Analysis of Bivariate Excess Losses

Transcription

1 by Janong Ren ABSTRACT The concept of ecess osses s wey use n rensurance an retrospectve nsurance ratng The mathematcs reate to t has been stue etensvey n the property an casuaty actuara terature However, t seems that the formuas for hgher moments of the ecess osses are not reay avaabe Therefore, n the frst part of ths paper, we ntrouce a formua for cacuatng the hgher moments, base on whch t s shown that they can be obtane recty from the Tabe of Insurance Charges (Tabe M) In the secon part of the paper, we ntrouce the concept of bvarate ecess osses It s shown that the jont moments of bvarate ecess osses can be compute through methos smar to the ones use n the unvarate case In aton, we prove eampes to ustrate possbe appcatons of bvarate ecess oss functons KYWORDS Rensurance, bvarate ecess oss, Tabe M VOLUM 0/ISSU CASUALTY ACTUARIAL SOCITY 95

2 Varance Avancng the Scence of Rsk Introucton The concept of ecess osses s wey use n rensurance an retrospectve nsurance ratng The mathematcs of t has been stue etensvey n the property an casuaty nsurance terature See, for eampe, Lee (988) an Hawe (0) The frst moment of the ecess osses has been tabuate nto the Tabe of Insurance Charges (Tabe M) for use n the NCCI retrospectve ratng pan Hgher moments of ecess osses can be use to measure the voatty of ecess osses However formuas for them are not reay avaabe n the property casuaty actuara terature One cou refer to Secton of Mccos (977) for some scussons In fact, the formuas for cacuatng hgher moments of ecess osses o est n the terature of stochastc orers, where the nth moment of ecess osses s name the nth orer stoposs transform (see, for eampe, Hürmann 000) Therefore, n the frst part of ths paper, we ntrouce the smpe formuas for cacuatng hgher moments of the ecess osses to the property casuaty actuara terature More mportanty, usng a etae numerca eampe, we show that the hgher moments can be obtane recty from Tabe M In the secon part of ths paper, we ntrouce the concept of bvarate ecess osses, whch has ts orgn n the reabty theory terature See, for eampe, Zahe (985) an Gupta an Sankaran (998) In the contet of stochastc orerng, Denut, Lefevre, an Shake (998) presente a formua for the jont moments of mutvarate ecess osses In ths paper, we show that the jont moments of bvarate ecess osses can be compute through methos smar to the ones use n the unvarate case We prove eampes to ustrate possbe appcatons of bvarate ecess oss functons The remanng parts of the paper are organze as foows Secton ntrouces formuas for hgher moments of ecess osses an show how they may be compute usng Tabe M Secton 3 presents the theory of bvarate ecess osses Secton 4 proves eampes an Secton 5 concues Unvarate ecess osses We begn by ntroucng some notatons an basc facts Premnares Let X be a ranom oss varabe takng non-negatve vaues an have cumuatve strbuton functon F an survva functon S Then the mte oss up to a retenton eve s efne by X 0 X f X f X > The oss n the ayer (, ) s efne by 0 f X X X0 X0 X f < X f X > The ecess oss over a mt s efne by X 0 f X ( X ) + X X0 X f X > It s we known that the epecte vaue of the mte oss s gven by (see, for eampe, Lee 988) ( ) ( ) X0 0 S u u () Due to the mportance of (), we net gve a short proof of t The metho use n the proof can be reay etene to the bvarate stuaton Proof: Frst of a, t s easy to verfy that X0 I( X > u) u, 0 where I() s an ncator functon that s equa to one when ts arguments are true an zero otherwse Then we have that [ ] [ ] X I( X > u) u 0 I( X > u) u 0 0 Suu ( ) 0 96 CASUALTY ACTUARIAL SOCITY VOLUM 0/ISSU

3 Because X X 0 X 0, we have for the ayere osses that X I( X > u) u + [ ( X ) + ] ( +! ) + [ ( X ) ] ( +! ) (7) an [ ] ( ) X For the ecess osses, S u u () [ ] ( ) X S u u (3) Hgher moments of ecess osses Hgher moments of the ecess oss X can be obtane usng the foowng Proposton Proposton Let an for, et Then R () X, (4) [ ] + R () R( u) u (5) R() [ ( X ) ], for (6)! The proof of the proposton was obtane n Denut, Lefevre, an Shake (998) an Hürmann (000), an t s ncue here for the competeness of ths paper Proof: We use mathematca nucton For, quaton (6) s true by efnton Assume that t s true for, then + R () R ( u) u [ ( X u) u + ]! ( X u) u! + If the strbuton of the uneryng oss X s known, then one cou compute [(X ) k ] for any nteger k usng Proposton teratvey More mportanty, we pont out that snce Tabe M n fact sts vaues of R (), one may compute R k (), k > recty from t recursvey, n a smar fashon as one wou compute R () from the survva functon S() Ths way, [(X ) k ], k can be obtane recty from Tabe M We net show the metho wth a numerca eampe ampe : Conser probem 4 of Brosus (00) Let X represent the oss rato for a homogeneous group of nsures; t s observe to have vaues 30%, 45%, 45% an 0%, respectvey Let Y X/ (X) be the corresponng entry ratos an thus take vaues 05, 075, 075, Tabe M, constructe usng the metho escrbe n Brosus (00), gves the mean ecess oss functon of Y: R ( r) Y r [ ] Then the secon moment of the ecess osses [(X ) ] may smpy be obtane by numercay ntegratng R (r) an mutpyng the resut by Reazng that R (r) s pecewse near between entry rato vaues, the numerca ntegraton s mpemente by R ( r) k 0 R( r+ k ) + R( r+ ( k+ ) ), where D s the nterva between entry rato vaues Tabe shows the etas of the cacuaton The thr coumn s the Tabe M charge, the fourth coumn (R n ayer), corresponng to an entry rato r, s cacuate by R ( r )+ R ( r + ), where D s the nterva VOLUM 0/ISSU CASUALTY ACTUARIAL SOCITY 97

4 Varance Avancng the Scence of Rsk Tabe Cacuatng hgher moments of ecess osses usng Tabe M ntry Rato (r) # of Rsks R (r) R n Layer R (r) [(Y r ) ] between entry ratos, whch s 05 n the eampe The ffth coumn (R (r)) s the cumuatve summaton of the fourth coumn The ast coumn s just the ffth one mutpe by Ths eampe shows the mportant fact that the hgher moments of the ecess osses can be obtane recty from Tabe M No other nformaton s neee! The secon moment of the ayere osses [(X ) ] s aso of nterest We have [( ) ] X X X [( ) ] [ ] [ ] ( ) ( ) [( )( )] X + X X X [ ( X ) ] + [ ( X ) ] X + X X [( )( )] [( ) ] [( ) ] [( )( )] X X X X (8) The frst two terms of (8) can be obtane from Tabe M, as shown n the prevous eampe The ast term can agan be obtane from Tabe M by appyng quaton () erve n Secton 3 3 Bvarate ecess osses Let (X, Y ) be a par of ranom oss ranom varabes wth jont strbuton functon F(, y) P(X, Y y) an jont survva functon S(, y) P(X >, Y > y) Smar to Formua () for the unvarate case, we have the foowng formua for the frst jont moment of the ayere osses X an Y y y Proposton 3 y XY S u, v vu (9) ( ) y Proof: As n (), frst notce that y y X Y I( X > u) u I( Y > v) v y Then we have I ( X > u) I ( Y > v) vu y y X Y [ I( > u) I( y > v) ] vu y y y Suvvu (, ) y Wth ths, the covarance between X an Y y y s gven by (, ) (, ) Cov X Y S u v vu y y S ( u) u S ( v) v, where S an S y enote the margna survva functon of X an Y respectvey A somewhat smar formua can be foun n Dhaene an Goovaerts (996) y y 98 CASUALTY ACTUARIAL SOCITY VOLUM 0/ISSU

5 As shown n Denut, Lefevre, an Mesfou (999), hgher jont moments of the bvarate ecess osses can be compute usng the foowng resut Proposton 3 Let y ( ) ( ) R, S u, v vu (0) an for (, j) > (, ), et j j y j y ( ), ( ), ( ) R, R u, u R, v v Then, j Rj(, y) ( X ) ( Y )!! () j Proof: We agan use mathematca nucton For j, the statement s true by Proposton 3 Assume that t s true for, j, then R+, j(, y) R, j( u, y) u j [ ( X u) Y u + ( y) ] +! j! j ( Y y) ( X u) u! j! + + X + j [ ( ) + ( Y y) ] + ( +! ) j! The ervaton for R,j+ (, y ) s symmetrc Smar to Secton, Proposton 3 can be use to construct a bvarate Tabe M to tabuate the jont moments of the bvarate ecess osses ampe 4 n the net secton proves an ustraton In the rest of ths secton, we show that Proposton 3 may she some ght on the jont moments of the amount n fferent ayers of a ranom oss To ths en, settng X Y, we have Suv (, ) P[ X > u, Y > v] P[ X > ma ( u, v) ] S ( ma ( u, v) ), where S () enotes the survva functon for X Then for two non-overappng ayers (, ) an (, ) of X wth, we have [ X X ] S( u, v) vu S ( v) vu ( ) X [ ] () As a resut, the covarance of X an X s gven by [ ] ( [ ]) [ ] Cov X X X X, (3) whch s quaton (39) of Mccos (977) As mentone n Secton, Formua () s usefu n computng the secon moment of ayere osses X In fact, appyng t to (8) yes [( X ) ] [( X ) ] [( X ) ] ( ) [ X ] (4) Notce that a three terms on the rght-han se of (4) can be obtane from Tabe M Another formua to compute the secon moment of the ayer osses s: [( ) ] X S( u, v) vu u u S( u, v) vu from whch we may wrte [( ) ] S( u) vu ( u ) S( u) u, X ( u ) S( u) u ( u ) R ( u) ( u ) R( u) u + R( u) u ( R () R ( ) ( ) R () ), (5) whch agrees wth quaton (4) VOLUM 0/ISSU CASUALTY ACTUARIAL SOCITY 99

6 Varance Avancng the Scence of Rsk 4 Numerca eampes In ths secton, we present three eampes In the frst eampe, we erve formuas for the jont moments of ecess osses for a bvarate Pareto strbuton In the secon eampe, we show that a bvarate Tabe M can be constructe to tabuate the covarances between ayers of osses from two nes of busnesses In the thr eampe, we appy the formuas erve heren to stuy the nteractons between per-occurrence an stop-osses mts when they coest n an nsurance pocy ampe : Bvarate Pareto Dstrbuton Foowng Wang (998), assume that there ests a ranom parameter L such that for,, X L are nepenent an eponentay strbute wth rate parameter /q Then the contona jont survva functon s gven by S (, ) e X, X Λλ λ + θ θ Assume that L foows a Gamma (α, ) strbuton wth moment generatng functon M L (t) ( t) α Then the uncontona strbuton of (X, X ) s a bvarate Pareto wth the jont survva functon α S (, y ) + + (6) θ θ As an etenson of unvarate Pareto strbutons, bvarate Pareto strbutons are usefu n moeng bvarate osses wth heavy tas From the jont survva functon (6), we have that y ( X Y ) α y + + y y θ θ θθ y + ++ ( α )( α ) θ θ α+ y + ++ θ θ α+ In aton, the net equatons are easy obtane an w be use n the foowng eampe an ( X Y ) ( ) X α+ θ ( α ) +, θ α+ θθ + + ( α )( α ) θ θ y, [( X ) ] ( ) + θ α θ θ+ ( α )( α ) θ α+ One mght woner how the epenence between (X )an(y ) vares wth the retenton eve For ustraton, we assume that α 3, q 5, q 0 an cacuate the correaton coeffcents between X an Y corr X (, Y ) ( X Y ) ( X ) ( Y ) ( ) ( ) Var X Var Y for fferent vaues of The reatonshp between the correaton coeffcents an the retenton eve s ustrate n Fgure It shows that for ths partcuar jont strbuton, the correaton coeffcent ecreases to some mt as the retenton eve ncreases Fgure The correaton between X Ç an Y Ç as a functon of Correaton between X an Y Lmt: 00 CASUALTY ACTUARIAL SOCITY VOLUM 0/ISSU

7 ampe : A bvarate Tabe M Ths eampe shows that a bvarate Tabe M can be constructe for the bvarate ecess osses n a smar way to the unvarate Tabe M Assume that one observes a sampe of a par of bvarate oss rato ranom varabes (X, Y) as shown n Tabe To compute the jont moments of the bvarate ecess of osses (X Y y ), we bascay nee to construct ther emprca jont survva functon an then numercay mpement the oube ntegraton n quaton (9) The etae steps are shown n a companon ce tabe ( e-tabe-ren-papers) The ce tabe s easy to use, for eampe, [X Y y ] s smpy gven by the vaue n coumn J an the row wth oss rato vaues an y If t s esre to cacuate the hgher jont moments of X an Y y, one can procee to o some more numerca ntegratons n the spreasheet ampe 3: Per-occurrence an stop-oss coverage Ths eampe foows the one n Secton of Homer an Cark (00) wth some mofcatons Assume that the sze of Workers Compensaton osses from a fctona arge nsure ABC, enote by Z, foow a Pareto strbuton wth the survva functon α ( ) S + θ, where α 3 an q $00,000 Assume that the number of osses N foows a negatve bnoma strbuton wth the probabty generatng functon (see, for eampe, Kugman, Panjer, an Wmot 0) where β 0 an r 5 r P ( z) ( β( z )), N Tabe Sampe of bvarate oss ratos X Y An nsurance company, XYZ, has been aske to prove a per-occurrence coverage of $50,000 ecess of 0 an then a stop-oss coverage on an aggregate bass of $500,000 ecess of As an actuary of XYZ, you are tryng to etermne an optma combnaton of 0 an, so that your objectve functon the rato between the epecte payments an the stanar evaton of the payments s mamze Notce that the epecte payments can be consere as a proy for the epecte unerwrtng profts assumng a rsk oang eve, an the stanar evaton of the payments of course may represent the rsk eve Therefore, the objectve functon bears some resembance to the Sharpe rato (Boe, Kane, an Marcus 00) use n portfoo anayss We ntrouce the foowng notatons to mathematcay escrbe the probem The monetary unt we use s n thousans of oars Let the amount that ABC has to pay per occurrence be enote by 0 ZA Z + Z Let the amount that XYZ has to pay per occurrence be enote by Let Z X Z V N Z X, enote the aggregate amount that XYZ pays for the per-occurrence coverage an et U N Z A, be the aggregate amount ABC pays after the peroccurrence coverage but before the stop-oss coverage Then the tota amount XYZ has to pay uner the nsurance treaty s gven by where W V + U, VOLUM 0/ISSU CASUALTY ACTUARIAL SOCITY 0

8 Varance Avancng the Scence of Rsk Our goa s to seect vaues of 0 an so that the objectve functon [W]/s W, where s W stans for the stanar evaton of W, s mamze To sove the probem, we cou appy the foowng steps: Assgn some vaues to 0 an Construct a matr contanng the jont probabty strbuton functon of (U, V) Ths can be obtane by appyng the bvarate fast Fourer transform (FFT) metho as propose n Homer an Cark (00) 3 Construct a matr for the jont survva functon, S (U,V), from the matr for the jont probabty functon obtane n step Construct two vectors contanng vaues for the margna survva functons S U an S V, respectvey 4 Construct vectors contanng vaues of the functons R () an R () for ranom varabes U an V by appyng quatons (4) an (5) to the corresponng survva functons S U an S V Then compute [V], [U ], [V ], an [(U ) ] usng quatons (6) an (5) 5 Construct a matr contanng vaues of the functon R from S (U,V) usng quaton (0) an compute [U V] by appyng () 6 Compute the mean an varance of W U + V usng quanttes obtane n steps 4 an 5, then [ W ] evauate the objectve functon σ 7 Repeat steps 6 for fferent vaues of 0 an an compare the vaues of the objectve functon Tabes 3, 4, an 5 shows vaues of [W], s W an [ W ] the objectve functon for some combnatons σ W of 0 an, respectvey It appears that when the peroccurrence entry pont 0 s ow an the stop-oss coverage entry pont s hgh, the objectve functon s mamze In aton, the tabes can be use to etect neffcent combnatons of 0 an For eampe, the ( 0, ) (50, 000) combnaton resuts n ower epecte osses but a hgher stanar eva- W Tabe 3 The epecte vaue of W (n thousans) 0 \ Tabe 4 The stanar evaton of W (n thousans) 0 \ Tabe 5 The rato of the mean an the stanar evaton of W 0 \ ton than the ( 0, ) (00, 500) combnaton Therefore, the former s neffcent 5 Concusons We frst showe that hgher moments of ecess osses may be obtane from Tabe M Then we showe that the jont moments of bvarate ecess osses can aso be obtane n a smar fashon These technques are usefu n rensurance an retrospectve nsurance ratng when osses from two sources of rsks are consere References Boe, Z, A Kane, an A Marcus, Investments, New York: McGraw-H/Irwn, 9th e, 00 Brosus, J, Tabe M Constructon, CAS am 8 Stuy Note, 00 0 CASUALTY ACTUARIAL SOCITY VOLUM 0/ISSU

9 Denut, M, C Lefevre, an M Mesfou, A Cass of Bvarate Stochastc Orerngs, wth Appcatons n Actuara Scences, Insurance: Mathematcs an conomcs 4(), 999, pp 3 50 Denut, M, C Lefevre, an M Shake, The S-Conve Orers among Rea Ranom Varabes, wth Appcatons, Mathematca Inequates an ther Appcatons (4), 998, pp Dhaene, J, an M J Goovaerts, Depenency of Rsks an Stop-Loss Orer, ASTIN Buetn 6(), 996, pp 0 Gupta, R P, an P G Sankaran Bvarate qubrum Dstrbuton an Its Appcatons to Reabty, Communcatons n Statstcs-Theory an Methos 7(), 998, pp Hawe, L J, The Mathematcs of cess Losses, Varance 6, 0, pp 3 47 Homer, D L, an D R Cark Insurance Appcatons of Bvarate Dstrbutons, 00 CAS Reserves Dscusson Papers, 00, p 83 Hürmann, W, Hgher Degree Stop-Loss Transforms an Stochastc Orers () Theory, Bätter er DGVFM 4(3), 000, pp Kugman, S A, H H Panjer, an G Wmot, Loss Moes: From Data to Decson, New York: Wey, 4th e, 0 Lee, Y-S, The Mathematcs of cess of Loss Coverages an Retrospectve Ratng: A Graphca Approach, Proceengs of the Casuaty Actuara Socety 75, 988, p 49 Mccos, R S, On the Theory of Increase Lmts an cess of Loss Prcng, Proceengs of the Casuaty Actuara Socety 64, 977 p 7 Wang, S, Aggregaton of Correate Rsk Portfoos: Moes an Agorthms, In Proceengs of the Casuaty Actuara Socety 85, 998, pp Zahe, H, Some New Casses of Mutvarate Survva Dstrbuton Functons, Journa of Statstca Pannng an Inference () 985, pp 7 88 VOLUM 0/ISSU CASUALTY ACTUARIAL SOCITY 03