CAE, May 5 London Eposure Rating and ILF Stean Bernegger, Dr. sc. nat., SAV ead Analytical Services & Tools Swiss Reinsurance Company Ltd
Tale o Contents / Agenda Introduction ISO Curves or US Casualty lines Wat is te Magic eind ILF? Eposure Curves derived rom ILF Appendi Formula Collection Stean Bernegger Eposure Rating and ILF CAE, May London
Introduction Wile Eposure Rating tecniques are quite advanced in L& and P&C sort tail lines (man made, nat cat, credit risk), tere is quite some potential or enancements in Casualty. Stean Bernegger Eposure Rating and ILF CAE, May London 3
Eposure Rating or Liaility Lines Te eposure rating metodology applied y Swiss Re in Casualty varies y market, line o usiness and product: Swiss Re is using internal "event ased" models and proprietary eposure curves were possile and applicale A road set o eposure curves derived rom industry claims data y ISO and oter providers is urtermore used or US eposures An Increased Limit Factor (ILF ) ased approac is used in te asence o alternatives Te ILF approac is quite simple ut it as several limitations Swiss Re is tus aiming to replace ILF y more advanced tecniques ISO as launced an initiative to epand its services eyond te US: Develop market and product speciic eposure curves ased on data provided y te industry Eplore i US curves can e adapted and e applied in oter markets Stean Bernegger Eposure Rating and ILF CAE, May London 4
Deinition and Representation o Eposure Curves Survival unction: Premium Function: F Simple representation covering a road dynamic range: ( y) dy ISO approac (mied eponentials): Alternative (mied US Paretos): w i e i w ( B ( B ) ) λ i i i + Q European ILF : Remark: ( ) ( ) ( ) + Tere is an implicit European Pareto Curve eind wit α ln(+)/ln() < Only a "pseudo" survival unction can tus e deined or ILF An implied local ILF can e derived or any given survival unction Stean Bernegger Eposure Rating and ILF CAE, May London 5
Eposure Curves ISO approac Curve is represented y a miture o eponential curves Typical ISO curve: Premises and Operations Liaility Remarks ISO its to claims eperience up to a lower tresold and etrapolates wit a Pareto tail up an upper tresold ISO typically uses 6 to ase curves per eposure curve Individual eponential curves are predominant witin a certain range only. Almost any overall sape can e itted wit a small numer o parameters Stean Bernegger Eposure Rating and ILF CAE, May London 6
ISO Curves or US Casualty lines ISO provides customers wit a road selection o eposure curves wic are applicale to most P&C lines o usiness in te US. Stean Bernegger Eposure Rating and ILF CAE, May London 7
ISO Eposure Curves Segmentation Curves are dierentiated y line o usiness, risk category and geograpy Some curves are urtermore dierentiated y size o risk Caliration and representation: Curves are itted to actual claims data up to a lower tresold o typically USD mio Curves are etrapolated wit a Pareto tail wit α typically in te range..4 Te resulting curves are represented wit te elp o typically 6- eponential unctions Te resulting curves can e applied to limits up to an upper tresold o typically USD- mio Te speciic representation enales an easy trending or distortion Survival proailities at given tresold levels as well as local and average ILF's or given ranges can easily e derived Stean Bernegger Eposure Rating and ILF CAE, May London 8
Structure o ISO Curves used in te US or Casualty Eposure Rating Stean Bernegger Eposure Rating and ILF CAE, May London 9
Typical ISO Curves: Premises and Operations Liaility Survival at '' level.7% -4.3% Pareto tail α.33.44 Avg ILF ' '' Avg ILF '' '' 9.8% - 4.% 4.8% - 4.% Stean Bernegger Eposure Rating and ILF CAE, May London
Typical ISO Curves: Products Liaility Survival at '' level.5% -.44% Pareto tail α.38-.4 Avg ILF ' '' Avg ILF '' '' 6.6% - 34.%.% - 9.9% Stean Bernegger Eposure Rating and ILF CAE, May London
Typical ISO Curves: Personal Liaility, Farm Owners Survival at '' level.3% -.86% Pareto tail α. -.34 Avg ILF ' '' Avg ILF '' '' 8.4% -.9% 9.3% - 6.5% Stean Bernegger Eposure Rating and ILF CAE, May London
Typical ISO Curves: Commercial Automoile / eavy, Etra-eavy Survival at '' level.4% -.8% Pareto tail α.5.33 Avg ILF ' '' Avg ILF '' '' 5.% - 3.6%.% -.3% Stean Bernegger Eposure Rating and ILF CAE, May London 3
Typical ISO Curves: Commercial Automoile / Zone Rated, Ligt & Medium, All Oter Survival at '' level.6% -.73% Pareto tail α.4.44 Avg ILF ' '' Avg ILF '' ''.9% - 9.% 8.% - 7.% Stean Bernegger Eposure Rating and ILF CAE, May London 4
Typical ISO Curves: Personal Automoile Survival at '' level.6% -.4% Pareto tail α.5.36 Avg ILF ' '' Avg ILF '' '' 9.6% - 7.6% 5.% -.5% Stean Bernegger Eposure Rating and ILF CAE, May London 5
Typical ISO Curves: Medical Malpractice Survival at '' level.6% - 5.% Pareto tail α..45 Avg ILF ' '' Avg ILF '' '' 7.5% - 59.% 6.3% - 8.3% Stean Bernegger Eposure Rating and ILF CAE, May London 6
ISO Eposure Curves Some conclusions Caracteristics o ISO curves: Te survival proaility at te USD mio tresold is correlated to te local ILF Te irst cart depicts te relationsip etween te survival proaility at USD mio and te average ILF in te USD ' - '' range Te second cart depicts te relationsip etween te survival proaility at USD mio and te average ILF in te USD '' '' range Stean Bernegger Eposure Rating and ILF CAE, May London 7
Wat is te Magic eind ILF? Te ILF approac is airly simple and widely used ut te underlying implicit matematical model unveils it's limitations. Stean Bernegger Eposure Rating and ILF CAE, May London 8
European ILF A ig level view Te ILF metodology is widely used in Casualty eposure rating in Europe / Asia Parameter selection is oten ased on est practice and epert judgement Te metodology is airly simple ut te applicaility is limited: Te implicit matematical model eind te ILF approac can easily e derived It is a European Pareto distriution wit < α < Te approac can not e applied to very low loss levels (e.g. or evaluating deductile credits) and it needs to e adapted to e applicale in a road dynamic range Te properties o tis Pareto distriution permit to analyze te limitations o te applicaility o te ILF approac. Stean Bernegger Eposure Rating and ILF CAE, May London 9
European ILF Deinition: Premium is increased y a constant percentage i douling te policy limit Eample ILF actor 3% Basis: Limit Premium -> 3 -> 4 69 Underlying implicit model: Te implicit Claims Severity can e represented y a Pareto Distriution Te loss-requency curve (LFC) is diverging at te origin Te mean owever eists or a inite limit Te mean does not eist or an ininite limit Practical implementation Speciic ILF-actors are oten deined or various ranges (actor is decreasing wit increasing policy limit) Stean Bernegger Eposure Rating and ILF CAE, May London
European ILF Constant ILF-actor. LFC can e represented y a single Pareto distriution Frequency 3% α.6 Loss Severity ILF-actor varying y range. LFC can e represented y discontinuous piecewise Pareto distriutions. Frequency 3% α.6 5% α.68 % α.74 Loss Severity Stean Bernegger Eposure Rating and ILF CAE, May London
Eample: ILF's or a selected ISO curve Implied ILF actors are derived or deined ranges Typical ISO curve: Premises and Operations Liaility Remarks An ILF implies a Pareto curve wit slope alpa < Multiple ILF actors are required to cover te entire dynamic range Te overall sape o te ILF representation is largely determined y te discontinuities (jumps) at te range oarders. Stean Bernegger Eposure Rating and ILF CAE, May London
Local ILF () and local Pareto α() Implied parameters derived or ininitesimal ranges Typical ISO curve: Premises and Operations Liaility Remarks ILF actors evaluated as a unction () o te loss or an ininitesimal range are called "local ILF " Similarly, a local Pareto α() can e derived. Tis is not identical to te Pareto alpa derived rom te ILF actor, te latter is conined to te - range! Stean Bernegger Eposure Rating and ILF CAE, May London 3
Wic are te Limitations o te ILF Approac? Te implied (pseudo) survival unction as a singularity at! > Te implied loss requency is ininite! Te premium unction is inite around ecause o alpa <! Te premium unction owever as a singularity at ininity ecause o alpa <! Te tail o a realistic severity distriution cannot e adequately represented y a Pareto distriution wit alpa <! Implications: A given ILF actor < cannot e used to deine te sape o te severity distriution or very small losses ( -> ) A given ILF actor > cannot e used to deine te sape o te severity distriution or very large losses ( -> ) A given ILF actor can tus only e used witin a conined range A local ILF actor as to e used at and a local actor as to e used or very large Stean Bernegger Eposure Rating and ILF CAE, May London 4
ow can tese Limitations e overcome? Approac commonly used in Europe/Asia: Use dierent actors or dierent ranges Te resulting implicit pseudo survival unction is continuous witin eac range (local Pareto pat wit alpa < ) ut jumps at te range oundaries Convert into gu eposure curves y preserving ILF properties: Derive empirical survival unctions up to a given tresold Etrapolate using a Pareto approac (or an alternative adequate distriution) up to te required maimum limit Represent te resulting overall survival unction y itting a single parametric distriution or y lending multiple ase distriutions An implicit local ILF as well as an average ILF or a given range can easily e derived rom te curve Switc to event ased approaces and avoid ILF at all Stean Bernegger Eposure Rating and ILF CAE, May London 5
Roadmap rom ILF to ILF ased Eposure Curves Issues Te implicitly used piecewise ILF Pareto distriutions cannot adequately represent te entire severity distriution: Discontinuity at te origin -> ininite requency! Mean not deined or unlimited coverage Te individual local ILF Pareto curves wit α < cannot adequately represent te underlying severity curve Te overall sape o te curve is deined y te structure o te ILF actors Solution Replace y a continuous distriution unction Introduce a inite requency at te origin Preserve te mean (ILF actors) witin eac given range Complete wit a reasonale tail (e.g. Pareto wit α >) Stean Bernegger Eposure Rating and ILF CAE, May London 6
European ILF 3% Replace te discontinuous piecewise Pareto LFC y a continuous LFC wit inite requency at te origin and y preserving te mean witin eac range. Frequency Frequency α.6 5% α.68 Callenge : Frequency o low severity claims % α.74 Loss Severity Callenge : Beaviour o ig severity claims Loss Severity Stean Bernegger Eposure Rating and ILF CAE, May London 7
Wic alternative Approaces are availale? Currently used ILF actors can e used as a asis to derive a set o continuous eposure curves wic preserve te main statistics (tere is not a unique solution) Epand scope o availale eposure curves: Curves migt e applicale in oter parts o te world eo modiiers migt e used to convert Selection migt e ased on implicit ILF Eplicit stocastic modelling o underlying risk and claims process (equivalent to event ased modelling in nat cat) Stean Bernegger Eposure Rating and ILF CAE, May London 8
Eposure Curves derived rom ILF Some general properties o ISO curves or oter sources can e used to derive speciic eposure curves on te asis o given ILF's. Stean Bernegger Eposure Rating and ILF CAE, May London 9
ILF ased Eposure Curve Prolem statement or te case o a single ILF actor iven: ILF actor and derived implicit ILF curve ' ILF () Relative Scale Survival Proaility Task: Fit a survival unction () wit given tail eaviour y preserving te mass witin eac range. ' ILF () 3 R 3 () R R R ( ) R R X ( ) X X X Stean Bernegger Eposure Rating and ILF CAE, May London 3
ILF ased Eposure Curve Cookook or converting a single ILF actor into an eposure curve: Deine ILF tresold: Deine ILF actor aove tresold: (* ) ( )*(+) Deine Pareto tail parameter: α (α ) Fit e.g. a two parametric curve elow tresold : p, p (p, p ) Deine one additional condition, e.g.: survival proaility at tresold: ( ) ensure monotonous irst derivative: ' ( ) ' ( ) Converting multiple ILF actors into an eposure curve : Deine ILF parameters or eac interval { { i, i+ }, i } Deine Pareto tail eyond upper tresold (as aove) Deine one additional condition (as aove) Fit a curve witin eac interval y preserving te interval mean and te oundary conditions Stean Bernegger Eposure Rating and ILF CAE, May London 3
ILF ased Eposure Curve Recipe or single ILF actor Survival Proaility 3 Model: () Body (td): Tail (Pareto) : Relative Scale R 3 Caliration Input parameters: α : Tresold α : Pareto tail parameter (> ) : Increased Limit Factor {, } : Survival proaility ( ) Constraint: < ma < ma 4 3 R Pareto tail: α R R α ( ) R ma ILF: R R X () ma(α) X R R 3 4 Tail survival: Epected Loss: R ( ) 3 R ma ( p ) d p Stean Bernegger Eposure Rating and ILF CAE, May London 3
Stean Bernegger Eposure Rating and ILF CAE, May London 33 Some model candidates ρ ρ Model : Power unction Model : Eponential unction: Model 3 : MBBEFD unction: e e + +
ILF derived Eposure Curve iven tresold survival Eample Tresold survival is given (derived rom ISO regression) Tresold: '' ILF: 5% Tresold survival: 8.3% Pareto tail : α.35 Stean Bernegger Eposure Rating and ILF CAE, May London 34
ILF derived Eposure Curve iven tresold survival Eample Tresold survival is given (derived rom ISO regression) Tresold: '' ILF: % Tresold survival:.34% Pareto tail : α.35 Stean Bernegger Eposure Rating and ILF CAE, May London 35
ILF derived Eposure Curve Continuous derivative Eample 3 Tresold survival is NOT given Tresold: '' ILF: 5% Pareto tail : α.5 Continuous derivative at tresold Stean Bernegger Eposure Rating and ILF CAE, May London 36
ILF derived Eposure Curve Continuous derivative Eample 4 Tresold survival is NOT given Tresold: '' ILF: % Pareto tail : α.5 Continuous derivative at tresold Power Function eposure curve Constraints or : 3.4% < < 58.6% ILF actor % is outside tis range Boundary condition cannot e met! Stean Bernegger Eposure Rating and ILF CAE, May London 37
Conclusions A single ILF actor cannot adequately represent te severity distriution Te tail o te distriution migt e represented y a Pareto model wit α >. Fitting a curve to te ody o te distriution y preserving te ILF properties is an improvement compared to te pure ILF approac. It can not overcome te undamental lack o inormation. Tere is no unique solution and at least one additional condition is needed, e.g.: survival proaility at a given tresold enorcing a continuous slope at te tresold Some properties o ISO curves migt also e applicale in areas outside te initial scope: tresold survival proaility (asolute or derived via ILF regression analysis) tail eaviour (Pareto alpa) Among te evaluated candidates, te MBBEFD unction appears to e most appropriate or modelling te ody o te survival unction: te model is covering te entire parameter space te resulting survival unctions sow a reasonale eaviour Stean Bernegger Eposure Rating and ILF CAE, May London 38
Appendi Formula Collection Stean Bernegger Eposure Rating and ILF CAE, May London 39
Stean Bernegger Eposure Rating and ILF CAE, May London Survival unction: Premium unction: Mied Eponential: Mied US Pareto: European ILF : 4 Deinition and Representation o Eposure Curves F dy y i i i i i w e w e w i i λ µ λ λ λ + + Q B w B B Q B w B B w i i Q i i i i Q i i i µ ln ln +
Stean Bernegger Eposure Rating and ILF CAE, May London Deinition: Premium unction: wit Pareto alpa: Pseudo survival unction: 4 Matematical properties o te ILF approac n n + + α + ln ln ln ln + α α α α α α () ( ) ( )
Matematical properties o te ILF approac Relationsip etween local ILF () and premium (): ( + ) ln( ) ln d d ln ( ) ( + ) ln( ) ln ( ) ( ) e ln ( + ( y) ) ln( ) y dy Relationsip etween local Pareto α() and te survival unction (): d d ln ( ) α α α( ) ( ) e α ( y ) y dy Stean Bernegger Eposure Rating and ILF CAE, May London 4
Stean Bernegger Eposure Rating and ILF CAE, May London 43 ILF ased Eposure Curve Pareto tail Model: + + α α α α α α α α ma + + α α α α α α α X X () () ma(α) Evaluation at tresold
Stean Bernegger Eposure Rating and ILF CAE, May London 44 ILF ased Eposure Curve Body itted wit Power Function + + ρ ρ ρ ρ ρ ρ ρ ρ Model: + + ρ ρ ρ ρ ρ ρ X X () () ma(α) Evaluation at tresold
ILF ased Eposure Curve Body itted wit Power Function Caliration: is known! Condition : ma ( ρ ) ρ + ( ρ + ) () Solution : ρ ma ma Constraint: Alternative caliration: is not known! Condition : Solution : < ma < ma ( α ) ( ρ ) ρ α + ( α + ) ma ( α ) ma ρ ( α + ) ma ma () X ρ α + ρ ma(α) X Constraint: ma < α + < ma Stean Bernegger Eposure Rating and ILF CAE, May London 45
ILF ased Eposure Curve Body itted wit Power Function Alternative caliration: Remarks Te Power unction is represented in a single/doule logaritmic scale. An aritrary scale is used or te vertical ais, te average "requency" or te '' '' layer is set to. Te Pareto tail parameter α determines te upper and lower range or te ILF parameter wic can e covered y te power unction. Stean Bernegger Eposure Rating and ILF CAE, May London 46
Stean Bernegger Eposure Rating and ILF CAE, May London 47 ILF ased Eposure Curve Body itted wit Eponential Function e e e e e e Model: + e e e e Evaluation at tresold
ILF ased Eposure Curve Body itted wit Eponential Function Caliration: is known! Condition : ( ) ma () Solution : ma ma + + e e ma ma(α) Constraint: < ma < ma () X X Alternative caliration: is not known! Condition : ( α ) ( ) Solution : ma + α ma + α + α ( e ) + e ma α Stean Bernegger Eposure Rating and ILF CAE, May London 48
ILF ased Eposure Curve Body itted wit Eponential Function Alternative caliration: Remarks Te eponential unction is represented in a single/doule logaritmic scale. An aritrary scale is used or te vertical ais, te average "requency" or te '' '' layer is set to. Te Pareto tail parameter α determines te maimum ILF, tere is no lower constraint or te ILF parameter, i.e. te eponential unction covers te entire parameter range. Stean Bernegger Eposure Rating and ILF CAE, May London 49
Stean Bernegger Eposure Rating and ILF CAE, May London 5 ILF ased Eposure Curve Body itted wit MBBEFD unction Model: + + + + ln ln ln ln + + ln ln
Stean Bernegger Eposure Rating and ILF CAE, May London 5 ILF ased Eposure Curve Body itted wit MBBEFD unction ln ln ln ln ln ln X X () () ma(α) Evaluation at tresold
ILF ased Eposure Curve Body itted wit MBBEFD unction Caliration: is known! Condition : ma ( ) () Solution : (solve or ) ( ) ln( ) ( ) ln( ) ma Constraint: Alternative caliration: is not known! Condition : Solution : (solve or ) < ma < ma ( α ) ( ) ( ) ln( ) ( ) α + α ln ( ) ln ln ( ) + α ln( ) ( ) + α ma () X ma(α) X ( ) ln( ) Constraint: ( ) > ln( ) α Stean Bernegger Eposure Rating and ILF CAE, May London 5
ILF ased Eposure Curve Body itted wit MBBEFD Function Alternative caliration: Remarks Te MBBEFD unction is represented in a single/doule logaritmic scale. An aritrary scale is used or te vertical ais, te average "requency" or te '' '' layer is set to. Te Pareto tail parameter α determines te maimum ILF, tere is no lower constraint or te ILF parameter, i.e. te MBBEFD unction covers te entire parameter range. Stean Bernegger Eposure Rating and ILF CAE, May London 53
Tank you
Legal notice Swiss Re. All rigts reserved. You are not permitted to create any modiications or derivatives o tis presentation or to use it or commercial or oter pulic purposes witout te prior written permission o Swiss Re. Altoug all te inormation used was taken rom reliale sources, Swiss Re does not accept any responsiility or te accuracy or compreensiveness o te details given. All liaility or te accuracy and completeness tereo or or any damage resulting rom te use o te inormation contained in tis presentation is epressly ecluded. Under no circumstances sall Swiss Re or its roup companies e liale or any inancial and/or consequential loss relating to tis presentation. Stean Bernegger Eposure Rating and ILF CAE, May London 55