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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 transform so 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?

?

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

The log-normal family

- A convenientdefinitionofthe log-normal model in the present context is
- as where
- 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 as is 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

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 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 lognormalNon 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

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)

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 modelwithoutzero claims

Non parametric

Log-normal, Gamma

The Pareto

Extreme value

Searching

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