07 Texas Klugman 28

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Credibility TheoryValuation Actuary Symposium

TS 28

Stuart KlugmanDrake University and Society of Actuaries

September 18, 2007

Stuart Klugman () Credibility Theory September 18 , 20 07 1 / 6 2

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Table of Contents

1 Credibility for Valuation Actuaries

2 A History of Credibility

3 Types of CredibilityLimited Fluctuation Credibility

Greatest Accuracy CredibilityCredibility Example

4 Credibility for Mortality Ratios

A Limited Fluctuation ApproachAn ExampleConclusionsA Non-credibility Approach


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Credibility for Valuation Actuaries



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Why do you care?

The NAIC proposed AG for VACARVM uses "credibility"

in several places:The deterministic assumptions to be used forprojections are to be the actuary’s Prudent BestEstimate. This means that they are to be set at theconservative end of the actuary’s con…dence intervalas to the true underlying probabilities for theparameter(s) in question, based on the availability

of relevant experience and its degree of credibility.Document the mathematics used to adjust mortalitybased on credibility and summarize the result of applying credibility to the mortality segments.


C dibili f V l i A i

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Why do you care?

More from the NAIC AG

The report section shall show any experience dataused to develop the assumptions and describe thesource, relevance and credibility of that data. If

relevant and credible

data was not used, the reportsection should discuss how the assumption isconsistent with the requirement that the assumptionis to be on the conservative end of the plausible

range of expected experience. The expectedmortality curves are then adjusted based on thecredibility of the experience used to determine theexpected mortality curve.


C dibilit f V l ti A t i

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Why are you here?

None of the above statements de…nes credibility.

None of the above statements advocates aparticular credibility method.



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Isn’t there an ASOP on credibility?

There is ASOP 25 - Credibility Procedures Applicable to

Accident and Health, Group Term Life, andProperty/Casualty Coverages. It does o¤er somede…nitions:

De…nitionCredibility is a measure of the predictive value in agiven application that the actuary attaches to aparticular body of data.

De…nitionFull credibility is the level at which the subjectexperience is assigned full predictive value based on a

selected con…dence interval.Stuart Klugman () Credibility Theory September 18 , 20 07 7 / 6 2


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ASOP

Some more from the ASOP:

The purpose of credibility is to blend informationfrom the subject experience and related experience.

There are various methods of credibility and severalshould be considered. A good method is reasonable,not materially biased (more from me later), is

practical, and balances responsiveness and stability.



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ASOP

Credibility requires informed judgment and is not aprecise mathematical process.

Classical credibility (sometimes called limited‡uctuation) is acceptable.

Empirical credibility (use the data but no underlyingmodel) is acceptable.



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ASOP

Bayesian credibility (sometimes called greatestaccuracy, uses a prior distribution and least squares(more from me later)) is acceptable.

Partial credibility can use the square root or

n/(n+k) rules.Once again, observe that there are no speci…c guidelinesand no formulas.

The purpose of this session is to …ll thesegaps and leave you with something you

can use.Stuart Klugman () Credibility Theory September 1 8, 2 00 7 1 0 / 6 2

A History of Credibility

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Stuart Klugman () Credibility Theory September 1 8, 2 00 7 1 1 / 6 2


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Introduction

Credibility as we know it today dates at least to 1914

(Mowbray, PCAS ) and by 1918 (Whitney, PCAS ) bothmethods used today were developed in the context of workers compensation insurance. Both begin with anexposure measure (such as lives or dollars) that is aproxy for the volume of data). Then:

Limited ‡uctuation - Somehow, determine when theexposure is enough to let the data speak for itself. If

not, weight the data using the square root of theratio of actuarial exposure to exposure needed forfull credibility.





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Introduction

Greatest accuracy - The weight is n/(n + k ) wheren is the exposure and k is somehow determined.

Full credibility is never achieved.

In both cases, the complement of the weight is appliedto what you would use if you had no data.



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Little progress until 1968

Both methods were used, but attempts to provide

statistical justi…cation were lacking. In 1950, ArthurBailey wrote (paraphrased by me):

While there seems to be some hazy logic behind the

method, it is too obscure to understand. The trained

statistician cries "absurd!" Actuaries admit they have

gone beyond anything proven mathematically. The only

thing they can do is demonstrate that in actual practice,

it works.History has shown that those who use credibility, even if they don’t understand it very well, do better than thosewho don’t.



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What does "do better" mean?

In any statistical estimation problem doing well, better,or best refers to the quality of the process employed, nota particular number obtained a particular time. A typicalmeasure of quality is mean squared error. This is the

squared di¤erence between the estimated and truevalues, averaged over all possible outcomes.Reminder:

Mean squared error = Variance + bias2



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Isn’t your speaker a trained statistician?

Yes, and my …rst encounter with credibility caused me tocry "absurd!"You and I learned that when estimating the mean, usethe sample mean. It is unbiased, consistent, sometimes

e¢cient. The credibility estimator is clearly biased(despite the ASOP). So how come it works?By surrendering some bias - variance is reduced and

hence it is possible to reduce mean squared error. This

can only work if we do this more than once so that

overall the biases will cancel.



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Modern results

In 1968 Hans Bühlmann (ASTIN Bulletin) used aBayesian least-squares argument to derive the n/(n + k )

version of greatest accuracy credibility.There has never been a legitimate derivation of limited‡uctuation credibility (but remember, it works).


Types of Credibility

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Types of Credibility Limited Fluctuation Credibility


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Types of CredibilityLimited Fluctuation



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Full Credibility

De…nition

Full Credibility is assigned to a data set when theprobability that the relative error in the estimate is lessthan r is at least p .

This is a con…dence interval approach in that it assignsfull credibility when a 100p % con…dence interval for therelative error has a width that is less than 2r . Forexample, we may want to be 90% con…dent that the

relative error is less than 5%. Then p = 0.9 andr = 0.05.Note - the exposure method (dollars, lives, etc) is notrelevant. What you use for an estimator is relevant.


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Three assumptions

1 There are enough terms in the sum for the CentralLimit Theorem to hold,

2 The amounts of insurance are not random, and3 The lives are independent and have the same value

of q .



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Example, continued

Then q̂ has a normal distribution with mean andvariance:

E (q̂ ) = ∑

1000 j =1 b j E (d j )

∑ 1000

j =1 b j = ∑

1000 j =1 b j q

∑ 1000

j =1 b j = q

Var (q̂ ) = ∑

1000 j =1 b

2 j Var (d j )

∑

1000

j =1 b j 2 =

∑ 1000

j =1 b 2

j

∑

1000

j =1 b j 2 q (1 q ).



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Example, continued

Let σ 2 be the variance and then,

Prq̂ q

q < 0.05 = Pr(0.05q < q̂ q < 0.05q )= Pr

0.05q σ

< Z <0.05q

σ

where Z has the standard normal distribution. We thenlook up this probability and see if it exceeds 0.9.





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Example, concluded

To calculate the probability, we need the value of q . It iscustomary to use q̂ . Suppose the outcome was asfollows:There were 200 policies with b = 10, 000 and 3 died,

300 with b = 25, 000 and 7 died, 400 with b = 50, 000and 8 died, and 100 with b = 100, 000 and 3 died.Then we have q̂ = 905, 000/39, 500, 000 = 0.022911and σ 2 = 2.2075

1012(0.022911)(0.977089)/(3.95

107)2 = 3.1673 105. The probability statementbecomes Pr(0.20355 < Z < 0.20355) = 0.1613 < 0.9and the data are not fully credible.



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Partial credibility

The weight, Z , is conventionally determined as follows.There are several justi…cations for this result, but all areunsatisfactory.

1 Determine the minimum exposure needed for fullcredibility.

2 The weight is the square root of the ratio of the

actual exposure to the exposure from step 1.



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Example

ExampleDetermine the weight for the previous example

The goal is to get the right hand term in the probability

statement to be 1.645. The equation to solve is:

1.645 = 0.05q

σ =

0.05q ∑ n

j =1 b j

hq (1 q ) ∑

n j =1 b 2 j

i1/2 .



E l d

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Example, continued

The sums go to n because this does not represent our

sample, it represents a sample that would deserve fullcredibility. We cannot work with this because theb -values are not known for a general sample. Assumethey are proportional to those in our sample. Dividing by

1,000 and substituting the sample value of q ,

1.645 = 0.05(0.022911)(39500n)

[0.022911(0.977189)(2.2075

109)n]1/2

= 0.0064365n1/2

n = 65, 318.



E l i d

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Example, continued

This leads to Z = (1000/65318)1/2

= 0.1237. There isan easier way to do this. The answer is simply0.20355/1.645. Also note that had we measuredexposure as dollars, the result would also be the same.

Our estimate now has a weight. What about thecomplement? Limited ‡uctuation credibility says to giveit to the quantity you would use if you had no data.Maybe that is an industry table, maybe it is yourcompany experience on a similar group. The methodo¤ers no advice.



Li it d ‡ t ti dibilit

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Limited ‡uctuation credibility

PROS:Good for experience rating, where there is a defaultpremium.

Simple to implement and understand.CONS:

Re‡ects only reliability of data, not of base rate.

May not have an obvious base rate.No sound statistical justi…cation.


Types of Credibility Greatest Accuracy Credibility

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Greatest Accuracy



Statistical model

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Statistical model

1 Each person, group, block (whichever applies) has adistribution that is governed by a parameter θ.

2 The parameter θ varies randomly from group togroup.

3 Based on n observations from a group, let m̂ be anestimator of the mean.

4 Pick m̂ to minimize E [ m̂

m(θ)]2 where m(θ) is

the true mean when the parameter is θ and theexpectation is taken over the data and θ.



Myths

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Myths

Myth #1 - This is a Bayesian analysis.No, the distribution of θ is not a prior distribution. It isnot an opinion. It is a true (though unobservable)

distribution.Myth #2 - This is a linear analysis where the answermust be Z x̄ + (1 Z )µ. The form of the answerdepends on the distributions over which the expectation

in step 4 is taken.



Bühlmann credibility

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This is the linear version. Start by assuming the answer

must be of the form in Myth #2. Then the value of Z that minimizes squared error is

Z = n

n + k k =

E [Var (X jθ)]Var [E (X jθ)]

Notes: As n increases Z increases. As the numerator of k increases, there is less credibility because the sample datais less reliable. As the denominator of k increases, thereis more credibility because µ is less likely to be useful.




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The numerator of k is called the process variance. It isessentially the same as the variance used in limited‡uctuation credibility and plays the same role.

The denominator is the new part. It measures how oneperson/group/block di¤ers from others. When datacomes from only one group, there may be no way toestimate this quantity.


Types of Credibility Credibility Example

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An Example



A baseball example

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A baseball example

Greatest accuracy credibility cannot be done in the

context of the previous example. The reason is that wehave no information about how q varies from group togroup. So, I will provide a non-actuarial example toillustrate this method.

ExampleAs of May 30, 2006, 77 National League batters had 175or more plate appearances. Their batting averages

ranged from Miguel Cabrera (.346) to Clint Barmes(.191). Estimate the 77 season-ending averages andcompare the answers to the end-of-season actualaverages.



Results using no credibility



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Results using no credibility

The traditional estimate is to use the sample mean(current batting average) to estimate the …nal average.Use a weighted squared error measure to see how we did.

The weights are the …nal number of at-bats.For this estimate, the result is 31.418.Of 40 who were above average to start, 30 had theiraverages drop. Of 37 who were below average to start,

27 had their averages increase.



Results using limited ‡uctuation credibility

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Results using limited ‡uctuation credibility

Using the sample mean and variance, 700 at-bats areneeded for full credibility.

The weighted squared error is 16.887.Gaming the system, we could identify the standard forfull credibility that would give the best result. It is 1,016at-bats and the squared error is 16.546.



Results using greatest accuracy credibility

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esu ts us g g eatest accu acy c ed b ty

A beta distribution was used to model how battingaverages vary from player to player. Method of momentsestimation set k as 653. For that value the weighted

squared error is 18.492. There are other (includingnonparametric) methods for estimating k .The optimal (post-results) value of 16.564 is achieved atk = 245.



Conclusions

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Using credibility helped considerably

The two methods performed about equally well

We could use greatest accuracy credibility becausewe had information about how batting averages varyfrom player to player.


Credibility for Mortality Ratios

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The estimator


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Let

m̂ = ∑

g i =1 ∑

ng j =1 b ij f ij d ij

∑ g

i =1∑

ng

j =1

b ij f ij q s

i

= ∑

g i =1 ∑


e

be the estimated mortality ratio. The denominator, e , isthe known expected number of deaths. With no datawould set m = 1 and use the standard table. Howcredible is m̂?


Credibility for Mortality Ratios A Limited Fluctuation Approach

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A Limited Fluctuation Approach



First two moments

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We …rst need the moments of the estimator:

µ = E ( m̂) = E

∑ g i =1 ∑


e

!

= ∑

g

i =1∑

ng

j =1

b ij f ij q i

e

σ 2 = Var ( m̂) = Var

∑

g i =1 ∑


e

!

= ∑

g i =1 q i (1 q i ) ∑ (b ij f ij )2

e 2 .



Estimating the moments

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It is less volatile to use the standard table values asadjusted by the morality ratio to estimate each q i :

µ̂ =m̂ ∑

g i =1 q

s i ∑

ng j =1 b ij f ij

e = m̂

σ̂ 2 = ∑

g i =1 m̂q

s i (1 m̂q s i ) ∑ (b ij f ij )2

e 2

Alternatives would be to use the sample q i values are thestandard table q s i values without adjustment.



Obtaining Z

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As noted earlier, the key is to standardize the variable

and thus look at (recalling that r is the error tolerance)

ẑ = r m̂

σ̂ .

If ẑ is below 1.645 (for 90% con…dence), thenZ = ẑ /1.645 else it is 1. Then the …nal value for themortality ratio is

Z m̂ + (1 Z )


Credibility for Mortality Ratios An Example

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Crediblity for Mortality Ratios

An Example



The data

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11,370 males from ages 70 through 100 who hadpurchased charitable trust annuities were studied (thanksto Don Behan for supplying the data, someapproximations were made for simpli…cation). The data

supplied counted life-years and number of deaths at eachage. Because individual data was not available, values of b and f in the formulas were both set to 1. There were782.67 expected deaths (US 2000 annuity table) and 744actual deaths for a mortality ratio of 0.9506.



Limited ‡utuation calculation

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The variance is 0.0011. Set the full credibility standardas being within 5% of the true ratio 95% of the time.The standardized variable isẑ = 0.05(0.9506)/

p 0.0011 = 1.432. This is below 1.96

and therefore full credibility cannot be granted.The partial credibility factor is Z = 1.423/1.96 = 0.7306and then the credibility estimate for the mortality ratio is

0.7306(0.9506) + 1 0.7306 = 0.9638.


Credibility for Mortality Ratios Conclusions

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Conclusions



The Good

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This method is easy to use. It is likely the datarequired are available and if not, a reasonableapproximation may be.

It is not terribly arbitrary (as long as bounds on thetolerance and probability are agreed upon).

It is credibility and thus "it works."



The Bad

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It is not as scienti…c as greatest accuracy. Somepurists may object.

It is somewhat arbitrary.

Why does 1 Z get to multiply 1?



The Ugly

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Maybe this is not a credibility problem after all.We are trying to estimate only one thing. We donot care if over all companies the error is reduced,

only the error for our company.Won’t there be times when 1 is not the rightstarting point? Maybe our block is better (or worse)for reasons that are independent of the data

(marketing, underwriting distrinctions, etc.)



Can greatest accuracy be used?

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Maybe - I am told that a stardard table is being preparedbased on experience from about 40 companies. That

may make it possible to estimate the variance needed toplace in the denomiator of k .Stay tuned.


Credibility for Mortality Ratios A Non-credibility Approach

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A Non-credibility Approach



What is the problem we really want to solve?

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ProblemWhat mortality ratio should we use when data are limited?

How about a two-step process?

1 If m̂ 1.96σ̂ includes 1, use the standard table.2 If the interval is below 1, use m̂ + 1.96σ̂ , if above,

use m̂ 1.96σ̂ Note - this is a nod to credibility - if there is evidencethat the true ratio is not 1, do not move all the way tothe observed ratio.



More on this idea

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Step 1 is simply a standard test of the null hypothesisthat the true ratio is 1. Regardless of the amount of data, if there is no evidence that the true ratio is not 1,

it makes sense to use the standard table.Step 2 notes that either if there is a lot of data (in whichcase σ̂ will be small) or if the observed ratio is a longway from 1, then an adjustment is appropriate.



Example re-visited

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The upper bound of the con…dence interval is

0.9506 + 1.96p

0.0011 = 1.0156

and therefore there is no reason to deviate from thestandard table.



The bottom line

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You know more about credibility than when youcame in.

You know that for setting mortality assumptions forprinciples based reserves, it is likely you are notfaced with a credibility problem.

But you are faced with a reliability of data problem.

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Documents

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