Non-life insurance mathematics. Nils F. Haavardsson, University of Oslo and DNB Skadeforsikring

Transcription

1 No-lfe surace mathematcs Nls F. Haavardsso, Uversty of Oslo ad DNB Skadeforskrg

2 Repetto clam se The cocept No parametrc modellg Scale famles of dstrbutos Fttg a scale famly Shfted dstrbutos Skewess No parametrc estmato Parametrc estmato: the log ormal famly Parametrc estmato: the gamma famly Parametrc estmato: fttg the gamma 2

3 Frequecy Clam severty modellg s about descrbg the varato clam se The cocept The graph below shows how clam se vares for fre clams for houses The graph shows data up to the 88th percetle 700 Clam se fre How does clam se vary? How ca ths varato be modelled? B Trucato s ecessary large clams are rare ad dsturb the pcture 0-clams ca occur because of deductbles Two approaches to clam se modellg o-parametrc ad parametrc 3

4 No-parametrc modellg ca be useful No parametrc modellg Clam se modellg ca be o-parametrc where each clam of the past s assged a probablty / of re-appearg the future A ew clam s the evsaged as a radom varable for whch Pr Zˆ,,..., Ths s a etrely proper probablty dstrbuto It s kow as the emprcal dstrbuto ad wll be useful Secto 9.5. Ẑ 4

5 No-parametrc modellg ca be useful Scale famles of dstrbutos All sesble parametrc models for clam se are of the form Z Z0, where 0 s a parameter ad Z0 s a stadarded radom varable correspodg to. The large the scale parameter, the more spread out the dstrbuto Ẑ Z Z Z 0, 0 ~ N, Z ~ N, 2 Z ~ N2,2 3 Z ~ N3,3 5

6 Fttg a scale famly Fttg a scale famly Models for scale famles satsfy Pr Z Pr Z / or F F 0/ F ad F 0/ 0 where are the dstrbuto fuctos of Z ad Z0. Dfferetatg wth respect to yelds the famly of desty fuctos df0 f f0, 0 where f0 d The stadard way of fttg such models s through lkelhood estmato. If,, are the hstorcal clams, the crtero becomes L, f0 log log{ f / }, whch s to be maxmed wth respect to ad other parameters. A useful exteso covers stuatos wth cesorg. 0 6

7 Fttg a scale famly Fttg a scale famly Full value surace: The surace compay s lable that the object at all tmes s sured at ts true value Frst loss surace The object s sured up to a pre-specfed sum. The surace compay wll cover the clam f the clam se does ot exceed the pre-specfed sum The chace of a clam Z exceedg b s, ad for b such evets wth lower bouds b,,bb the aalogous jot probablty becomes { 0 F0 b / F0 b / } x... x{ F b / }. Take the logarthm of ths product ad add t to the log lkelhood of the fully observed clams,,. The crtero the becomes L, f0 log log{ f0 / } log{ F0 complete formato for objects fully sured b b cesorg to the rght for frst loss sured / }, 7

8 Shfted dstrbutos Shfted dstrbutos The dstrbuto of a clam may start at some treshold b stead of the org. Obvous examples are deductbles ad re-surace cotracts. Models ca be costructed by addg b to varables startg at the org;.e. where Z0 s a stadarded varable as before. Now Pr Z Pr b Z0 Pr Z0 b Example: Re-surace compay wll pay f clam exceeds NOK Z Z 0 The payout of the surace compay Total clam amout Currecy rate for example NOK per EURO, for example 8 NOK per EURO 8

9 Skewess as smple descrpto of shape Skewess A major ssue wth clam se modellg s asymmetry ad the rght tal of the dstrbuto. A smple summary s the coeffcet of skewess skew Z where E Z 3 Negatve skewess Postve skewess Negatve skewess: the left tal s loger; the mass of the dstrbuto Is cocetrated o the rght of the fgure. It has relatvely few low values Postve skewess: the rght tal s loger; the mass of the dstrbuto Is cocetrated o the left of the fgure. It has relatvely few hgh values 9

10 The radom varable that attaches probabltes / to all clams of the past s a possble model for future clams. Expectato, stadard devato, skewess ad percetles are all closely related to the ordary sample versos. For example Furthermore, Thrd order momet ad skewess becomes s Z sd Z Z E Z E Z s, ˆ ˆ Pr ˆ ˆ ˆ var No-parametrc estmato 0. ˆ Pr ˆ Z Z E ˆ} { ˆ ˆ skewẑ ad ˆ ˆ Z sd Z Z Ẑ No parametrc estmato

11 The log-ormal famly Parametrc estmato: the log ormal famly A coveet defto of the log-ormal model the preset cotext s 2 / 2 as Z Z 0 where Z0 e for ~ N0, Mea, stadard devato ad skewess are E Z, sdz e 2, skew Z e 2 2 e 2 see secto 2.4. Parameter estmato s usually carred out by otg that logarthms are Gaussa. Thus Y 2 log Z log / 2 ad whe the orgal log-ormal observatos,, are trasformed to Gaussa oes through y=log,,y=log wth sample mea ad varace y ad, the estmates of become s y ad 2 log ˆ / 2 ˆ y, ˆ s y or ˆ e 2 y s /2y, ˆ s y.

12 The Gamma famly Parametrc estmato: the gamma famly The Gamma famly s a mportat famly for whch the desty fucto s f x / It was defed Secto 2.5 as s the stadard Gamma wth mea oe ad shape alpha. The desty of the stadard Gamma smplfes to f x Mea, stadard devato ad skewess are x x e e x / x, x 0, where x x x, Z G x 0, where x 0 0 e e where G ~ Gamma dx dx E Z, sdz /, skewz 2/ ad there s a covoluto property. Suppose G,,G are depedet wth. The G ~ Gamma G ~ Gamma... f G... G G... 2

13 The Gamma famly Parametrc estmato: fttg the gamma The Gamma famly s a mportat famly for whch the desty fucto s f x / x e x / x, x 0, where x It was defed Secto 2.5 as s the stadard Gamma wth mea oe ad shape alpha. The desty of the stadard Gamma smplfes to f x x e x x, Z G x 0, where x 0 0 e e where G ~ Gamma dx dx 3

14 The Gamma famly 4 Parametrc estmato: fttg the gamma f L G Z f 0 0 / log log / log / log log log log log / / log log log log / / log log log log / log log, log log log log

15 Example: car surace No parametrc Log-ormal, Gamma Hull coverage.e., damages o ow vehcle a collso or other sudde ad uforesee damage Tme perod for parameter estmato: 2 years Covarates: Car age Rego of car ower Tarff class Bous of sured vehcle Gamma wthout ero clams the best model 5

16 QQ plot Gamma model wthout ero clams No parametrc Log-ormal, Gamma 6

17 No parametrc Log-ormal, Gamma Results tarff class 250,0 % 200,0 % ,0 % 00,0 % 50,0 % Rsk years Dfferece from referece, gamma model Dfferece from referece, Gamma model wthout ero clams ,0 % 7

18 No parametrc Log-ormal, Gamma Results bous 40,0 % 20,0 % ,0 % 80,0 % 60,0 % 40,0 % 20,0 % Rsk years Dfferece from referece, gamma model Dfferece from referece, Gamma model wthout ero clams 0 70,00 % 75,00 % Uder 70% 0,0 % 8

19 No parametrc Log-ormal, Gamma Results rego 40,0 % 20,0 % ,0 % 80,0 % 60,0 % 40,0 % 20,0 % 0,0 % Rsk years Dfferece from referece, gamma model Dfferece from referece, Gamma model wthout ero clams 9

20 No parametrc Log-ormal, Gamma Results car age 20,0 % ,0 % ,0 % Rsk years ,0 % 40,0 % 20,0 % Dfferece from referece, gamma model Dfferece from referece, Gamma model wthout ero clams 0 <= 5 years 5-0 years 0-5 years >5 years 0,0 % 20

21 Overvew Importat ssues Models treated Currculum Durato lectures What s drvg the result of a olfe surace compay? surace ecoomcs models Lecture otes 0,5 Posso, Compoud Posso How s clam frequecy modelled? ad Posso regresso Secto EB,5 How ca clams reservg be modelled? Cha ladder, Berhuetter Ferguso, Cape Cod, Note by Patrck Dahl 2 How ca clam se be modelled? Gamma dstrbuto, logormal dstrbuto Chapter 9 EB 2 How are surace polces prced? Geeraled Lear models, estmato, testg ad modellg. CRM models. Chapter 0 EB 2 Credblty theory Buhlma Straub Chapter 0 EB Resurace Chapter 0 EB Solvecy Chapter 0 EB Repetto 2

22 The ultmate goal for calculatg the pure premum s prcg Pure premum = Clam frequecy x clam severty total clam amout Clam severty umber of clams Clam frequecy umber of clams umber of polcy years Parametrc ad o parametrc modellg secto 9.2 EB The log-ormal ad Gamma famles secto 9.3 EB famles secto 9.4 EB methods secto 9.5 EB for the model secto 9.6 EB 22

23 dstrbuto No parametrc Log-ormal, Gamma dstrbutos, troduced Secto 2.5, are amog the most heavytaled of all models practcal use ad potetally a coservatve choce whe evaluatg rsk. Desty ad dstrbuto fuctos are / f ad F -, 0. / / Smulato ca be doe usg Algorthm 2.3:. Iput alpha ad beta 2. Geerate U~Uform 3. Retur X = betau^^-/alpha- Pareto models are so heavy-taled that eve the mea may fal to exst that s why aother parameter beta must be used to represet scale. Formulae for expectato, stadard devato ad skewess are E Z, sdz 2 /2, - 2 skewz 2 /2 3 vald for alpha>, alpha>2 ad alpha >3 respectvely. 23

24 dstrbuto No parametrc Log-ormal, Gamma The meda s gve by med Z 2 / The expoetal dstrbuto appears the lmt whe the rato / s kept fxed ad. There s ths sese overlap betwee the Pareto ad the Gamma famles. The expoetal dstrbuto s a heavy-taled Gamma ad the most lghttaled Pareto ad t s commo to regard t as a member of both famles Lkelhood estmato model was used as llustrato Secto 7.3, ad lkelhood estmato was developed there Cesored formato s ow added. Suppose observatos are two groups, ether the ordary, fully observed clams,.., or those oly to kow to have exceeded certa thresholds b,..,b but ot by how much. The log lkelhood fucto for the frst group s as Secto 7.3 log / log, 24

25 dstrbuto No parametrc Log-ormal, Gamma whereas the cesored part adds cotrbuto from kowg that Z>b. The probablty s Pr Z b b / ad the full lkelhood becomes or L, log / log{pr Z Complete formato Ths s to be maxmsed wth respect to much the same as Secto 7.3. b } - b log b log / log b ad b / / Cesorg to the rght, a umercal problem very 25

26 Over-threshold uder Pareto 26 0.,, Pr Pr b F b f f b F b F b F b Z b Z b Z b Z Z b Oe of the most mportat propertes of the Pareto famly s the behavour at the extreme rght tal. The ssue s defed by the over-threshold model whch s the dstrbuto of Zb=Z-b gve Z>b. Its desty fucto s The over-threshold desty becomes Pareto: / / / / / b b b b b f b Pareto desty fucto The shape alpha s the same as before, but the parameter of scale has ow chaged to Over-threshold dstrbutos preserve the Pareto model ad ts shape. The mea s gve by alpha must exceed b b b b b Z Z E b b No parametrc Log-ormal, Gamma

27 Add the umerator f The exteded Pareto famly / / / to the Pareto desty fucto, ad t reads where,, 0 whch defes the exteded Pareto model. Shape s ow defed by two parameters ad, ad ths creates useful flexblty. The desty fucto s ether decreasg over the postve real le f theta <= or has a sgle maxmum f theta >. Mea ad stadard devato are / 2 E Z ad sdz 2 Pareto desty fucto whch are vald whe alpha > ad alpha>2 respectvely whereas skewess s 2 / 2 2 skew Z 2, 3 provded alpha > 3. These results verfed Secto 9.7 reduce to those for the ordary Pareto dstrbuto whe theta=. No parametrc Log-ormal, Gamma 27

28 Samplg the exteded Pareto famly A exteded Pareto varable wth parameters G Z where G ~ Gamma, G2 ~ Gamma G 2,, ca be wrtte Here G ad G2 are two depedet Gamma varables wth mea oe. The represetato whch s provded Secto 9.7 mples that /Z s exteded Pareto dstrbuted as well ad leads to the followg algorthm: Algorthm 9. The exteded Pareto sampler. Iput,, ad / 2. Draw G ~ Gammatheta 3. Draw G2 ~ Gammaalpha 4. Retur Z <- etta G/G2 No parametrc Log-ormal, Gamma 28

29 methods No parametrc Log-ormal, Gamma Large clams play a specal role because of ther mportace facally The share of large clams s the most mportat drver for proftablty volatlty «The larger clam the greater s the degree of radomess» But experece s ofte lmted How should such stuatos be tackled? Theory Pareto dstrbutos are preserved over thresholds If Z s cotuous ad ubouded ad b s some threshold, the Z-b gve Z>b wll be Pareto as b grows to fty!!..ok. How do we use ths? How large does b has to be? 29

30 methods No parametrc Log-ormal, Gamma Our target s Zb=Z-b gve Z>b. Cosder ts tal dstrbuto fucto F b Fb Pr Zb where F PrZ F b b ad let Yb Z b / b where s a scale parameter depedg o b. We are assumg that Z>b, ad Yb s the postve wth tal dstrbuto Pr Yb b Fb b y. The geeral result says that there exsts a parameter alpha ot depedg o b ad possbly fte ad some sequece betab such that F y P y; b b as b where y P y; y e,, 0 The lmt P y; s the tal dstrbuto of the Pareto model whch shows that Zb becomes Pareto, b as b. Both the shape alpha ad the scale parameter betab deped o the orgal model but oly the latter vares wth b. 30

31 methods No parametrc Log-ormal, Gamma The decay rate ca be determed from hstorcal data Oe possblty s to select observatos exceedg some threshold, mpose the Pareto dstrbuto ad use lkelhood estmato as explaed Secto 9.4. We wll revert to ths A alteratve ofte referred to the lterature of extreme value s the Hll estmate Start by sortg the data ascedg order... ad take ˆ log where - p Here p s some small, user-selected umber. The method s o-parametrc o model s assumed We may wat to use ˆ as a estmate of a Pareto dstrbuto mposed over the threshold b ad would the eed a estmate of the scale parameter b. A practcal choce s the 2 2 ˆ b where / ˆ

32 The etre dstrbuto through mxtures No parametrc Log-ormal, Gamma Assume some large clam threshold b s selected The there are may values the small ad medum rage below ad up to b ad few above b How to select b? Oe way: choose some small probablty p ad let = teger-p ad let b= Aother way: study the percetles Modellg may be dvded to separate parts defed by the threshold b 0 f Z b Z Ib Zb IbZb where Ib Cetral Rego plety of data Extreme rght tal data s scarce Modellg the cetral rego: o-parametrc emprcal dstrbuto or some selected dstrbuto.e., log-ormal gamma etc Modellg the extreme rght tal: The result due to Pckads suggests a Pareto dstrbuto, provded b s large eough But s b large eough?? Other dstrbutos may perform better, more about ths Secto 9.6. f Z b 32

33 No parametrc Log-ormal, Gamma 00 % percetles fre 95 % 90 % 85 % 80 % 75 % 70 % 65 % 60 %

34 No parametrc Log-ormal, Gamma 00 % percetles water 95 % 90 % 85 % 80 % 75 % 70 % 65 % 60 %

35 No parametrc Log-ormal, Gamma 00 % percetles other clam types 95 % 90 % 85 % 80 % 75 % 70 % 65 % 60 %

36 No parametrc Log-ormal, Gamma 00,0 % Share of total cost fre 90,0 % 80,0 % 70,0 % 60,0 % 50,0 % Share of total cost fre 40,0 % 30,0 % 20,0 % 0,0 % 0,0 % 80 % 82 % 84 % 86 % 88 % 90 % 92 % 94 % 96 % 98 % 00 %

37 No parametrc Log-ormal, Gamma 00,0 % Share of total cost water 90,0 % 80,0 % 70,0 % 60,0 % 50,0 % Share of total cost water 40,0 % 30,0 % 20,0 % 0,0 % 0,0 % 80 % 82 % 84 % 86 % 88 % 90 % 92 % 94 % 96 % 98 % 00 %

38 00,0 % Share of total cost other 90,0 % 80,0 % 70,0 % 60,0 % 50,0 % Share of total cost other 40,0 % 30,0 % 20,0 % 0,0 % 0,0 % 80 % 82 % 84 % 86 % 88 % 90 % 92 % 94 % 96 % 98 % 00 %

39 00,0 % Share of total cost all 90,0 % 80,0 % 70,0 % 60,0 % 50,0 % Share of total cost all 40,0 % 30,0 % 20,0 % 0,0 % 0,0 % 80 % 82 % 84 % 86 % 88 % 90 % 92 % 94 % 96 % 98 % 00 %

40 No parametrc Log-ormal, Gamma 00,0 % 90,0 % 80,0 % 70,0 % 60,0 % 50,0 % 40,0 % Share of total cost fre Share of total cost water Share of total cost other Share of total cost all 30,0 % 20,0 % 0,0 % 0,0 % 80 % 82 % 84 % 86 % 88 % 90 % 92 % 94 % 96 % 98 % 00 %

41 No parametrc Log-ormal, Gamma fre water other all 99, % ,2 % ,3 % ,4 % ,5 % ,6 % ,7 % ,8 % ,9 % ,0 %

42 Frequecy No parametrc Log-ormal, Gamma 700 Fre up to 88 percetle B

43 Frequecy No parametrc Log-ormal, Gamma 80 Fre above 88th percetle B

44 More Frequecy No parametrc Log-ormal, Gamma 2 Fre above 95th percetle B

45 for the model How s the fal model for clam se selected? Tradtoal tools: QQ plots ad crtero comparsos Trasformatos may also be used see Erk Bølvke s materal No parametrc Log-ormal, Gamma 45

46 No parametrc Log-ormal, Gamma Descrptve Statstcs for Varable Skadeestmat Number of Observatos 85 Number of Observatos Used for Estmato 85 Mmum Maxmum Mea Stadard Devato

47 No parametrc Log-ormal, Gamma Model Selecto Table Dstrbuto Coverged -2 Log Lkelhood Selected Burr Yes 5808 No Log Yes 5807 No Exp Yes 587 No Gamma Yes 5799 No Igauss Yes 5804 No Pareto Yes 5874 No Webull Yes 5799 Yes

48 No parametrc Log-ormal, Gamma Dstrbuto -2 Log Lkelhood All Ft Statstcs Table AIC AICC BIC KS AD CvM Burr ,4053 3, ,5630 Log , ,7004 0,69457 Exp ,7846 4, ,54964 Gamma ,2029 * 2,8879 0,4546 Igauss , , ,94893 Pareto ,49278,8209,989 Webull 5799 * 5803 * 5803 * 5809 *, ,64769 * 0,428 *

49 No parametrc Log-ormal, Gamma

50 No parametrc Log-ormal, Gamma

51 No parametrc Log-ormal, Gamma

52 No parametrc Log-ormal, Gamma Descrptve Statstcs for Varable Skadeestmat Number of Observatos 5 Number of Observatos Used for Estmato 5 Mmum Maxmum Mea Stadard Devato

53 No parametrc Log-ormal, Gamma Model Selecto Table Dstrbuto Coverged -2 Log Lkelhood Selected Burr Yes 3590 No Log Yes 3586 No Exp Yes 370 No Gamma Yes 3587 No Igauss Yes 3586 Yes Pareto Yes 3763 No Webull Yes 3598 No

54 No parametrc Log-ormal, Gamma Dstrbuto -2 Log Lkelhood All Ft Statstcs Table AIC AICC BIC KS AD CvM Burr , , ,0544 Log , , ,08734 Exp ,4643 7,7328 3,44525 Gamma ,5905 * 0,4652 * 0,04250 * Igauss 3586 * 3590 * 3590 * 3595 * 0, ,6707 0,096 Pareto ,264 25,2499 5,09728 Webull ,88778,224 0,0294

55 No parametrc Log-ormal, Gamma

56 No parametrc Log-ormal, Gamma

57 No parametrc Log-ormal, Gamma

58 for the model No parametrc Log-ormal, Gamma Ca we do better? Does t exst a more geerc class of dstrbuto wth these dstrbutos as specal cases? Does ths geerc class of dstrbutos outperform the selected model the two examples fre above 90th percetle ad fre above 95th percetle? 58