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Non-life insurance mathematics

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Non-lifeinsurancemathematics

Nils F. Haavardsson, University of Oslo and DNB Skadeforsikring

The result drivers ofinsuranceeconomics:

Numberofclaims

Ik

Ik+1

Ik-1

tk-2

tk

tk+1

tk=T

t0=0

tk-1

- What is rare can be describedmathematically by cutting a given time period T into K small pieces ofequallengthh=T/K
- On shortintervalsthechanceof more thanoneincident is remote
- Assumingno more than 1 event per intervalthecount for theentireperiod is
- N=I1+...+IK ,whereIj is either 0 or 1 for j=1,...,K

- Ifp=Pr(Ik=1) is equal for all k and eventsare independent, this is an ordinaryBernoulli series

where

- Assumethat p is proportional to h and set

is an intensitywhichapplies per time unit

Client

Policies and claims

Policy

Insurableobject

(risk)

Claim

Insurance cover

Cover element

/claim type

- The graph shows claimfrequency for all covers for motor insurance
- Noticeseasonalvariations, due to changingweatherconditionthroughouttheyears

Howvaries over theportfoliocanpartially be described by observablessuch as age or sex oftheindividual (treated in Chapter 8.4)

Therearehoweverfactorsthat have impactonthe risk whichthecompanycan’t know muchabout

- Driver ability, personal risk averseness,
This randomenesscan be managed by makinga stochastic variable

This extensionmay serve to captureuncertaintyaffecting all policy holders jointly, as well, such as alteringweatherconditions

The modelsareconditionalonesofthe form

Let which by double rules in Section 6.3 imply

Now E(N)<var(N) and N is no longer Poissondistributed

Policy level

Portfoliolevel

What is a fair priceofinsurance policy?

The model (Section 8.4 EB)

An example

Why is a regressionmodelneeded?

Repetitionofimportantconcepts in GLM

Before ”Fairness” wassupervised by theauthorities (Finanstilsynet)

- To someextentcommon tariffs betweencompanies
- The market wascontrolled
During 1990’s: deregulation

Now: free market competitionsupposed to give fairness

According to economictheorythere is noprofit in a free market (in Norway general insurance is cyclical)

Hence, thepriceequalstheexpectedcost for insurer

Note: costofcapitalmay be includedhere, butnoadditionalprofit

The fair price

The model

An example

Whyregression?

Repetitionof GLM

Main component is expected loss (claimcost)

The average loss for a large portfoliowill be close to themathematicalexpectation (by thelawof large numbers)

So expected loss is the basis oftheprice

Variesbetweeninsurancepolicies

Hencethe market pricewillvarytoo

The addotherincome and costs, incl administrative cost and capitalcost

The fair price

The model

An example

Whyregression?

Repetitionof GLM

Toohighpremium for somepoliciesresults in loss ofgoodpolicies to competitors

Toolowpremium for somepoliciesgivesinflowofunprofitablepolicies

This willforcethecompany to charge a fair premium

The fair price

The model

An example

Whyregression?

Repetitionof GLM

How to findtheexpected loss ofeveryinsurance policy?

Wecannotpriceindividualpolicies (why?)

- Thereforepoliciesaregrouped by rating variables
Rating variables (age) aretransformed to ratingfactors (age classes)

Ratingfactorsare in most cases categorical

The fair price

The model

An example

Whyregression?

Repetitionof GLM

- The idea is to attributevariation in to variations in a setofobservable variables x1,...,xv. Poissonregressjon makes useofrelationshipsofthe form

The fair price

(1.12)

- Why and not itself?
- The expectednumberofclaims is non-negative, where as thepredictoronthe right of (1.12) can be anythingonthe real line
- It makes more sense to transformso thattheleft and right side of (1.12) are more in line witheachother.
- Historical data areofthefollowing form
- n1 T1 x11...x1x
- n2 T2 x21...x2x
- nnTn xn1...xnv

- The coefficients b0,...,bvareusuallydetermined by likelihoodestimation

The model

An example

Whyregression?

Repetitionof GLM

Claimsexposurecovariates

- In likelihoodestimation it is assumedthatnj is Poissondistributedwhere is tied to covariates xj1,...,xjv as in (1.12). The densityfunctionofnj is then

The fair price

- or

The model

An example

- log(f(nj)) above is to be added over all j for thelikehoodfunction L(b0,...,bv).
- Skip themiddle terms njTj and log (nj!) sincetheyareconstants in thiscontext.
- Thenthelikelihoodcriterionbecomes

Whyregression?

Repetitionof GLM

(1.13)

- Numerical software is used to optimize (1.13).
- McCullagh and Nelder (1989) provedthat L(b0,...,bv) is a convexsurfacewith a single maximum
- Thereforeoptimization is straight forward.

The fair price

The model

An example

Whyregression?

Repetitionof GLM

- The modelbecomes

- for l=1,...,5 and s=1,2,3,4,5,6,7.
- To avoidover-parameterizationputbbus age(5)=bdistrict(4)=0 (thelargestgroup is often used as reference)

The fair price

The model

An example

Whyregression?

Repetitionof GLM

The fair price

The model

An example

Whyregression?

Repetitionof GLM

The fair price

The model

An example

Whyregression?

Repetitionof GLM

The fair price

The model

An example

Whyregression?

Repetitionof GLM

- In zon 4 (marked as 9 in thetables) thefit is ok
- There is much more data in thiszonthan in theothers
- Wemaytry to re-groupzon, into 2,3,7 and other

The fair price

The model

An example

Whyregression?

Repetitionof GLM

- Zon 9 (4,1,5,6) still has the best fit
- The otherarebetter – butaretheygoodenough?
- Wetry to regroup bus age as well, into 0-1, 2-3 and 4.

The fair price

The model

An example

Whyregression?

Bus age

Bus age

Repetitionof GLM

Bus age

Bus age

- Zon 9 (4,1,5,6) still has the best fit
- The otherare still better – butaretheygoodenough?
- May be there is not enoughinformation in thismodel
- May be additionalinformation is needed
- The final attempt for now is to skip zon and relysolelyon bus age

The fair price

The model

An example

Whyregression?

Repetitionof GLM

Bus age

Bus age

Bus age

Bus age

- From thegraph it is seenthatthefit is acceptable
- Hypothesis 1: Theredoes not seem to be enoughinformation in the data set to provide reliable estimates for zon
- Hypothesis 2: there is anothersourceofinformation, possiblyinteractingwithzon, thatneeds to be takenintoaccountifzon is to be included in themodel

The fair price

The model

An example

Whyregression?

Repetitionof GLM

Bus age

The variables in themultiplicativemodelareassumed to work independent ofoneanother

This may not be the case

Example:

- Auto model, Poissonregressionwith age and gender as explanatory variables
- Young males drive differently (worse) thanyoungfemales
- There is a dependencybetween age and gender
This is an exampleof an interactionbetweentwo variables

Technicallytheissuecan be solved by forming a newratingfactorcalledage/genderwithvalues

- Young males, youngfemales, older males, older femalesetc

The fair price

The model

An example

Whyregression?

Repetitionof GLM

There is not enough data to pricepoliciesindividually

What is actually happening in a regressionmodel?

- Regressioncoefficientsmeasuretheeffectceterisparibus, i.e. when all other variables are held constant
- Hence, theeffectof a variable can be quantifiedcontrolling for theother variables
Whytakethe trouble ofusing a regressionmodel?

Why not pricethepoliciesonefactor at a time?

The fair price

The model

An example

Whyregression?

Repetitionof GLM

The fair price

The model

An example

- ”One factor at a time” gives 6.1%/2.6% = 2.3 as themileagerelativity
- But for eachVehicle age, theeffect is close to 2.0
- ”One factor at a time” obviouslyoverestimatestherelativity – why?

Whyregression?

Repetitionof GLM

The fair price

The model

- New vehicles have 45% oftheirduration in lowmileage, while old vehicles have 87%
- So, the old vehicles have lowerclaimfrequenciespartly due to less exposure to risk
- This is quantified in theregressionmodelthroughthemileagefactor
- Conclusion: 2.3 is right for High/Lowmileageif it is theonlyfactor
- Ifyou have bothfactors, 2.0 is the right relativity

An example

Whyregression?

Repetitionof GLM

Hull coverage (i.e., damagesonownvehicle in a collisionor othersudden and unforeseendamage)

Time period for parameter estimation: 2 years

Covariates:

- Driving length
- Car age
- Region ofcarowner
- Tariff class
- Bonus ofinsuredvehicle
2 modelsaretested and compared – Gamma and lognormal

Non parametric

Log-normal, Gamma

The Pareto

Extreme value

Searching

The model is evaluatedwithrespect to fit, result, validationofmodel, type 3 analysis and QQ plot

Fit: ordinaryfitmeasuresareevaluated

Results: parameter estimatesofthemodelsarepresented

Validationofmodel: the data material is split in two, independentgroups. The model is calibrated (i.e., estimated) onone half and validatedontheother half

Type 3 analysisofeffects: Doesthefitofthemodelimprovesignificantly by includingthespecificvariable?

QQplot:

Non parametric

Log-normal, Gamma

The Pareto

Extreme value

Searching

Tariff class

Bonus

Region

Driving Length

Car age

Type 3 analysisofeffects: Doesthefitofthemodelimprovesignificantly by includingthespecific variable?

ExponentialdispersionModels (EDMs)

- FrequencyfunctionfYi (eitherdensity or probabilityfunction)
- For yi in the support, else fYi=0.
- c() is a function not dependingon
- C
- twicedifferentiablefunction
- b’ has an inverse
- The setofpossible is assumed to be open

The fair price

The model

An example

Whyregression?

Repetitionof GLM

- ClaimfrequencyYi=Xi/Tiwhere Ti is duration
- NumberofclaimsassumedPoissonwith
- LetC
- Then
- EDM with

The fair price

The model

An example

Whyregression?

Repetitionof GLM

...is not a parametricfamilyofdistributions (like Normal, Poisson)

...is rather a classofdifferentsuch families

The function b() speficieswhichfamilywe have

The idea is to derive general results for all families withintheclass – and use for all

The fair price

The model

An example

Whyregression?

Repetitionof GLM

By usingcumulant/moment-generatingfunctions, it can be shown (seeMcCullagh and Nelder (1989)) that for an EDM

- E
- Ee
This is why b() is calledthecumulantfunction

The fair price

The model

An example

Whyregression?

Repetitionof GLM

Recallthat is assumed to exist

Hence

The variancefunction is defined by

Hence

The fair price

The model

An example

Whyregression?

Repetitionof GLM

Distribution Normal Poisson Gamma Binomial

The fair price

The model

An example

Note: Gamma EDM has stddeviationproportional

to , which is much more realisticthanconstant (Normal)

Whyregression?

Repetitionof GLM

Withinthe EDM class, a familyofprobabilitydistributions is uniquelycharacterized by itsvariancefunction

The fair price

The model

Proof by professor Bent Jørgensen, Odense

An example

Whyregression?

Repetitionof GLM

Let c>0

IfcYbelongs to same distributionfamily as Y, thendistribution is scale invariant

Example: claimcostshouldfollowthe same distribution in NOK, SEK or EURO

The fair price

The model

An example

Whyregression?

Repetitionof GLM

If an EDM is scale invariant then it has variancefunction

This is alsoproved by Jørgensen

This definestheTweediesubclassofGLMs

In pricing, suchmodelscan be useful

The fair price

The model

An example

Whyregression?

Repetitionof GLM

The fair price

The model

An example

Whyregression?

Repetitionof GLM

- A general link function g()
- Linear regression:identity link
- Multiplicativemodel:log link
- Logisticregression:logit link

The fair price

The model

An example

Whyregression?

Repetitionof GLM

Generalized linear models:

- Yifollows an EDM:
- Meansatisfies

The fair price

The model

MultiplicativeTweediemodels:

- YiTweedie EDM:
- Meansatisfies

An example

Whyregression?

Repetitionof GLM