# Non-life insurance mathematics - PowerPoint PPT Presentation

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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

Policy holder

Insurance company

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

• 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

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