Non life insurance mathematics
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Non-life insurance mathematics. Nils F. Haavardsson, University of Oslo and DNB Skadeforsikring. Key ratios – claim frequency. The graph shows claim frequency for all covers for motor insurance Notice seasonal variations , due to changing weather condition throughout the years.

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Non-life insurance mathematics

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Non-lifeinsurancemathematics

Nils F. Haavardsson, University of Oslo and DNB Skadeforsikring


Key ratios – claimfrequency

  • The graph shows claimfrequency for all covers for motor insurance

  • Noticeseasonalvariations, due to changingweatherconditionthroughouttheyears


The model (Section 8.4)

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

(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

Claimsexposurecovariates


Introduction to reserving

Non-lifeinsurance from a financialperspective:

for a premium an insurancecompanycommitsitself to pay a sum if an event has occured

Contractperiod

retrospective

prospective

Policy holder

signs up for an

insurance

During thedurationofthe policy, someof

thepremium is earned, some is unearned

  • How muchpremium is earned?

  • How muchpremium is unearned?

  • Is theunearnedpremiumsufficient?

Premium reserve, prospective

Policy holder

payspremium.

Insurance company

starts to earnpremium

During thedurationofthe policy, claimsmight or might not occur:

  • How do wemeasurethenumber and sizeofunknownclaims?

  • How do weknowifthe reserves onknownclaimsaresufficient?

Claims

reserve,

retrospective

Accident

date

Reporting

date

Claims

payments

Claimsclose

Claims

reopening

Claims

payments

Claimsclose


Imagine youwant to build a reserve risk model

Therearethreeeffectsthatinfluencethe best estimat

and theuncertainty:

•Paymentpattern

•RBNS movements

•Reporting pattern

Up to recentlytheindustry has basedmodelonpaymenttriangles:

Whatwillthefuturepaymentsamount to?

?


Overview


The ultimate goal for calculatingthe pure premium is pricing

Pure premium = Claimfrequency x claimseverity

Parametric and non parametricmodelling (section 9.2 EB)

The log-normal and Gamma families (section 9.3 EB)

The Pareto families (section 9.4 EB)

Extreme valuemethods (section 9.5 EB)

Searching for themodel (section 9.6 EB)


Claimseveritymodelling is aboutdescribingthevariation in claimsize

  • The graphbelow shows howclaimsizevaries for fire claims for houses

  • The graph shows data up to the 88th percentile

  • How do we handle «typicalclaims» ? (claimsthatoccurregurlarly)

  • How do we handle largeclaims? (claimsthatoccurrarely)

Non parametric

Log-normal, Gamma

The Pareto

Extreme value

Searching


Claimseveritymodelling is aboutdescribingthevariation in claimsize

  • The graphbelow shows howclaimsizevaries for water claims for houses

  • The graph shows data up to the 97th percentile

  • The shapeof fire claims and water claimsseem to be quite different

  • Whatdoesthissuggestaboutthe drivers of fire claims and water claims?

  • Anyimplications for pricing?

Non parametric

Log-normal, Gamma

The Pareto

Extreme value

Searching


The ultimate goal for calculatingthepure premium is pricing

  • Claimsizemodellingcan be parametricthrough families ofdistributionssuch as the Gamma, log-normal or Pareto with parameters tuned to historical data

  • Claimsizemodellingcanalso be non-parametricwhereeachclaimziofthepast is assigned a probability 1/n of re-appearing in thefuture

  • A newclaim is thenenvisaged as a random variable for which

  • This is an entirely proper probabilitydistribution

  • It is known as theempiricaldistribution and will be useful in Section 9.5.

Non parametric

Log-normal, Gamma

The Pareto

Sizeofclaim

Client behavourcanaffectoutcome

  • Burglar alarm

  • Tidyship (maintenanceetc)

  • Garage for thecar

Bad luck

  • Electric failure

  • Catastrophes

  • House fires

Extreme value

Where do wedraw

the line?

Searching

Here we sample from theempiricaldistribution

Here weusespecial

Techniques (section 9.5)


Example

Non parametric

Empirical

distribution

Log-normal, Gamma

  • The thresholdmay be set for example at the 99th

    percentile, i.e., 500 000 NOK for thisproduct

  • The threshold is sometimescalledthelargeclaimsthreshold

The Pareto

Extreme value

Searching


Scale families ofdistributions

  • All sensible parametricmodels for claimsizeareofthe form

  • and Z0 is a standardized random variable corresponding to .

  • This proportionality os inherited by expectations, standard deviations and percentiles; i.e. ifareexpectation, standard devation and -percentile for Z0, thenthe same quantities for Z are

  • The parameter canrepresent for exampletheexchange rate.

  • The effectof passing from onecurrency to anotherdoes not changetheshapeofthedensityfunction (iftheconditionabove is satisfied)

  • In statistics is known as a parameter ofscale

  • Assumethe log-normal modelwhere and are parameters and . Then . Assumewerephrasethemodel as

  • Then

Non parametric

Log-normal, Gamma

The Pareto

Extreme value

Searching


Fitting a scalefamily

  • Models for scale families satisfy

  • wherearethedistributionfunctionsof Z and Z0.

  • Differentiatingwithrespect to z yieldsthefamilyofdensityfunctions

  • The standard wayof fitting suchmodels is throughlikelihoodestimation. If z1,…,znarethehistoricalclaims, thecriterionbecomes

  • which is to be maximizedwithrespect to and other parameters.

  • A usefulextension covers situationswithcensoring.

  • Perhapsthesituationwheretheactual loss is only given as somelowerbound b is most frequent.

  • Example:

    • travel insurance. Expenses by loss oftickets (travel documents) and passportarecovered up to 10 000 NOK ifthe loss is not covered by anyoftheotherclauses.

Non parametric

Log-normal, Gamma

The Pareto

Extreme value

Searching


Fitting a scalefamily

  • The chanceof a claim Z exceeding b is , and for nbsucheventswithlowerbounds b1,…,bnbtheanalogous joint probabilitybecomes

  • Takethelogarithmofthisproduct and add it to the log likelihoodofthefullyobservedclaims z1,…,zn. The criterionthenbecomes

Non parametric

Log-normal, Gamma

completeinformation

censoring to the right

The Pareto

Extreme value

Searching


Shifteddistributions

  • The distributionof a claimmay start at sometreshold b insteadoftheorigin.

  • Obviousexamplesaredeductibles and re-insurancecontracts.

  • Models can be constructed by adding b to variables starting at theorigin; i.e. where Z0 is a standardized variable as before. Now

  • and differentiationwithrespect to z yields

  • which is thedensityfunctionof random variables with b as a lower limit.

Non parametric

Log-normal, Gamma

The Pareto

Extreme value

Searching


Skewness as simple descriptionofshape

  • A major issuewithclaimsizemodelling is asymmetry and the right tailofthedistribution. A simple summary is thecoefficientofskewness

  • The numerator is thethird order moment. Skewnessshould not dependoncurrency and doesn’tsince

  • Skewness is often used as a simplifiedmeasureofshape

  • The standard estimateoftheskewnesscoefficient from observations z1,…,zn is

Non parametric

Log-normal, Gamma

The Pareto

Extreme value

Searching


Non-parametricestimation

  • The random variable that attaches probabilities 1/n to all claimsziofthepast is a possiblemodel for futureclaims.

  • Expectation, standard deviation, skewness and percentilesare all closelyrelated to theordinary sample versions. For example

  • Furthermore,

  • Third order moment and skewnessbecomes

  • Skewnesstends to be small

  • No simulatedclaimcan be largeerthanwhat has beenobserved in thepast

  • These drawbacks implyunderestimationof risk

Non parametric

Log-normal, Gamma

The Pareto

Extreme value

Searching


TPL

Non parametric

Log-normal, Gamma

The Pareto

Extreme value

Searching


Hull

Non parametric

Log-normal, Gamma

The Pareto

Extreme value

Searching


The log-normal family

  • A convenientdefinitionofthe log-normal model in the present context is

  • aswhere

  • Mean, standard deviation and skewnessare

  • seesection 2.4.

  • Parameter estimation is usuallycarriedout by notingthatlogarithmsareGaussian. Thus

  • and whenthe original log-normal observations z1,…,znaretransformed to Gaussianonesthrough y1=log(z1),…,yn=log(zn) with sample mean and variance , theestimatesofbecome

Non parametric

Log-normal, Gamma

The Pareto

Extreme value

Searching


Log-normal sampling (Algoritm 2.5)

Input:

Draw

Return

Non parametric

Log-normal, Gamma

The Pareto

Extreme value

Searching


The lognormalfamily

  • Different choiceof ksi and sigma

  • The shapedependsheavilyon sigma and is highlyskewedwhen sigma is not tooclose to zero

Non parametric

Log-normal, Gamma

The Pareto

Extreme value

Searching


The Gamma family

  • The Gamma family is an importantfamily for whichthedensityfunction is

  • It wasdefined in Section 2.5 asis the standard Gamma withmeanone and shapealpha. The densityofthe standard Gamma simplifies to

  • Mean, standard deviation and skewnessare

  • and there is a convolutionproperty. Suppose G1,…,Gnareindependentwith

  • . Then

Non parametric

Log-normal, Gamma

The Pareto

Extreme value

Searching


Exampleof Gamma distribution

Non parametric

Log-normal, Gamma

The Pareto

Extreme value

Searching


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

Example: carinsurance

Non parametric

Log-normal, Gamma

The Pareto

Extreme value

Searching


The modelsarecomparedwithrespect to fit, results, validationofmodel, type 3 analysis and QQ plots

Fit: ordinaryfitmeasuresarecompared

Results: parameter estimatesofthemodelsarecompared

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 includingthespecific variable?

Comparisonsof Gamma and lognormal

Non parametric

Log-normal, Gamma

The Pareto

Extreme value

Searching


Comparisonof Gamma and lognormal - fit

Gamma fit

Lognormalfit

Non parametric

Log-normal, Gamma

The Pareto

Extreme value

Searching


Comparisonof Gamma and lognormal – type 3

Non parametric

Log-normal, Gamma

Gamma fit

Lognormalfit

The Pareto

Extreme value

Searching


QQ plot Gamma model

Non parametric

Log-normal, Gamma

The Pareto

Extreme value

Searching


QQ plot log normal model

Non parametric

Log-normal, Gamma

The Pareto

Extreme value

Searching


Non parametric

Log-normal, Gamma

The Pareto

Extreme value

Searching


Non parametric

Log-normal, Gamma

The Pareto

Extreme value

Searching


Non parametric

Log-normal, Gamma

The Pareto

Extreme value

Searching


Non parametric

Log-normal, Gamma

The Pareto

Extreme value

Searching


Non parametric

Log-normal, Gamma

The Pareto

Extreme value

Searching


Non parametric

Log-normal, Gamma

The Pareto

Extreme value

Searching


None ofthemodelsseem to be perfect

Lognormalbehavesworst and can be discarded

Canwe do better?

Wetry Gamma once more, nowexludingthe 0 claims (about 17% oftheclaims)

  • Claimswherethe policy holder has noguilt (other party is to blame)

Conclusions so far

Non parametric

Log-normal, Gamma

The Pareto

Extreme value

Searching


Comparisonof Gamma and lognormal - fit

Gamma fit

Gamma without zero claimsfit

Non parametric

Log-normal, Gamma

The Pareto

Extreme value

Searching


Comparisonof Gamma and lognormal – type 3

Non parametric

Log-normal, Gamma

Gamma fit

Gamma without zero claimsfit

The Pareto

Extreme value

Searching


QQ plot Gamma

Non parametric

Log-normal, Gamma

The Pareto

Extreme value

Searching


QQ plot Gamma modelwithoutzero claims

Non parametric

Log-normal, Gamma

The Pareto

Extreme value

Searching


Non parametric

Log-normal, Gamma

The Pareto

Extreme value

Searching


Non parametric

Log-normal, Gamma

The Pareto

Extreme value

Searching


Non parametric

Log-normal, Gamma

The Pareto

Extreme value

Searching


Non parametric

Log-normal, Gamma

The Pareto

Extreme value

Searching


Non parametric

Log-normal, Gamma

The Pareto

Extreme value

Searching


Non parametric

Log-normal, Gamma

The Pareto

Extreme value

Searching


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