07 texas klugman 28
TRANSCRIPT
-
8/16/2019 07 Texas Klugman 28
1/62
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
http://find/http://goback/
-
8/16/2019 07 Texas Klugman 28
2/62
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
Stuart Klugman () Credibility Theory September 18 , 20 07 2 / 6 2
http://find/
-
8/16/2019 07 Texas Klugman 28
3/62
Credibility for Valuation Actuaries
Credibility for Valuation Actuaries
Stuart Klugman () Credibility Theory September 18 , 20 07 3 / 6 2
http://find/
-
8/16/2019 07 Texas Klugman 28
4/62
Credibility for Valuation Actuaries
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.
Stuart Klugman () Credibility Theory September 18 , 20 07 4 / 6 2
C dibili f V l i A i
http://find/
-
8/16/2019 07 Texas Klugman 28
5/62
Credibility for Valuation Actuaries
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.
Stuart Klugman () Credibility Theory September 18 , 20 07 5 / 6 2
C dibilit f V l ti A t i
http://find/
-
8/16/2019 07 Texas Klugman 28
6/62
Credibility for Valuation Actuaries
Why are you here?
None of the above statements de…nes credibility.
None of the above statements advocates aparticular credibility method.
Stuart Klugman () Credibility Theory September 18 , 20 07 6 / 6 2
Credibility for Valuation Actuaries
http://find/
-
8/16/2019 07 Texas Klugman 28
7/62
Credibility for Valuation Actuaries
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
Credibility for Valuation Actuaries
http://find/
-
8/16/2019 07 Texas Klugman 28
8/62
Credibility for Valuation Actuaries
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.
Stuart Klugman () Credibility Theory September 18 , 20 07 8 / 6 2
Credibility for Valuation Actuaries
http://find/
-
8/16/2019 07 Texas Klugman 28
9/62
Credibility for Valuation Actuaries
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.
Stuart Klugman () Credibility Theory September 18 , 20 07 9 / 6 2
Credibility for Valuation Actuaries
http://find/
-
8/16/2019 07 Texas Klugman 28
10/62
Credibility for Valuation Actuaries
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
http://find/
-
8/16/2019 07 Texas Klugman 28
11/62
y y
A History of Credibility
Stuart Klugman () Credibility Theory September 1 8, 2 00 7 1 1 / 6 2
A History of Credibility
http://find/
-
8/16/2019 07 Texas Klugman 28
12/62
y y
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.
Stuart Klugman () Credibility Theory September 1 8, 2 00 7 1 2 / 6 2
A History of Credibility
http://find/http://goback/
-
8/16/2019 07 Texas Klugman 28
13/62
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.
Stuart Klugman () Credibility Theory September 1 8, 2 00 7 1 3 / 6 2
A History of Credibility
http://find/
-
8/16/2019 07 Texas Klugman 28
14/62
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.
Stuart Klugman () Credibility Theory September 1 8, 2 00 7 1 4 / 6 2
A History of Credibility
http://find/
-
8/16/2019 07 Texas Klugman 28
15/62
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
Stuart Klugman () Credibility Theory September 1 8, 2 00 7 1 5 / 6 2
A History of Credibility
http://find/
-
8/16/2019 07 Texas Klugman 28
16/62
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.
Stuart Klugman () Credibility Theory September 1 8, 2 00 7 1 6 / 6 2
A History of Credibility
http://find/
-
8/16/2019 07 Texas Klugman 28
17/62
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).
Stuart Klugman () Credibility Theory September 1 8, 2 00 7 1 7 / 6 2
Types of Credibility
http://find/
-
8/16/2019 07 Texas Klugman 28
18/62
Types of Credibility
Stuart Klugman () Credibility Theory September 1 8, 2 00 7 1 8 / 6 2
http://find/http://goback/
-
8/16/2019 07 Texas Klugman 28
19/62
Types of Credibility Limited Fluctuation Credibility
-
8/16/2019 07 Texas Klugman 28
20/62
Types of CredibilityLimited Fluctuation
Stuart Klugman () Credibility Theory September 1 8, 2 00 7 2 0 / 6 2
Types of Credibility Limited Fluctuation Credibility
http://find/
-
8/16/2019 07 Texas Klugman 28
21/62
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.
Stuart Klugman () Credibility Theory September 1 8, 2 00 7 2 1 / 6 2
http://find/
-
8/16/2019 07 Texas Klugman 28
22/62
Types of Credibility Limited Fluctuation Credibility
-
8/16/2019 07 Texas Klugman 28
23/62
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 .
Stuart Klugman () Credibility Theory September 1 8, 2 00 7 2 3 / 6 2
Types of Credibility Limited Fluctuation Credibility
http://find/
-
8/16/2019 07 Texas Klugman 28
24/62
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 ).
Stuart Klugman () Credibility Theory September 1 8, 2 00 7 2 4 / 6 2
Types of Credibility Limited Fluctuation Credibility
http://find/
-
8/16/2019 07 Texas Klugman 28
25/62
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.
Stuart Klugman () Credibility Theory September 1 8, 2 00 7 2 5 / 6 2
Types of Credibility Limited Fluctuation Credibility
http://find/http://goback/
-
8/16/2019 07 Texas Klugman 28
26/62
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.
Stuart Klugman () Credibility Theory September 1 8, 2 00 7 2 6 / 6 2
Types of Credibility Limited Fluctuation Credibility
http://find/
-
8/16/2019 07 Texas Klugman 28
27/62
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.
Stuart Klugman () Credibility Theory September 1 8, 2 00 7 2 7 / 6 2
Types of Credibility Limited Fluctuation Credibility
http://find/
-
8/16/2019 07 Texas Klugman 28
28/62
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 .
Stuart Klugman () Credibility Theory September 1 8, 2 00 7 2 8 / 6 2
Types of Credibility Limited Fluctuation Credibility
E l d
http://find/
-
8/16/2019 07 Texas Klugman 28
29/62
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.
Stuart Klugman () Credibility Theory September 1 8, 2 00 7 2 9 / 6 2
Types of Credibility Limited Fluctuation Credibility
E l i d
http://find/
-
8/16/2019 07 Texas Klugman 28
30/62
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.
Stuart Klugman () Credibility Theory September 1 8, 2 00 7 3 0 / 6 2
Types of Credibility Limited Fluctuation Credibility
Li it d ‡ t ti dibilit
http://find/
-
8/16/2019 07 Texas Klugman 28
31/62
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.
Stuart Klugman () Credibility Theory September 1 8, 2 00 7 3 1 / 6 2
Types of Credibility Greatest Accuracy Credibility
http://find/
-
8/16/2019 07 Texas Klugman 28
32/62
Types of Credibility
Greatest Accuracy
Stuart Klugman () Credibility Theory September 1 8, 2 00 7 3 2 / 6 2
Types of Credibility Greatest Accuracy Credibility
Statistical model
http://find/
-
8/16/2019 07 Texas Klugman 28
33/62
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.
Stuart Klugman () Credibility Theory September 1 8, 2 00 7 3 3 / 6 2
Types of Credibility Greatest Accuracy Credibility
Myths
http://find/
-
8/16/2019 07 Texas Klugman 28
34/62
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.
Stuart Klugman () Credibility Theory September 1 8, 2 00 7 3 4 / 6 2
Types of Credibility Greatest Accuracy Credibility
Bühlmann credibility
http://find/
-
8/16/2019 07 Texas Klugman 28
35/62
Bühlmann credibility
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.
Stuart Klugman () Credibility Theory September 1 8, 2 00 7 3 5 / 6 2
Types of Credibility Greatest Accuracy Credibility
Bühlmann credibility
http://find/
-
8/16/2019 07 Texas Klugman 28
36/62
Bühlmann credibility
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.
Stuart Klugman () Credibility Theory September 1 8, 2 00 7 3 6 / 6 2
Types of Credibility Credibility Example
http://find/
-
8/16/2019 07 Texas Klugman 28
37/62
Types of Credibility
An Example
Stuart Klugman () Credibility Theory September 1 8, 2 00 7 3 7 / 6 2
Types of Credibility Credibility Example
A baseball example
http://find/
-
8/16/2019 07 Texas Klugman 28
38/62
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.
Stuart Klugman () Credibility Theory September 1 8, 2 00 7 3 8 / 6 2
Types of Credibility Credibility Example
Results using no credibility
http://find/http://goback/
-
8/16/2019 07 Texas Klugman 28
39/62
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.
Stuart Klugman () Credibility Theory September 1 8, 2 00 7 3 9 / 6 2
Types of Credibility Credibility Example
Results using limited ‡uctuation credibility
http://find/
-
8/16/2019 07 Texas Klugman 28
40/62
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.
Stuart Klugman () Credibility Theory September 1 8, 2 00 7 4 0 / 6 2
Types of Credibility Credibility Example
Results using greatest accuracy credibility
http://find/
-
8/16/2019 07 Texas Klugman 28
41/62
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.
Stuart Klugman () Credibility Theory September 1 8, 2 00 7 4 1 / 6 2
Types of Credibility Credibility Example
Conclusions
http://find/
-
8/16/2019 07 Texas Klugman 28
42/62
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.
Stuart Klugman () Credibility Theory September 1 8, 2 00 7 4 2 / 6 2
Credibility for Mortality Ratios
http://find/
-
8/16/2019 07 Texas Klugman 28
43/62
Credibility for Mortality Ratios
Stuart Klugman () Credibility Theory September 1 8, 2 00 7 4 3 / 6 2
http://find/
-
8/16/2019 07 Texas Klugman 28
44/62
Credibility for Mortality Ratios
The estimator
-
8/16/2019 07 Texas Klugman 28
45/62
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 ∑
ng j =1 b ij f ij d ij
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?
Stuart Klugman () Credibility Theory September 1 8, 2 00 7 4 5 / 6 2
Credibility for Mortality Ratios A Limited Fluctuation Approach
http://find/
-
8/16/2019 07 Texas Klugman 28
46/62
Credibility for Mortality Ratios
A Limited Fluctuation Approach
Stuart Klugman () Credibility Theory September 1 8, 2 00 7 4 6 / 6 2
Credibility for Mortality Ratios A Limited Fluctuation Approach
First two moments
http://find/
-
8/16/2019 07 Texas Klugman 28
47/62
We …rst need the moments of the estimator:
µ = E ( m̂) = E
∑ g i =1 ∑
ng j =1 b ij f ij d ij
e
!
= ∑
g
i =1∑
ng
j =1
b ij f ij q i
e
σ 2 = Var ( m̂) = Var
∑
g i =1 ∑
ng j =1 b ij f ij d ij
e
!
= ∑
g i =1 q i (1 q i ) ∑ (b ij f ij )2
e 2 .
Stuart Klugman () Credibility Theory September 1 8, 2 00 7 4 7 / 6 2
Credibility for Mortality Ratios A Limited Fluctuation Approach
Estimating the moments
http://find/
-
8/16/2019 07 Texas Klugman 28
48/62
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.
Stuart Klugman () Credibility Theory September 1 8, 2 00 7 4 8 / 6 2
Credibility for Mortality Ratios A Limited Fluctuation Approach
Obtaining Z
http://find/
-
8/16/2019 07 Texas Klugman 28
49/62
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 )
Stuart Klugman () Credibility Theory September 1 8, 2 00 7 4 9 / 6 2
Credibility for Mortality Ratios An Example
http://find/
-
8/16/2019 07 Texas Klugman 28
50/62
Crediblity for Mortality Ratios
An Example
Stuart Klugman () Credibility Theory September 1 8, 2 00 7 5 0 / 6 2
Credibility for Mortality Ratios An Example
The data
http://find/
-
8/16/2019 07 Texas Klugman 28
51/62
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.
Stuart Klugman () Credibility Theory September 1 8, 2 00 7 5 1 / 6 2
Credibility for Mortality Ratios An Example
Limited ‡utuation calculation
http://find/
-
8/16/2019 07 Texas Klugman 28
52/62
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.
Stuart Klugman () Credibility Theory September 1 8, 2 00 7 5 2 / 6 2
Credibility for Mortality Ratios Conclusions
http://find/
-
8/16/2019 07 Texas Klugman 28
53/62
Crediblity for Mortality Ratios
Conclusions
Stuart Klugman () Credibility Theory September 1 8, 2 00 7 5 3 / 6 2
Credibility for Mortality Ratios Conclusions
The Good
http://find/
-
8/16/2019 07 Texas Klugman 28
54/62
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."
Stuart Klugman () Credibility Theory September 1 8, 2 00 7 5 4 / 6 2
Credibility for Mortality Ratios Conclusions
The Bad
http://find/
-
8/16/2019 07 Texas Klugman 28
55/62
It is not as scienti…c as greatest accuracy. Somepurists may object.
It is somewhat arbitrary.
Why does 1 Z get to multiply 1?
Stuart Klugman () Credibility Theory September 1 8, 2 00 7 5 5 / 6 2
Credibility for Mortality Ratios Conclusions
The Ugly
http://find/
-
8/16/2019 07 Texas Klugman 28
56/62
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.)
Stuart Klugman () Credibility Theory September 1 8, 2 00 7 5 6 / 6 2
Credibility for Mortality Ratios Conclusions
Can greatest accuracy be used?
http://find/
-
8/16/2019 07 Texas Klugman 28
57/62
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.
Stuart Klugman () Credibility Theory September 1 8, 2 00 7 5 7 / 6 2
Credibility for Mortality Ratios A Non-credibility Approach
http://find/
-
8/16/2019 07 Texas Klugman 28
58/62
Crediblity for Mortality Ratios
A Non-credibility Approach
Stuart Klugman () Credibility Theory September 1 8, 2 00 7 5 8 / 6 2
Credibility for Mortality Ratios A Non-credibility Approach
What is the problem we really want to solve?
http://find/
-
8/16/2019 07 Texas Klugman 28
59/62
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.
Stuart Klugman () Credibility Theory September 1 8, 2 00 7 5 9 / 6 2
Credibility for Mortality Ratios A Non-credibility Approach
More on this idea
http://find/
-
8/16/2019 07 Texas Klugman 28
60/62
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.
Stuart Klugman () Credibility Theory September 1 8, 2 00 7 6 0 / 6 2
Credibility for Mortality Ratios A Non-credibility Approach
Example re-visited
http://find/
-
8/16/2019 07 Texas Klugman 28
61/62
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.
Stuart Klugman () Credibility Theory September 1 8, 2 00 7 6 1 / 6 2
Credibility for Mortality Ratios A Non-credibility Approach
The bottom line
http://find/
-
8/16/2019 07 Texas Klugman 28
62/62
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.
Stuart Klugman () Credibility Theory September 1 8, 2 00 7 6 2 / 6 2
http://find/