The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators

Transcription

1 The Insurance Rsk n the SST and n Solvency II: Modelng and Parameter Estmators ASTIN Colloquum 1-4 June 009, Helsnk Alos Gsler

2 Introducton SST and Solvency II common goal: to nstall a rsk based solvency regulaton solvency captal requred (SCR) should depend on the rsks a company has on ts book SST 004: standard SST model developed and frst feld test 008: all Swss companes have to calculate the SST fgures 011: SST SCR wll be n force Solvency II 007: SII Framework Drectve Proposal adopted by the EU Commsson 008: 4 th quanttatve mpact study 01: "orgnal" schedule to put the regulaton nto force schedule under dscusson Subject of ths presentaton: non-lfe nsurance rsk modelng and parameter estmators ASTIN Colloquum, 1-4 June 009, Helsnk / A. Gsler / The Insurance Rsk n the SST and n Solvency II: Modelng and Parameter Estmators

3 The Insurance Rsk Non-Lfe Insurance Rsk non-lfe nsurance rsk = next years techncal result CY PY TR = P K C C E ( P K E C C E C ) C expected techncal result CY CY CY PY where P K CY C PY C = earned premum, = admnstratve costs, = total clam amount current year (CY), = total clam amount prevous years (PY) = CDR (CDR = clams development result) segmented nto lnes of busness (lob) =1,,...,I ; 3 ASTIN Colloquum, 1-4 June 009, Helsnk / A. Gsler / The Insurance Rsk n the SST and n Solvency II: Modelng and Parameter Estmators

4 Modelng n SST: Insurance Rsk CY clam amount CY, C s splt nto C CY n "normal clam" amount CY, b and C "bg clam amount" analytcal nsurance rsk model CY, n CY, b PY modelng of ( C, C, C ) descrbes adequately realty except for extraordnary stuatons scenaros complements analytcal model to take nto account extraordnary stuatons; to take nto account extraordnary stuatons; by means of scenros SC k, k=1,,...,k, charactersed by face amounts c k wth occurrence probabltes p k. 4 ASTIN Colloquum, 1-4 June 009, Helsnk / A. Gsler / The Insurance Rsk n the SST and n Solvency II: Modelng and Parameter Estmators

5 Modelng n SST: Insurance Rsk Rsk measure n the SST 99% expected shortfall SCR for nsurance rsk [ ] SCR = ES TR ns 99%. 5 ASTIN Colloquum, 1-4 June 009, Helsnk / A. Gsler / The Insurance Rsk n the SST and n Solvency II: Modelng and Parameter Estmators

6 Modelng n SST: normal clam amount CY Model assumpton ( ) T Condtonal on Θ = Θ, Θ, CY, n C 1 s compound Posson; Θ1, Θ are random factors wth expected value 1 ndcatng how much next year's "true underlyng" clam frequency and the "true underlyng" expected clam severty wll devate from ther a pror expected values due to thngs lke weather condtons, change n economc envronment, change n legslaton, etc. Θ (, ) = Θ Θ T 1 s the "rsk characterstcs" of next year for lob ASTIN Colloquum, 1-4 June 009, Helsnk / A. Gsler / The Insurance Rsk n the SST and n Solvency II: Modelng and Parameter Estmators 6

7 Modelng n SST: normal clam amount CY CY, n C X =, where P varance structure from model assumptons follows that σ where P, param = E C CY, n σ fluct, : = Var( X ) = σ +, ν ( ) Var( ) σ Var Θ + Θ, param 1, υ ( Y ) ( ) σ, = CoVa + 1. fluct pure rsk premum; and where CoVa Y ν ( υ ) ( ) w λ () = = = the coeffcent of varaton of the clam severtes, a pror expected number of clams. 7 ASTIN Colloquum, 1-4 June 009, Helsnk / A. Gsler / The Insurance Rsk n the SST and n Solvency II: Modelng and Parameter Estmators

8 Modelng n SST: normal clam amount CY aggregaton over lob CY, n the varance of X = C / P s calculated by assumng a correlaton matrx T ( ) ( j) Corr( X X ) R = Corr X, X ( R, =, ) CY CY j => σ ( ) = 1 T := Var X ( W R W ) CY CY CY P, where ( P1σ1 Pσ Pσ ) W =,,,. CY I I T 8 ASTIN Colloquum, 1-4 June 009, Helsnk / A. Gsler / The Insurance Rsk n the SST and n Solvency II: Modelng and Parameter Estmators

9 Modelng n SST: bg clam amount CY Model Assumptons, ) for each lob the bg clam amount C CY b s compound Possondstrbuton wth (essentally) Pareto-dstrbuted clam szes ), C CY b, = 1,,, I are ndependent I CY, b CY, b => C s agan compound Posson wth = C = 1 λ I n b b b λ = λ = λ, F = b = 1 = 1 λ F. 9 ASTIN Colloquum, 1-4 June 009, Helsnk / A. Gsler / The Insurance Rsk n the SST and n Solvency II: Modelng and Parameter Estmators

10 Modelng n SST: normal and bg clam amount CY lob and standard parameters normal and bg clam amount CY 10 ASTIN Colloquum, 1-4 June 009, Helsnk / A. Gsler / The Insurance Rsk n the SST and n Solvency II: Modelng and Parameter Estmators

11 Modelng n SST: clam amount PY Reserve rsk (clam amount PY) L = outstandng clams labltes at 1.1. for lob, R = best estmate of L per 1.1. = best estmate reserve, PY 31.1., PY R = PA + R = best estmate of L per 31.1., R Y =. R note that C = R R PY Model Assumptons t s assumed that τ. τ : = Var( Y ) = τ + R, param fluct, 11 ASTIN Colloquum, 1-4 June 009, Helsnk / A. Gsler / The Insurance Rsk n the SST and n Solvency II: Modelng and Parameter Estmators

12 Modelng n SST: clam amount PY current standard parameters for PY-rsks 1 ASTIN Colloquum, 1-4 June 009, Helsnk / A. Gsler / The Insurance Rsk n the SST and n Solvency II: Modelng and Parameter Estmators

13 Modelng n SST: clam amount PY aggregaton over lob the varance of Y = R / R s calculated by assumng a correlaton matrx T ( Y ) ( j) Corr( Y Y ) R = Corr, Y ( R, =, ) PY PY j current standard SST assumpton Y, =1,,...,I, are ndependent,.e. R PY = dentty matrx. => τ 1 I : = Var( Y) = R τ R = 1 Dscusson on correlaton assumpton current standard SST assumpton s questonable; reason: calendar year effects affectng several lob smultaneously; an obvous example of s clams nflaton. 13 ASTIN Colloquum, 1-4 June 009, Helsnk / A. Gsler / The Insurance Rsk n the SST and n Solvency II: Modelng and Parameter Estmators

14 Modelng n SST: combned normal clam amount CY + clam amount PY Notatons CY, n + CY, n + = + C R PX RY S, = =, = C R Z V +, + P R P R P + R C + R P X + RY S C R, Z, V P R. CY, n CY, n = + = = = + P + R P + R Model assumpton It s assumed that s lognormal dstrbuted wth S [ ] = WCY WCY ES P + R, Var( S ) = R. WPY WPY T 14 ASTIN Colloquum, 1-4 June 009, Helsnk / A. Gsler / The Insurance Rsk n the SST and n Solvency II: Modelng and Parameter Estmators

15 Modelng n SST: combned normal clam amount CY + clam amount PY Correlaton matrces: R T X X RCY RCY, PY = Corr, =, Y Y RCY, PY RPY where R = Corr, CY, PY T ( X Y ) current standard SST assumpton current year clams and prevous year clams are uncorrelated, that s R = 0. CY, PY => Var ( Z ) = P σ + R ( P + R ) τ 15 ASTIN Colloquum, 1-4 June 009, Helsnk / A. Gsler / The Insurance Rsk n the SST and n Solvency II: Modelng and Parameter Estmators

16 Modelng n SST: correlaton CY and PY; convoluton wth bg clams Dscusson on correlaton assumpton between CY and PY current standard SST assumpton s questonable; reason: calendar year effects affectng the CY-year clam amount as the prevous years' clam amounts of several lob smultaneously; an example of such a calendar year effect s clams nflaton; Convoluton wth bg clam,, The dstrbuton of T = C CY n CY b PY can be calculated by + C + C convolutng the lognormal dstrbuton of CY, n PY C wth the + C CY, b compound Posson dstrbuton of => dstrbuton F before scenaros C 16 ASTIN Colloquum, 1-4 June 009, Helsnk / A. Gsler / The Insurance Rsk n the SST and n Solvency II: Modelng and Parameter Estmators

17 Modelng n SST: scenaros Model Assumptons: ns Scenaros SC k, k=1,,...,k, are characterzed by face amounts c k and occurrence probabltes p k. It s assumed that only one of the scenaros can occur wthn the next year (mutual excluson of scenaros). Remark: The "excluson assumpton" s not such a bg restrcton as t seems, snce one s free n defnng the scenaros. One can always defne new scenaros combnng two already exstng scenaros. Dstrbuton after scenaros,, dstrbuton functon of T = C CY n + C CY b + C PY + SC ns : K ( ) k ( k) 0 k 0 F x = p F x c, where p = 1 p and c = 0. k= 0 k= 1 K 17 ASTIN Colloquum, 1-4 June 009, Helsnk / A. Gsler / The Insurance Rsk n the SST and n Solvency II: Modelng and Parameter Estmators

18 Modelng n SII: Insurance rsk General to compare wth SST: only one regon, company s workng n; SCR for non-lfe nsurance rsk s named SCR nl n solvency II (SII). SII also consders CY-rsk (named premum rsk) and PY-rsk called reserve rsk. For CY-rsk : no dstncton s made between normal and bg clams. In addton: CAT-rsks, manly thought for natural perl rsks. Characterzed by face amounts smlar to the scenaro rsks n the SST. SII provdes formulas how to calculate the SCR and not models. Models presented here = models leadng to the formulas n SII to calculate the SCR. 18 ASTIN Colloquum, 1-4 June 009, Helsnk / A. Gsler / The Insurance Rsk n the SST and n Solvency II: Modelng and Parameter Estmators

19 Modelng n SII: Insurance rsk Notaton X CY C R = (loss rato CY), Y =, P R σ ( X ) τ Var( Y ) = Var, =, where P R R = = = premum reserve per 1.1. "a posteror reserves" per 31.1.of L. 19 ASTIN Colloquum, 1-4 June 009, Helsnk / A. Gsler / The Insurance Rsk n the SST and n Solvency II: Modelng and Parameter Estmators

20 Modelng n SII: premum (CY) rsk calculaton premum rsk per lob where ( ) σ = α σ + 1 α σ,, nd, M α σ σ = = credblty weght, standard "market" parameter, M, n n 1 P j P j nd, = Xj X X = Xj n 1 j= 1P j= 1P Model assumpton CY-rsk (premum rsk) Nether σ M, nor the credblty weght depend on the sze of the company => model assumpton: Var X = σ Model assumpton PY-rsk (reserve rsk) model assumpton: Var =. ( ) wth. ( ). ( ) τ y 0 ASTIN Colloquum, 1-4 June 009, Helsnk / A. Gsler / The Insurance Rsk n the SST and n Solvency II: Modelng and Parameter Estmators

21 Modelng n SII: premum + reserve rsk premum + reserve rsk per lob 1 Z = ( PX + RY ), where V = P + R. V ( X Y ) ρ, Assumpton: Corr, = = 50% ( Z ) => ϕ = Var = CY PY correlaton and aggregaton ( Pσ ) + ρ Pσ Rτ + ( Rτ ) CY, PY :. V ( Z Zj) = ρj ρj assumpton: Corr,, gven standard parameters V Z Z Z VV ϕ ϕ I I j j = => ϕ = Var( ) = ρ, j = 1 V, j= 1 V 1 ASTIN Colloquum, 1-4 June 009, Helsnk / A. Gsler / The Insurance Rsk n the SST and n Solvency II: Modelng and Parameter Estmators

22 Modelng n SII: premum + reserve rsk mplcatons and dscusson of correlaton assumptons Corr must hold for any company => ( Z Z ) ρ Corr( X Y ) ρ,, =,, = = 50%. j j CY PY ( ) ( ) ( ) Corr X, X = Corr Y, Y = Corr Z, Z = ρ, j j j j correlaton between lob result from calendar year effects affectng several lob smultaneously. To assume the same correlaton matrx for X and for Y s questonable, snce the calendar year effect for CY- and PY-rsks mght not be the same or mght have a dfferent mpact. Corr( X, Yj) for j depend on the volumes and dffcult to nterpret ASTIN Colloquum, 1-4 June 009, Helsnk / A. Gsler / The Insurance Rsk n the SST and n Solvency II: Modelng and Parameter Estmators

23 Modelng n SII: formula to calculate SCR lob and parameters 3 ASTIN Colloquum, 1-4 June 009, Helsnk / A. Gsler / The Insurance Rsk n the SST and n Solvency II: Modelng and Parameter Estmators

24 Modelng n SII: premum + reserve rsk formula for SCR premum + reserve rsk SCR pr + res ( ( ) ϕ ) 1 Φ + exp log( 1) = V 1 ϕ + 1 VVaR ( ) mean = Ψ where Φ ( x) = standard normal dstrbuton. [ ] Var( ) ϕ Ψ= logormal dstrbuted r.v. wth E Ψ = 1 and Ψ =, ( ) ( Ψ ) = Ψ ( Ψ) VaR VaR E mean ASTIN Colloquum, 1-4 June 009, Helsnk / A. Gsler / The Insurance Rsk n the SST and n Solvency II: Modelng and Parameter Estmators

25 Modelng n SII: premum + reserve rsk model assumpton behnd ths formula S E[ S ] has the same dstrbuton as V ( Ψ 1, ) where has a lognormal dstrbuton wth E Ψ = 1 and Var Ψ =. Ψ [ ] ( ) ϕ remarks and dscusson [ ] [ ] but contrary to the SST: ( ) ( ) S E S = V Z E Z s aproxmated by V Ψ 1. E[ Z ] 1 (usually smaller than 1). S [ ] => s modeled by a lognormal dstrbuton wth mean ES, but wth a varance whch s dfferent from Var [ S ] 5 ASTIN Colloquum, 1-4 June 009, Helsnk / A. Gsler / The Insurance Rsk n the SST and n Solvency II: Modelng and Parameter Estmators

26 Modelng n SII: premum + reserve rsk [ ] Comparson of 99.5% VaR of Z E Z and 1 for E Z = Ψ [ ] 85%. 6 DAV Scentfc Day , Berln / A. Gsler / The Insurance Rsk n the SST and n Solvency II: Modelng and Parameter Estmators

27 Modelng n SII: cat rsks and total nsurance rsk SCR for CAT rsks SCR CAT K = c k = 1 k. total SCR for nl-nsurance rsk SCR = SCR + SCR nl CY + PY CAT. model assumptons behnd these formulas The cat rsks CAT, = are ndependent and normally k k 1,,, I dstrbuted wth VaR CAT c ( ) = k k. Same assumpton for aggregatng the cat rsks and the other nsurance rsks. 7 ASTIN Colloquum, 1-4 June 009, Helsnk / A. Gsler / The Insurance Rsk n the SST and n Solvency II: Modelng and Parameter Estmators

28 Modelng : Summary STT and SII "parametrzed" models; SII: factor model; STT dstrbuton based model; rsk measure: STT 99% expected shortfall, SII 99.5% VaR varance assumptons CY- und PY-rsks (for r.v. X and Y): STT: parameter rsk and random fluctuaton rsk, where the latter s nversely proportonal to the weght (sze of the company); SII: CY- and PY-rsks not dependent on the sze of the company CY rsk: STT dstngushes between "normal clams" and "bg clams". No such dstncton n SII. Correlaton Assumptons (current state): SST: no correlatons between lob for the reserve rsks and no correlatons between CYund PY-rsks; SII: same correlaton between lob for CY- and PY-rsks; SST as well as SII assumptons not fully satsfactory. 8 ASTIN Colloquum, 1-4 June 009, Helsnk / A. Gsler / The Insurance Rsk n the SST and n Solvency II: Modelng and Parameter Estmators

29 Modelng : Summary SST: scenaros for extraordnary stuatons can be taken nto account n a natural way n the dstrbuton calculaton; SII: CAT-rsks modeled smlar to scenaros n the SST; however aggregaton of cat-rsks and wth CY/PY-rsks questonable SST: fnal product s a dstrbuton, from whch the SCR s calculated; SII: fnal product s one fgure, the SCR. Results (AXA-Wnterthur) wth current standard parameters: SCR ns hgher n SII than n SST; splt between CY- und PY-rsks: SII: ca 5% CY and 75% PY SST: ca 7% CY and 73% PY 9 ASTIN Colloquum, 1-4 June 009, Helsnk / A. Gsler / The Insurance Rsk n the SST and n Solvency II: Modelng and Parameter Estmators

30 Parameter Estmators: SII parameters straghtforward estmators σ n 1 P = X X j ˆ ( j ), n 1 j = 1 P n 1 R Y Y j ˆ = ( j ), n 1 j = 1 R τ Remarks: can overestmate the rsk n case of "strong" busness cycles n the observaton perod; often underestmates the reserve rsks because of "smoothng" effects n the reserves σˆ τˆ 30 ASTIN Colloquum, 1-4 June 009, Helsnk / A. Gsler / The Insurance Rsk n the SST and n Solvency II: Modelng and Parameter Estmators

31 Parameter Estmators: SST parameters Random fluctuaton rsk CY CoVa ( ( ν ) Y ) σ, = CoVa + 1. fluct j = 1 N 1 j Nj υ= 1 ( ( ν ) Y ) j Y Y n long-tal lob: above estmator underestmates the CoVa n recent accdent years 31 ASTIN Colloquum, 1-4 June 009, Helsnk / A. Gsler / The Insurance Rsk n the SST and n Solvency II: Modelng and Parameter Estmators

32 Parameter Estmators: SST parameters parameter rsk CY T specfc lob; each year j characterzed by Θ = Θ, Θ ; r.v. belongng to dfferent years are ndependent and Θ1, Θ,, 1 ΘJ are..d. => ( 1 ) j j j σ σ E X = 1, = + +, fluct ˆfluct j Var( X j ) σparam σparam ν j P j fulfll the assumptons of the Bü-Straub credblty model => estmator J w ( j ) J ˆ σ J j fluct ˆ param = c X X, J 1 j = 1 w n σ where 1 I I 1 w ( ( ) 1, ˆ ) w υ c = σ fluct = CoVa Y + 1, I = 1 w w n = observed number of clams. 3 ASTIN Colloquum, 1-4 June 009, Helsnk / A. Gsler / The Insurance Rsk n the SST and n Solvency II: Modelng and Parameter Estmators

33 Parameter Estmators: SST parameters parameter rsk CY (contnued) snce σ Var Θ + Var Θ param one can, alternatvely to the estmator gven before, estmate the two components separately based on the observed clam frequences and the observed clam szes. Here agan one can use a credblty procedure. more detals: see paper ( ) ( ) 1 33 ASTIN Colloquum, 1-4 June 009, Helsnk / A. Gsler / The Insurance Rsk n the SST and n Solvency II: Modelng and Parameter Estmators

34 Parameter Estmators: SST parameters Estmaton of the Pareto parameters for bg clam CY ML-estmator (adjusted for unbasedness) b n ˆ 1 Y ν ϑ = ln n 1ν = 1 c 1 wth E ( ˆ) ˆ 1 ϑ = ϑ, CoVa ϑ =. n Number of observed bg clams often rather small; combne ndvdual estmate wth market wde estmate; ML-estmators fulfll Bü-Straub cred. assumptons => credblty estmator ˆcred ϑ = αϑˆ + (1 α) ϑ0 where n, standard value from the SST, ϑ ( ). 0 = κ = CoVa Θ n 1+ κ ( ) 5%, n=16 Example: CoVa Θ = => gve a credblty weght of 3% to your ndvdual estmate 34 ASTIN Colloquum, 1-4 June 009, Helsnk / A. Gsler / The Insurance Rsk n the SST and n Solvency II: Modelng and Parameter Estmators

35 Parameter Estmators: SST parameters reserve rsk reserve rsk should be valuated wth reservng technques; well known: Mack's mse of the ultmate for chan ladder reservng method; for solvency purposes one needs the one-year reserve rsk; the formula can be found n Bühlmann and alas (009); In Solvency we are nterested n the one n a century adverse reserve events. What scenaros come to our mnd: for nstance a hyper-nflaton or a bg change n legslaton. These are "calendar-year" events not observed n the trangles and not captured by standard reservng methods. => the reserve rsk resultng from standard reservng methods are not suffcent for solvency purposes and should be supplemented by reserve scenaros. 35 ASTIN Colloquum, 1-4 June 009, Helsnk / A. Gsler / The Insurance Rsk n the SST and n Solvency II: Modelng and Parameter Estmators

36 Parameter Estmators: SST parameters reserve rsk (contnued) For small and medum szed companes the observed fgures n a development trangle mght fluctuate a lot. It would be helpful f one could combne ndustry wde patterns wth the one evaluated wth the data of the ndvdual company. For chan ladder a credblty method was developed of how one could combne the nformaton ganed from the two sources: ndvdual data and ndustry wde nformaton. The dea s to estmate the age-to-age factors by credblty technques. For more nformaton see Gsler-Wüthrch (008). 36 ASTIN Colloquum, 1-4 June 009, Helsnk / A. Gsler / The Insurance Rsk n the SST and n Solvency II: Modelng and Parameter Estmators

37 References Bühlmann, H., De Felce M., Gsler, A., Morcon F., Wüthrch, M.V. (009). Recursve Credblty Formula for Chan Ladder Factors and the Clam Development Result. Forthcomng n the ASTIN Bulletn. Gsler, A., Wüthrch, M.V. (008).Credblty for the Chan Ladder Reservng Method. ASTIN Bulletn 38/, Gsler, A. (009). The Insurance Rsk n the SST and n Solvency II: Modellng and Parameter Estmaton. ASTIN Colloquum n Helsnk. Merz, M., Wüthrch M.V. (008). Modellng the clams development result for solvency purposes. CAS Forum, Fall 008, ASTIN Colloquum, 1-4 June 009, Helsnk / A. Gsler / The Insurance Rsk n the SST and n Solvency II: Modelng and Parameter Estmators