Modelling Lifetime Dependence for Older Ages using a Multivariate Pareto Distribution

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1 Modellng Lfetme Dependence for Older Ages usng a Multvarate Pareto Dstrbuton Danel H Ala Znovy Landsman Mchael Sherrs 3 School of Mathematcs, Statstcs and Actuaral Scence Unversty of Kent, Canterbury, Kent CT 7NF, UK Department of Statstcs, Unversty of Hafa Mount Carmel, Hafa 3905, Israel CEPAR, Rsk and Actuaral Studes, UNSW Busness School UNSW, Sydney NSW 05, Australa DRAFT ONLY DO NOT CIRCULATE WITHOUT AUTHORS PERMISSION Abstract In order to solate the longevty component n lfe-beneft products, we focus our attenton on deferred annutes These products are drven by older age mortalty, where lttle s known about potental dependence structures We propose to nvestgate a multvarate Pareto dstrbuton, whch wll allow us to explore a varety of applcatons, from large portfolos of standard annutes to jont-last survvor annuty products for couples In past work, t has been shown that even a lttle dependence between lves can lead to much hgher uncertanty Therefore, the ablty to assess and ncorporate the approprate dependence structure wll sgnfcantly mprove the prcng and rsk management of deferred annuty products Keywords: Longevty Rsk, Lfetme Dependence, Multvarate Pareto Dstrbuton dhala@kentacuk landsman@stathafaacl 3 msherrs@unsweduau

2 Introducton The study of lfetme dependence s hghly mportant n actuaral scence We consder a pool of lves where the ndvdual lfetmes follow a Pareto dstrbuton The dependence among the lves s determned by the nature of the multvarate dstrbuton We consder a multvarate constructon of the type II Pareto dstrbuton such that the correlaton between lves s governed by the Pareto shape parameter The nature of the problem s determned by the sze of the pool For example, for a pool of sze two, an applcaton of ths model wll help determne the prcng and rsk management of jont annuty products On the other hand, where the pool ncludes a natonal cohort, an applcaton of ths model wll help quantfy systematc longevty rsk Both ends of the spectrum are hghly relevant to ether prvate nsurance or publc polcy In the work of Ala et al 03, 05a,b, lfetme dependence modellng was consdered for members of the exponental dsperson famly, specfcally for the Tweede subclass Dependence was nduced va a common stochastc component, rather than governed parametrcally The Pareto dstrbuton represents an nterestng and relevant dstrbuton for modellng heavy-taled data It s chosen here to more accurately model old-age dependence, whether the applcaton of nterest s a jont-last survvor annuty or a pool of deferred annuty products The ssue of dependence has also been studed n Dhaene et al 000, Denut et al 00, Denut 008 and Dhaene and Denut 007, among others Snce the focus s on older age-mortalty, lfetmes are necessarly left-truncated Ths represents a non-trval ssue wth respect to parameter calbraton; one that we nvestgate on two fronts The frst of whch consders matchng observed wth theoretcal moments, and the second, observed wth theoretcal quantles mnmum and maxmum Organzaton of the paper: In Secton we ntroduce some basc notaton and provde some results for the unvarate Pareto dstrbuton The multvarate Pareto dstrbuton s ntroduced n Secton 3, where we derve certan mean and varance as well as mnmum and maxmum results These results are necessary for establshng parameter estmaton procedures, whch wll be consdered n subsequent sectons We provde some temporary conclusons n Secton 4 Notaton and the Type II Pareto Dstrbuton Notaton We begn by provdng some notaton concernng moments We denote wth k X and µ k X the k th, k Z +, raw and central theoretcal moments of random varable X, respectvely k X E[X k ], µ k X E[X X k ] The raw sample moments for random sample X X,, X n are gven by a k X X k, k Z + n

3 The raw sample moments of an dentcally dstrbuted sample are unbased estmators of the correspondng raw moments of X E[a k X] k X Fnally, adjusted second central sample moment s gven by m X X a X n The adjusted central sample moment of an ndependent and dentcally dstrbuted sample s an unbased and consstent estmator of the correspondng central moment of X E[ m X] µ X The Type II Pareto Dstrbuton We consder the type II Pareto dstrbuton wth scale and shape parameters > 0 and, respectvely The densty functon s gven by fy y +, y > 0 The survval functon s gven by F y The raw moments of nterest are gven by Y Y or, generally, for k Z + and > k, The varance s gven by µ Y y, y > 0, >,, > k Γ k k Y Γk + Γ, > 3 Mean and Varance for the Truncated Pareto Theorem Consder Y dstrbuted type II Pareto, Defne the assocated truncated random varable Y Y Y > The mean and varance of Y are gven by Y +, µ Y + 3

4 Proof F y; denotes the survval functon of a type II Pareto dstrbuton wth shape parameter Y F y y + dy Applyng partal fractons produces Y F y y dy + F y dy F F ; F ; F F F ; F + y + dy Y F F F + F y y y y + y + + dy y dy F y F ; F ; F F dy F F dy y dy

5 We presently use the fact that µ Y Y Y µ Y A Multvarate Pareto Dstrbuton We now consder a multvarate constructon of the type II Pareto dstrbuton Scale and shape parameters are gven by > 0 and, respectvely Let Y Y,, Y n be an n-dmensonal multvarate Pareto dstrbuton; the survval functon s gven by n F y y, where y y,, y n It s known that the margnal dstrbuton of Y,,, n follows a unvarate type II Pareto dstrbuton wth parameters and Furthermore, the dependence structure of the margnals s characterzed by the parameter ; that s, the correlaton between Y and Y j, for j s gven by / We provde some detal; for Y Y,, Y n multvarate Pareto, the covarance of Y and Y s gven by CovY, Y E[Y Y ] E[Y ]E[Y ] 3 Mean, Varance and Covarance Results We presently consder mean, varance, and covarance results for the margnal dstrbutons after applyng truncaton to the multvarate dstrbuton Note that ths s dfferent from consderng truncaton on a subset of the multvarate dstrbuton only For example, one may consder mean and varance results on the margnal dstrbuton when t alone s truncated, or even covarance results when the two margnals n queston are truncated Incdentally, we acheve the latter results as a by-product of multvarate truncaton by trvally allowng n and n To avod confuson, we ntroduce precse notaton Let Y Y,, Y n be the multvarate dstrbuton of nterest Let n be an n-dmensonal vector where each entry takes value Then, let Y Y Y > Theorem Consder Y Y,, Y n Multvarate P areto, wth survval functon denoted F y;, Defne the assocated truncated multvarate dstrbuton 5

6 Y Y Y > The mean and varance of Y are gven by Y µ Y + n +, + n The covarance between Y and Y j, j remans Cov Y, Y j, but the correlaton between Y and Y j, j s now gven by Corr Y, Y j + n Proof The densty of the multvarate dstrbuton s found by approprately dfferentatng the jont survval functon fy n n F y y y y 3 y n The truncated margnal densty s found by, frst, ntegratng ths jont densty; snce we are dealng wth a truncated multvarate dstrbuton, lower ntegraton ndces are set to And second, by normalzng wth constant F Note that the survval functon of the n-dmensonal jont Pareto evaluated at pont, F, s equvalent to the survval functon of a unvarate Pareto evaluated at pont n, F n For completeness, whenever F takes a sngle argument, a unvarate Pareto survval functon, otherwse, a multvarate Pareto survval functon, s mpled We consequently have that Y F n y dy y n + Apply partal fractons to obtan Y F n y n n y +n + dy Fnally, apply substtuton z y n and recognze that ntegrals are scaled survval functons of Pareto dstrbutons Y F n n z n z + dz F n F n; n F n n n n n + + n + 6

7 Apply a smlar approach to obtan the second raw moment Y Y F n y dy y n + Apply partal fractons and substtuton z y n Y F n n z n z + n z + dz F n; n F n; F n + n F n Ths mples Y / n n n + n n [ ] + + n [ ] + [ ] n + n + Rewrte the above as a quadratc of to obtan Y + n + + n + n + To derve the varance, we agan use the fact that µ Y Y Y Usng a common denomnator of, the expresson reduces very ncely to the one gven above To derve the covarance, we requre E[ Y Y ] Agan, we take expectaton wth respect to the the jont densty After ntegratng out the remanng n varables, we have E[ Y Y ] F n y y + dy dy y y +n + Although fndng an expresson for ths term s more complcated, t s based on the same prncples as before; we provde some detals Let z y y + n and 7

8 z y + n E[ Y Y ] + F n + F n + F n y y y [ y y +n dy y y +n + dy y +n z + z + dz dy y +n + y +n + ]dy y +n Havng dealt wth y, collect the y terms, notng the presence of y [ + y E[ Y Y ] F n y +n y n + y +n + y y +n + ]dy + Apply partal fractons and pull out scaled Pareto survval functons [ + E[ Y Y ] F n n z n z n n + z z + + z n n ] z + z + dz [ + F n F n; n F n; n n F n; F n + n F n; F n; + n ] + F n The rato of two Pareto survval functons reduces dependng on the dfference n shape parameters Collect terms based on these ratos, usng common denomnator 8

9 ; a lot of terms cancel out! [ n E[ Y Y ] + + n n n n n + [ n + n Rewrte as a quadratc n to obtan n n + ] + E[ Y Y ] + n + + n + n + Notce the smlarty of ths expresson wth that of Y In order to derve the covarance, we now take E[ Y Y ], rather than Y, and subtract Y Cov Y, Y E[ Y Y ] Y CovY, Y Clearly the varance of the margnal from the truncated multvarate dstrbuton dffers from the varance of the margnal from the un-truncated dstrbuton Hence, we obtan a dfferent correlaton coeffcent, one that goes to zero as n ncreases Corr Y, Y + n ] Remark It s convenent to note that Y E[ Y Y ] 3 Mnmum and Maxmum Results + n We presently consder the mnmum and maxmum element of our n-dmensonal truncated multvarate Pareto dstrbuton wth shape and scale parameters and As before, we have Y Y,, Y n and Y Y Y > Let Y mn Y and Y n max Y It s easy to demonstrate that Y follows a Pareto dstrbuton wth shape and scale /n, and hence that Y follows a truncated Pareto dstrbuton wth the same parameters In some detal, we have Pr[Y > y] Pr[Y > y,, Y n > y] F y,, y F ny 9 ny

10 Therefore, adjustng the scale parameter by /n results n a Pareto survval functon Furthermore, t s rrelevant whether you ether: fnd the mnmum of a truncated multvarate Pareto, or truncate the mnmum of an un-truncated multvarate Pareto Both lead to the same result, the latter beng more convenent We may apply Theorem to obtan the mean and varance of Y Y µ Y /n +, /n + For the maxmum, we have a less straght-forward result We start wth the dstrbuton functon of the maxmum of the truncated multvarate Pareto Pr[ < Y y] Pr[ Y n y] Pr[Y n y Y > ] Pr[Y > ] F n + n n F y F n n F y F n Dfferentate to fnd the densty f Y n y n F n y The expectaton s gven by n E[ Y n ] + y dy F n y + n + F n y n + F n n + F F n n + F F n n F + / + F n +, y > dy y + F ; F F F n Remark It s nterestng to note that when 0, we obtan the followng n / E[Y n ] + n + n! + γ, where γ s Euler s constant 0

11 It s also of nterest to know the varance of the truncated maxmum For ths, we begn wth the second raw moment Y n n + y dy F n y + n + F n y y + y dy + n + F ; F n F ; + F n + F + F n n + F + F n n + F + + F n n F F n Consequently, we have that µ Y n n F + F n n F + F n + + / + Remark 3 It s nterestng to note that when 0, we obtan the followng n µ Y n + / n / + n + n + n + n + [ ] n n + + [ n ] n The frst square-bracketed term appears to converge; t s equal to for n 600, reachng 6 at n 0 The second square-bracketed term appears to go to nfnty, but very slowly, for n 600, the term s 35, for n 0, t s 57

12 4 Concluson Lfetme dependence s studed usng a multvarate constructon of the type II Pareto dstrbuton We am to apply ths model to nvestgate older age mortalty, specfcally for jont-last survvor annutes and portfolos of deferred annuty products Gven the nature of the data, parameter estmaton technques need to ncorporate lefttruncaton We derve the necessary results for two estmaton procedures, one that uses mean-varance results, a second based on mnmum-maxmum We am to test the performance of both procedures usng smulaton We antcpate that one wll perform better for a large collecton of small pools of lves, say, one thousand jontlast survvor polces; the other for a small collecton of large pools of lves, say, ten portfolos of fve hundred deferred annutes References Ala, D H, Landsman, Z, and Sherrs, M 03 Lfetme dependence modellng usng a multvarate gamma dstrbuton Insurance: Mathematcs and Economcs, 53, Ala, D H, Landsman, Z, and Sherrs, M 05a A multvarate Tweede lfetme model: Censorng and truncaton Workng Paper Ala, D H, Landsman, Z, and Sherrs, M 05b Multvarate Tweede lfetmes: The mpact of dependence To appear n Scandnavan Actuaral Journal Denut, M 008 Comonotonc approxmatons to quantles of lfe annuty condtonal expected present value Insurance: Mathematcs and Economcs, 4, Denut, M, Dhaene, J, Le Bally de Tlleghem, C, and Teghem, S 00 Measurng the mpact of a dependence among nsured lfelengths Belgan Actuaral Bulletn,, 8 39 Dhaene, J and Denut, M 007 Comonotonc bounds on the survval probabltes n the Lee-Carter model for mortalty projectons Journal of Computatonal and Appled Mathematcs, 03, Dhaene, J, Vanneste, M, and Wolthus, H 000 A note on dependences n multple lfe statuses Bulletn of the Swss Assocaton of Actuares,, 9 34