NCCI.InformationalExhibits

Reading: National Council on Compensation Insurance, Circular CIF-2018-28, 06/21/2018. Informational Exhibits 1 — 3.

Synopsis: This is the second of two articles on the NCCI Circular CIF-2018-28 reading. It covers the second part of the circular which is a series of informational exhibits that explain how the NCCI derives their aggregate excess loss factors on demand for their retrospective rating plan. The first part of the NCCI Circular reading is available at NCCI.Circular.

Study Tips

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Estimated study time: 4 hours (not including subsequent review time)

BattleTable

This is a new reading and due to the CAS no longer publishing past exams there are no prior exam questions available. At BattleActs we feel the main things you need to know (in rough order of importance) are:

Questions are held out from most recent exam. (Use these to have a fresh exam to practice on later. For links to these questions see Exam Summaries.)

reference	part (a)	part (b)	part (c)	part (d)
Currently no prior exam questions

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In Plain English!

Overview

There are three informational exhibits which detail the derivation of the aggregate excess loss factors for both the on-demand factors and those provided in the pre-tabulated countrywide tables.

In terms of testability, the first exhibit and the last part of the third exhibit are probably more realistic to draw material from. If you're reading the source as well, note there is a lot of overlap between these exhibits and the NCCI.Circular wiki article - Alice has done her best to avoid duplicating any content here so you can study efficiently.

Informational Exhibit 1

This exhibit deals with how the NCCI produces the aggregate loss distribution used to create the Aggregate Excess Loss Factors. The key idea is to start with separate distributions for the claim counts and claim severity before merging them using Panjer's Algorithm to create an aggregate loss distribution.

Claim Count Distribution

The NCCI models claim counts using a negative binomial distribution. The distribution is specified (parameterized) by its mean, E[N], and Variance-to-Mean function, VtM. The Variance-to-Mean function allows you to compute the variance by multiplying the mean and variance-to-mean function together.

Notation The NCCI uses a superscript ^PC to mean "per-claim" and a superscript ^PO to mean "per-occurrence". Remember, an accident is an occurrence and a single occurrence may result in several claims if multiple parties were harmed. You need to keep track of whether your calculations are on a "per-claim" or "per-occurrence" basis as it is necessary to carefully convert between the two at times.

Irrespective of the use of per-claim or per-occurrence in general you should perform calculations at the state/hazard group level and then sum the results across all state/hazard group combinations for a risk.

Per-Claim Basis

Let E[N]^PC be the sum of the expected number of claims for the policy over all state/hazard groups. Use this value for the mean, E[N].

The negative binomial variance is the product of E[N]^PC and VtM^PC. Here, VtM^PC is the Variance-to-Mean function evaluated at E[N]^PC.

Question: What does the Variance-to-Mean function look like, what properties does it have, and how do I find it?

Solution: The Variance-to-Mean function is defined as [math]VtM(x, A, B) = \displaystyle\begin{cases}1+m\cdot x & \mbox{if } x\leq k \\ A\cdot x^B & \mbox{if } x\gt k \end{cases}[/math]. The Variance-to-Mean function must satisfy the following: It must be continuous, and Its first derivative must also be continuous, i.e. the function is smooth. The above conditions determine the values of m and k in the Variance-to-mean formula as follows: The slope of the linear function, m, is expressed in terms of A, B and k via [math]m=\displaystyle\frac{A\cdot k^B -1}{k}[/math]. The transition point, k, is given by [math]k=[A(1-B)]^\frac{-1}{B}[/math]. So the Variance-to-Mean function is entirely specified by A and B. Due to the above conditions, the transition point k is called the tangent point and denoted by E[N]TP. This is because the slope of the line and the slope of the power curve are equal at this point. The NCCI determines the Variance-to-Mean function by fitting it to empirical data so in the exam you would be given the fixed values of A and B.

Per-Occurrence Basis

When working on a per-occurrence basis the claim count distribution is still parameterized using E[N]=E[N]^PO and Variance-to-Mean function VtM^PO. However, it's necessary to carefully convert from the per-claim basis to per-occurrence basis by dividing E[N]^PC by the per-occurrence constant to get E[N]=E[N]^PO. The per-occurrence constant is determined empirically by dividing the number of claims by the number of occurrences.

Similarly, an adjustment to VtM^PC is needed to calculate VtM^PO. It is trickier because you need to form a negative binomial per-occurrence distribution whose probability of zero occurrences is the same as the probability of zero claims under the per-claim negative binomial distribution. (Can't have a claim without an occurrence!)

Panjer's Recursive Algorithm

Alice: "Panjer's Algorithm is also covered briefly in [Clark.AggModels#Recursive Calculation of Aggregate Distribution (Panjer's Recursive Algorithm)||Clark's section on aggregate models]]. You should compare the material there against here to reinforce your learning. This section places emphasis on the NCCI key points of Panjer's Algorithm."

Informational Exhibit 2

This exhibit gives you seven tables of aggregate excess loss factors for the state of Alaska, one for each of the seven hazard groups. The tables assume there is no loss limit and all exposures are from a single hazard group.

Each table is grouped by entry ratio (rows) from 0.2 to 2 in increments of 0.2 and then from 2 to 10 in increments of 1. The columns are the expected number of claims ([math]E[N][/math]).

Alice: "This is a really weird exhibit to have in the study kit as none of the other materials in the Circular refer to these tables. Make sure you know the rest of the NCCI circular material really well and then if they test you on these particular tables you should be able to logic your way through it."

Informational Exhibit 3

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