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8/10/2019 Froehlich Reinsurance-Pricing
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Reinsurance-Pricing
Prof.Dr. Michael Frhlich
Hochschule RegensburgFakultt fr Informatik und Mathematik
Risk Management and Reinsurance, VGH Versicherungen
17.01.2013
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1 Introduction
2 Experience Pricing - BC methods and examples
3 Exposure Pricing (Casualty and Property)
4 Probabilistic Methods (Frequency / Severity models)
5 Conclusion
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1. Introduction
Prof.Dr. Michael Frhlich (HS Regensburg) Reinsurance-Pricing 17.01.2013 3 / 23
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1. IntroductionIn the following we exclusively considerper risk Excess of Losstreaties (per risk XLs). This is a specialnon-proportional
reinsurance contract: The distribution of each loss betweencedant (insurer)andreinsurerisnt proportional but depends onthe respective loss size.
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1. IntroductionIn the following we exclusively considerper risk Excess of Losstreaties (per risk XLs). This is a specialnon-proportional
reinsurance contract: The distribution of each loss betweencedant (insurer)andreinsurerisnt proportional but depends onthe respective loss size.
Priority (Deductible) = Retention of the cedantandLimit
(Coverage) = maximum excess-loss for the reinsurerare themost important characterizing parameters of an XL.
Prof.Dr. Michael Frhlich (HS Regensburg) Reinsurance-Pricing 17.01.2013 3 / 23
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1. IntroductionIn the following we exclusively considerper risk Excess of Losstreaties (per risk XLs). This is a specialnon-proportional
reinsurance contract: The distribution of each loss betweencedant (insurer)andreinsurerisnt proportional but depends onthe respective loss size.
Priority (Deductible) = Retention of the cedantandLimit
(Coverage) = maximum excess-loss for the reinsurerare themost important characterizing parameters of an XL.
Common description of an (per risk) XL:
Limit after / in excess of / xs Priority,
e.g. 3.000.000 EUR xs 1.000.000 EUR.
Prof.Dr. Michael Frhlich (HS Regensburg) Reinsurance-Pricing 17.01.2013 3 / 23
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1. IntroductionIn the following we exclusively considerper risk Excess of Losstreaties (per risk XLs). This is a specialnon-proportional
reinsurance contract: The distribution of each loss betweencedant (insurer)andreinsurerisnt proportional but depends onthe respective loss size.
Priority (Deductible) = Retention of the cedantandLimit
(Coverage) = maximum excess-loss for the reinsurerare themost important characterizing parameters of an XL.
Common description of an (per risk) XL:
Limit after / in excess of / xs Priority,
e.g. 3.000.000 EUR xs 1.000.000 EUR.XL Pricing= XL Quotation = Calculation of XL Net Premium =Calculation ofXL Risk Premium=Expected XL-loss amountfor the forthcoming year =(Expected XL severity) * (Expected XL frequency).
Prof.Dr. Michael Frhlich (HS Regensburg) Reinsurance-Pricing 17.01.2013 3 / 23
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1. Introduction
Prof.Dr. Michael Frhlich (HS Regensburg) Reinsurance-Pricing 17.01.2013 4 / 23
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1. Introduction
Reinsurance is all about peak risks or risk volatility on:
Severity, Claims frequency or both (Aggregate loss amount)
Prof.Dr. Michael Frhlich (HS Regensburg) Reinsurance-Pricing 17.01.2013 4 / 23
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1. Introduction
Reinsurance is all about peak risks or risk volatility on:
Severity, Claims frequency or both (Aggregate loss amount)
Mathematics and Statistics help:
To decompose information available and to analyze eachcomponent.
To recompose results into a synthesis.
Translate results into economic and business interpretations.
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1. Introduction
Reinsurance is all about peak risks or risk volatility on:
Severity, Claims frequency or both (Aggregate loss amount)
Mathematics and Statistics help:
To decompose information available and to analyze eachcomponent.
To recompose results into a synthesis.
Translate results into economic and business interpretations.Pricing Approach needs to be:
Accurate and reproducible (even if little information). Flexible (quantitative and qualitative information) Easy to communicate and to disclose.
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1. Introduction
Reinsurance is all about peak risks or risk volatility on:
Severity, Claims frequency or both (Aggregate loss amount)
Mathematics and Statistics help:
To decompose information available and to analyze eachcomponent.
To recompose results into a synthesis.
Translate results into economic and business interpretations.Pricing Approach needs to be:
Accurate and reproducible (even if little information). Flexible (quantitative and qualitative information) Easy to communicate and to disclose.
Ideal World:
Stable portfolio, claims frequency, average loss amount. Stable underwriter / actuary judgement.
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2. One year Burning Cost of accident year i
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2. One year Burning Cost of accident year i
LetNibe the number of losses to the treaty per accident yeari.
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2. One year Burning Cost of accident year i
LetNibe the number of losses to the treaty per accident yeari.
LetLi,kbe thekth loss to the treaty in accident yeari.
Prof.Dr. Michael Frhlich (HS Regensburg) Reinsurance-Pricing 17.01.2013 5 / 23
O B i C f id i
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2. One year Burning Cost of accident year i
LetNibe the number of losses to the treaty per accident yeari.
LetLi,kbe thekth loss to the treaty in accident yeari.
LetPibe the Volume (e.g. premium, number of policies,
aggregate sum insureds, exposed premium) of accident yeari.
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2 O B i C f id i
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2. One year Burning Cost of accident year i
LetNibe the number of losses to the treaty per accident yeari.
LetLi,kbe thekth loss to the treaty in accident yeari.
LetPibe the Volume (e.g. premium, number of policies,
aggregate sum insureds, exposed premium) of accident yeari.
The Burning Cost BCiof accident yeariis defined as
BCi :=
Nik=1
Li,k
Pi .
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2 BC lti l l ti S l d fi iti
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2. BC multi-year calculation: Several definitionsSame estimator over a longer period = weighted average
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2 BC lti l l ti S l d fi iti
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2. BC multi-year calculation: Several definitionsSame estimator over a longer period = weighted average
Givennaccident years. LetNibe the number of losses to the
treaty per accident yeari.
Prof.Dr. Michael Frhlich (HS Regensburg) Reinsurance-Pricing 17.01.2013 6 / 23
2 BC m lti ear calc lation Se eral definitions
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2. BC multi-year calculation: Several definitionsSame estimator over a longer period = weighted average
Givennaccident years. LetNibe the number of losses to the
treaty per accident yeari.LetLi,kbe thekth loss to the treaty in accident yeari.
Prof.Dr. Michael Frhlich (HS Regensburg) Reinsurance-Pricing 17.01.2013 6 / 23
2 BC multi year calculation: Several definitions
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2. BC multi-year calculation: Several definitionsSame estimator over a longer period = weighted average
Givennaccident years. LetNibe the number of losses to the
treaty per accident yeari.LetLi,kbe thekth loss to the treaty in accident yeari.
LetPibe the Volume (e.g. premium, number of policies,aggregate sum insureds, exposed premium) of accident yeari.
Prof.Dr. Michael Frhlich (HS Regensburg) Reinsurance-Pricing 17.01.2013 6 / 23
2 BC multi year calculation: Several definitions
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2. BC multi-year calculation: Several definitionsSame estimator over a longer period = weighted average
Givennaccident years. LetNibe the number of losses to the
treaty per accident yeari.LetLi,kbe thekth loss to the treaty in accident yeari.
LetPibe the Volume (e.g. premium, number of policies,aggregate sum insureds, exposed premium) of accident yeari.
The Burning Cost BC of the accident years 1 tonis defined as
BC:=n
i=1
Nik=1
Li,k
Pi=
ni=1
Pin
i=1Pi
BCi
.
Prof.Dr. Michael Frhlich (HS Regensburg) Reinsurance-Pricing 17.01.2013 6 / 23
2 BC: Split between frequency and average cost:
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2. BC: Split between frequency and average cost:
BC=n
i=1
Pin
i=1Pi
NiPi
Ni
k=1Li,k
Ni
.
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2 BC: Split between frequency and average cost:
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2. BC: Split between frequency and average cost:
BC=n
i=1
Pin
i=1Pi
NiPi
Ni
k=1Li,k
Ni
.
Pin
i=1Pi
are weights to consider portfolio growth development.
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2 BC: Split between frequency and average cost:
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2. BC: Split between frequency and average cost:
BC=n
i=1
Pin
i=1Pi
NiPi
Ni
k=1Li,k
Ni
.
Pin
i=1Pi
are weights to consider portfolio growth development.
NiPi
represents the claims frequency in the accident yeari.
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2 BC: Split between frequency and average cost:
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2. BC: Split between frequency and average cost:
BC=n
i=1
Pin
i=1Pi
NiPi
Ni
k=1Li,k
Ni
.
Pin
i=1Pi
are weights to consider portfolio growth development.
NiPi
represents the claims frequency in the accident yeari.N
i
k=1Li,k
Nirepresents the average loss to the treaty in the ayi.
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2 Burning Cost improving evaluation
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2. Burning Cost improving evaluation
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2. Burning Cost improving evaluationAccuracy: Portfolio risk volume could be modeled by informationother than premium:
Exposed Premium (i.e.=ratable premium) Number of risks
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2. Burning Cost improving evaluationAccuracy: Portfolio risk volume could be modeled by informationother than premium:
Exposed Premium (i.e.=ratable premium) Number of risks Total sum insureds
Weights can be allocated to each accident year to reflect datareliability, portfolio changes etc.
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2. Burning Cost improving evaluationAccuracy: Portfolio risk volume could be modeled by informationother than premium:
Exposed Premium (i.e.=ratable premium) Number of risks Total sum insureds
Weights can be allocated to each accident year to reflect datareliability, portfolio changes etc.
Trends adjustments on:
Claims Frequency Average Claim Amounts ...and to reflect portfolio specificities.
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2. Burning Cost improving evaluationAccuracy: Portfolio risk volume could be modeled by informationother than premium:
Exposed Premium (i.e.=ratable premium) Number of risks Total sum insureds
Weights can be allocated to each accident year to reflect datareliability, portfolio changes etc.
Trends adjustments on:
Claims Frequency Average Claim Amounts ...and to reflect portfolio specificities.
As with all BC analysis, prior to applying historic losses to the XL,these must be trended for claims inflation, developed forIBNR/IBNER and adjusted on-level for changes in exposurebetween the loss years.
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3. Exposurequotierung in der Sachversicherung
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p q g g
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p q g g
In der Sachversicherung sind Versicherungsvertrge in der Regel
auf PML-Basis gegeben. Ein PML kann als Mass fr die Hheeines Risikos angesehen werden. Die Aufteilung derOriginalprmie erfolgt in diesem Fall anhand sogenannterExposurekurven.
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p q g g
In der Sachversicherung sind Versicherungsvertrge in der Regel
auf PML-Basis gegeben. Ein PML kann als Mass fr die Hheeines Risikos angesehen werden. Die Aufteilung derOriginalprmie erfolgt in diesem Fall anhand sogenannterExposurekurven.
Fr die verschiedenen Geschftssegmente bzw. Risikoartenstehen diverse Exposurekurven zur Verfgung. Die richtigeAuswahl der zu dem Portefeuille passenden Kurven hatwesentlichen Einfluss auf die korrekte Kalkulation der Prmie.
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p q g g
In der Sachversicherung sind Versicherungsvertrge in der Regel
auf PML-Basis gegeben. Ein PML kann als Mass fr die Hheeines Risikos angesehen werden. Die Aufteilung derOriginalprmie erfolgt in diesem Fall anhand sogenannterExposurekurven.
Fr die verschiedenen Geschftssegmente bzw. Risikoartenstehen diverse Exposurekurven zur Verfgung. Die richtigeAuswahl der zu dem Portefeuille passenden Kurven hatwesentlichen Einfluss auf die korrekte Kalkulation der Prmie.
Die in der Praxis am hufigsten verwendeten Exposurekurvenentstehen durch die Auswertung der Schadenhistorie grosserPortefeuilles.
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In der Sachversicherung wird davon ausgegangen, dass Risiken
der gleichen Risikogruppe dieselbe Verteilung
G(w) =P(Yiw) =P( XiPMLi
w)
der SchadengradeYi := XiPMLi besitzen.
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In der Sachversicherung wird davon ausgegangen, dass Risiken
der gleichen Risikogruppe dieselbe Verteilung
G(w) =P(Yiw) =P( XiPMLi
w)
der SchadengradeYi := XiPMLi besitzen.
Es werden oft seit Jahrzehnten die gleichen Exposurekurvenverwendet. So entstanden zum Beispiel die von Peter Gasserentwickelten Swiss Re Kurven aus Schadenstatistiken der Jahre
1959-1967. Das beste Argument fr die fortdauernde Anwendungdieser Kurven ist die Tatsache, dass sie sich in der Praxis seitjeher bewhrt haben - und das sogar weltweit.
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Der RV ermittelt also lediglich einen Anteil an der vom EVkalkulierten Originalprmie. Dies birgt die Gefahr, dass beiunzureichender Originalprmie auch eine nicht ausreichendeRckversicherungsprmie verlangt wird. Um dies zu vermeiden,kann der RV die berechnende Prmie mit dem Faktor SQ
(erwarteteSchadenquote) korrigieren.
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Der RV ermittelt also lediglich einen Anteil an der vom EVkalkulierten Originalprmie. Dies birgt die Gefahr, dass beiunzureichender Originalprmie auch eine nicht ausreichendeRckversicherungsprmie verlangt wird. Um dies zu vermeiden,kann der RV die berechnende Prmie mit dem Faktor SQ
(erwarteteSchadenquote) korrigieren.In der vorstehenden Formel wird anstelle der PMLs der einzelnenRisiken (sind aufgrund der Klassenbildung nicht bekannt) die
jeweilige PML-Bandmitte gewhlt und somit die Annahme
getroffen, dass die PMLs der einzelnen Risiken innerhalb einesBandes einer Gleichverteilung folgen.
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Stetige Darstellung der Exposurekurven nach Bernegger
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Stetige Darstellung der Exposurekurven nach Bernegger
Ursprnglich sind Exposurekurven als diskrete Kurven gegeben,und die Zwischenwerte werden interpoliert. Stefan Berneggergelang es, eine einparametrische, stetige Funktionenschar zufinden, die sich mit den bekannten Exposurekurven erstaunlichgut deckt. Die Funktion ist definiert durch
Gb,g
(x) =
x, g=1 v b=0ln(1+(g1)x)
ln(g) , b=1 , g>1
1
bx
1b, bg=1 , g>1ln( (g1)b+(1gb)b
x
1b )
ln(gb) , b>0 , b=1 , bg=1 , g>1.
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Die beiden Parameterbundglassen sich durch den Parameterceindeutig beschreiben:
bc=b(c) =e3,10,15(1+c)c gc=g(c) =e
(0,78+0,12c)c.
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Die beiden Parameterbundglassen sich durch den Parameterceindeutig beschreiben:
bc=b(c) =e3,10,15(1+c)c gc=g(c) =e
(0,78+0,12c)c.
Bei der Implementierung kann man sich auf den letzten Ast derFunktion beschrnken, da alle plausiblen Eingabewerte des
Parameterscfr Exposurekurven in diesem Fall abgedecktwerden.
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Die beiden Parameterbundglassen sich durch den Parameterceindeutig beschreiben:
bc=b(c) =e3,10,15(1+c)c gc=g(c) =e
(0,78+0,12c)c.
Bei der Implementierung kann man sich auf den letzten Ast derFunktion beschrnken, da alle plausiblen Eingabewerte des
Parameterscfr Exposurekurven in diesem Fall abgedecktwerden.
Die vier Kurven, die durch die Parameterc {1.5, 2,3,4}definiertsind, entsprechen nach Bernegger nherungsweise den Swiss Re
Kurven 1-4. Diese Werte sind zwar einfach zu merken, nochbesser im Sinne der kleinsten Abweichungsquadrate werden dievier Kurven allerdings durch die Parameterc {1.64,1.94,2.88,3.88}beschrieben.
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3. Exposurequotierung in der Haftpflichtversicherung
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Die Haftpflichtversicherung (Liability, Casualty) gehrt im
Gegensatz zur Sachversicherung der Longtail-Branche an. DieSchadenaufwnde werden, anders als in der Shorttail-Branche,nicht direkt nach Eintritt des Schadens bekannt.
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3. Exposurequotierung in der HaftpflichtversicherungAusgehend von einem Basispolicenlimit v zugehriger
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Ausgehend von einem Basispolicenlimitv0, zugehrigerBasisprmieb(v0) =b0und Zuschlagsfaktorzfr eineVerdopplung der Deckungssumme ergibt sich fr alle Policenlimitsv>0 die berhmteFormel von Riebesell:
b(v) =b0(1 +z)log2(
vv0)
=b0(v
v0)log2(1+z).
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3. Exposurequotierung in der HaftpflichtversicherungAusgehend von einem Basispolicenlimit v zugehriger
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Ausgehend von einem Basispolicenlimitv0, zugehrigerBasisprmieb(v0) =b0und Zuschlagsfaktorzfr eineVerdopplung der Deckungssumme ergibt sich fr alle Policenlimitsv>0 die berhmteFormel von Riebesell:
b(v) =b0(1 +z)log2(
vv0)
=b0(v
v0)log2(1+z).
Wre Riebesells Prmienfunktionbals Kollektives Modelldarstellbar, msste fr alle Policenlimitsvdie Gleichung
b(v) =E(N) E(min(X, v))
erfllt sein, und fr den Schadenentlastungskoeffizienten desRisikosigelte dann:
ri(p) = E(min(X,p))E(min(X, vi))
= b(p)
b(vi)= (
p
vi)log2(1+z).
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3. Exposurequotierung in der Haftpflichtversicherung
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3. Exposurequotierung in der Haftpflichtversicherung
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Die Darstellung von Riebesells Prmienfunktion als Kollektives
Model fhrt jedoch zu Widersprchen. Dennoch wurde die Formelaus Mangel an Alternativen ber Jahrzehnte angewendet.
Die Schadenentlastungskurven, die sich durch RiebesellsPrmienfunktionbergeben, knnen wie die im vorherigen Kapitelvorgestellten Exposurekurven interpretiert werden. Sie besitzendie gleichen Eigenschaften, und die Prmienberechnung erfolgtanschliessend analog.
Mack und Fackler fanden 2003 einen Weg, die Prmienfunktionbim Kollektiven Modell darzustellen, so dass Riebesells Sytem
daher als einfache theoretische fundierte Mglichkeit derHaftpflicht-Exposurequotierung angesehen werden kann.
Prof.Dr. Michael Frhlich (HS Regensburg) Reinsurance-Pricing 17.01.2013 16 / 23
4. XL-Frequency / XL-Severity - Collective Model
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Definition of theCollective Modelin respect of an XL:
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4. XL-Frequency / XL-Severity - Collective Model
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Definition of theCollective Modelin respect of an XL:
LetNbe a discrete random variable modelling the XL-lossfrequency of given XL in a one-year period andX1,X2, ...,Xncontinuous random variables modelling the single XL-lossseverities of given XL in an one-year period, i.i.d and independent
fromN.Then we get for the random variableS:=
Ni=1
Xi
E(S) =E(N) E(Xi).
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4. Severity Modeling, Pareto distribution
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Useful Properties:
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4. Severity Modeling, Pareto distribution
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Useful Properties:
Fat tail distribution properties (mean-excess functionf(P) =E(X P |X>P)is linear and increases strictly).
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4. Severity Modeling, Pareto distribution
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Useful Properties:
Fat tail distribution properties (mean-excess functionf(P) =E(X P |X>P)is linear and increases strictly).
Risk characteristics defined by one real parameter. MaximumLikelihood estimation for the parameteralways possible in caseof given loss history.
Prof.Dr. Michael Frhlich (HS Regensburg) Reinsurance-Pricing 17.01.2013 18 / 23
4. Severity Modeling, Pareto distribution
U f l P i
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Useful Properties:
Fat tail distribution properties (mean-excess functionf(P) =E(X P |X>P)is linear and increases strictly).
Risk characteristics defined by one real parameter. MaximumLikelihood estimation for the parameteralways possible in caseof given loss history.
Easy to interpret and communicate.
Prof.Dr. Michael Frhlich (HS Regensburg) Reinsurance-Pricing 17.01.2013 18 / 23
4. Severity Modeling, Pareto distribution
U f l P ti
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Useful Properties:
Fat tail distribution properties (mean-excess functionf(P) =E(X P |X>P)is linear and increases strictly).
Risk characteristics defined by one real parameter. MaximumLikelihood estimation for the parameteralways possible in caseof given loss history.
Easy to interpret and communicate.
Graphical assessment:P(X>x) = (x0/x).
Prof.Dr. Michael Frhlich (HS Regensburg) Reinsurance-Pricing 17.01.2013 18 / 23
4. Severity Modeling, Pareto distribution
U f l P ti
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Useful Properties:
Fat tail distribution properties (mean-excess functionf(P) =E(X P |X>P)is linear and increases strictly).
Risk characteristics defined by one real parameter. MaximumLikelihood estimation for the parameteralways possible in caseof given loss history.
Easy to interpret and communicate.
Graphical assessment:P(X>x) = (x0/x).
Benchmark exists per LoB and / or markets (experienced valuesfor), e.g.
Property personal lines: =1.5 1.9 (depending on sum insured). MTPL line of business: = 2.0 2.5 (depending on markets).
Prof.Dr. Michael Frhlich (HS Regensburg) Reinsurance-Pricing 17.01.2013 18 / 23
4. Severity Modeling, more sophisticated distributions
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Generalized Beta distribution, e.g. applied in the RMS-Model
Prof.Dr. Michael Frhlich (HS Regensburg) Reinsurance-Pricing 17.01.2013 19 / 23
4. Severity Modeling, more sophisticated distributions
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Generalized Beta distribution, e.g. applied in the RMS-Model
Many distributions are special cases of the GB distribution.
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4. Severity Modeling, more sophisticated distributions
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Generalized Beta distribution, e.g. applied in the RMS-Model
Many distributions are special cases of the GB distribution.Generic: Covers a large family of distribution functions.
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4. Severity Modeling, more sophisticated distributions
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Generalized Beta distribution, e.g. applied in the RMS-Model
Many distributions are special cases of the GB distribution.Generic: Covers a large family of distribution functions.
Complex: 1+4 parameters, which are not easy to interpret.
P(X x) =B(,, (x x0)
(x x0) + ).
B(,,) = (+ )
()()
0
u1(1 u)1du.
Prof.Dr. Michael Frhlich (HS Regensburg) Reinsurance-Pricing 17.01.2013 19 / 23
4. Frequency Modeling, Poisson or Negative Binomial?How volatile is the claims frequency?
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Prof.Dr. Michael Frhlich (HS Regensburg) Reinsurance-Pricing 17.01.2013 20 / 23
4. Frequency Modeling, Poisson or Negative Binomial?How volatile is the claims frequency?
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Poisson:Easy to parameterise and addtive iro convolution
P(a) +P(b) =P(a+b). But variance equal to the mean - holdonly in fire insurance on a property book (maybe...).
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4. Frequency Modeling, Poisson or Negative Binomial?How volatile is the claims frequency?
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Poisson:Easy to parameterise and addtive iro convolution
P(a) +P(b) =P(a+b). But variance equal to the mean - holdonly in fire insurance on a property book (maybe...).
Neg. Binomial:Generalisation of Poisson, longer tail and greatervariance. When modelling working-XL mono-line liability using NBwith variance of around three times the mean wouldnt be unusual.
If Var(X)E(X) 1: Poisson distribution P().
Prof.Dr. Michael Frhlich (HS Regensburg) Reinsurance-Pricing 17.01.2013 20 / 23
4. Frequency Modeling, Poisson or Negative Binomial?How volatile is the claims frequency?
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Poisson:Easy to parameterise and addtive iro convolution
P(a) +P(b) =P(a+b). But variance equal to the mean - holdonly in fire insurance on a property book (maybe...).
Neg. Binomial:Generalisation of Poisson, longer tail and greatervariance. When modelling working-XL mono-line liability using NBwith variance of around three times the mean wouldnt be unusual.
If Var(X)E(X) 1: Poisson distribution P().
If Var(X)E(X) >1: Neg. Binomial = Poisson distribution P() with
(=
E(X)
;=
Var(X)
E(X) 1).
Prof.Dr. Michael Frhlich (HS Regensburg) Reinsurance-Pricing 17.01.2013 20 / 23
4. Frequency Modeling, Poisson or Negative Binomial?How volatile is the claims frequency?
P i E i d dd i i l i
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Poisson:Easy to parameterise and addtive iro convolution
P(a) +P(b) =P(a+b). But variance equal to the mean - holdonly in fire insurance on a property book (maybe...).
Neg. Binomial:Generalisation of Poisson, longer tail and greatervariance. When modelling working-XL mono-line liability using NBwith variance of around three times the mean wouldnt be unusual.
If Var(X)E(X) 1: Poisson distribution P().
If Var(X)E(X) >1: Neg. Binomial = Poisson distribution P() with
(=
E(X)
;=
Var(X)
E(X) 1).
Variance and Expected Loss frequency need to be avolume-weighted estimation.
Prof.Dr. Michael Frhlich (HS Regensburg) Reinsurance-Pricing 17.01.2013 20 / 23
4. Monte Carlo Simulation
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Prof Dr Michael Frhlich (HS Regensburg) Reinsurance-Pricing 17 01 2013 21 / 23
4. Monte Carlo Simulation
Generalized Quantil-function:
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Generalized Quantil function:
F
1(u) :=inf{xR|F(x)u}, u[0,1].
Prof Dr Michael Frhlich (HS Regensburg) Reinsurance-Pricing 17 01 2013 21 / 23
4. Monte Carlo Simulation
Generalized Quantil-function:
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Generalized Quantil function:
F
1(u) :=inf{xR|F(x)u}, u[0,1].
Simulation-Theorem:Let Ube anU(0, 1)-distributed randomvariable andF :RRa distribution function of a randomvariableX. ThenY :=F1(U)is a real random variable with
distribution functionF.
Prof Dr Michael Frhlich (HS Regensburg) Reinsurance-Pricing 17 01 2013 21 / 23
4. Monte Carlo Simulation
Generalized Quantil-function:
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Generalized Quantil function:
F
1(u) :=inf{xR|F(x)u}, u[0,1].
Simulation-Theorem:Let Ube anU(0, 1)-distributed randomvariable andF :RRa distribution function of a randomvariableX. ThenY :=F1(U)is a real random variable with
distribution functionF.
Pareto example:
F(x) =u=1 (xmin
x )
x=F1(u) = xmin
(1 u)1
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5. ConclusionFind the balance between model complexity and realism
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Long tail business would need
Actuarial and underwriting judgements are complementary to
mathematical approaches.
Prof Dr Michael Frhlich (HS Regensburg) Reinsurance-Pricing 17 01 2013 22 / 23
5. ConclusionFind the balance between model complexity and realism
Main advantages of simple mathematical models: Focus on trends
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a a a ag s s p a a a s s sdetection and calibration; economic interpretation (and control of
assumptions); communication.
Long tail business would need
Sophisticated mathematical approach for individual IBNR andIBNER.
Actuarial and underwriting judgements are complementary to
mathematical approaches.
Prof Dr Michael Frhlich (HS Regensburg) Reinsurance-Pricing 17 01 2013 22 / 23
5. ConclusionFind the balance between model complexity and realism
Main advantages of simple mathematical models: Focus on trends
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g pdetection and calibration; economic interpretation (and control of
assumptions); communication.Mixing simple models ensures accuracy and ease ofinterpretation.
Long tail business would need
Sophisticated mathematical approach for individual IBNR andIBNER.
Taking into account superimposed inflation for personal injury.
Actuarial and underwriting judgements are complementary to
mathematical approaches.
Prof Dr Michael Frhlich (HS Regensburg) Reinsurance-Pricing 17 01 2013 22 / 23
5. ConclusionFind the balance between model complexity and realism
Main advantages of simple mathematical models: Focus on trends
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g pdetection and calibration; economic interpretation (and control of
assumptions); communication.Mixing simple models ensures accuracy and ease ofinterpretation.
Long tail business would need
Sophisticated mathematical approach for individual IBNR andIBNER.
Taking into account superimposed inflation for personal injury.
Risk premium calculation on present value basis (investmentgains).
Actuarial and underwriting judgements are complementary to
mathematical approaches.
Prof Dr Michael Frhlich (HS Regensburg) Reinsurance-Pricing 17 01 2013 22 / 23
Insofern sich die Stze der
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Insofern sich die Stze der
Mathematik auf die Wirklichkeitbeziehen, sind sie nicht sicher, undinsofern sie sicher sind, beziehen siesich nicht auf die Wirklichkeit.
Albert Einstein
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