ABSTRACT KEYWORDS. Proportional hazard premium principle, subexponential distributions, bootstrap, 1. INTRODUCTION

Transcription

1 APPLYING THE PROPORTIONAL HAZARD PREMIUM CALCULATION PRINCIPLE BY MARIA DE LOURDES CENTENO AND JOAO ANDRADE E SILVA ABSTRACT I this pape we discuss the applicatio of the popotioal hazad pemium calculatio piciple. I Sectio 2 we popose a method to calculate the popotioal hazad pemium of a compoud isk whe the seveity distibutio is subexpoetial. I Sectio 3 we use the empiical distibutio to calculate the pemium whe the popotioal hazad piciple is applied, which leads to a systematic udeestimatio of the pemium. Afte studyig the bias of the pemium calculated usig this o-paametic appoach we take advatage of the bootstap techique with subsamplig to educe it. KEYWORDS Popotioal hazad pemium piciple, subexpoetial distibutios, bootstap, subsamplig. 1. INTRODUCTION As it is well kow the popotioal hazad pemium piciple (PH pemium piciple), itoduced by Wag (1995), satisfies popeties which make of it a vey attactive pemium fom the theoetical poit of view, see e.g. Wag (1996) ad Adade e Silva ad Ceteo (1998). Howeve its use depeds o the complete kowledge of the distibutio of the aggegate claims amout. I pactical tems this meas that oe must fit a distibutio to the data set o use the empiical distibutio fuctio. Whe the aggegate claims amout follows a compoud distibutio the calculatio of the PH pemium based o the paametic appoach ca aise some poblems. This will be discussed i the followig sectio. The o-paametic appoach is moe appealig, fom the pactical poit of view, but ca lead to a sigificat udeestimatio of the pemium. I subsectio 3.1 we calculate the bias of the pemium based o the empiical distibutio fo some distibutios, amely the expoetial, Paeto ad uifom. We also study the ate of covegecy of the bias i the expoetial case. ASTIN BULLETIN, Vol. 35, No. 2, 25, pp

2 41 M.L. CENTENO AND J. ANDRADE E SILVA I subsectio 3.2 we show how to use the bootstap techique with subsamplig to educe the bias of the estimato of the pemium based o the empiical distibutio. We also pefom two simulatio studies to give some isight to this techique. 2. APPLYING THE PH TRANSFORM TO COMPOUND DISTRIBUTIONS IN THE PARAMETRIC MODEL Let Y, the aggegate claims amout, be a oegative adom vaiable, with distibutio fuctio F(y). Let S(y)=1 F(y) be the suvival fuctio. The PH pemium piciple assigs to the distibutio F(y) the pemium p = 3 # ^S^yhh dy, (1) whee, with 1, is isk avesio idex. Whe usig the collective isk model, Y is equal to X X N, whee N is the umbe of claims occued i a give peiod ad {X i } i =1,2,... ae the idividual claims amouts which ae assumed to be oegative i.i.d. adom vaiables ad idepedet of N. I this case Y has a compoud distibutio ad the compay ofte estimates claim fequecy ad claim seveity sepaately. Vaious umeical techiques ae available to calculate the compoud distibutio, beig Paje s ecusio fomula the most well kow. Whe usig the ecusio fomula we will have to stop somewhee the calculatios. Whe the idividual claims amout has distibutio G with ulimited suppot, i.e. G(x)<1, fo all x >, ad it is vey skewed, the choice of the value whee to stop the calculatios to obtai a easoable estimate of the pemium is a citical aspect 3 of the model. Whe X follows a heavy tail distibutio # ^S^yhh dy ca be t of a expessive size eve fo lage values of t. I this case, whe the pobability geeatig fuctio of N is aalytic i the eighbouhood of 1 (which happes both fo the Poisso ad the egative biomial case) ad G is subexpoetial the (see Embechts et al. (1997), pp ) 1 F(x) ~E[N ](1 G(x)), x $. (2) Hece fo heavy tail seveity distibutios we will appoximate the pemium by t 3 * # # t 1 p * S y / 1 dy E N / 1 1 G y / = ^ ^ hh + ^ h ^ - ^ hh dy, (3) whee S*(y) is the appoximatio to S(y) obtaied by Paje s ecusio fomula, ad t is a suitable high value. Example 1. Suppose that a actuay has aived to the followig estimates fo a isk: the umbe of claims geate tha a give obsevatio poit d = 1K is Poisso distibuted with mea l d =6 ad the size of each claim geate tha d is Paeto with paamete a = Let us coside a excess of loss eisuace

3 THE PROPORTIONAL HAZARD PREMIUM CALCULATION PRINCIPLE 411 teaty. Fo a laye l xs m, with m dadl +, the aggegate claims amout ae compoud Poisso with expected umbe of claims l =(d/m) a l d ad seveity distibutio a m 1- _ m + xi if # x < l G] xg = * (4) 1 if x $ l. Usig =.925, Table 1 shows the PH tasfom pemium, as pecetage of the subject eaed pemium, SEP = $1,K, associate to the followig layes: 1. $4K xs $1K 2. $5K xs $5K 3. $9K xs $1K 4. the excess ove $1,K This example fo the limited layes was cosideed by Wag (1998). Fo the calculatio of the pemium of the ulimited laye we have used (3) with t = $1 5 K, fo which the diffeece betwee the aggegate suvival fuctio S *(t) ad l(1 G(t)), is The step used i the aithmetizatio of the seveity distibutio to pefom the ecusio was h = $1K. The easo to chose such a high value fo t i fomula (3) is elated to the vey small value of a. If istead of $1 5 K we had used $1 4 K fo t, the esult of applyig fomula (3) with = 1 would be 2.86% of SEP, compaed to a theoetical value of 2.9%. Usig t = $1 5 K the two tems i the ight had side of (3) ae % ad.155% fo = 1 ad 3.298% ad.358% fo =.925. Note that the elative weight of the secod tem iceases with ad is fa fom egligible i spite of the high values of t. 3. APPLYING THE PH TRANSFORM TO THE EMPIRICAL DISTRIBUTION 3.1. The bias of the pemium Let F (y) be the empiical distibutio fuctio of Y based o a adom sample of size,(y 1,...,Y ), let S (y) be the coespodig empiical suvival fuctio ad TABLE 1 PH-TRANSFORM PREMIUMS Laye l xs m Pue Pemium p* as pecetage of SEP as pecetage of SEP $4K xs $1K 6.% 6.384% $5K xs $5K 1.183% 1.48% $9K xs $1K 7.183% 7.742% xs $1,K 2.9% 3.388%

4 412 M.L. CENTENO AND J. ANDRADE E SILVA p # ^S ^yhh dy (5) 3 =, the pemium estimato. Although S (y) is a ubiased estimato of S(y),p give by (5) is a biased estimato of p, sice 3 3 E 7p A = # E 8^S ^yhh B dy # # 7E ^S ^yhha dy = p by Jese s iequality, with the iequality beig stict uless = 1. Let Y k: be the k-th ode statistic i a sample of size. The 1 y < Y1 : S ^ yh = *] - kg /, Yk : # y < Yk + 1 :,,..., -1 y $ Y, : which implies that - 1 p k =! - b Yk + 1 : - Yk :, l ] g (6) k = whee Y : /. Let F k: (y) be the distibutio fuctio of the k-th ode statistic, i.e. F k: (y) = P{Y k: y}. Give that - F k: (y) =! c F^y 1 - F y, m^ hh ^ ^ hh the = k 1 F k: (y) =1 F k+1: (y) c k m^ F ^ y hh ^ 1 - F ^ y hh k - k ad itegatig we obtai 3 EY Y. k F y k 1 F y - k 6 dy k + 1 : - k = # c m^ ^ hh ^ - ^ hh (7) The, usig (6) ad (7) we get E 7p A = = - 1! k =! k 3 - k - k b F y S y dy l c k m # ^ ^ hh ^ ^ hh k k -1 - k + 1 b l c k -1 m # ^F^yhh ^S^yhh dy (8) which is, whe F is absolutely cotiuous fo x >, equivalet to E p k 1 1 = A!b l c k -1 m # ] 1 - xg x k k + 1 1, f S x dx -1 7 ] ga (9)

5 THE PROPORTIONAL HAZARD PREMIUM CALCULATION PRINCIPLE 413 whee f (x) =F (x). Expessios (8) ad (9) ca be used to calculate the bias of p B(p )=E [p ] p. Fo some distibutios (ot i the compoud case) the bias is easily calculated, as it is the case whe Y is expoetial, Paeto o uifom, as ca be see i the followig examples. Example 2. Let Y be expoetial distibuted, i.e. S(y) =exp( qy), y > (q > ). I this case f [S 1 (x)] = qx, ad usig (9) we get, afte some calculatios, ad = - 1! 1 E 7p A q k - B p 1 / / k _ i = e - o q. (1)! Example 3. Let Y be Paeto distibuted, i.e. S(y) = ` case f [S 1 (x)] = ab 1 x 1+a, ad usig (9) we get, E 7p A = ba -1 -!! k b b + y G] k - ag k! G] + 1- ag a j, y > (a, b > ). I this ad! / a B p 1 / G] k - 1 g _ i = b = 1 /! k -. a k! G] + 1- ag a - G (11) Example 4. Whe Y is uifomly distibuted i (,1) we get ad E p 1 7 A = ] 1g B p 1 _ i = ] + 1g! 1 / +! 1 / k 1 / k 1 / Fo the expoetial distibutio it is possible to elate the ode of the bias with the sample size, as stated i the esult that follows. We wee ot able of developig simila esults fo othe distibutios, but we ae stogly coviced that the ode of the bias is smalle fo the uifom ad bigge fo the Paeto. Result. Whe Y follows a expoetial distibutio B(p ) coveges to zeo at the same ate as.

6 414 M.L. CENTENO AND J. ANDRADE E SILVA Poof. Noticig that / / =! ] k -xg dx, # ad usig (1) we have B_ p i = q 7] k -xg - k A dx.! # As (k x) 1 is a iceasig fuctio of x fo k 1 ad < x < 1, we have that 1 1! # # / / ] k -xg - k Adx # 7] 1-xg -1Adx! ] k -1g - k A = - k = 2 #. O the othe had as Ò 1 [(k x) 1 k 1 ]dx is stictly positive we ca coclude that the left had side of (12) coveges to a positive value A as goes to ifiity. Cosequetly lim B(p ) = q 1 A. " Coectig the bias via bootstappig As we have see, the distotio of the empiical distibutio leads to a udeestimatio of the pemiums. We use the bootstap techique, see e.g. Efo ad Tibshiai (1993), to coect, at least patially, the bias of that estimato. The bootstap techique cosists, as it is well kow, o the esamplig of the oigial data set. The bootstap estimato of the bias of p,is with % B _ p i = p * p p * = M 1 b M! = 1 p *b whee M is the umbe of bootstap samples ad p*b is calculated usig the b- th bootstap sample. Hece the bootstap estimato of the pemium p is % p4 = p B _ p i. (13) Usually the esamplig is made with eplacemet ad usig bootstap samples of size (i.e. the same size as the oigial sample). Howeve i this case, as

7 THE PROPORTIONAL HAZARD PREMIUM CALCULATION PRINCIPLE 415 FIGURE 1: B (p4) as fuctio of / B(p ) is always egative, we ca impove, i piciple, the esults by usig bootstap samples of size <. The easo fo the bias to be egative is the udeestimatio of the ight tail of the distoted distibutio. The use of a subsamplig i the bootstap method iteds to eplicate that behaviou ad cosequetly to obtai a bette bias coectio tha the oe obtaied with the full sample. Figue 1 shows how the bias of the coected estimato give by (3) vaies with the sample size (measued as a pecetage of ). The figue has bee daw usig a simulatio appoach (see simulatio 1 below) fo a Paeto distibutio with a = 4, sample size of 5 obsevatios, = 1.2 ad usig 2 eplicas ad 2 bootstaps samples fo each eplica. The shape of the cuve is vey simila fo all the othe situatios: it stats, whe =, with a egative value fo the mea of the obseved bias of the bootstap pemiums, which we deote B(p4), the thee is a optimal value of whee B(p4), is almost zeo, ad becomes positive whe is vey small. The questio is how to choose, idepedetly of the distibutio. To give some isight to the poblem we pefomed two simulatio studies. I these studies we also use the Jackkife, as a alteative to the bootstap, to coect the bias. As we will see the Jackkife pefoms bette tha full samplig bootstap, but we ca do bette by usig a suitable level of subsamplig.

8 416 M.L. CENTENO AND J. ANDRADE E SILVA Simulatio 1. This case is based o two families of distibutios, the Paeto ad the Gamma. I the Paeto case we have cosideed that the paamete a takes the value 2, 3 o 4, oigiatig what we call Paeto 2, Paeto 3 ad Paeto 4 espectively. The othe paamete was chose such that the distibutio has mea equal to 1, i.e. we have assumed that the suvival fuctio fo the Paeto a is S x a 1 a = - ] g b x + a -1 l a, x >. Fo the gamma distibutio we have kept the same mea ad assumed that the vaiace was equal to 1 (expoetial) o 2. We have cosideed that the oigial sample size is eithe 1, 5 o 1. Tables 2 ad 3 give the bootstap ad Jackkife esults fo = 1.2 ad = 1.15 espectively, with M = 2 ad usig 2 eplicas. As we have metioed, the bias of the pemium based o the empiical distibutio (colum labelled B(p ), ad calculated usig expessios (8), (1) ad (11), fo the gamma, expoetial ad Paeto distibutios) ca be of a cosideable size, amely fo heavy tail distibutios whe is small ad is high. Fo istace fo the Paeto 2, with = 1 ad = 1.2, the absolute value of the bias is 8.4% of the theoetical pemium. The aveage bias of the pemiums based o the 2 eplicas (befoe the bootstap is pefomed), labelled B, is i those cases still a bit fa fom B(p ), but it was out of ou computig facilities to coside a big eough umbe of eplicas i the simulatio study. The followig 1 colums show the aveage bias fo the bootstap pemiums. The bootstap with =, oly coects the bias patially, sice afte that coectio we still obseve a egative bias i all the situatios of ou example. As we ca see fom Tables 2 ad 3, the patte show i Figue 1 is chaacteistic of the behaviou of the bias as fuctio of /. The optimal popotio / depeds, of couse, o the distibutio, o ad o. As a ule / should icease with ad decease with. Fo istace, fo = 1.2, a bootstap with a esamplig of / = 4% pefoms, i aveage, bette tha the bootstap with full esamplig (ad cetaily bette tha with o bootstap), at least fo the sample sizes cosideed i the study. The followig colum, labelled JN, shows the aveage bias fo the Jackkife pemiums, i.e. we use the Jackkife techique istead of bootstap to coect the bias of the pemiums. As we ca see the Jackkife pefoms bette tha the full samplig bootstap but, usig subsamplig, we ca obtai a smalle bias with the bootstap. The last colum of the tables, povides the aveage bias of the pemiums if we have used the maximum likelihood estimatos of the paametes. Fo the Paeto 2, the figues ae ot peseted because, fo a lage umbe of samples (2 fo = 1.2 ad 11 fo = 1.15), we got a maximum likelihood estimate â smalle tha. I some othe situatios the estimate was vey close to, which would imply a extemely high pemium. Fo the Paeto 3 ad Paeto 4 we did t get ay sample with a maximum likelihood estimate smalle tha, but we got values ot vey fa fom, which explai the positive values of the bias i those cases. The maximum likelihood pefoms well (i associatio with

9 THE PROPORTIONAL HAZARD PREMIUM CALCULATION PRINCIPLE 417 the PH pemium calculatio piciple) if the distibutio does ot have a vey heavy tail, o if the sample size is big eough. As a coclusio, ad fo values of aoud we ca say that use of the bootstap techique with a esamplig popotio size of 4% of the oigial sample size povides good esults i geeal, but whe we kow that we ae dealig with heavy tail loss distibutios, we could use a smalle esamplig size. Table 4 is equivalet to Table 2 but with M = 5. As it ca be see, the esults ae quite simila. Hece M = 2 seems eough. Simulatio 2. The goal of the secod simulatio study is to aalyse the behaviou of ou pocedues whe the aggegate claims amout ae geeated accodig to a compoud Poisso o egative biomial distibutios. Fo the idividual claims amout we have cosideed the thee Paeto s ad the expoetial distibutios of the fome example. The expected umbe of claims was set at 1. As i the pevious simulatio study = 1.2 ad = 1.15, M = 2, we use 2 eplicas ad is equal to 1, 5 o 1. Tables 5 ad 6 efe to the compoud Poisso case ad ae simila to tables 2 ad 3, with the exceptios of B(p ) ad ML, which ae ot cosideed ow. Colum labelled p* was obtaied usig (3), with t equal to 1, 926 ad 297 whe a equal to 2, 3 ad 4 espectively, values fo which the P{X > t}=1 8. Fo the expoetial case we oly cosideed the fist tem of (3), with t = 9. As expected with l = 1 the pemiums associated to the compoud distibutio ae geate tha fo the coespodig Paeto s. The esults ae vey simila to the fome case. We obtaied the same behaviou fo the bias as a fuctio of /, ad agai we would ecommed a esamplig of / = 4%, but if we kew that we wee dealig with heavy tail loss distibutios, we could use a smalle esamplig size. The esults fo the simulatios with M = 5 give vey simila values, easo why we do ot povide them hee. Tables 7 ad 8 efe to the egative biomial case, with = 1.2 ad with the vaiace of the umbe of claims equal to 1.5 ad 1.2 espectively. As we ca see the esults lead to the same coclusio. ACKNOWLEDGMENTS This eseach was suppoted by Fudaçao paa a Ciêcia e a Tecologia - FCT/ POCTI. We ae also gateful to ou colleague Raul Bás fo assistig us with the computatio pogams. We also thak a aoymous efeee fo seveal suggestios which helped to impove the pape. REFERENCES ANDRADE E SILVA, J. ad CENTENO, M.L. (1998) Compaig isk adjusted pemiums fom the eisuace poit of view. ASTIN Bulleti, 28,

10 418 M.L. CENTENO AND J. ANDRADE E SILVA EMBRECHTS, P., KLÜPPELBERG C., MIKOSCH, T. (1997) Modellig extemal evets. Spige-Velag, Beli. EFRON, B. ad TIBSHIRANI, R. (1993) A Itoductio to the Bootstap. Chapma ad Hall, Lodo. WANG, S. (1995) Isuace picig ad iceased limits atemakig by popotioal hazads tasfoms. Isuace: Mathematics ad Ecoomics, 17, WANG, S. (1996) Pemium calculatio by tasfomig the laye pemium desity, ASTIN Bulleti, 26, WANG, S. (1998) Implemetatio of PH-tasfoms i atemakig, Poceedigs of the Casualty Actuaial Society, LXXXV, MARIA DE LOURDES CENTENO CEMAPRE, ISEG, Techical Uivesity of Lisbo, Rua do Quelhas, 2, Lisbo, Potugal lceteo@iseg.utl.pt

11 THE PROPORTIONAL HAZARD PREMIUM CALCULATION PRINCIPLE 419 TABLE 2 BOOTSTRAP RESULTS, 2 BOOTSTRAPS, = 1.2 Distibutio p B (p) B B (p4) fo / equal to 1% 9% 8% 7% 6% 5% 4% 3% 2% 1% JN ML Paeto Paeto Paeto Gamma Expoetial Paeto Paeto Paeto Gamma Expoetial Paeto Paeto Paeto Gamma Expoetial

12 42 M.L. CENTENO AND J. ANDRADE E SILVA TABLE 3 BOOTSTRAP RESULTS, 2 BOOTSTRAPS, = 1.15 Distibutio p B (p) B B (p4) fo / equal to 1% 9% 8% 7% 6% 5% 4% 3% 2% 1% JN ML Paeto Paeto Paeto Gamma Expoetial Paeto Paeto Paeto Gamma Expoetial Paeto Paeto Paeto Gamma Expoetial

13 THE PROPORTIONAL HAZARD PREMIUM CALCULATION PRINCIPLE 421 TABLE 4 BOOTSTRAP RESULTS, 5 BOOTSTRAPS, = 1.2 Distibutio p B (p) B B (p4) fo / equal to 1% 9% 8% 7% 6% 5% 4% 3% 2% 1% JN ML Paeto Paeto Paeto Gamma Expoetial Paeto Paeto Paeto Gamma Expoetial Paeto Paeto Paeto Gamma Expoetial

14 422 M.L. CENTENO AND J. ANDRADE E SILVA TABLE 5 BOOTSTRAP RESULTS FOR COMPOUND POISSON DISTRIBUTIONS, = 1.2 Seveity p * B B (p4) fo / equal to JN Distibutio 1% 9% 8% 7% 6% 5% 4% 3% 2% 1% Paeto Paeto Paeto Expoetial Paeto Paeto Paeto Expoetial Paeto Paeto Paeto Expoetial

15 THE PROPORTIONAL HAZARD PREMIUM CALCULATION PRINCIPLE TABLE 6 BOOTSTRAP RESULTS FOR COMPOUND POISSON DISTRIBUTIONS, = 1.15 Seveity p * B B (p4) fo / equal to JN Distibutio 1% 9% 8% 7% 6% 5% 4% 3% 2% 1% Paeto Paeto Paeto Expoetial Paeto Paeto Paeto Expoetial Paeto Paeto Paeto Expoetial

16 424 M.L. CENTENO AND J. ANDRADE E SILVA TABLE 7 BOOTSTRAP RESULTS FOR COMPOUND NEGATIVE BINOMIAL DISTRIBUTIONS, VARIANCE = 1.5, = 1.2 Seveity p * B B (p4) fo / equal to JN Distibutio 1% 9% 8% 7% 6% 5% 4% 3% 2% 1% Paeto Paeto Paeto Expoetial Paeto Paeto Paeto Expoetial Paeto Paeto Paeto Expoetial

17 THE PROPORTIONAL HAZARD PREMIUM CALCULATION PRINCIPLE TABLE 8 BOOTSTRAP RESULTS FOR COMPOUND NEGATIVE BINOMIAL DISTRIBUTIONS, VARIANCE = 1.2, = 1.2 Seveity p * B B (p4) fo / equal to JN Distibutio 1% 9% 8% 7% 6% 5% 4% 3% 2% 1% Paeto Paeto Paeto Expoetial Paeto Paeto Paeto Expoetial Paeto Paeto Paeto Expoetial