mathbf{\xi_{s}}\ =$$ factor for state $$s$$

• $$\mathbf{\eta_{hg}}\ =$$ factor for hazard group $$h$$ specific to claim group $$g$$

• $$\mathbf{\psi_{sg}}\ =$$ factor for interaction between state $$s$$ and claim group $$g$$

• $$\mathbf{\omega_{sh}}\ =$$ factor for interaction between state $$s$$ and hazard group $$h$$

• $$\mathbf{\rho_{r}}\ =$$ factor for policy period $$r$$.

The parameters $$\psi_{{sg}}$$ and $$\omega_{{sh}}$$ are credibility weights (or “shrunk”) using multilevel modeling. This is sometimes referred to as partial pooling.

Additionally, $$\eta_{{hg}}$$ is assumed to have a structure described later in this section.

The following are notes on each parameter in the equation:

• The log of the payroll $$(\delta_{{shr}})$$ is known from data and serves as the exposure base.

• The policy period factor $$(\rho_{r})$$ is fixed for all other factors and serves to account for differences between policy periods, such as benefit levels and trend in frequency per payroll.

• The state-specific parameters $$(\xi_{s},\ \psi_{{sg}},\ \omega_{{sh}})$$ account for the overall state variation separately from the state-to-state variation of individual claim groups and hazard groups, especially considering credibility.

• The state factor $$(\xi_{s})$$ is estimated very accurately, since every claim for all the modeled claim groups contributes to its estimation. Using this factor to account for state differences separately from state variation that interacts with claims groups or hazard groups allows for the interaction effects to be estimated more accurately. This is analogous to reducing the variance between groups and has the effect of shrinking the state–claim group $$(\psi_{{sg}})$$ and state–hazard group $$(\omega_{{sh}})$$ factors toward 1.0.
• The claim group factor $$(\gamma_{g})$$ accounts for the base frequency per payroll in each claim group.

• The hazard group factor ($$\eta_{hg}$$) differs by claim group and has the following structure:

$\eta_{{hg}} = \left\{ \begin{matrix} {\widehat{\eta}}_{h1} & \text{if}{\ g\ }\text{is fatal} \\ {\widehat{\eta}}_{h1} \cdot \alpha_{1} & \text{if}{\ g\ }\text{is PT} \\ {\widehat{\eta}}_{h2} & \text{if }g\text{ is not likely} \\ {\widehat{\eta}}_{h2} \cdot \alpha_{2} & \text{if}{\ g\ }\text{is likely}. \\ \end{matrix} \right.\$

• This structure reflects the results of an inspection of empirical hazard-group relativities by claim-group relativities, where we found that while they were more extreme for certain claim groups, the different claim groups varied similarly across hazard groups. For this purpose, we chose to group fatal and PT together, and likely and not-likely together.

Exhibit D.1 shows sample values for the calculation of the claim counts for an NCCI state. Note that medical-only claim counts are determined directly using reported data, and PT claim counts are calculated via a separate procedure.

Exhibit D.1.Sample calculation of expected claim counts by claim group and hazard group for an NCCI state
 (1) Payroll in $millions ($$e^{\delta_{{shr}}}$$) Hazard Group Policy Period A B C D E F G 5/1/11–4/30/12 953 3,388 15,369 3,376 6,203 1,978 597 5/1/10–4/30/11 944 3,290 15,104 3,293 5,940 1,854 627 5/1/09–4/30/10 921 3,245 14,332 3,142 5,695 1,862 593 5/1/08–4/30/09 921 3,217 14,080 3,072 5,450 1,813 570 5/1/07–4/30/08 911 3,288 14,546 3,119 5,678 1,885 568 (2) Factors for policy period ($$\rho_{r}$$) Policy Period Factors for Policy Period (ρr ) 5/1/11–4/30/12 1.000 5/1/10–4/30/11 1.051 5/1/09–4/30/10 1.089 5/1/08–4/30/09 1.102 5/1/07–4/30/08 1.205 (3) Adjusted payroll = (1) x (2) Hazard Group A B C D E F G Adjusted payroll ($ millions) 5,060 17,885 79,893 17,402 31,495 10,224 3,214 (4) State factor ($$\xi_{s})$$ State Factor (ξs ) 0.900 (5) Factor for interaction between state and hazard group ($$\omega_{{sh}}$$) Hazard Group A B C D E F G Relativity 1.293 1.139 1.112 0.984 0.961 0.820 0.787 (6) Claim group frequency ($$\gamma_{g}$$) Claim Group Claims per \$ Million Payroll (γg ) Fatal 0.00032 Likely PP/TT 0.05770 Not likely PP/TT 0.29446 (7) Factor for hazard group specific to claim group ($$\eta_{{hg}}$$) Hazard Group Claim Group A B C D E F G Fatal 1.000 0.984 0.769 2.459 3.224 10.672 16.531 Likely PP/TT 1.000 0.741 0.371 0.699 0.628 1.390 1.313 Not likely PP/TT 1.000 0.743 0.374 0.701 0.631 1.386 1.310 (8) Factor for interaction between state and claim group ($$\psi_{{sg}}$$) Claim Group Claim Group Factor (ψg ) Fatal 1.261 Likely PP/TT 0.900 Not likely PP/TT 0.881 (9) Expected number of claims by claim group and hazard group = (3) x (4) x (5) x (6) x (7) x (8) Hazard Group Claim Group A B C D E F G Fatal 2.341 7.172 24.440 15.075 34.933 32.010 14.972 Likely PP/TT 306 705 1,539 560 889 544 155 Not likely PP/TT 1,528 3,535 7,756 2,806 4,459 2,714 775

Note: Claim counts for the fatal claim group are shown to three decimal places.

Appendix D.2. Severity Model

The expected severity is assumed to follow a gamma distribution.

The model can be written as:

$\log\left( \mu_{{ghr}} \right) = \gamma_{g} + \xi_{s} + \eta_{{hg}} + \psi_{{sg}}$

where:

• $$\mathbf{\gamma_{g}}\ =$$ base severity for claim group $$g$$

• $$\mathbf{\xi_{s}}\ =$$ factor for state $$s$$

• $$\mathbf{\eta_{hg}}\ =$$ factor for hazard group $$h$$ specific to claim group $$g$$

• $$\mathbf{\psi_{sg}}\ =$$ factor for interaction between state $$s$$ and claim group $$g$$

The parameter $$\psi_{{sg}}$$ is credibility weighted (or “shrunk”) using multilevel modeling. This is sometimes referred to as partial pooling.

Additionally, $$\eta_{{hg}}$$ is assumed to have a structure described later in this section.

The developed, on-leveled, and trended empirical average severities at the claim-group–hazard-group–state level follow a gamma distribution with parameters adjusted to reflect the reduction in variance associated with an increase in number of claims.

The following are notes on each parameter in the equation:

• The base severity $$(\gamma_{g})$$ can be thought of as an intercept and is analogous to the idea of a base rate in a rating system.

• The state-specific parameters $$(\xi_{s},\ \psi_{{sg}})$$ account for the overall state variation separate from the state-to-state variation of individual claim groups, especially considering credibility. This is similar to the claim count model.

• The state factor $$(\xi_{s})$$ is estimated very accurately, since every claim for all the modeled claim groups for a state contributes to its estimation. Using this factor to account for state differences separately from state variation that interacts with claims groups or hazard groups allows for the interaction effect to be estimated more accurately. This is analogous to reducing the variance between groups and has the effect of shrinking the state–claim group $$(\psi_{{sg}})$$ factor toward 1.0.
• The hazard group factor ($$\eta_{hg}$$) differs by claim group but is common for all states and has the following structure across the claim groups:

${\eta_{{hg}} = \eta_{h} \cdot \alpha_{g}.}$

This structure reflects the results of an inspection of empirical hazard-group relativities by claim-group relativities, where we found that they were more extreme for certain claim groups, but the different claim groups varied similarly across hazard groups. This is similar to the structure for claim counts, but all claim groups in the severity model have a common shape ($$\eta_h$$) and vary only by a common multiple ($$\alpha_g$$) that reflects the magnitude of the shape.

Exhibit D.2 shows sample values for the calculation of the severities for an NCCI state. Note that medical-only severities are determined directly using reported data, and PT severities are calculated via a separate procedure.

Exhibit D.2.Sample calculation of expected severity by claim group and hazard group for an NCCI state
 (1) Base severity for claim group ($$\gamma_{g}$$) Claim Group Base Severity for Claim Group (γg ) Fatal 285,559 Likely PP/TT 91,090 Not likely PP/TT 25,518 (2) State factor ($$\xi_{s}$$) State Factor (ξs ) 0.946 (3) Factor for hazard group specific to claim group ($$\eta_{{hg}}$$) Hazard Group Claim Group A B C D E F G Fatal 1.000 1.097 1.122 1.209 1.281 1.382 1.436 Likely PP/TT 1.000 1.259 1.334 1.604 1.852 2.235 2.454 Not likely PP/TT 1.000 1.215 1.276 1.491 1.683 1.973 2.136 (4) Factor for interaction between state and claim group ($$\psi_{{sg}}$$) Claim Group Claim Group Factor (ψg ) Fatal 0.700 Likely PP/TT 1.366 Not likely PP/TT 1.046 (5) Expected severity by claim group and hazard group = (1) x (2) x (3) x (4) Hazard Group Claim Group A B C D E F G Fatal 189,207 207,468 212,336 228,690 242,346 261,457 271,611 Likely PP/TT 117,736 148,227 157,043 188,869 218,061 263,128 288,954 Not likely PP/TT 25,262 30,691 32,226 37,664 42,528 49,845 53,949

Appendix E. Treatment of PT Claims

Exhibits E.1 and E.2 show sample values for the calculations of the claim counts (exhibit E.1) and severities (exhibit E.2) for the PT claim group for an NCCI state.

Exhibit E.1.Sample calculation of expected claim counts by hazard group for the PT claim group for an NCCI state
 (1) State claim count (base period 5/1/2000 to 4/30/2005). These values are calculated using the same process shown in exhibit D.1, but these values include the PT claim group and use an older time period. Hazard Group Claim Group A B C D E F G Fatal 3.710 14.647 36.657 21.374 50.941 48.294 29.459 PT 5.013 19.045 53.456 22.731 50.082 33.133 16.509 Likely PP/TT 447 1,125 2,232 850 1,401 919 283 Not likely PP/TT 2,013 5,036 9,873 3,810 6,263 4,176 1,287 Total non-PT lost-time 2,463 6,176 12,141 4,681 7,715 5,143 1,599 Note: Claim counts for the fatal and PT claim groups are shown to three decimal places. (2) Initial proportion of PT claim count to total non-PT lost-time $$= \frac{(1)_{{PT}}}{(1)_{Total\ Non - PT\ Lost - Time}}$$ Hazard Group A B C D E F G Proportion 0.00203 0.00308 0.00440 0.00486 0.00649 0.00644 0.01032 (3) Fitted state claim counts. These values are taken from the final claim counts shown in exhibit D.1. The total non-PT lost-time claim count is calculated. Hazard Group Claim Group A B C D E F G Fatal 2.341 7.172 24.440 15.075 34.933 32.010 14.972 Likely PP/TT 306 705 1,539 560 889 544 155 Not likely PP/TT 1,528 3,535 7,756 2,806 4,459 2,714 775 Total non-PT lost-time 1,836 4,247 9,319 3,381 5,383 3,290 945 (4) Estimated PT claim count = $$(2) \times (3)_{Total\ Non - PT\ Lost - Time}$$ Hazard Group A B C D E F G PT claim count 3.737 13.098 41.032 16.417 34.942 21.197 9.755
Exhibit E.2.Sample calculation of expected severities by hazard group for the PT claim group for an NCCI state
 (1) State PT severity (base period 5/1/2000 to 4/30/2005). These values are calculated using the same process as shown in exhibit D.2, but these values include the PT claim group and use an older time period. Only the values for the PT claim group are shown here, although data for all lost-time claim groups are used in the model to produce these PT values. Hazard Group Claim Group A B C D E F G PT 590,710 817,530 896,879 1,043,220 1,257,983 1,526,118 1,780,806 (2) Calculation of trend factors Trend Stage Annual Indemnity Trend Annual Medical Trend Trend Period Start Date Trend Period End Date Number of Years Indemnity Trend Factor Medical Trend Factor First stage 1.050 1.067 5/15/2003 5/15/2010 7.005 1.407 1.575 Second stage 1.020 1.030 5/15/2010 4/1/2017 6.885 1.146 1.226 (3) Combined trend factors = first stage trend x second stage trend Indemnity Trend Factor Medical Trend Factor Combined trend factor 1.613 1.931 (4) Selected on-level factor. The on-level factor reflects changes in PT benefit levels between the base period and the effective time period. Indemnity Medical On-Level factor 1.101 1.127 (5) PT indemnity/medical split. The PT indemnity/medical split is calculated using developed PT loss dollars in the base period. Indemnity Medical PT loss weight 0.231 0.769 (6) Combined trend and on-level factors $$\quad \ \ Total\ = (3)_{{Indemnity}} \times (4)_{{Indemnity}} \times (5)_{{Indemnity}} + (3)_{{Medical}} \times (4)_{{Medical}} \times (5)_{{Medical}}$$ Indemnity Medical Total Combined Trend and On-Level Factors 1.776 2.175 2.083 (7) Estimated PT severity $$= (1) \times (6)_{{Total}}$$ Hazard Group Claim Group A B C D E F G PT 1,230,525 1,703,019 1,868,314 2,173,161 2,620,538 3,179,099 3,709,645

Appendix F. Excess Ratio Curve for Losses including ALAE

Let s denote the ratio of the loss-only severity to the loss-including-ALAE severity, for the claim group and hazard group in a given state. For a fixed loss limit $$L$$, let $$r$$ be the corresponding entry ratio when the limit is viewed as a pure loss and $$\widehat{r} = sr$$ be the entry ratio when that limit is viewed as applying to a loss that includes ALAE. For each claim group $$i$$, let $$E_{i}$$ be the excess ratio function for the state on a pure loss basis and let $${\widehat{E}}_{i}$$ be the excess ratio function for the state on a loss-with-ALAE basis, with both $${\widehat{E}}_{i}$$ and $$E_{i}$$ being functions of the applicable entry ratio. $${\widehat{E}}_{i}$$ is calculated using the same formula structure as $$E_{i}$$ but with different parameter values. Let $${ALA}E_{{State}}$$ and $${ALA}E_{{CW}}$$ denote state and countrywide ALAE percentages, respectively. Then the formula for the claim group component excess ratio for loss with ALAE is:

\begin{align} E_{i}^{{ALAE}}\left( \widehat{r} \right)= &{Min}\biggl( {Max}\bigl( E_{i}\left( \widehat{r} \right) \\ &+ \left( \frac{{ALAE}_{{State}} - 1}{{{AL}AE}_{{CW}} - 1} \right)\left( {\widehat{E}}_{i}\left( \widehat{r} \right) - E_{i}\left( \widehat{r} \right) \right), \\ &{sE}_{i}(r) \bigr),1 - s + {sE}_{i}(r) \biggr). \end{align}

The appropriate value to use for $${ALA}E_{{CW}}$$ is 1.127. The value to use for $${ALA}E_{{State}}$$ is the ALAE factor appropriate for the state and time period.

Note: The excess ratio curve for losses with ALAE can be viewed as the weighted sum of two excess ratio functions:

\begin{align} &E_{i}\left( \widehat{r} \right) + \left( \frac{{ALAE}_{{State}} - 1}{{ALAE}_{{CW}} - 1} \right)\left( {\widehat{E}}_{i}\left( \widehat{r} \right) - E_{i}\left( \widehat{r} \right) \right) \\ &= \left( 1 - \frac{{ALAE}_{{State}} - 1}{{ALAE}_{{CW}} - 1} \right)E_{i}\left( \widehat{r} \right) + \frac{{ALAE}_{{State}} - 1}{{ALAE}_{{CW}} - 1}{\widehat{E}}_{i}\left( \widehat{r} \right) \end{align}

that is then subject to a lower bound of $${sE}_{i}(r)$$ that corresponds to the case where the additional ALAE has no contribution to the excess and is also subject to an upper bound of $$1 - s + {sE}_{i}(r)$$ that corresponds to the case where the additional ALAE has full contribution to the excess.

Because the excess ratio function for losses with ALAE is determined by this formulaic adjustment of excess ratios, this construction does not provide a parametric claim severity distribution function.

Finally, for the loss limit $$L$$, the overall excess ratio for loss with ALAE is the loss-weighted average:

$E^{{ALAE}}(L) = \sum_{i}^{}{{\widehat{\omega}}_{i} \cdot E_{i}^{{ALAE}}\left( \widehat{r} \right)}$

where the $${\widehat{\omega}}_{i}$$ denote the weights that correspond to itemizing the losses including ALAE for the state and hazard group into claim groups.

Appendix G. Calculation of Severities including ALAE

Exhibit G.1.Sample calculation of expected severities including ALAE by claim group and hazard group for an NCCI state
 (1) Expected loss on a loss-only basis, calculated by multiplying expected claim counts by expected severities (fatal, likely PP/TT, and not-likely PP/TT from exhibit D.1; PT values from exhibit E.1; medical-only values calculated as reported from unit data) Hazard Group Claim Group A B C D E F G Fatal 443,014 1,487,960 5,189,480 3,447,497 8,465,864 8,369,227 4,066,563 PT 4,598,375 22,306,722 76,660,356 35,676,057 91,568,062 67,386,530 36,186,006 Likely PP/TT 36,027,211 104,500,008 241,689,517 105,766,864 193,855,977 143,141,729 44,787,898 Not likely PP/TT 38,600,488 108,491,174 249,946,987 105,685,608 189,630,436 135,278,756 41,810,279 Medical only 10,506,947 27,324,096 56,680,683 19,513,664 30,287,788 16,593,726 4,609,704 (2) State ALAE percentage (from state rate or loss cost filing) State ALAE Percentage 0.116 (3) Countrywide ALAE relativities by claim group Claim Group Countrywide ALAE Relativity Fatal 0.0590 PT 0.0782 Likely PP/TT 0.1188 Not likely PP/TT 0.1132 Medical only 0.1320 Total 0.1067 (4) Ratio of state ALAE to countrywide ALAE $$= \frac{(2)}{(3)_{{Total}}}$$ Ratio of State ALAE to Countrywide ALAE 1.087 (5) State ALAE percentage by claim group$$= (3) \times (4)$$ Claim Group Countrywide ALAE Relativity Fatal 0.0641 PT 0.0850 Likely PP/TT 0.1292 Not likely PP/TT 0.1231 Medical only 0.1435 (6) Expected loss including ALAE by claim group and hazard group$$\ = (1) \times \lbrack 1 + (5)\rbrack$$ Hazard Group Claim Group A B C D E F G Fatal 471,430 1,583,402 5,522,346 3,668,627 9,008,886 8,906,050 4,327,403 PT 4,989,311 24,203,149 83,177,708 38,709,090 99,352,806 73,115,458 39,262,393 Likely PP/TT 40,680,292 117,996,668 272,904,839 119,427,146 218,893,376 161,629,148 50,572,463 Not likely PP/TT 43,350,917 121,842,808 280,707,098 118,691,970 212,967,598 151,927,045 46,955,725 Medical only 12,014,749 31,245,245 64,814,654 22,313,975 34,634,242 18,975,011 5,271,220 (7) Expected number of claims