Werner12.Credibility

Reading: BASIC RATEMAKING, Fifth Edition, May 2016, Geoff Werner, FCAS, MAAA & Claudine Modlin, FCAS, MAAA Willis Towers Watson

Chapter 12: Credibility

Pop Quiz

Which of these are ITV (Insurance-to-Value) initiatives:

• insurers offer GRC (Guaranteed Replacement cost)

→ allows replacement cost to exceed the policy limit if the property is 100% insured to value

• insurers use more sophisticated property estimation tools

→ so insurers know more accurately whether a property is appropriately insured

• Ian-the-Intern brings you coffee when you get to work in the mornings

→ so you can examine policy forms with more energy and focus

Click for Answer 

Study Tips

This chapter begins with some material on the theory of credibility that has been covered extensively in prior exams. Since it's all review, you can get through the "Intro" section pretty quickly.

The new material for you is on methods for finding complements of credibility. There are a lot of them, 10 altogether, but the source text is well-organized and the wiki follows the organization of the text. Most of the methods are not difficult but Harwayne's method is an exception and you'll need to allocate a full day to learning it (not including review.)

The ranking table lists this chapter as Bottom 20% according to how frequently it comes up on the exam. Credibility does comes up however in many problems where you have to calculate an indication, but it's usually just a small part of the problem hence the low ranking.

A specific problem on Harwayne's method, for example, comes up far less often but when it does, it's usually worth quite a few points. That means that unless you are very pressed for time, you've got to take the time to learn it. And learn it solidly because it's very confusing and easy to mess up when you're under pressure on the exam.

In any case, the single most important thing to memorize is the 6 desirable qualities of a complement of credibility. Each of the 10 methods is then evaluated according to these 6 qualities but if you understand the methods, you don't necessarily have to memorize how each method performs individually. You can figure it out based on common sense. You should at least know however, what the best and worst methods are, and this is covered in the wiki article.

Estimated study time: 3-4 days (not including subsequent review time)

BattleTable

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

• desirable qualities for a complement of credibility
• Harwayne's method - for calculating the complement of credibility
• trended present rates - for calculating the complement of credibility
• limits analysis - for calculating the complement of credibility

Let C of C stand for Complement of Credibility

reference	part (a)	part (b)	part (c)	part (d)
E (2019.Fall #3)	trended present rates - calculate C of C	C of C - first $ ratemaking
E (2019.Fall #12)	Harwayne's method - calculate C of C	current rate as C of C - advantages
E (2019.Spring #7)	Werner08.Indication	loss ratio method - credibility-weighted	disadvantage - using competitor rates
(2018.Spring #2)	Excel Practice Problems
(2018.Spring #4)	Excel Practice Problems
E (2017.Spring #9)	Harwayne's method - calculate C of C	Harwayne's method - appropriateness	C of C - desirable qualities
E (2015.Fall #3)	role of credibility - in ratemaking	C of C - desirable qualities
E (2015.Spring #11)	trended present rates 1 - calculate C of C
E (2014.Fall #8)	C of C - recommend	C of C 1 - use in indication
E (2014.Spring #7)	capped losses - calculate C of C in layer	limits analysis - calculate C of C in layer	limits analysis - disadvantages

¹ This problem is part of a much larger problem on calculating the indicated rate change.

Full BattleQuiz

In Plain English!

Introduction to Credibility

Necessary Criteria for Measures of Credibility

Intuitively, credibility is a measure of predictive value attached to a particular body of data. The necessary criteria for measures of credibility was covered on earlier exams but it doesn't hurt to review them quickly.

• Let Z a measure of credibility
• Let Y be number of claims

Then the 3 criteria are:

• 0 ≤ Z ≤ 1
• Z is an increasing function of Y (first derivative of Z with respect to Y is positive)
• Z increases at a decreasing rate (second derivative of Z with respect to Y is negative)

You shouldn't be specifically asked to list these criteria on the exam but it's good to keep them in mind.

General Methods for Measuring Credibility

Classical Credibility

For classical credibility, the definition of full credibility is when:

• there is probability p that the observed value will differ from the expected value by less than k

We're going to skip the theoretical derivation and simply state the final formula for full credibility:

For example, if p = 90% and k = 0.05 then we look up (p+1)/2 = 95% on the normal distribution table. The value of 95% corresponds to a standard deviation of 1.645 so that

E(Y) = (1.645 / 0.05)² = 1,082

If the number of claims Y ≥ 1,082, then our data is fully credible and Z = 1.0. If Y < 1,082 then the formula for the credibility Z is the square root rule:

And if we had Y = 500, the result for our example is:

Z = (500 / 1,082)^½ = 0.680

Then our final estimate of of whatever quantity we're interested in is:

estimate (Classical)   =   Z x (observed value)   +   (1 - Z) x (related experience)

Part (a) of the follwing exam problem required you to do the above calculation to find to the credibility of the given data. It's part of a larger problem where you had to calculate the indicated rate change, but you should take a few minutes to do just the credibility part.

E (2019.Fall #3)

If you'd like a little more practice, here's a table from the text. You don't have to do the lookup on the normal distribution table but make sure you understand how to calculate columns (4) & (6). Note: Column (6) is for when exposures are used to define the full credibility standard instead of number of claims. In this case, you divide the numbers of claims by the frequency (as shown in the table.)

The last thing you need to know about classical credibility is...

Question: identify advantages (3) and disadvantages (2) of the classical credibility approach

advantages

• commonly used and generally accepted (by regulators)
• required data is readily available
• easy to compute

disadvantages

• makes simplifying assumptions (Ex: no variation in size of loss)
• doesn't consider quality of estimator compared to the latest observation and therefore judgment is required

Buhlmann

An important difference between Buhlmann and classical credibility is how the estimator is calculated. Instead of related experience, we substitute prior mean as shown below.

estimate (Buhlmann)   =   Z x (observed value)   +   (1 - Z) x (prior mean)

Of course, Z is calculated differently as well. It's less likely you'll be asked to use Buhlmann credibility on the exam, but you should know the formulas. Suppose

• N is the number of observations
• EVPV is the Expected Value of the Process Variance
• VHM is the Variance of the Hypothetical Mean

Then

• Z   =   N / (N + K)

where

• K   =   EVPV / VHM

As an incredibly simple example: (Even Ian-the-Intern thinks it's easy.)

• observed pure premium is $300 based on 55 observations
• EVPV is 2.50
• VHM is 0.40.
• prior mean is $280

Then

• K   =   6.25
• Z   =   55 / (55 + 6.25)   =   0.898

And our estimate of the pure premium is

• estimate (Buhlmann)   =   0.898 x $300 + (1 - 0.898) x $280   =   $298.0

And we'll finish up with this:

Question: identify advantages (1) and disadvantages (2) of the Buhlmann credibility approach

advantages

• generally accepted (less commonly used than classical credibility however)

disadvantages

• EVPV and VHM may be difficult to compute
• makes simplifying assumptions (Ex: risk parameters and risk process do not shift with time)

Bayesian

Bayesian credibility is not discussed in enough detail for you to be able to do the calculation. You do not calculate a Z-value. Instead, you use Baye's theorem to incorporate new information into the prior estimate.

Desirable Qualities of a Complement of Credibility

Recall the formula for the classical estimate from earlier:

estimate (Classical)   =   Z x (observed value)   +   (1 - Z) x (related experience)

Desirable qualities of a complement of credibility listed in the text pertain to related experience. Strictly speaking, the qualities listed below apply only when classical credibility is used. Buhlmann credibility is supposed to use the prior mean, but this is not always done in practice. Sometimes Buhlmann uses related experience. Anyway...

Question: identify desirable qualities for a complement of credibility [Hint: AU-SALE - Australian Sale - slip another shrimp on the barbie!]

• Accurate - low error variance around the future expected losses being estimated
• Unbiased - complement should not be routinely higher or lower than the observed experience (which is different from being "close" to observed experience)

–

• Statistically independent from the base statistic - otherwise an error in the base statistic could be compounded
• Available - should be easy to obtain
• Logical relationship to base statistic - easier for a third party to understand (Ex: regulator)
• Easy to compute (so that even Ian-the-Intern can do it!)

Just memorize this list. You had to know it for the following 2 exam problems. You can take a look at them but you can't fully answer them yet because you need to know Harwayne's method for the first problem, and for the second problem, some knowledge of first-dollar methods for coming up with a complement of credibility from later in this chapter.

E (2017.Spring #9) part (c)

E (2015.Fall #3) Part (b)

And here's the introductory quiz.

mini BattleQuiz 1

Method for Developing Complements of Credibility: FIRST DOLLAR RATEMAKING

First dollar ratemaking is for products that cover claims from the first dollar of loss (or after some small deductible) up to some limit. Examples of first-dollar products are personal auto and homeowners. We generally use historical losses for the base statistic then select a complement from one of these 6 choices:

The first 2 methods don't come up very often:

• Loss Costs of a Larger Group that Include the Group being Rated
• Loss Costs of a Larger Related Group

These 3 methods tend to appear on the exam:

• Rate Change from the Larger Group Applied to Present Rates
• Harwayne’s Method
• Trended Present Rates

This method is terrible and is used only for new and/or small companies:

• Competitors’ Rates

Harwayne's method can be a little tricky. Trended present rates may require you to calculate trends or trend periods for losses which you can review by clicking the links. The other 4 methods don't generally require calculations as that would be given information. You just have to know when each is appropriate. You'll need to keep in mind AU-SALE, the desirable qualities of a complement of credibility.

Loss Costs of a Larger Group that Include the Group being Rated

Typical example: multi-state personal auto insurer uses countrywide data as a complement to the state being reviewed.

Countrywide data is higher volume thus more credible while still having a connection to the particular state. Let's evaluate countrywide data as a complement according to AU-SALE.

• Accurate?

→ probably not - if an insurer is active in many, there are likely real differences in loss costs between states so countrywide data would only be accurate if the state being reviewed is an "average" state

• Ubiased?

→ probably not - countrywide data will likely either be consistently high or consistently low unless the state being reviewed is an "average" state

• Statistically independent?

→ probably not - because the state being reviewed is part of the countrywide total - the bigger the state, the less independent

• Available?

YES! - because it's the company's own data

• Logical relationship to base

YES! - because countrywide data includes this state and the data is all from the same company

• Easy to calculate

YES! - there is nothing to calculate - the countrywide data should be available in the company's database

Loss Costs of Larger Group that Include the Base Group: scores 3 out of 6 desirable qualities - not a great method

Loss Costs of a Larger Related Group

Typical example: homeowners insurer uses "contents" loss data from house-owners to complement "contents" loss data for condo-dwellers

Loss experience for house-owners could be higher volume and more credible in a region where single-family houses predominate. House-owners and condo-owners are still related because they are both homes. Let's evaluate house-owners data as a complement according to AU-SALE.

• Accurate?

→ probably not - homeowners probably have more contents so loss experience would be different (higher)

• Ubiased?

→ depends - there likely is bias on the high side but if the bias is consistent, it can be corrected

• Statistically independent?

YES! - because house-owners data and condo-owners are separate

• Available?

→ depends - in this example it is available, but in general a larger related group may not be available

• Logical relationship to base

YES! - because it's a related group (same coverage, same company)

• Easy to calculate

YES! - there is nothing to calculate - the house-owners data should be available in the company's database

Loss Costs of Larger Related Group: scores 4 out of 6 desirable qualities (assuming depends counts for ½) - better than the previous method but still not great

Rate Change from the Larger Group Applied to Present Rates

Typical example: homeowners insurer uses "contents" loss experience rate change from house-owners to complement "contents" loss experience rate change for condo-dwellers

Did you spot the difference in the above example from the previous section. Both pertain to house-owners and condo-owners but here we use the rate change instead of the actual loss data. Using the rate change turns out to be a very good complement. Let's evaluate this complement according to AU-SALE.

• Accurate?

YES! - same coverage, same company, so likely to be accurate over the long-term assuming rate changes are relatively small

• Ubiased?

YES! - same coverage, same company, so over time the average rate change for house-owners is likely to be the same as for condo-owners

• Statistically independent?

YES! - because house-owners data and condo-owners are separate

• Available?

YES! - in this example it is available, but in general a larger related group may not be available

• Logical relationship to base

YES! - because it's a related group '(same coverage, same company)

• Easy to calculate

YES! - there is nothing to calculate - the house-owners data should be available in the company's database

Rate Change from Larger Group: scores 6 out of 6 desirable qualities - this is a very good method when used properly

And here's a nice simple example from Werner:

Harwayne’s Method

Typical Example: Harwayne’s Method is used when the subject experience and related experience have significantly different distributions, and the related experience requires adjustment before it can be blended with the subject experience.

For Harwayne's method, we're first going to look at Alice's example and then evaluate it according to the 6 desirable qualities of a complement of credibility. Warning: It's very easy to get all discombobulated when doing Harwayne's method. Go slowly. Go carefully. You might even come up with a way organizing your solution that you like better than Alice's.

Here's the question. (Note that it's really 2 questions but using the same data)

Here's the solution to the first question: State A, Class 1.

Here's the solution to the second question: State C, Class 2.

Ok, going through that was probably not a fun experience. Before I give you the practice problems, let's evaluate Harwayne's method as a complement according to AU-SALE.

• Accurate?

YES! - uses data from multiples segments so process variance should be minimized as long as volume in those segments is high enough

• Ubiased?

YES! - because it adjusts for distributional differences between classes

• Statistically independent?

YES! - base data and subject data come from different segments of data (even though it's the same company)

• Available?

YES! - usually available

• Logical relationship to base

YES! - but might be hard to explain because of the complexity of the method

• Easy to calculate

No - calculations are time-consuming and complicated

Harwayne: scores 5 out of 6 desirable qualities - Harwayne is almost as using the good as rate change from larger group applied to smaller group

Here are 2 practice problems just like the example. (It's really 4 practice problems altogether.)

Practice: 2 Harwayne's Method problems

Trended Present Rates

Typical example: when no larger group exists to use as a complement

The formula for the complement C is:

Now, we need to discuss the terms in this formula. It's a lot easier than Harwayne's method but you still have to be careful when applying it.

• present rate

- this is known because it's the "base"

- the present rate is being adjusted/trended to find a suitable complement for this base

• loss trend factor

- changes in cost levels may have occurred between the time the current rates were implemented and the time of the review

- this change has to be accounted for by trending the current rate

- for the trend period, use the original target effective date of the current rates for the "trend from" date

• (prior indicated loss cost) / (loss cost implemented with last review)

- this ratio is required because the implemented rate change is not always the same as the indicated rate change

- the complement should be based on the original indicated rate change and this ratio makes that adjustment

Once you have all the pieces, it's just a matter of substituting into the formula. You might need to review how to find the loss trend period. If so, just click the link in the previous sentence.

Here's a simple example. Suppose you're given:

• present average rate = $300
• selected loss trend = 4%
• indicated rate change from last review = 8%, and the target effective date was April 1, 2020
• implemented rate change from last review = 5%, and the actual effective date was May 15, 2020
• proposed effective date of next rate change is April 1, 2022

Then we calculate the complement using the above formula:

• C   =   $300 x (1.04)² x 1.08 / 1.05   =   $334

Note that to calculate the trend period of 2 years, we used the original target date and the proposed effective date. The actual effective date of the prior change is irrelevant. Let's evaluate trend present rates as a complement according to AU-SALE.

• Accurate?

YES! - provided current rates were based on high-volume data (so that process variance is low)

• Ubiased?

YES! - because pure trended costs are unbiased (i.e. no updating for recent experience)

• Statistically independent?

depends - unfortunately, "trended present rates" probably fails this test due to overlapping experience periods between the data current rates were based on the the data proposed rates are based on (maybe current rates were based on data from 2017-2019 and proposed rates are based on data from 2018-2020)

• Available?

YES! - unless your company has a very disorganized filing system the current rates should always be available :-)

• Logical relationship to base

YES! - obvious (it's related to the base by the above formula), and super-easy to explain to muggles like regulators :-)

• Easy to calculate

YES! - yup, pretty easy, if you remember how to calculate the loss trend period

Trended Present Rates: scores 5 out of 6 desirable qualities - PRETTY GOOD! - or maybe 5½ if depends counts for ½ because statistical independence might sometimes be satisfied

Competitors’ Rates

Typical example: new or small companies with small volumes of data often find their own data too unreliable for ratemaking.

Competitors likely have higher volume if they've been in existence for a longer period. The competitors’ data would then be more credible and have less process error but the only quality they possess from AU-SALE is Statistical independence.

There are many ways a competitor's data would not be appropriate for a complement. A competitor may have different U/W guidelines, mix of business, limits, deductibles, profit provision, and potentially others. In other words, it would not have a logical relationship to the base. It's a terrible method. New and/or small companies use it because they have no other choice.

Competitors' Rates: scores 1 out of 6 desirable qualities - a terrible method!

Summary

mini BattleQuiz 2

Harwayne's method:

mini BattleQuiz 3

Trended Present Rates:

mini BattleQuiz 4

Misceallaneous:

mini BattleQuiz 5

Full BattleQuiz

POP QUIZ ANSWERS

Below are 2 of the 5 ITV initiatives from Pricing - Chapter 11 - Special Classification. Ian-the-Intern is not an ITV initiative.

• insurers offer GRC (Guaranteed Replacement cost)

→ allows replacement cost to exceed the policy limit if the property is 100% insured to value

• insurers use more sophisticated property estimation tools

→ so insurers know more accurately whether a property is appropriately insured

And the other 3 from Pricing - Chapter 11 - Special Classification are:

• insurers educate customers:

→ coverage is better if F/V is closer to 100% (for both insurer and insureds)

• insurers inspect property and use indexation clauses

→ more information allows insurers to better price a policy

→ indexation clauses ensure the face value of a policy keeps up with the value of the home

• insurers use a coinsurance clause

→ assigns a penalty if the coinsurance requirement is not met