Construction and Evaluation of Actuarial Models. Rajapaksha Premarathna

Transcription

1 Costructo ad Evaluato of Actuaral Models Raapaksha Premaratha

2 Table of Cotets Modelg Some deftos ad Notatos...4 Case : Polcy Lmtu...4 Case : Wth a Ordary deductble....5 Case 3: Maxmum Covered loss u wth a polcy deductble d u.7 Case 4: Cosurace factor where 0 ad/or flato rate r..7 Other Cocepts.8 Bous Paymets..8 Frachse Deductble.9 Compoud Dstrbutos.9 Stop Loss Isurace.0 Model Estmato Revew of Estmators.. No-Parametrc Emprcal Pot Estmato. Case : Emprcal estmato from a radom sample wth complete dvdual data Case : Emprcal Estmato from Grouped Data Case 3: Estmato from Cesored ad trucated data 3 The Kapla-Meer/Product Lmt Estmator.3 Kapla Meer Approxmato for Large data Sets 4 Varace of Survval Probablty Estmates 4 Cofdece Iterval for Survval Probablty Estmates...6 Method of Momets 6 Method of Percetle Matchg 6

3 Maxmum Lkelhood Estmato (MLE) Defto.7 Maxmum Lkelhood Estmato for Complete Data (No trucato or Cesorg) 7 Lkelhood fucto for Loss data wth polcy lmt u (rght cesored data)..7 Lkelhood fucto for Loss data wth Polcy Deductble d (left trucated data).7 Lkelhood fucto for Loss data wth Polcy Deductble d ad Maxmum Covered Lossu...8 Maxmum Lkelhood of Expoetal Dstrbuto wth parameter.8 Maxmum Lkelhood Estmato Shortcuts for Dstrbutos Exam C Table (gve o Trucato or Cesorg)...8 Maxmum Lkelhood Estmato for Grouped Data..0 Credblty Lmted Fluctuato Credblty Theory Full Stadard of Credblty for Compoud dstrbutos... Full Credblty Stadard for Posso Radom Varable N (Number of Clams)..3 Partal Credblty 3 Predctve Probablty..4 Bayesa Credblty Shortcuts 5 Buhlma Credblty 7 The Buhlma Straub model.7 Emprcal Bayes Credblty Methods 8 Emprcal Bayes Estmato for Buhlma model (Equal Sample Sze).9 Emprcal Bayes Estmato for the Buhlma Straub Model (Uequal Sample Sze) 9 Sem parametrc Emprcal Bayesa Credblty 30 3

4 Modelg Some deftos ad Notatos Groud up loss the actual loss amout pror to modfcatos. The loss radom varable s deoted. We geerally assume 0. Cost per loss the amout pad by surer. Ths cludes the zero paymets. Deoted Y L Cost per paymet the amout pad by surer whch cludes oly the o-zero paymets made by the surer. It s also called the excess loss radom varable or the left trucated ad shfted varable. Deoted Yp Severty dstrbuto the dstrbuto loss amout or the cost to the surer Frequecy dstrbuto the dstrbuto of the umber of losses, or amout pad per ut tme Polcy Lmt Maxmum amout pad by surace polcy for a sgle loss, deotedu. If there s a deductble the polcy lmt s u d Ordary Deductble for loss amouts below deductble, d the surer pays 0 ad for loss amouts above d the surer pays the dfferece of the loss amout ad deductble. Maxmum Covered Loss the amout u for whch o addtoal beefts are pad. Deoted u Case : Polcy Lmt u u Amout pad by surer = u m(, u) u u Note: us also referred to as Lmted loss radom varable Therefore the expected value s: u E u x f x dx u F u (for cotuous) 0 E u x p x u F u (for dscrete) x u u E u F x dx Sx( x) dx Also 0 0 u (for dscrete or cotuous) 4

5 u k k k E u x f x dx u F u Ad (for cotuous) 0 Var u E u E u Case : Wth a Ordary deductble Amout pad by surer = cost per loss = left cesored ad shfted radom varable 0 d YL ( d) max( d,0) d d d Therefore the expected values are: ( ) E Y E d x d f x dx (for cotuous) L L ( ) ( ) E Y E d x d p x d x u (for dscrete) E YL E ( d) E E d F x dx Sx( x) dx d d (for dscrete or cotuous) L L L Var d E d E d Var Y E Y E Y Also ote that E YL E ( d) E E d d E E d Whe cosderg the expected cost per paymet E d d E Y p e( d) EY P E YL E ( d) E E d P d F d F d EY P E ( d) F ( d) d x d f x dx d f x dx (for cotuous) 5

6 EY P E ( d) F d x u ( x d) p x x u p x (for dscrete) EY P d F x dx Sx( x) dx d (for dscrete or cotuous) F d S d The expected cost per paymet s also referred to as the mea excess loss, or the mea resdual loss or mea resdual lfetme. Here are some shortcuts that wll be useful the examato:. Gve s a uform dstrbuto o 0, ad a ordary deductble d s appled. The: d E Yp ad d E YP ad the varace s Var Yp 3 d. Gve s a expoetal dstrbuto wth mea ad a ordary deductble d s appled. The: EY p ad E Y ad the varace s Var Y P p 3. Gve s a Pareto dstrbuto wth parameters ad ad a ordary deductble d s appled ( ). The: d The pdf of YP s also Pareto wth parameters ad d. Therefore, EY P ad P d E Y 4. Gve s a sgle parameter Pareto dstrbuto wth parameters ad ad a ordary deductble d s appled ( ). If d the EY EY E d ad Var Y Var Y Var If d P L they P has two parameter Pareto dstrbuto wth parameters ad The varace of cost per loss wth a deductble d s P P P p L Var d d Var Y E Y E Y E d d E d d d 6

7 Note that E Y ( ) L E d E Y P E d d P d F d Case 3: Maxmum Covered loss u wth a polcy deductble d Therefore the Cost per Loss YL s 0 d YL d d u u d u d u The expected cost per loss s: u E Y E u E d x d f x dx u d F u F x dx L d d The secod momet for cost per loss therefore s: u u L d u E Y E u E d d E u E d x d f x dx u d F u The expected cost per paymet s: EY P L F d E Y E u E d F d The secod momet of cost per paymet s: EY L EY P F d Case 4: Cosurace factor where 0 ad/or flato rate r Wth maxmum covered loss u ad deductble d the amout pad by the surer s (o flato): 0 d YL d d u u d u The expected cost per loss s: 7

8 L E Y E u E d F x dx u d The expected cost per paymet s: EY P L EY E u E d F d F d Wth flato the expected cost per loss s: u d EYL re E r r Ad the expected cost per paymet s: EY P u d r E E EY L r r d d F F r r Other Cocepts The Loss Elmato rato s E d E Bous Paymets If there s a bous for loss amouts less tha a specfc lmtu Bous = u 0 u 0 u Therefore the expected bous paymet s EBous u E u ad f the bous s equal to a fracto c of the amout by whch the loss s less tha u the EBous c u E u 8

9 Frachse Deductble Frachse deductble s whe the surer pays the full amout a deductble deoted d. Therefore 0 d Amout pad by surer = d Therefore the Expected cost per loss s x f xdx E d d F d Expected cost per paymet = E Y P d EY E d d F d E d L d P d F d F d Compoud Dstrbutos Termology ad the N Is the umber of clams or the clam cout radom varable. The dstrbuto s called the clam cout dstrbuto or frequecy dstrbuto. Is the sgle or dvdual loss radom varable whose dstrbuto s kow as the severty dstrbuto S N ad s the aggregate loss per perod ad has a compoud dstrbuto N,,,, N are mutually depedet radom varables ES EN E Var E S N Var N E ad E Var S N ENVar If t s a compoud Posso dstrbuto S ad the frequecy dstrbuto s Posso wth mea the E ad Var S E E S The dstrbuto of S f N s the ab,, class We kow that PN k pk ad PS fs. Also P( x) f x P S x f x S b p a b p f x a f f x a f 0 x 0 S x 9

10 The dstrbuto of S f N s the ab,,0 class P S x f x The dstrbuto of S f N s Posso Stop Loss Isurace S x x f x f f x S S x b a f fs x x a f 0 If a deductble s appled to aggregate losses the surace paymet s the aggregate loss excess of the deductble. The stop loss surace paymet s 0 S d MaxS d,0 S d S S d S d S d The expected value stop loss surace paymet s called the et stop loss premum where E S d E S E S d Also b d b a E S d E S a d a FS a E S a E S b b a b a where ad E S d E S d F d a d b S for d 0 0

11 Model Estmato Revew of Estmators s a ubased estmator of ad has the same mea as E E Var N ths s the varace of the sample mea For a parameter the estmator s ubased f E ˆ The Bas of the parameter estmator s Bas ˆ E ˆ E ˆ Bas ˆ Var ˆ The Mea Square error or MSE s No-Parametrc Emprcal Pot Estmato The radom varable ca be a loss radom varable or a falure tme radom varable. It ca be dscrete or cotuous. A falure tme radom varable descrbes the tme utl a partcular eve happes. Sample formato for estmatg the radom varable s avalable the oe of the followg ways:. A radom sample of depedet dvdual observatos. Grouped data: the rage of the radom varable s dvded to a seres of tervals, (, )(, ),...,(, )(, ) c0 c0 c cr cr cr ad the umber of observatos a terval c 3. Cesored or trucated data Case : Emprcal estmato from a radom sample wth complete dvdual data (, c ) s If the exact values of observatos x, x,..., x (where x s a loss amout gve the data s a loss dstrbuto or t s tmes of death or falure gve t s survval dstrbuto) the data s cosdered to be complete. A probablty of s assged for each x. If there are k dstct umercal values such that these k values or ordered from smallest to largest as y y... y wth s =umber of observatos equal to y ad s s... s.

12 The emprcal dstrbuto probablty fucto s umber of x's that are equal to y s p( y) The emprcal dstrbuto fucto s F t umber of x 's t The emprcal survval fucto s S t F t The rsk set at umber of x's t y s deoted r, where r. If there are s deaths at tme y so there are r sat rsk at secod death tme y. If there are s deaths at tme thrd death tme. The Nelso-Aelo estmate of the cumulatve hazard fucto s 0 t x s H t x t x,,3,..., k r k s xk t r y so there are r s s at rsk at The Nelso-Aelo estmate of the survval fucto s S x e H x, ad the Nelso-Aelo estmate of the dstrbuto fucto s F x S x e H x I order to fd the smoothed emprcal estmate of the 00p-th percetle p use the followg steps.. Order the sample values from smallest to largest. g g. Fd a teger g such that p 3. p s foud by lear terpolato p g p x g p g x g Case : Emprcal Estmato from Grouped Data The emprcal estmate of the mea of s c c r uformly dstrbuted. The emprcal estmate of the k-th momet s:. We assume the loss amouts are r k k c c k c c

13 Case 3: Estmato from Cesored ad trucated data A trucated observato s data pot that s ot observed. Left trucato s trucato below (deductble). A cesored observato s a observato that s observed to occur, but whose value s ot kow. Rght cesorg s cesorg from above (polcy lmt). Data descrpto If dvdual s a left trucated data pot who has a value d that satsfes y d y, the we add that dvdual to the rsk set r for the ext death pot y ad dvdual m s rght cesored data pot who has a value u m that satsfes y u m y, the we remove that dvdual from the rsk set r for the ext data pot y. Ths s smlar to the followg: the umber of dvduals who have r r s y d y y u y where s s m the umber of deaths at death pot If trucated or cesored observato tme s the same as death tme y removed after the deaths at death pot Therefore: y ad t oly affects the rsk set r umber of 's< umber of 's< umber of 's< umber of 's umber of u's umber of d's r d y x y u y or r x y y y The Kapla-Meer/Product Lmt Estmator 0t y s S t y t y,,3,..., k r k s or 0 t yk r If z deoted the largest observato the data set. Therefore whe estmatg. S t 0. S t S z y that dvdual s added or St fort z: 3

14 3. S t S z t/ z (geometrc exteso approxmato) Kapla Meer Approxmato for Large data Sets Frst choose a sequece of tme pots say c 0 c... ck. For a terval( c, c ], the umber of ucesored observed deaths s deoted x ; the umber of rght cesored observatos s deoted ad d deotes the umber of left trucated observatos. The umber at rsk at tme 0, r 0 d 0. The umber at rsk for tme terval ( c, c ] s r d x u 0 0 The product lmt estmate for the survval probablty to the pot c s x x u 0 r0 r r A varato o the Kapla Meer/Product Lmt large approach s defed by the followg factors: P0 0 ad the umber at rsk at tme P d u x 0 Varace of Survval Probablty Estmates c s r P d u If there s o cesorg or trucato gve dvdual data the emprcal estmate of the survval fucto umber of deaths that occur after tme x Y x S x where x =the umber of survvors to tme x Also the estmator s a ubased ad cosstet estmator of Var S x S x S x S x ad the varace s For grouped data for data pots wth tervals the form ( c0, c ],( c, c],...,( c, c ],( c, ) the varace of Var S S x x s c c c c Var Y x c Var m c c x c Cov Y, m x 4

15 Ad c x x c S x S c S c c c c c Where x = value betwee terval ( c, c ] S c Y umber of survvors at tme c m umber of deaths terval ( c, c ] Var Y S c S c Var m S c S c S c S c Cov Y, m S c S c S c The estmate for the desty fucto the terval c varace of the estmator s Var f x c s S c S c f x c c c c S c S c S c S c The Greewood s Approxmato of the estmated varace of the product lmt estmator s s Var ˆ S y S y r r s The estmated varace of the Nelso Aale estmate of the cumulatve hazard fucto H y s ˆ ˆ s Var H y r ad the 5

16 Cofdece Iterval for Survval Probablty Estmates For a estmator ˆ for a parameter the 95% lear cofdece terval for s ˆ.96 var ˆ The lower lmt for the 95% log trasformed cofdece terval for s.96 Var ˆ S S U t t where U exp Stl S t St s /U S t ad the upper lmt The lower lmt for the 95% log trasformed cofdece terval for Ĥ t.96 U where U exp Var ˆ Hˆ t Hˆ t Note:.96 s foud usg the ormal dstrbuto table provded Method of Momets Ht s Ĥt U ad the upper lmt s For a dstrbuto defed terms of r parameters,,..., r the method of momets estmator of the parameter values s foud by solvg the r equatos: theoretcal -th momet = emprcal -th momet, =,,., r If the estmator has oly oe parameter,the solve for from the equato theoretcal dstrbuto frst momet = emprcal dstrbuto frst momet If the dstrbuto has two parameters ad the we solve the followg equatos, E emprcal estmate of E ad E emprcal estmate of E or theoretcal dstrbuto varace = emprcal dstrbuto varace. Method of Percetle Matchg Gve a radom sample or a terval grouped data sample ad a dstrbuto wth r parameters, choose r percetle pots p,..., pr ad set the dstrbuto p yth percetle equal to the emprcal estmate for the p th percetle. The r parameter values are foud by solvg the system of equatos. 6

17 Maxmum Lkelhood Estmato (MLE) Defto Maxmum Lkelhood Estmato s used to estmate the parameters a parametrc dstrbuto. We are tryg to fd the dstrbuto parameters that would maxmze the desty or the probablty of the data set occurrg. Frst we create the lkelhood fucto For dvdual data L f x ; r ; ; L where s the parameter beg estmated. for a radom sample x, x,..., x ad for grouped data L F c F c for r tervals where terval ( c, c ] has observatos Maxmum Lkelhood Estmato for Complete Data (No trucato or Cesorg) Use the followg steps to fd the maxmum lkelhood estmato. Fd L. Fd log lkelhood l l L d l d 4. Solve for 3. Set 0 Lkelhood fucto for Loss data wth polcy lmt u (rght cesored data) m The lkelhood fucto s L f x ; F u; where m s the umber of lmt paymets equal to u (losses greater thau ) ad there are paymets below the lmt. Lkelhood fucto for Loss data wth Polcy Deductble d (left trucated data) Loss data ca be avalable forms. Isurace paymets >0 deoted y, y,..., y k. Actual loss amouts greater tha the deductble, x, x,..., x k Ths meas that x y d therefore L k k f x ; f y d; F d; F d; 7

18 Lkelhood fucto for Loss data wth Polcy Deductble d ad Maxmum Covered Loss u If there observed paymets y, y,..., ythat satsfy 0 y u d ad loss amouts x, x,..., xwhere x y d the lkelhood fucto s: L m f x ; F u; f y d; F u; m m F d; F d; Where m s the amout of observed lmt paymets equal to u d therefore there wll be m correspodg losses u Maxmum Lkelhood of Expoetal Dstrbuto wth parameter For complete dvdual data wthout trucato or cesorg the MLE estmator for parameter s the sample mea x x for a radom sample of observatos x, x,..., x For a expoetal dstrbuto wth a data set wth m lmt paymets ad polcy lmt u the MLE of the MLE for the mea of s ˆ x mu total of all paymet amouts umber of o cesored paymets For a expoetal dstrbuto wth a polcy deductble the MLE for the mea of groud up loss s (gve data avalable was surace paymets y, y,..., y k ) ˆ y total of all surace paymet amouts umber of surace paymets Maxmum Lkelhood Estmato Shortcuts for Dstrbutos Exam C Table (gve o Trucato or Cesorg) For a radom sample x, x,..., x of the followg dstrbutos: m For a verse expoetal Dstrbuto wth parameter the MLE of s x For a Pareto dstrbuto wth parameters where, s gve the MLE of s l x l 8

19 For a Webull Dstrbuto wth parameters where, s gve the MLE for s ˆ For a Iverse Pareto dstrbuto wth parameters where, s gve the MLE of s l x l x For a Iverse Webull Dstrbuto wth parameters where, s gve the MLE for s ˆ x For a Normal Dstrbuto wth mea ad varace the MLE of s ˆ x,the sample mea. For a Normal Dstrbuto wth mea ad varace the MLE of form of the sample varace. For a Logormal Dstrbuto wth parameters ad s ˆ x ˆ the MLE of s ˆ l x mea. For a Normal Dstrbuto wth mea ad varace the MLE of,the based form of the sample varace. For a Gamma dstrbuto wth parameters where, s gve the MLE of s ˆ x,the based,the sample s ˆ l x ˆ For a Iverse Gamma dstrbuto wth parameters where, s gve the MLE of s ˆ x For a Posso dstrbuto wth parameter the MLE of s ˆ k observatos s 0... k0 k x where the total umber of For a Bomal Dstrbuto wth parameters mqf, the sample varace s larger tha the sample mea, the the bomal dstrbuto s ot a good ft for the data. If m s kow or gve for a data set 9

20 0,,... m he momet estmate ad the MLE of q are both m kk k 0 total umber of heads qˆ m m total umber of co tosses k 0 k Maxmum Lkelhood Estmato for Grouped Data The data gve s grouped to 4 categores Category : data value x that has o trucato or cesorg Category : data value u, o deductble but polcy lmt u Category 3: data value x before a deductble d ad o polcy lmt Category 4: polcy lmt paymet u dwth deductble d ad maxmum covered loss u C the sum of x 's Category C the sum of u 's Category C3 the sum of x d's Category 3 C4 the sum of u d's Category 4 umber of data pots Category,,3,4 The MLE for a expoetal dstrbuto wth mea s ˆ C C C C For a sgle parameter Pareto dstrbuto wth parameters where, s gve: x Category : z l u Category : v l x Category 3: w l d 0

21 u Category 4: y l d 3 The MLE of s ˆ C C C C 3 4 where the C ad factors are defed the same as earler For a Webull Dstrbuto wth parameters where, s gve: Category : Category : z x,for x that s ot cesored or trucated v u,for u that s rght cesored(lmt paymet) ad ot trucated(o deductble) The MLE for s ˆ C C ad f the data s separated to four categores: Category : Category : z x,for x that s ot cesored or trucated v u,for u that s rght cesored(lmt paymet) ad ot trucated(o deductble) Category 3: w x d Category 4: y u d The MLE for s ˆ C C C C 3 3 4

22 Credblty Gve a radom varable from a radom sample,,..., the goal of credblty theory s to estmate the mea of Lmted Fluctuato Credblty Theory If the radom sample beg aalyzed s W ad there s depedet observatos W, W,..., W avalable. Also the mea of W s ad the varace s whe P W k P s satsfed where k s some fracto of. Rage parameter k: usually k = 0.05 Probablty Level P: usually P = 0.90 Full credblty stadard s satsfed whe P W k P s satsfed the full credblty stadard s satsfed Oce P ad k are chose we fd a value y such that Py Z y P where Z s the stadard ormal dstrbuto. Therefore f P 0.90, the y.645 The chose 0 y k Therefore, for a radom varable W, full credblty s gve to W f the followg codtos are satsfed Var( W ) (square of coeffcet of varato) where s the umber of EW ( ) observatos of W. 0 0 Var W. The sum of all observed W values 0 EW Full Stadard of Credblty for Compoud dstrbutos Let compoud dstrbuto radom varable be S. S has two compoets N (Frequecy) ad Y (Severty). Severty s a o-egatve radom varable that ca be cotuous or dscrete. Usually S represets aggregate clams (per perod) whle N represets umber of clams (per perod or per polcy holder) ad Y represets sze of clam.

23 We kow that mea of varace of S s ES ENEY Var S Var N E Y E N Var Y Therefore ad Var S. Number of observatos of S eeded E S Var S. Sum of all observed S s 0 E S 0 Var S E N Var S 3. Total umber of observed clams 0 0 E S E Y E N If S has a compoud Posso dstrbuto.e. N s Posso wth mea, N ad Y are mutually depedet ad S has compoud Posso wth mea for S s EY therefore the stadard of full credblty Var S Var Y 0. Number of observatos of S eeded ES EY 0 Var S Var Y. Sum of all observed S s 0 0 EY ES EY 3. Total umber of observed clams Var S E N 0 0 Var Y ES EY Full Credblty Stadard for Posso Radom Varable N (Number of Clams). Number of observed values of N eeded = umber of perods eeded. Total umber of clams eeded 0 Var( N) 0 EN ( ) 0 Partal Credblty The credblty premum P ZW ( Z) M where W s the sample mea ad M s the maual premum. Z s called the credblty factor where Z f o avalable f o eeded for full credblty 3

24 For example to satsfy codto the partal credblty factor s Z umber of observatos avalable umber of observatos eeded for full credblty P A P A B P A B P A B P B P A B P B ad PB P A B P A B P B A PA P A B P B P A B P B Predctve Probablty P B A P B C P C A P B C P C A E Y P Y C P C P Y C P C E Y B P Y C P C B P Y C P C B m E Y E Y C P C ad m E Y B E Y C P C B The tal assumpto for the dstrbuto (wth parameter ) s called the pror dstrbuto ad the pdf/pf s deoted. The dstrbuto ca be cotuous or dscrete. The model dstrbuto s a codtoal dstrbuto (gve ) wth pdf/pf f x. For a data set of radom observed values from dstrbuto of ad a specfc, the model dstrbuto s,,, f x x x f x f x f x f x The Jot dstrbuto of ad has pf/pdf f x f x,,,,,, f, x x x f x f x f x The margal dstrbuto of s f x f x f x, x,, x f x f x f x ad for a data set ad for a data set (for cotuous ) ad,,, f x x x f x f x f x d (for dscrete ) 4

25 The posteror dstrbuto of gve xhas pdf/pf x f x x,, f Gve datax, x,, x, the predctve dstrbuto of has pdf/pf,,,,,,,,,,,, f x x x x f x x x x d for cotuous ad f x x x x f x x x x Bayesa Credblty Shortcuts for dscrete. If model dstrbuto s expoetal wth mea ad the pror dstrbuto verse gamma wth parameters the:, Whe a sgle data value s gve the mea of margal dstrbuto of s ad the posteror dstrbuto s verse gamma wth parameters ad x ad the x predctve mea s Whe there are data values the posteror dstrbuto s verse gamma wth parameters x ad x ad the predctve mea s. If model dstrbuto s Posso wth mea ad the pror dstrbuto gamma wth parameters the:, Whe a sgle data value s gve the mea of margal dstrbuto of s egatve bomal wth r ad ad the posteror dstrbuto s gamma wth parameters x ad Whe there are data values the posteror dstrbuto s gamma wth parameters x ad I both cases the predctve dstrbuto s egatve bomal wth r x ad ad the predctve mea s the same as the mea of the posteror dstrbuto If the pror dstrbuto the pror dstrbuto becomes expoetal wth the margal dstrbuto becomes geometrc. 3. If the model dstrbuto s bomal wth parameters m, q ad the pror dstrbuto s beta wth parameters a,b, the: 5

26 Whe a sgle data value s gve the posteror dstrbuto s beta wth parameters a+x ad b+m-x Whe data values are gve the posteror dstrbuto s beta wth parameters a x ad b m x ad the predctve mea wll be m (posteror mea) 4. If model dstrbuto s verse expoetal wth parameter ad the pror dstrbuto gamma wth parameters the:, Whe a sgle data value s gve the margal dstrbuto of s verse Pareto wth r the same ad the posteror dstrbuto s gamma wth parameters ad x ad the predctve mea s x ad Whe there are data values the posteror dstrbuto s gamma wth parameters ad x 5. If the model dstrbuto s Normal wth mea ad varace ad the pror dstrbuto s Normal wth mea ad varace the: For a sgle data value of x, the posteror dstrbuto s ormal wth mea x ad varace x For data values the posteror dstrbuto for s Normal wth mea ad varace. Also the predctve mea s the same as the posteror mea. 6. If the model dstrbuto s Uform wth o the terval 0, ad the pror dstrbuto s sgle parameter Pareto wth parameters the:, If there are observatos x, x,, x ad M = max( x, x,, x, ) the the posteror dstrbuto s sgle parameter Pareto wth ad M the Bayesa premum s M 6

27 Buhlma Credblty The tal structure for Buhma credblty s the same as Bayesa credblty model. Therefore the model dstrbuto s a codtoal dstrbuto (gve ) wth pdf/pf f x The tal assumpto for the dstrbuto (wth parameter )s called the pror dstrbuto ad the pdf/pf s deoted. The dstrbuto ca be cotuous or dscrete. Uder Buhlma credblty the codtoal dstrbutos of (depedet ad detcally dstrbuted). Therefore: E s the hypothetcal mea Var v s the process varace s gve s cosdered to be..d E E E E s the pure premum or collectve premum Var E Var a s the varace of the hypothetcal mea VHM E Var E v v s the expected process varace or EPV Also Var v a From ths we calculate the Buhlma Credblty Premum to be Z ( Z) where v k. Z s called the Buhlma Credblty factor. If a 0 the Z 0. a Z k where The Buhlma Straub model The dfferece betwee the orgal Buhlma model ad the Buhlma Straub model s that the codtoal varaces of gve mght ot be the same. Therefore for a gve measurg v exposure m where m m m... m the process varace Var m E E s the pure premum or collectve premum Var a s the varace of the hypothetcal mea VHM 7

28 E v v s the expected process varace or EPV Var a m Also Emprcal Bayes Credblty Methods v Our obectve s stll to apply the Buhlma or Buhlma-Straub models to determe the credblty premum based o observed clam data oly usg the followg formato:. Isurace Portfolo has r polcy holders where,,3..., r. For polcy holder data o exposure perods s avalable where,, For polcyholder ad exposure perod there are m exposure uts wth a average observed clam of per exposure ut 4. The total clam observed for polcyholder exposure perod s m ad the total clam observed for polcy holder all exposure perods s 5. The total umber of exposure uts for polcyholder s m m m 6. The average observed clams per exposure ut for polcyholder s 7. The total umber of exposure uts for all polcyholders s m m 8. The average clam per exposure perod for all polcyholders s m r m m r total observed clams for all polcyholders all perods m total umber of exposure perods for all polcyholders 9. Polcyholder has rsk parameter varable where each s d. 0. E ad Var v m. E, Var a ad E v v. The credblty premum for the ext exposure perod for polcyholder s Z ( Z ) where Z m v m a 8

29 Emprcal Bayes Estmato for Buhlma model (Equal Sample Sze) Sce... r ad m uder the Buhlma model use the followg formato to fd the premum:. ad. The estmated pror mea ˆ r r r r r 3. The estmated process varace s r vˆ ˆ v r r vˆ where r vˆ a r 4. The estmated varace of the hypothetcal mea s ˆ 5. m Z vˆ m aˆ ad f aˆ 0 the Zˆ 0 vˆ aˆ Emprcal Bayes Estmato for the Buhlma Straub Model (Uequal Sample Sze) Use the followg formato to fd premum:. ˆ. r vˆ m r r aˆ m vˆ( r ) m 3. r m m 4. m Z vˆ m aˆ vˆ aˆ 9

30 Sem parametrc Emprcal Bayesa Credblty Cosder a case where represets the umber of clams for a perod of tme. Use the followg formato to fd the premum:. E. Var v 3. v 4. Var v a 5. From ths we calculate the Buhlma Credblty Premum to be Z ( Z) where Z k where v k. Z s called the Buhlma Credblty factor a 30