Fisher.ExpRating

Reading: Fisher, G., et al., "Individual Risk Rating Study Note" CAS Study Note Version 3 October 2019, Chapter 1

Synopsis: This is a fairly straight forward introduction to experience rating. Pay attention to the definitions, particularly for the quintiles test.

Study Tips

This reading can be thought of in three pieces. The first part is a quick grounding in general experience rating theory. The next part is a deeper dive into experience rating credibility - this can be tested in detail but doesn't come up that often. The last part is the Quintiles and Efficiency Tests which are tested heavily.

Estimated study time: 16 Hours (not including subsequent review time)

BattleTable

Based on past exams, the main things you need to know (in rough order of importance) are:

• Quintiles and Efficiency Tests - be able to apply the tests and interpret the results.
• Experience Modifications - be able to calculate modification factors, understand and interpret the results of experience rating.
• Split rating, credibility

Questions from the Fall 2019 exam are held out for practice purposes. (They are included in the CAS practice exam.)

reference	part (a)	part (b)	part (c)	part (d)
E (2018.Fall #9)	Efficiency Test - calculate	Interpret test results
E (2017.Fall #10)	Schedule Mod
E (2017.Fall #11)	Efficiency Test - calculate	Interpret test results
E (2016.Fall #9)	Experience Mod - understand	Experience Mod - understand	Experience Mod - understand
E (2016.Fall #11)	Quintiles Test - calculate	Experience Rating - understand
E (2015.Fall #10)	Split Rating - understand	Split Rating - credibility	Credibility - understand
E (2015.Fall #11)	Quintiles Test - calculate	Experience Rating - understand
E (2015.Fall #12)	Credibility - understand	Efficiency Test - calculate	Experience Rating - understand
E (2014.Fall #9)	Experience Mod - calculate
E (2013.Fall #9)	Experience Rating - calculate	Experience Rating - understand
E (2013.Fall #10)	Experience Rating - understand	Experience Rating - understand
E (2012.Fall #11)	Experience Rating - understand	Experience Rating - understand
E (2012.Fall #16)	Experience Rating - understand	Experience Rating - calculate	Experience Rating - calculate and interpret

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In Plain English!

Overview

Experience rating is the use of an insured's past loss experience to determine rates for a future exposure period. An experience rating plan is the set of rules, definitions, formulas, etc. which are needed to calculate such a rate. An experience rating plan produces a multiplicative factor called the experience modification or e-mod. The experience modification factor corresponds to the individual insured and is used to adjust the manual premium which is the starting point premium for a whole class of insureds. Schedule rating goes hand in hand with experience rating and is used to reflect changes to the insured which are not yet seen in the underlying experience. Schedule rating produces a schedule modification.

The standard premium is the manual premium multiplied by the product of both the experience modification factor and schedule modification factor. If the schedule factor is non-unitary, then the manual premium multiplied by the experience modification factor is known as the modified premium. The standard premium is then the modified premium multiplied by the schedule modification factor. The first chapter of the Fisher paper assumes the schedule modification factor equals 1.000.

A debit mod is when the experience modification satisfies: [math]\mbox{e-mod}\gt 1.0[/math]. This implies the insured's experience is worse than average for its class.

A credit mod is when the experience modification satisfies: [math]\mbox{e-mod}\lt 1.0[/math]. This implies the insured's experience is better than average for its class.

The experience modification facilitates a comparison of the insured's expected loss relative to other risks in its class. Experience rating quantifies how much an insured's past loss experience is predictive of its future loss experience and uses this to rate prospectively with greater accuracy.

The Experience Rating Off-balance is the standard premium divided by the manual premium.

Question: What are some advantages of experience rating?

Solution:

1. Account for differences between risks within a class, such as
   • Locations
   • Equipment
   • Management styles
   • Use of materials
   • Safety processes and level of training.
2. Account for differences due to variables which are difficult, impractical, or impossible to measure using rating variables.
3. Additional refinement - more useful when there are few rating classes or classes have a broad range.

Experience rating is useful because not all risks fit neatly into a class. A risk may have unique operations, or the rating system may lack sufficient refinement to group appropriately. In fact, if the rating classification plan is highly refined so there is limited variation between risks within a class then experience rating is not very useful.

Question: What are the goals of experience rating?

Solution:

1. Greater risk equity: Increase fairness by charging rates that better match the expected loss(es) of the insured.
   • The size of the charge based on historical losses must reflect the extent it impacts future losses rather than attempting to recoup prior losses.
2. Create a safety incentive by placing a cost on losses. This adds a reason for the insured to prevent or mitigate losses.
   • Examples of measures an insured could be incentivized to introduce include: hiring a safety manager, developing a return to work program, etc.
3. Improve market competition: By better matching rate to risk, insurers should be encouraged to insure more risks as they can charge a more appropriate premium.

Gary Venter in his article "Experience Rating – Equity and Predictive Accuracy" says "... to the extent that the loss experience is indicative of true differences from the classification average, it appears equitable to charge for it." The experience modification is intended to reflect future loss experience, not to penalize or reward past experience.

Question: What are two things an effective experience rating plan should achieve?

Solution:

1. Identify differences between otherwise similar risks.
   • For instance, for Commercial General Liability policies for shopping centres then the experience plan should identify distinguishing characteristics such as location or size.
2. Adjust for the differences between risks in the same rating class.
   • For instance, if two shopping centres in the same class are the same size but one is an urban location and the other a suburban location then the experience rating plan should compensate for the difference.

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Credibility for Experience Rating

Experience rating treats each risk as its own risk class, so how much credibility should be given to the insured's individual experience when determining their premium?

While there are several methods for determining credibility, such as classical/limited fluctuation, Bayesian, or Bühlmann, experience rating only uses Bühlmann credibility.

Let [math]Z=\frac{E}{E+K}[/math] where K is a constant defined by the ratio [math]\frac{\mbox{Expected Process Variance}}{\mbox{Variance of Hypothetical Means}}[/math], and E is the size of the risk (level of expected losses). Process variance (EPV) is purely random and comes from the underlying stochastic process. The Variance of Hypothetical Means (VHM) is variation resulting from an individual risk being (slightly) different from the other risks in its class. The complement of credibility is the experience of the class containing the risk. The credibility assigned, Z, represents how much of the experience is due to the difference between risks within a class.

Question: What is the aim of using credibility weighting with experience rating?

Solution:

Aim is not to penalize an individual for random risk but encourage ownership of risk which is due to differences between the risk and other risks in its class.

The experience modification factor, M, may be expressed as [math]M=\frac{ZA+(1-Z)E}{E}=\ldots =\frac{A+K}{E+K}[/math], where A is the actual loss experience for the individual, and E is the expected loss experience for the class containing the individual.

Alice: "Remember the credibility, Z, must still satisfy the following three conditions:" [math] 0\le Z \le 1[/math] [math]\frac{\mathrm{d}}{\mathrm{d}E}(Z) \ge 0 [/math] (larger risks are more credible) [math]\frac{\mathrm{d}}{\mathrm{d}E}(\frac{Z}{E}) \lt 0 [/math] (credibility per exposure decreases as the size of risk increases)

Credibility Issues:

Maximum Single Loss (MSL)	This is the amount at which individual large losses are capped at when they are included in the calculation of a risk's experience.
Minimum and Maximum Adjustments	The experience modification factor is often subject to a maximum or minimum value to make sure the rating adjustment isn't too extreme.

The loss experience of larger risks receive greater credibility than that of smaller risks. Similarly, the maximum single loss and loss adjustments also vary with the size of the risk.

The size of the risk may be measured in various ways, such as manual premium, expected loss, expected number of claims, or using an exposure base such as sales receipts for general liability.

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Split Loss Plans

These plans separate individual claims from a risk's experience into two layers known as the primary and excess layers based on a split point such as $5,000 (see table below).

Claim Number	Incurred Loss Amount	Primary Loss Amount	Excess Loss Amount
1	1365	1365	0
2	6094	5000	1094
3	3438	3438	0
4	11179	5000	6179
5	2695	2695	0
6	5000	5000	0
7	52431	5000	47431

Loss distributions are often skewed by heavy tails, the best estimates of the expected loss often come from transforming or normalizing the distribution; the log transformation is often used. However, this can lead to complicated rating algorithms. Split rating is a good compromise between rating accuracy and simplicity. A split plan can be thought of as a linear approximation to assigning credibility to the log of the loss amount. They normally give more weight to the smaller portion of the losses.

We can calculate an expected value for the primary and excess losses. The primary layer can be viewed as the frequency of the loss experience, and the excess layer considered as the severity of the loss experience as follows. If the actual primary loss exceeds the expected loss then there must be more small claims than expected, i.e. a higher frequency. A large claim only contributes a fixed amount (loss is truncated above in the primary layer) so a small number of large claims shouldn't drive the primary loss above the expected. If the actual excess loss exceeds the expected then the large claim size must have been higher than thought, i.e. higher severity. This is because small claims won't pierce the layer, so a lot of small claims can't have this impact.

The National Council on Compensation Insurance (NCCI) Workers' Compensation experience rating plan is a split loss plan (see NCCI.ExperienceRating). It works better than a total or limited loss plan possibly because workers' compensation loss distributions have heavy tails. In other words, we can split the claim count uncertainty out from the severity uncertainty. The latter is driven by relatively speaking a handful of large major permanent partial (PP), permanent total (PT), and Fatal claims (see Couret.Venter). The claim count uncertainty is parameter risk, whereas the severity uncertainty is process risk.

Split loss plans require two measures of credibility, one for the primary layer (Z_p) and the other for the excess layer (Z_e). We'll use a subscript p for the primary layer and subscript e for the excess layer. We'll also use A to represent actual losses, and E to represent expected losses. We can write [math]A=A_p+A_e[/math] and [math]E=E_p+E_e[/math]. With this setup, the experience modification formula for a split rating plan is

[math] \begin{align} M &= \frac{Z_pA_p+(1-Z_p)E_p +Z_eA_e+(1-Z_e)E_e}{E}\\ &= 1+Z_p\frac{A_p-E_p}{E}+Z_e\frac{A_e-E_e}{E} \end{align} [/math].

The credibilities are given by [math]Z_p=\frac{E}{E+K_p}[/math] and [math]Z_e=\frac{E}{E+K_e}[/math].

The NCCI experience rating plan uses an equivalent formula that describes the components in terms of "weighting" and "ballast".

The NCCI experience modification factor is

[math] \begin{align} M &= \frac{A_p+(1-w)E_e+B+wA_e}{E_p+(1-w)E_e+B+wE_e}\\ &=\frac{A_p+(1-w)E_e+B+wA_e}{E+B}, \end{align} [/math]

where w is the excess loss weighting factor and B is the ballast value.

This is covered in much more detail in NCCI.ExperienceRating.

Schedule Rating

Schedule rating involves determining credits and debits which modify an insured's rates to reflect their individual risk based on characteristics not used in experience rating. It is normally used for workers' compensation and in commercial lines. An underwriter subjectively determines the credits and debits.

Plans which use schedule rating generally specify ranges for the modifications, such as "Training of the workforce: 5% credit to 5% debit". A well trained workforce might receive up to a 5% credit whereas a poorly trained one may receive say a 4% debit. All of the schedule debits and credits are summed together to get a total schedule debit or credit which may be capped. For an example of the categories and ranges used in a schedule rating plan, see the section on schedule rating within the ISO Rating plan

Question: Describe the overlap between experience and schedule rating

Solution:

If a risk made a recent change that should impact its future loss experience but is not reflected in its historical loss experience then apply a schedule credit/debit. For example: Hire a new safety manager who enforces workplace safety. This should reduce future losses so apply a schedule credit. However, if there had always been a safety manager then the loss experience already reflects the presence of a safety manager so it's not appropriate to use a schedule credit as it would double count.

Question: When could it be appropriate to use both schedule and experience rating?

Solution:

If a risk is too small to be fully credible then a small schedule credit/debit could also be given. Following on from the previous example, if there had always been a safety manager at a low credibility risk, the safety manager credit would be smaller than the credit if the safety manager position was newly created.

Pop Quiz    :-o

Briefly describe the following: Manual Premium Modified Premium Standard Premium Click for Answer

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The Quintiles Test

Important Note: For exam questions from 2016 and earlier, what was called the Dorweiler Test is now called the Quintiles Test. Further, what was called the Efficiency Test is no longer on the exam. So... for 2016 and earlier, if you run into the Efficiency Test, you can ignore that exam question. That test is no longer used. If you see the Dorweiler Test, then mentally substitute Quintiles Test. Post 2016, you have only two tests: a Quintiles Test (old Dorweiler Test) and Efficiency Test. The Efficiency Test is a totally new test and unrelated to the old Efficiency Test.

Here's an example of a straight forward exam question on this topic. Don't try to solve it for now - Alice has a set of solutions to help you break this problem down.

E (2018.Fall #9)

The Quintiles Test

1. Rank order the set of risks by the size of their experience modifications: rank from lowest to highest.
2. Collapse the risks into five (5) groups (this is the quintile part).
   • Compute the average experience modification. This is the weighted average of the experience modifications in the quintile, where the manual premiums are the weights.
3. Calculate the manual loss ratio for each quintile. That is, [math]\frac{\mbox{loss}}{\mbox{manual premium}}[/math].
4. Calculate the standard loss ratio for each group. That is, [math]\frac{\mbox{loss}}{\mbox{(average experience mod)}\cdot\mbox{(manual premium)}}[/math].
5. Observe any trends in the manual or standard loss ratios across the quintiles.

If premiums are not available then use [math]\frac{\mbox{actual losses}}{\mbox{expected losses}}[/math] for the manual loss ratio, and use [math]\frac{\mbox{actual losses}}{\mbox{modified expected losses}}[/math] for the standard loss ratio. Here, modified expected losses means the experience modification multiplied by the expected losses.

Let's see this in practice:

Applying the Quintiles Test in 2018.Fall #9

Here are two practice problems to test your understanding:

2 practice problems like 2018.Fall #9

Interpreting the results of a Quintiles Test

Dispersion is the difference between the maximum and minimum value. We can look at the dispersion of the manual loss ratio and the standard loss ratio, each tells us something different.

Question: What does an upward trend in the manual loss ratio signify?

Solution:

If the manual loss ratio increases as the average modification factor increases then the experience rating plan identifies differences between risks.

Question: If the experience rating plan adjusts for differences between risks, what should the standard loss ratios look like?

Solution:

If the experience rating plan is working properly, the standard loss ratios should be less dispersed than the manual loss ratios. There should also be no noticeable trend in the standard loss ratios.

Question: What does a downward trend in the standard loss ratio signify?

Solution:

Risks with the best historical loss experience now have higher loss ratios than risks with worse past loss experience. This means the experience rating plan gives too much credibility to the risk's actual loss experience. The risks with the lowest modifications get more credit than their past loss history is predictive of future loss, and risks with the highest modifications get penalized too much.

Question: What does an upward trend in the standard loss ratio signify?

Solution:

The experience rating plan does not give enough credibility to the risk's actual loss experience. Good risks don't get enough credit while bad risks are not penalized enough.

Here's another two practice problems for you to try interpreting the results of the quintiles test.

2 more practice problems like 2018.Fall #9

Summary

Ideally the standard loss ratios should be equal or at least close to flat. We should also see a wide difference in manual loss ratios which would indicate the experience rating plan is good at identifying differences in risks.

Let's give this a try:

Evaluating the results of a Quintiles Test

And here are two practice problems.

2 practice problems like 2011.Fall #16

The Efficiency Test

The efficiency test is used to compare between experience rating plans. For each experience rating plan, first perform the Quintiles Test. Then calculate the ratio of the sample variance of the standard loss ratio to the sample variance of the manual loss ratio. The variances are calculated across all five quintiles. The ratio is known as the efficiency test statistic.

The plan with the lower efficiency test statistic (lower variance ratio) is "better". It does a better job of making risks of differing loss experience more equally desirable - i.e. does a better job of matching rate to risk.

Let's see this in practice:

Alice's Efficiency Test for 2018.Fall #9 with two practice problems

Important! The quintiles test is a qualitative way of comparing experience rating plans whereas the efficiency test is a way of quantifying the effectiveness of the plans.

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Pop Quiz Answers

Manual Premium	This is the premium calculated based on the criteria found in the rating manual. It is the premium before applying either experience or schedule rating.
Modified Premium	This is the manual premium multiplied by the experience modification factor.
Standard Premium	This is the modified premium multiplied by the schedule rating factor. If schedule rating is not used then this is the same as the modified premium.

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