A Heavy Traffic Approach o Modelig Large Life Isurace Porfolios Jose Blache ad Hery Lam Absrac We explore a ew framework o approximae life isurace risk processes i he sceario of pleiful policyholders, via a boom-up approach. Give he isurace corac srucure, we aggregae he balace of idividual policy accous, ad derive a approximaig Gaussia process wih compuable correlaio srucure. The mehodology is borrowed from heavy raffic heory i he lieraure of may-server queues, ad ivolves he so-called fluid ad diffusio approximaios. Our framework is differe from idividual risk model by akig io accou he ime dimesio ad he specific policy srucure icludig he premium paymes. I is also differe from classical risk heory by buildig he risk process from micro-level coracs ad parameers isead of assumig claim ad premium processes ourigh. As a resul, our approximaig process behaves differely depedig o he issued corac srucure. We also illusrae he flexibiliy of our approach by formulaig a fiie-horizo rui problem ha icorporaes acuarial reserve i he cosideraio. The sudy of risk processes is a ceral opic i acuarial sciece. Amog he lieraure, he majoriy focuses o he calculaio of rui probabiliy, as well as he opimal corol of premiums, reisurace levels ad ivesme allocaio. These quesios have bee sudied uder a variey of sochasic seigs, from he classical Cramer-Ludberg approximaio o diffusio processes. The ceral heme is ha radom-walk-ype models, wih a egaively drifed premium process ad a jump process of claims, provide a rich framework o allow pley of exesios, modificaios ad problem formulaios see, for example, Asmusse 2 for survey o rui probabiliy calculaios, ad Schmidli 28 for he couerpar i sochasic corol problems. I his paper we ake a differe view from he exisig lieraure. Raher ha focusig o he compuaio of risk-relaed quaiies, we explore he quesio of he cosrucio of risk process iself. The approach we use is boom-up: Give he srucure ad parameers of he idividual isurace coracs, how does he risk process of he isurer look like o a aggregae scale? Naurally, he risk process uder his framework is he sum of all he idividual accous i.e. he balaces of policyholders who eered corac wih he isurer over ime. For acuaries, his pois o he sadard oe-period idividual ad collecive risk models. However, hese sadard models do o cosider he ime dimesio. This i ur also resrais he power of such models o capure he specific corac srucure ivolved e.g. he premium paymes. I his regard, our work ca be see as a geeralizaio of he sadard risk models o a process-level approximaio. Of course, mere summaio of all idividual accous migh ed up geig a upleasa process ha is hardly compuable. To ackle his issue, we borrow echiques i so-called heavy raffic heory i he lieraure o may-server queues. The basic idea is ha uder he assumpio of large umber of policyholders, oe ca approximae he fucioals of hese policyholders sauses usig fluid ad diffusio approximaios. I he saisics lieraure, hese correspod o sochasic-process versio of Law of Large Numbers ad Ceral 1
2 Limi Theorems. Wih he sheer scale of major isurace compaies, he assumpio of pleiful policyholders is sesible, ad so hese approximaio echiques ca be used. As we will see, hese heavy raffic approximaio would he lead o Gaussia process ha is as aalyzable as may sadard processes used i he curre risk heory lieraure. I paricular, he correlaio srucure of his Gaussia process is explicily compuable give he corac srucure see Secio 4. To illusrae our argume o racabiliy, we formulae a fiie-horizo rui problem based o our Gaussia approximaio see Secio 3. We disiguish our coribuio from classical risk heory ad sadard acuarial risk models i a few ways. Firs, our model explais how idividual isurace policies lead o cerai feaures of he aggregae risk process. The cosrucio of our risk process depeds iricaely o he premium ad beefi srucure of sigle policies. This meas ha differe ypes of isurace, such as whole life isurace, erm life, edowme ec. would lead o differe correlaio srucure of our resulig Gaussia process. This is i sharp coras o he curre model i risk heory, where premium ad claim processes are modeled separaely, each as a drifed radom walk or is varias or marked poi process. This feaure ca poeially provide a framework o aalyze he effec of corac srucure o he firm-wide risk level. Secod, our model allows aurally he icorporaio of acuarial reserve i our approximaio. Ideed, he fiie-horizo rui problem ha we formulae i Secio 3 will ivolve he calculaio of prospecive reserve. Third, sice correlaio is explicily compuable, his provides a way o capure he flucuaio of our approximaig process over ime, which ca be poeially applicable o dyamically moiorig mismach o he isurer s balace shee wih regard o saisical error. I a more orgaized fashio, we summarize our coribuios as follows: A Uder he assumpio of large umber of policyholders, we cosruc he fluid limi ad diffusio limi for he aggregae risk processes. As we meioed, hese correspod o fucioal Law of Large Numbers ad Ceral Limi Theorem respecively i he saisics commuiy; hroughou he paper we mosly use he former ermiology o alig wih he queueig lieraure, bu will also use he laer ierchageably whe ecessary. The risk processes ha we are ieresed iclude he isurer s cash level, liabiliies, ad per basis reserve level. These will be discussed i Secio 2. We prove ha hese risk processes ca be approximaed by Gaussia processes wih cerai correlaio srucures. B Usig he heory of Gaussia processes, we illusrae how our resul ca be used o approximae rui probabiliies. We model rui as he siuaio i which he liabiliies surpasses he asses plus he iiial capial wihi a give ime horizo see Secio 3.1. This highlighs he flexibiliy of our mehodology i icorporaig reserve calculaio, ad also he depedecy o he uderlyig isurace coracs. I paricular, we apply our resuls o several commo ypes of isurace. C Our diffusio approximaio shows how, uder he Equivalece Priciple, he beefi reserve arises as he fluid limi of he empirical cash level per basis a ay poi i ime see Secio 2. These resuls, we believe, provide a useful perspecive io he basic coceps uderlyig he defiiio of beefi reserve; see he discussio followig Theorem 1. D We compue he correlaio srucures of our limiig processes, hereby showig heir racabiliy. I paricular, we illusrae how our approach allows o evaluae ad compare he auocorrelaio as a fucio of ime of risk processes wih differe isurace ypes; see Secio 4.
3 Le us emphasize ha our purpose i applicaios such as B ad C is o illusrae he coceps behid our ideas, ad hece he models we are usig i his paper are basic. There are ceraily may pracical cosideraios o make he model more realisic. We shall lis ou hese geeralizaios ad more realisic exesios ha we believe are worh pursuig i Secio 5. I erms of mehodology, as aforemeioed, we will ivoke primarily he machiery i heavy raffic heory i.e. fluid ad diffusio approximaios i he queueig lieraure. The ideas dae back o Kigma 1961, 1962 for sigle-server queues, ad hey sill cosiue a acive research area amog queueig heoriss see he sadard surveys of Whi 22 ad Billigsley 1999 for isace. Uder fairly mild assumpios, he ools sigificaly simplify ad sigle ou he impora elemes of he sysem dyamics of ieres, ad provide approximae soluios o may impora performace measures i our coex, he rui probabiliy meioed i B cosiues oe such example. More precisely, he resuls i his paper relae o he aalysis of so-called mayserver queues, which have bee subsaially sudied i rece years. I hese queueig sysems, cusomers arrive ad elici service for a radom amou of ime, as log as here are available servers. Whe he umber of servers is ifiie, every cusomer ca sar service righ a arrival. Coecig o our work, policyholders ca be hough of as cusomers i he queueig sysem. While he feaure of arrivals is o our focus i his paper, he deah ime of policyholders is aalogous o he ed of service, ad hece he approximaio echique is raslaable. Some releva refereces o he opic iclude Pag ad Whi 21 ad Decreusefod ad Moyal 28, which focuses o ifiie-server models, Halfi ad Whi 1981, Kaspi ad Ramaa 21 ad Reed 29, which sudy fiie bu large umber of servers i differe proporio or so-called regime o he umber of cusomers, Puhalskii ad Reima 2 ha sudy queues wih muliclass cusomers, ad Dai, He ad Tezca 21 o queues wih reegig. The commo heme of all hese work is he heavy raffic echique beig applicable o various feaures of he queues. Fially, we discuss wo papers ha use similar approach ad highligh our differece. Oe is a rece workig paper by Besusa ad El Karoui 29, who proposes a microsrucural approach o model populaio dyamics o capure moraliy/logeviy risk. Their moivaio is differe from ours: isead of buildig our moraliy disribuio microsrucurally, we make commo assumpios o moraliy; isead, our focus is o how his moraliy assumpio, uder he ieracio wih he corac srucure, beefi level ad premium calculaio, leads o a macroscopic flucuaio of oal asses, liabiliies ad oher acuarial quaiies. Secodly, we oe ha diffusio approximaio has bee ivoked by Igehar 1969 i arguig he use of Browia moio i modelig isurace risk process. However, he maiaied a Cramer-Ludberg framework by assumig compoud Poisso claims ad cosaly drifed premium, ad showed ha uder cerai scalig heir differece coverges o a diffusio process. Corac srucure, relaio bewee premium ad beefi, ad acuarial reserve ec. were o cosidered i his work. The orgaizaio of his paper is as follows. I Secio 2 we lay ou our model assumpios ad defie he key quaiies ha we approximae. Secio 3 is devoed o he saeme of our mai resul ad is discussio. Secio 4 relaes o applicaios, such as rui probabiliy compuaios ad he ideificaio of he auocorrelaio srucure of our approximaig Gaussia processes. Fially, Secio 5 cosiues a appedix, which is divided io wo pars. The firs par discusses basic facs abou heavy raffic limi heorems ad gives he proof of our mai resul; he secod par coais a discussio o he simulaio mehodology ha is used o geerae various examples i his paper.
4 1 Model, Assumpios ad Basic Quaiies We cosider a porfolio of idepede policyholders a ime. For simpliciy assume ha policyholders have he same profile i.e. ideical moraliy disribuio. This assumpio is made maily for simpliciy, he exesios of which will be discussed laer i he paper. Also we assume a cosa rae of ieres. We use he followig oaios hroughou he paper: δ: cosa rae of ieres coiuous compoudig X i : he deah ime of he i-h policyholder he X i s are idepede ad ideically disribued iid f, F, F : desiy, disribuio ad survival fucios of X i T : upper limi of he suppor for he deah ime i.e. T sup{ > : F < 1} P : accumulaed premium payme discoued a ime if he policyholder dies a B: beefi payme discoued a ime if he policyholder dies a ; oe ha he case where beefis are paid a imes oher ha deah such as regular bous prior o deah ca be merely redefied as a deducio i accumulaed premium payme, ad hece is also covered i our framework I addiio, we make he followig echical assumpios ha are commoly used: Assumpio 1. We assume ha f > for all, T, wih T <. Assumpio 2. Defie H : P B. We assume ha H is o-decreasig ad ha H < while HT >. Moreover, we assume ha P ad B are coiuously differeiable ad have bouded firs derivaives, ad hece so is H. Assumpio 1 is aural i he seig of life isurace, which is he focus of his paper. Assumpio 2 is aliged wih he pracice ha premiums are paid prior o beefi. For example, i he case of whole life isurace wih coiuous level premium payme p ad beefi b, P pe δs ds p1 e δ /δ ad B be δ. This isurace saisfies Assumpio 2 uder he Equivalece Priciple Bowers e. al. 1997. Fially, we oe ha he moooiciy assumpio is oly eeded i he opimizaio problem iroduced i rui calculaio; see Secio 3.1. We ow look a some basic quaiies of ieres ha are relaed o he policyholders wih assumpios described above. To keep our discussio simple for illusraio, hroughou he paper we will focus o his seig. There are may aural exesios, such as he arrivals of policyholders over ime ad muli-profile muli-produc busiess lies. These will be lef for fuure exploraio, ad we oe a relaed paper by Blache ad Lam 211 ha discusses he sceario of policyholder arrivals. Le N be he umber of deahs before ime. Wih he oaio above, we wrie N IX i. 1 i1
5 Similarly, we wrie N for he umber of survivig policyholders a ime T, amely N N T N N. 2 Our resuls ivolve he followig hree basic quaiies of ieres, all of which ca be expressed i erms of 1 ad 2 above. For coveiece, we ame hese quaiies as Cash Process, Reserve Process ad Average Cash Process respecively: Cash Process We defie he Cash Process as he prese value a ime of he oal accumulaed cash geeraed by all idividual accous, excludig iiial surplus. We deoe i by C : C : e δ [P X i BX i IX i + P IX i >. i1 Observe ha we ca wrie more ealy as C e δ [ HsdN s + P N. We also defie m o be he mea of he cash coribuio from a idividual accou over ime i.e. m E[P X i BX i IX i + P IX i > 3 [ e δ Hsfsds + P F. The Equivalece Priciple idicaes ha oe should selec he premium level i such a way ha he oal i.e. up o he ed of he ime horizo acuarial e prese value of he premiums is equal o ha of he beefis paid see Bowers e al 1997. I our oaio, assumig he validiy of he Equivalece Priciple amous o sayig ha mt e δ Hsfsds. Reserve Process The acuarial reserve a ime of a give corac is he amou of capial ha he isurace compay should se aside for fuure coigecies, defied by he expeced prese value of he corac s fuure e cos. I oher words, i is he differece of he acuarial e prese value a ime of he beefis o be paid ad he premiums o be eared. This defiiio is used uder he prospecive mehod Bowers e. al. 1997. We deoe V as he acuarial reserve. I mahemaical erms, his is V : e δ Bs P s P f sds e [P δ Hsf sds where f s fs X i > fs/ F. If he Equivalece Priciple holds, oe ca also compue V usig he rerospecive mehod Bowers e. al. 1997, hereby obaiig [ e δ Hsfsds + P F V. 4 F If he Equivalece Priciple is used, we also call V he beefi reserve.
6 Isurace compay mus reflec he oal reserves i heir balace shees as liabiliy. We defie he Reserve Process a ime, deoed by C, as he sum of he acuarial reserves from all survivig policies. Hece C : N V N e δ We also defie he relaed quaiy m as which as we shall see is he fluid limi of C as. Bs P s P f sds N e [P δ Hsf sds. m e [P δ F Hsfsds, 5 Average Cash Process As meioed earlier, a ime, isurace compay mus recogize he liabiliies refleced by he oal reserves of he survivig policyholders. Those liabiliies are o be faced, ideally, wih he geeraed cash from he pas. This moivaes associaig a Average Cash Process o each survivig policyholder, which we deoe by V. This quaiy divides up he accumulaed cash equally amog he curre survivors. I mahemaical erms, i is [ e δ HsdN s + P N V : C N N. As we shall sudy, uder he Equivalece Priciple, he process V flucuaes aroud V. I he ex secio we will describe our mai resuls ivolvig limi heorems ad approximaios o hese key quaiies. 2 Mai Resul I order o describe our resuls we eed o recall he defiiio of Browia bridge, a impora process obaied ou of codiioig he value of Browia moio a ime T. We iroduce W, 1 as our oaio for a Browia bridge. I urs ou ha W is equal i disribuio o W W, where W is a Browia moio. I is also he uique Gaussia process wih mea ad covariace fucio CovW s, W s1, s. This implies ha we ca wrie he ideiies i disribuio for whole sochasic processes W F D W F F W 1 D fsdw s F fsdw s 6 See, for example, Seele 21 ad Karazas ad Shreve 28. We are ow ready o sae ad discuss our resuls. They are formulaed i erms of weak covergece i a useful opology o spaces of fucios, called he Skorokhod opology. The discussio of his opology ad is prelimiary heorems will be discussed i Secio 6. Our mai resul provides a joi approximaio o he Cash Process, Reserve Process ad Average Cash Process. The proof is give i Secio 6.
7 Theorem 1. Assume ha he Equivalece Priciple holds ad herefore ha he ideiy 4 is i force. Regardig C, C, V as elemes i D[, T D[, T D[, T ɛ for ay ɛ, T equipped wih Skorokhod produc opology, we have ha C /, C /, V / m, m, V 7 as. Moreover, C / m, C / m, V V as. e δ [ W F e δ [ HsdW F s P W F, 8 Hsf sds P, e δ [ HsdW F s P W F F + V F W F The ɛ > i he heorem is o avoid zero divider a ime T. The approximaio i 8 suggess ha whe is large, he Cash Process ca be approximaed by C m + [ e δ HsdW F s P W F 9 Simulaeously, we have ha he Reserve Process admis he approximaio C m + W F e δ [ Hsf sds P, 1 ad ha V V + 1 { e δ F [ HsdW F s P W F + V } F W F 11 The flucuaio aroud he average i he firs wo processes is smalles a he wo eds of he ime horizo, amely, a ime ad a T, sice we kow for sure ha here are ad decremes respecively; he flucuaios become larger i he middle of he ime rage. The maximum flucuaio of he accumulaed cash process will occur a a ime which is characerized i Secio 3.1. The approximaio 8 is joi i fucio space, so haks o he coiuous mappig priciple Theorem 2 i he appedix, we ca approximae he disribuio of a whole coiuous fucioal of he sample pahs C ad C. This is precisely he sigificace of he previous resul. As a paricular applicaio, we will show i he ex secio how o exploi he coiuous mappig priciple o esimae rui probabiliies uder differe ypes of life isurace coracs. Also i he ex secio we will provide closed formulae for he joi correlaio of he limiig Gaussia processes i he righ had side of 8; hereby fully characerizig he whole asympoic disribuio of asses ad liabiliies across ime.
8 The approximaio dicaed by he hird compoe, amely V, provides a lik bewee our sochasic formulaio for a large pool of policyholders ad he classical reserve evaluaio V. I also provides suppor for he use of he Equivalece Priciple from a micro-srucural perspecive. I paricular, we show ha uder he Equivalece Priciple he idividual cash accous flucuaes aroud he beefi reserve as he umber of policyholders icreases. Moreover, he resul provides a Ceral Limi Theorem correcio. We evisio ha our resuls i his secio are poeially useful i evaluaig i pracice wheher he differece bewee asses ad liabiliies o he balace shee is wihi ormal saisical error, alhough such applicaio ceraily requires beig able o iclude oher sylized feaures such as ivesmes i risky asses ad so forh, which we pla o ivesigae i he fuure. Figure 1 depics a joi sample pah of he approximaig process of C ad C respecively. Here we use he followig assumpios: 1, δ.1, T 1, uiform deah disribuio over [, T, ad a whole life isurace policy wih b 1; he premium p is calculaed accordig o he Equivalece Priciple. value 1 2 3 asse liabiliy..2.4.6.8 1. ime Figure 1: Approximaig C i red ad C i blue Figure 2 shows V ad a approximaig sample pah of V give by 11 as a process ceered a V. Here we use he same assumpios specified for Figure 1. We explai how o impleme he simulaio procedure i Secio 6.2 based o he approximaios 9, 1 ad 11 3 Applicaios ad Examples Prevailig isurace pracice calculaes reserve based o he mahemaical expecaio of cash flows i.e. he acuarial e prese value of fuure premiums mius beefis o idividual basis.
9 cumulaed cash per survival reserve value..5 1. 1.5..2.4.6.8 1. ime Figure 2: Approximaig V i red ad deermiisic rajecory of V i blue However, he overall realized liabiliy of he compay is really give by he aggregaio of he realized idividual prese values of he premiums mius he e prese values of he paid beefis. Cosiderig he process 9 we derived i he previous secio as he flucuaio of overall asses, a ieresig problem would be o aalyze he mismach bewee he overall realized liabiliy process ad he asse process. More precisely, whe he size of asses are below he sauory reserve requireme, we say ha a rui occurs. Because he heavy raffic limi is Gaussia, ad he heory of Gaussia processes is well developed, oe ca approximae such rui probabiliy easily. 3.1 Rui Probabiliies Here we formulae a rui problem based o reserve requireme. Suppose ha prospecive mehod are employed o se up required reserve o he balace shee. Bakrupcy he occurs wheever he oal asse falls shor of he liabiliy, plus iiial surplus. More precisely, le he iiial surplus be U ha is scaled wih. The ierpreaio of he scalig is aural as a compay wih large umber of policyholders i he sysem will aurally sar wih a large iiial amou of capial requireme; his is precisely he iiial surplus. We defie U U e δ o be he value a ime of he iiial surplus. Rui occurs if U + C C < assumig a cosa rae of ivesme ieres. Uder a fiie-ime formulaio, he rui probabiliy is give by P U + C C < for some [, T This formulaio differs from he classical seig maily i wo aspecs: 1 he risk processes C ad C deped o he srucure of he isurace coracs raher ha separae modelig of
1 premium ad claim processes; 2 he per-basis reserve ha resembles he acual pracice of he isurace compay ca be icorporaed aurally io our framework. We make wo mai assumpios i our formulaio. Firs, we assume he Equivalece Priciple for calculaig premiums, due o marke compeiio. Uder his assumpio he process C C is esseially ceered. Secod, we assume he surplus is scaled as U u for some u >. Noe ha oher scalig of U would lead o differe approximaios. For example, if U is of order, he raher ha usig our diffusio-ype approximaio i he previous secios, oe would have o ur o large deviaios asympoic. The deails of such will be repored i Blache ad Lam 211, which also preses asympoics ad simulaio desig for esimaig rui problem uder policyholder arrivals. By Theorems 1 ad 2, we have C C m m [ e δ Hsfsds + Hsfsds Noe ha uder he Equivalece Priciple his process will be ideically zero. To fid he flucuaio of his process, agai we use Theorem 2, scale by ad ge C C m m [ [ e δ HsdW F s P W F W F e δ Hsf sds P e δ [ HsdW F s W F Hsf sds Now he rui probabiliy is wrie as as, where P rui P U + C C < for some [, T C C P sup > ue δ X P T sup X > u T 12 1 + o 1 13 HsdW F s W F Hsf sds 14 This follows from 12 ad he fac ha X D X. The ex figure depics a sample pah of he approximaig e asse process C C ad he deermiisic rajecory of U. The approximaig e asse correspods o he simulaio oupu of Figure 1. We use he iiial surplus u 5. Noe ha rui happes a aroud ime.58 i his sceario. The probabiliy 13 ypically does o have closed-form soluio, alhough a fair amou is kow for sharp asympoics of he maximum of a Gaussia process see for isace, Husler ad
11 value 15 1 5 5 1 15 2 e asse egaive iiial surplus..2.4.6.8 1. ime Figure 3: Approximaig C C i red ad deermiisic rajecory of U i blue Pierbarg 1999 ad Dieker 25; hese approximaios deped o he local correlaio srucure which is obaied i he ex secio. Here we will prese a formal approximaio ha is popular i he coex of Gaussia queues see Chaper 5 i Madjes 27 ad is easy o develop. The approximaio works well for large values of u. Of course, oe has o be careful whe we use his approximaio because our diffusio limi is esablished assumig ha u is O 1. The Gaussia approximaio, however, sill remais valid for he ails if u is allowed o grow as a a sufficiely slow speed. I he presece of a large deviaios resul, which ca be derived i our curre seig, i suffices o le u : u i such a way ha u o 1/2. This is wha is kow as moderae deviaios scalig see Chaper 8 i Gaesh e. al. 24. This is he ype of asympoic evirome ha we have i mid whe we use our Gaussia approximaio for ail probabiliies. We have P sup X > u T sup P X > u 15 T We ow aalyze he righ had side of 15. Usig 6, X ca be show o be equal i disribuio
12 o Hs fsdw s Hsfsds fsdw s Hsf sds + F Hsf sds dw F s + Hs Hs D N, σ 2 fsdw s fsdw s Hsf sds F Hsf sds dw F s + W 1 Hsf sds Hsf sds where he secod equaliy comes from he Equivalece Priciple, ad σ 2 + 2 Hs Hs 2 Hvf vdv fsds + Hs 2 fsds 2 Hvf vdv fsds 2 + Hsf sds + 2 by Io s isomery. Hece Hsf sds Hsfsds 2 Hsf sds Hsf sds Hsfsds W 1 2 Hsfsds + Hsf sds F 2 Hsf sds 2 Hsf sds F 2 Hs 2 fsds + F Hsf sds 16 P sup X > u T sup P N, σ 2 > u 17 T O he oher had, he upper boud of he rui probabiliy is provided by Borell s iequaliy see, for example, Adler 199 P sup X > u T e Cu 1 2 u2 /σ 2 where argmax T σ 2 ad C is a cosa depedig o E sup T X. Togeher, 17 ad 18 give he asympoic resul 1 lim u u 2 log P sup X > u 1 T 2σ 2 18
13 To fid, oe merely differeiaes 16 o ge dσ 2 d f 2 Hsfsds 2 HsfsdsHf + H F F 2 f [ 2 f H 2 2H Hsf sds Hsf sds [ f H 1 + 2 [ Hsf sds H + 2 1 Hsf sds 19 Noe ha f > for all, T. Sice H is o-decreasig, we have ad so H 1 + 2 Hsf sds H Hsf sds < for all [, T. Also, by he Equivalece Priciple ad ha H < ad is coiuous, we have H + 2 1 eighborhood of i.e., ɛ for some ɛ >. O he oher had, sice ad ha HT > ad is coiuous, we have H + 2 1 Hsf sds < for a pucured Hsf sds H Hsf sds > for a pucured eighborhood of T i.e. T ɛ, T for some ɛ >. These lead o he coclusio ha here is a global maximum i he ierior of he domai i.e., T. To solve for he global maximum, oe ca umerically solve for he zeros of H + 2 1 Hsf sds. The he global maximizer is eiher he zero or he discoiuous poi of dσ 2 /d ha gives he highes value of σ 2. Example 1: Suppose X i follows uiform disribuio o [, T. Also assume whole life isurace wih coiuous level premium p ad beefi b i.e. H p1 e δ /δ be δ. By he Equivalece Priciple we ca calculae p bδ1 e δt δt 1 + e δt ad so H + 2 1 Hsf sds p δ p δ + b e δ + 2 1 [ p δ 1 p δt δ + b e δ e δt We assume he values T 1, b 1 ad δ.1. The.414. Noe ha σ 2 1 [ p 2 T δ 2 2p p δ 2 δ + b 1 e δ + 1 p 2 2δ δ + b 1 e 2δ 1 [ p T δ 1 p 2 δt δ + b e δ e δt Pluggig i, we have σ.256. Example 2: Suppose X i follows he same disribuio as above, wih same values of T ad b. However, le us cosider a icreasig premium rae pe µ where µ < δ. Le µ.5. The he
14 Equivalece Priciple gives I his case ad H + 2 1 p δ µ + 2 1 p H bδ µ2 δ Hsf sds p δ µ e δ µ be δ [ p δ µ We ge ha.416. Noe ha 1 e δt δ µ 1 + e δ µt p δ µ p δ µ e δ µ be δ p 1 δ µ 2 T e δ µ e δ µt b δt e δ e δt σ 2 [ 1 p 2 T δ µ 2 + p 2 2δ µ 3 1 e 2δ µ + b2 2δ 1 e 2δ 2p2 δ µ 3 1 e δ µ 2pb δδ µ 1 2pb e δ + δ µ2δ µ 1 e 2δ µ + 1 T [ p δ µ p 1 δ µ 2 T e δ µ e δ µt b 2 δt e δ e δt Pluggig i, we ge σ.519. Example 3: Assume he same seig as he las wo examples, bu his ime wih erm life isurace wih eor l < T. Le l.5. The ad The Equivalece Priciple gives Noe ha ad H + 2 1 { p δ p p P B { p1 e δ δ p1 e δl δ for l for > l { be δ for l for > l bδ1 e δl lδ 1 + e δl + T lδ1 e δl { p H δ p δ + b e δ p δ 1 e δl for l for > l Hsf sds δ + b e δ + [ 2 1 p l δ T p δ + b 1 δt e δ e δl + T l p T δ 1 e δl 2 p δ 1 e δl for l for > l
15 We ge ha.5. Noe ha σ 2 1 T [ p 2 δ 2 2p + 1 T 1 T [ p 2 l δ 2 p δ 2 δ + b 1 e δ + p δ + b 2 1 [ p δ p δ 2 δ + b 1 e δl + p δ + b 2 1 2p + 1 p 2 T 1 e δl 2 δ 2 Pluggig i, we ge σ.679. 2δ 1 e 2δ l T p δ + b 1 δt e δ e δl + T l T p 2 δ 1 e δl for l 2δ 1 e 2δl + l p2 1 e δl 2 δ 2 for > l We see ha boh icreasig premium srucure wih rae.5 ad erm life a ime.5 have σ 2 larger ha whole life isurace. I oher words, implemeig whole life isurace gives he isurer a beer risk profile. A ieresig quesio would be he ype of isurace policy, say P, ha miimizes σ 2 P. I his case, we have o solve he problem of miimizig max T σ 2 P over a fixed family of isurace policies P. This quesio will be explored i fuure work. Remark 1. Suppose we relax he ideical profile assumpio for policyholders as discussed i Secio 1 ad we replace his assumpio by a disribuio of policyholder ypes. If he disribuio is discree, he all our resuls i his paper sill hold, excep ha raher ha usig Gaussia process drive by oe Browia moio, he Gaussia process will be a mixure of Gaussia processes each drive by a idepede Browia moio. σ 2 ca be foud similarly bu he opimizaio problem will be less liear. O he oher had, if he ype of disribuio is coiuous, he he limiig process will ivolve Browia shee. This seems o iroduce furher echicaliies ha are herefore lef o fuure work. 4 Correlaio Srucure Our model also provides a framework for sudyig emporal correlaios of he risk processes. The Gaussia aure of he limis we have discussed allows easy compuaio. As a illusraio, cosider he processes C ad C i 8. As aforemeioed, hey ca be ierpreed as he asses ad liabiliies of he isurace compay. Their variaces as well as emporal ad cross correlaios ca be foud easily as follows: Temporal covariaces for Cash Process ad Reserve Process: CovC, C [ e δ+ Hs P Hs P fsds Cov C, C e δ+ F F Hs P f sds Hs P fsds Hs P f sds Hs P fsds
16 Cross emporal covariace bewee Cash Process ad Reserve Process: CovC, C [ e δ+ F Hs P fsds These i paricular give: Hs P fsdsi > Hs P f sds Variace for Cash Process ad Reserve Process: V arc V ar C e 2δ [ [ e 2δ F F 2 Hs P 2 fsds Hs P fsds 2 Hs P f sds Cross covariace bewee Cash Process ad Reserve Process: CovC, C e 2δ F Hs P fsds From hese oe ca calculae he emporal ad cross correlaios CorrC, C Corr C, C CorrC, C CovC, C V arc V arc Cov C, C V ar C V ar C CovC, C V arc V ar C Hs P f sds Now cosider he e asse process uder he Equivalece Priciple give by X defied i
17 14. We have CovX, X Hs + + + T Hsf sds Hsf sds Hsf sds Hs 2 fsds + F Hs 2 fsds Hvf vdv Hs T Hsf sds Hs Hs T Hsf sds Hsf sds Hvf v fsds Hvf vdv fsds Hvf vdv fsds Hsf sds Hsf sds by he Equivalece Priciple i he las equaliy, ad so i paricular V arx which recovers he value of σ 2 i 16. 5 Exesios Hsfsds + Hsf sds Hsfsds Hsfsds + 2 F Hsf sds Hsf sds 2 Hs 2 fsds + F Hsf sds We emphasize ha he curre work serves as a firs aemp o iroduce heavy raffic approach i modelig large life isurace porfolios o he sample pah level. Regardig he sochasic compoe, especially i modelig rui, he bigges limiaio of he curre work is he igorace of he dyamic arrival process of policyholders. Whe such arrivals are prese, he risk process will be a fucioal of a uderlyig ifiie-server queue, i which he service imes are he deah imes of he arrivig policyholders. Such cosideraio will be oe of our key fuure research direcios. As we discussed i he previous secio, aoher impora relaxaio is he ideical profile assumpio. Whereas a discree mixure of policyholders is sraighforward, echicaliy arises whe he mixure is coiuous. Besides, several oher direcios of exesios ca be pursued. A few possible ad impora exesios are: 1 explorig more complicaed policy srucures e.g. ui-liked producs 2 modelig he ieres rae as a marke risk ad sochasically chagig 3 icorporaig operaioal cos ad oher expeses 4 allowig ime-varyig correlaio amog policyholders e.g. Markov-modulaed arrival rae ad deah disribuio 5 relaxig he Equivalece Priciple assumpio ad allowig safey loadig ec.
18 6 Appedix: Techical Developme ad Moe Carlo Simulaio Mehodology This appedix is divided io wo pars. We firs provide he proofs of our mai resuls, which require a quick review of some basic facs o heavy raffic ad weak covergece heory. The secod par of his appedix cocers he implemeaio of he simulaios show i he paper. 6.1 Review of Weak Covergece ad Proofs of Mai Resuls Before we provide he proof of our mai resuls, le us review some resuls o weak covergece heory i fucio spaces. The we proceed wih our proofs. 6.1.1 Review of Weak Covergece Resuls Defie C[, T as he se of all coiuous fucios o [, T equipped wih he uiform meric, deoed by d. Tha is, d x, y sup x y. T We also defie D[, T o be he se of all cadlag lef coiuous wih righ limi fucios o [, T, ad we equip he space wih he sadard Skorokhod meric, which we shall deoe by d J. I paricular, if we le A be he se of all sricly icreasig coiuous fucios ha map [, T io iself we have ha d J x, y if sup x λ y λ A T see, for example, Billigsley 1999. I order o quickly have a grasp of he Skorokhod opology, i is easy o show ha x x i he d J meric if ad oly if here exiss a sequece of elemes λ A such ha x λ x i he d meric. The uiform or Skorokhod opologies i a produc space such as C[, T C[, T or D[, T D[, T are defied as he sum of he correspodig merics i each projecio. I paricular, for isace, if x 1, x 2 ad y 1, y 2 are elemes i C[, T C[, T, he we defie he produc uiform meric as d Π x 1, x 2, y 1, y 2 d x 1, x 2 + d y 1, y 2. Eirely aalogous cosideraios ad defiiios apply o he Skorokhod produc meric. We deoe for weak covergece of probabiliy measure. A useful characerizaio of weak covergece is ha for a sequece of probabiliy measures P, P P o he space C[, T respecively D[, T if ad oly if gdp gdp for ay bouded, coiuous fucio g o C[, T respecively D[, T. The uiform ad Skorokhod opologies are se o defie coiuiy for g. Equivalely, we say ha a sequece of sochasic processes o C[, T respecively D[, T, Y Y, if ad oly if EgY EgY for ay bouded coiuous fucio g defied o C[, T respecively D[, T. This characerizaio will be useful for our developme. The mai reaso for developig weak covergece resuls i spaces of fucios is give by he coiuous mappig priciple, which allows o derive furher approximaio resuls by expressig quaiies of ieres such as he Cash Process ad Reserve Process defied i Secio 1 as fucios of a suiable process. A saeme of he coiuous mappig heorem is give ex see, for example, Billigsley 1999:
19 Theorem 2. Le h : D[, T S be measurable ad D h be he se of is discoiuiies i Skorokhod opology. If he sequece of sochasic processes i D[, T, Y Y, ad P Y D h, he hy hy. I paricular, if Y C[, T, he same codiios ad resuls hold by defiig D h o be he se of discoiuiy i he uiform opology. Moreover, he heorem holds for produc of D[, T spaces respecively C[, T. By carefully choosig he coiuous fucioals such as maximal, iegral ec. we will be able o obai hady covergece resuls. Alhough he Skorokhod meric is raher explici, he reducio o checkig coiuiy i he d meric whe Y is coiuous which is he secod saeme of he heorem makes some of our calculaios easier. This reducio comes from he fac ha if x x i he d J meric ad if x is i C[, T he x x i he d meric. We shall also use he followig sadard resul of he weak covergece of empirical process io Browia bridge see, for example, Billigsley 1999 ad Dudley 1999. This ca be summarized as: Theorem 3. For ay i.i.d. radom variables X i, i 1,..., wih disribuio F x suppored o [, T, defie F 1 P X i ad regardig F as elemes of D[, T equipped wih he Skorokhod opology, we have i1 F F 2 ad F F W F 21 where W is sadard Browia bridge o [, 1 as defied i he discussio prior o 6. The limis i 2 ad 21 are well-kow i saisics ad probabiliy see for example Dudley 1999. I he queueig lieraure, hese ypes of resuls are kow as fluid ad diffusio limis. Fluid limi refers o approximaio by deermiisic rajecory ad hece is ame. Diffusio limi refers o approximaio by diffusio process, i his case drive by Browia bridge. These covergece resuls ad heir various exesios serve as buildig blocks of oher more complicaed limi approximaios. 6.1.2 Proofs of Mai Resuls Proof of Theorem 1. Our sraegy is o show ha processes C / m, C / m, ad V V ca be expressed each as coiuous fucios of he same uderlyig process. Therefore, because of he form of he produc merics, we will have joi covergece i he produc opology. We he ca rea each of he hree processes separaely. We firs cocerae o C / m. By Theorem 3 ad 1, we have N / F ad Z : N F W F 22
2 o D[, T. Noe ha 22 describes he process for he fracio of deahs over ime as. Nex cosider he iegral Hsdrs. 23 Noe ha by Assumpio 2 Hs has a coiuous ad bouded firs derivaive, so for r C[, T wih bouded quadraic variaio, iegraio by pars gives Hsdrs Hr Hr H srsds 24 We ca exed he domai of defiiio of 23 o he whole of C[, T by 24. To ease oaio deoe as he uiform orm so ha d x, y x y. Noe ha for r 1, r 2 C[, T, Hsdr 1 s Hsdr 2 s 2 H + H s ds r 1 r 2 which shows coiuiy of C Hsdrs o r C[, T. Now wrie [ e δ N s N T Hsd + P N Coiuiy of iegral 23 ad he fac ha elemeary operaios are coiuous o C[, T yields ha [ e δ Hsdrs + P rt r is coiuous o r C[, T. Sice F is coiuous ad lies i C[, T, Theorem 2 ad 3 cocludes he firs compoe of 7. For he firs compoe of 8, wrie C [ m e δ HsdZ s + P Z T Z ad 8 follows by similar argume, oig ha Z T ZT. The reame of he process processes C is eirely similar. Firs, i is clear from Theorem 3 ha C N T N F e δ [P [ e δ P Hsf sds m. Hsf sds Moreover, a sraighforward applicaio of he coiuous mappig priciple ad Theorem 3 followig similar coiuiy argumes as hose give earlier yields C m [ Z T Z e [P δ Hsf sds W F e δ P Hsf sds.
21 Fially, for V oe ha [ V e δ HsdN s/ N T / N / + P A direc applicaio of Theorem 2 allows us o coclude he hird compoe of 7. Nex wrie V V V V F N + V F N V e δ N s Hsd + P N N / e δ HsdZ s + P Z T Z N T / N / HsdF s P F Sice F is deermiisic i follows ha he followig weak covergece resul Z, N W F, F V N N / F V N T / N / Z T Z 25 follows joily i D[, T D[, T. Sice he limiig processes are coiuous, i suffices o check ha 25 is a coiuous mappig from W F, F o C[, T ɛ 2 o R, by Theorem 2. The argume proceeds as i he aalysis of C give earlier ad we coclude our resul. 6.2 Simulaio Mehodology We lay ou he simulaio mehodology we use o geerae he graphs i his paper. The few ad elemeary seps i he mehodology advocaes our use of heavy raffic approximaio. Noe ha all he processes we iroduced so far are elemeary fucios of HsdW F s, W F 26 i he poiwise sese. So we will discuss how o geerae a pah of hese quaiies. More precisely, we will geerae a discreized versio of his pah a ime pois, 1,..., m T. Defie Y ad Noe ha Y i Y i 1 + i Y i i i 1 HsdW F s Y i 1 + HsdW F s, i i 1 HsdW F s i 1,..., m i i 1 HsfsdsW 1, i 1,..., m by 6. Our simulaio algorihm is he as follows. Firs geerae W F 1,..., W F m, where W F i W F i 1 + N, F i F i 1, i 1,..., m
22 For coveiece le he realizaios be W F i x i. We have immediaely ha W F i x i F i x m for i 1,..., m. Usig he ierpreaio of Browia bridge as he codiioal process give he ed pois of sadard Browia moio, we have, give W F i 1 x i 1 ad W F i x i, W F, [ i 1, i is equal i disribuio o x i 1 + F F i 1 F i F i 1 x i x i 1 + W F F i 1 F F i 1 F i F i 1 W F i F i 1 where W is a sadard Browia moio. Moreover, give he values of W F i x i, i 1,..., m, {W F } i 1 i are idepede porios of sample pahs, ad hece i i 1 HsdW F s i x i x i 1 Hsfsds i 1 F i F i 1 + i i 1 Hsd W F s i Hsfsds W F i F i 1 i 1 F i F i 1 We he have where ad σi 2 i i i 1 HsdW F s R i : Nµ i, σ 2 i µ i i x i x i 1 Hsfsds i 1 F i F i 1 2 i Hs 2 1 fsds Hsfsds i 1 i 1 F i F i 1 Therefore, o simulae 26, we firs oupu x i, i 1,..., m, ad he codiioal o x i, i 1,..., m, i Y i, W F i Y i 1 + R i x m Hsfsds, x i F i x m i 1 for i 1,..., m. Refereces 1. Adler, R. 199, A Iroducio o Coiuiy, Exrema, ad Relaed Topics for Geeral Gaussia Processes, Vol. 12, Isiue of Mahemaical Saisics Lecure Noes-Moograph Series. 2. Besusa, H. ad El Karoui, N. 29, Microscopic modellig of populaio dyamic : A aalysis of logeviy risk, prepri. 3. Billigsley, P. 1999, Covergece of Probabiliy Measures, Wiley. 4. Blache, J. ad Lam, H. 211, Imporace samplig for acuarial reserve aalysis uder a heavy raffic model. workig paper.
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