Measuring tail dependence for collateral losses using bivariate Lévy process. Jiwook Jang

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1 Measurig ail depedece fr cllaeral lsses usig bivariae Lévy prcess Jiwk Jag Acuarial Sudies, Uiversiy f New Suh Wales, Sydey, NSW 252, Ausralia, j.jag@usw.edu.au Absrac I pracice, isurace cmpaies face cllaeral lsses, fr example, i wrldwide, ce a srm r earhquake arrives, i brigs abu damages i prperies, mrs ad ierrupi f busiesses. I ccurred a cuple f lsses simulaeusly frm he Wrld Trade Cere (WTC) caasrphe. Hwever i has bee develped a applicable mdel fr isurace cmpaies measure schasic depedece fr hese cllaeral lsses. The aim f his paper is measure (upper) ail depedece fr cllaeral lsses, emplyig bivariae Lévy prcess, i.e. bivariae cmpud Piss prcess wih a cpula, as isurace idusry is mre ccered wih depedece bewee exreme values. I rder derive a explici expressi f ji Laplace rasfrm f cllaeral lsses, we use a member f Farlie-Gumbel-Mrgeser cpula wih expeial margis. Iversi f ji Fas Furier rasfrm baied frm he ji Laplace rasfrm f cllaeral lsses is used calculae he cefficies f (upper) ail depedece umerically. We als prvide he figures f he ji disribui f cllaeral lsses ad heir curs a each value f he parameer i a Farlie-Gumbel-Mrgeser cpula. Keywrds: Schasic depedece fr cllaeral lsses; bivariae Lévy prcess; Farlie-Gumbel-Mrgeser cpula; he cefficie f (upper) ail depedece; iversi f Fas Furier rasfrm. 1. Irduci Pracically isurace lsses have depedece bewee heir cmpes, e.g. depedece bewee lsses, depedece bewee lss arrivals ad depedece bewee lsses ad heir arrivals. S i requires fr us emply a mulivariae mdel wih depedece. Sice cpulae araced academics ad praciiers, mulidimesial mdels have bee develped capure depedece bewee cmpes f isurace risks (Lidskg ad McNeil 23; Pfeifer ad Nešlehvá 24; Chavez-Demuli e al. 25 ad Bäuerle ad Grübel 25). Ms f hese papers deal wih cvariace, liear crrelai ad simulai f depede lsses. I cras, i his paper, we deal wih measurig (upper) ail depedece f lsses usig bivariae Lévy prcess, i.e. bivariae cmpud Piss prcess wih a cpula, as isurace idusry is mre ccered wih depedece bewee exreme values. We are examiig w lsses ha are ccurrig cllaerally wih depedece. I rder measure he depedece bewee lsses, we may als csider usig he Lévy cpula, ha ca be fud i C ad Takv (24), raher ha usig a rdiary cpula. Igrig he ieres rae, le us assume ha isurace cmpay is experiecig depede lsses frm e specific evesuchasfld, widsrm, hail, earhquake ad errris aack. Fr example, WTC caasrphe brugh abu lsses f prperies, lsses f vehicles, lsses due busiess ierrupis, ec. u f e rigger. S fr bivariae risk case, we ca mdel P = N i=1 P = N i=1 X i, Y i (1.1) where is he al lsses arisig frm risk ype 1, is he al lsses arisig frm risk ype 2 ad N is he al umber f cllaeral lsses up ime. X i ad Y i,i=1, 2,, are he lss amus, which are assumed depede wih a cpula fuci C (H (x),h(y)) (x >, y > ) where H(x) be he ideically disribui fuci f X ad H(y) be he ideically disribui fuci f Y. Fr deails cpulae, we refer yu Nels (1998). We assume ha he cllaeral lss arrival prcess N fllws a Piss prcess wih lss frequecy rae µ. I is als assumed ha is idepede f X i ad Y i. We emply he Farlie-Gumbel-Mrgeser family cpula, ha is give by 1

2 C(u, v) =uv + θuv(1 u)(1 v), (1.2) where u [, 1], v [, 1] ad θ [ 1, 1], capure he depedece f cllaeral lsses f X ad Y. I rder bai he explici expressi f he fuci F (x, y), ha is a w-dimesial disribui fuci wih margis H (x) ad H(y), welex ad Y be expeial radm variables, i.e. H (x) =1 e αx (α >, x>) ad H(y) =1 e βy (β >, y>), he he ji disribui fuci F (x, y) is give by F (x, y) = C(1 e αx, 1 e βy ) = 1 e βy e αx + e αx βy + θe αx βy θe αx 2βy θe 2αx βy + θe 2αx 2βy. (1.3) ad is derivaive is give by df (x, y) = dc(1 e αx, 1 e βy ) = (1+θ) αβe αx βy 2θαβe αx 2βy 2θαβe 2αx βy +4θαβe 2αx 2βy. (1.4) We examie (upper) ail depedece f cllaeral lsses X ad Y as isurace cmpaies ccers are exreme lsses i pracice. S we adp he cefficie f (upper) ail depedece, λ U, used by mbrechs, Lidskg ad McNeil (23), ½ ¾ lim P >G 1 (u) u%1 >G 1 (u) = λ U (1.5) prvided ha he limi λ U [, 1] exiss, where G L (1) ad. ad G L (2) are margial disribui fucis fr I rder evaluae (1.5), we eed bai he ji disribui f he aggregae lsses ad, where heir idividual lsses are ccurrig cllaerally. Ufruaely, i is kw ha i is pssible fr us bai he ji disribui he aggregae lsses explicily. S i seci 2, we derive he geeral frm f he ji Laplace rasfrm f he disribui f he aggregae lsses expressed wih a cpula fuci, applyig he piecewise deermiisic Markv prcesses hery. Based his geeral frm f he ji Laplace rasfrm, we bai he explici expressi f he ji Laplace rasfrm f he disribui f he aggregae lsses usig (1.3). Seci 3 prvides he explici expressi f he cvariace ad liear crrelai bewee ad a ime. I seci 4, we illusrae he calculais f he cefficie f (upper) ail depedece usig he ji Fas Furier rasfrm. We als prvide he figures f he ji disribui f cllaeral lsses ad heir curs a each value f θ. Seci 5 ccludes. 2. The ji Laplace rasfrm f he disribui f aggregae lsses The piecewise deermiisic Markv prcesses hery develped by Davis (1984) is a pwerful mahemaical l fr examiig -diffusi mdels. Frm w, we prese defiiis ad impra prperies f ad wih he aid f piecewise deermiisic Markv prcesses hery (Dassis ad mbrechs 1989; Rlski e al ad Dassis ad Jag 23). This hery is used derive he geeral frm f he ji Laplace rasfrm f he disribui ³ f aggregae lsses ad. The geerar f he prcess,, acig a fuci f l (1),l (2), belgig is dmai is give by ³ Af l (1),l (2), = f + µ Z Z f ³ ³ l (1) + x, l (2) + y, dc(h (x),h(y)) f l (1),l (2), (2.1) where f :(, ) (, ) < + (, ). Fr f l (1),l (2), belg he dmai f he geerar A, i is sufficie ha f l (1),l (2), is differeiable w.r..l (1),l (2), fr all l (1),l (2), ad ha 2

3 Z Z f ³ ³ l (1) + x, l (2) + y, dc(h (x),h(y)) f l (1),l (2), <. Le us fid a suiable marigale i rder derive he ji Laplace rasfrm f he disribui f ad. Lemma 2.1 Csiderig csas ν ad ξ, ³ exp ν ³ exp ξ exp µ R R is a marigale where ĉ (ν, ξ) = e νx e ξy dc(h (x),h(y)). Z {1 ĉ (ν, ξ)} ds (2.2) Prf. Frm (2.1), f l (1),l (2), has saisfy Af =fr f l (1),l (2), be a marigale. f l (1),l (2), =exp νl (1) exp ξl (2) e B() we ge he equai ad he slui is Seig B ()+µ {ĉ (ν, ξ) 1} = (2.3) Z B () =µ {1 ĉ (ν, ξ)} ds (2.4) by which he resul fllws. Usig he marigale baied i Lemma 2.1, we ca easily bai he geeral frm f he ji Laplace rasfrm f he disribui f ad a ime, ³ e νl(1) e ξl(2),l(2) =exp ν ³ exp ξ exp µ Fr simpliciy, we assume ha =ad =,heiisgiveby e νl(1) e ξl(2) =exp µ Z Z {1 ĉ (ν, ξ)} ds. (2.5) {1 ĉ (ν, ξ)} ds. (2.6) I rder bai he explici expressi f he ji Laplace rasfrm f he disribui f ad a ime, le us use he ji disribui fuci F (x, y) drive by (1.3), he i is give by If θ =, he we have = exp e νl(1) µ e νl(1) e ξl(2) ½ (αξ + βν + νξ)(2α + ν)(2β + ξ) θαβ νξ (α + ν)(β + ξ)(2α + ν)(2β + ξ) e ξl(2) =exp µ ¾. (2.7) ½ ¾ (αξ + βν + νξ), (2.8) (α + ν)(β + ξ) which is he case ha w lsses X ad Y ccur same ime frm a sharig lss frequecy rae µ, bu heir sizes are idepede each her. I will be f ieres fid mre explici expressis f he ji Laplace rasfrm f he disribui f ad a ime, usig her cpulae ad her margis H (x) ad H(y). Hwever i is bvius ha we will be able derive explici frms f he ji Laplace rasfrm f he disribui f ad if we apply heavy-ailed disribuis fr margi H (x) ad H(y). 3

4 If we se ξ =, he he Laplace rasfrm f he disribui f is give by ½ µ ¾ e νl(1) ν =exp µ (2.9) α + ν ad if we se ν =, he he Laplace rasfrm f he disribui f is give by ½ µ ¾ e ξl(2) ξ =exp µ, (2.1) β + ξ which are he Laplace rasfrm f he disribui f he cmpud Piss prcess wih expeial lss sizes. Due he depedece f cllaeral lsses f X ad Y wih sharig lss frequecy rae µ, i is bvius ha e νl(1) e ξl(2) 6= e νl(1) e ξl(2). (2.11) If lss X ccurs wih is frequecy rae µ (x) ad lss Y ccurs wih is frequecy rae µ (y) respecively ad everyhig is idepede each her, we ca easily derive he explici expressi f he ji Laplace rasfrm f he disribui f ad a ime, i.e. e νl(1) e ξl(2) = = exp e νl(1) ½ µ ν µ (x) α + ν e ξl(2) ¾ ½ µ ¾ ξ exp µ (y). (2.12) β + ξ If we se µ = µ (x) = µ (y), i.e. frequecy rae fr lss X ad Y are jus he same, he (2.12) becmes e νl(1) e ξl(2) = e νl(1) ½ = exp µ = exp µ e ξl(2) ¾ µ ½ ν exp α + ν ½ (αξ + βν +2νξ) (α + ν)(β + ξ) µ ¾ ξ µ β + ξ ¾. (2.13) quai (2.13) lks similar (2.8) as lss size X ad Y are idepede ad heir frequecy raes are he same. Hwever he ji Laplace rasfrm f he disribui f ad a ime expressed by (2.13) are he case ha hey are ccurrig idepedely, cllaerally like (2.8). 3. Cvariace ad liear crrelai f cllaeral lsses Le us examie he cvariace ad liear crrelai bewee ad a ime. Differeiaig (2.7) w.r.. ν ad ξ ad se ν =ad ξ =, he we ca easily derive he ji expecai f ad a ime, i.e. = µ2 αβ 2 + µ µ 1+ θ. (3.1) αβ 4 Als frm (2.9) ad (2.1) we ca easily derive he expecai f ad ad a ime, i.e. = µ α (3.2) = µ. (3.3) β The higher mmes f ad a ime ca be baied by differeiaig i furher, i.e. 4

5 ad Var Var The cvariace bewee ad a ime is give by Cv(, )= = 2µ. (3.4) α2 = 2µ 2. (3.5) β = µ µ 1+ θ. (3.6) αβ 4 As liear crrelai (r Pears s crrelai) has bee ms ppularly used i pracice as a measure f depedece, ³ we prese he expressi f he liear crrelai cefficie fr ad a ime, deed by ρ,, ³ ρ, Cv(, µ ) 1+ θ = r 4 r Var Var = q αβ q = 4+θ 2µ 8. (3.7) Le us w illusrae he calculais f he cvariace ad liear crrelai bewee ad a ime. xample 3.1 The parameer values used calculae he cvariace ad liear crrelai usig (3.6) ad (3.7) are µ =4, α =1, β =.5, =1. Frm (3.6) ad (3.7), he calculais f cvariace ad liear crrelai bewee ad a ime are shw i Table 3.1 ad Table 3.2 respecively. 2µ α 2 β 2 Table 3.1. θ Cv(, ) Table 3.2. ³ θ ρ, Tail depedece f cllaeral lsses Le us lk a hw he ji Laplace rasfrm derived i Seci 2 ca be used evaluae he cefficie f (upper) ail depedece, λ U. As i is pssible fr us bai he ji disribui f ad explicily, we iver he ji Fas Furier rasfrm baied frm he ji Laplace rasfrm f cllaeral lsses apprximae he cefficie f (upper) ail depedece (Caslema 1996; Gzalez ad Wds 22 ad Gzalez e al. 24). The belw figures are he ji disribui f cllaeral lsses ad heir curs a each value f θ. Nw le us illusrae he calculais f he cefficie f (upper) ail depedece usig (1.5). xample 4.1 The parameer values used apprximae he cefficie f (upper) ail depedece are µ =4, α =1, β =.5, =1. Frm (2.9) ad (2.1), he mea f he aggregae lsses arisig frm risk ype 1 is give by 5

6 Figure 1: The ji disribui f cllaeral lsses wih θ =1 Figure 2: The cur f he ji disribui f cllaeral lsses wih θ =1 6

7 Figure 3: The ji disribui f cllaeral lsses wih θ =.5 Figure 4: The cur f he ji disribui f cllaeral lsses wih θ =.5 7

8 Figure 5: The ji disribui f cllaeral lsses wih θ = Figure 6: The cur f he ji disribui f cllaeral lsses wih θ = 8

9 Figure 7: The ji disribui f cllaeral lsses wih θ =.5 Figure 8: The cur f he ji disribui f cllaeral lsses wih θ =.5 9

10 Figure 9: The ji disribui f cllaeral lsses wih θ = 1 Figure 1: The cur f he ji disribui f cllaeral lsses wih θ = 1 1

11 = µ α =4 ad he mea f he aggregae lsses arisig frm risk ype 2 is give by = µ β =8. Usig Malab, he calculais f mea f aggregae lsses arisig frm risk ype 1 ad 2 respecively are shw i Table 4.1. θ Table Usig Malab, he calculais f he cefficies f (upper) ail depedece fr cllaeral lsses are shw i Table 4.2, Table 4.3, Table 4.4 ad Table 4.5 usig he differe VaR a 9%, 95%, 99% ad 99.9%. Table 4.2 θ P > 7.84, P > 7.84 > > = P > > where P > 7.84 = P > =.1 Table 4.3 θ P > 9.37, P > 9.37 > > = P > > where P > 9.37 = P > =.5 11

12 Table 4.4 θ P > 12.61, P > > > = P > > where P > = P > =.1 Table 4.5 θ P > 16.81, P > > > = P > > where P > = P > =.1 5. Cclusi I his paper, we examied (upper) ail depedece fr cllaeral lsses usig bivariae cmpud Piss prcess as isurace idusry is mre ccered wih depedece bewee exreme values. The cvariace ad liear crrelai f cllaeral lsses were als sudied. I rder measure depedece bewee lsses, we used a member f Farlie-Gumbel-Mrgeser cpula wih expeial margis. mplyig piecewise deermiisic Markv prcesses hery ad marigale apprach, we derived he explici expressi f he ji Laplace rasfrm f he disribui f cllaeral lsses. Iversi f ji Fas Furier rasfrm baied frm he ji Laplace rasfrm was used calculae he cefficies f (upper) ail depedece umerically. We als prvided he figures f he ji disribui f cllaeral lsses ad heir curs a each value f he parameer i a Farlie-Gumbel-Mrgeser cpula. Fr furher research, firsly we ca examie iverig aalyical frms f he ji Laplace rasfrm f he disribui f lsses, usig her cpulae ad her margis, i paricular heavy-ailed disribuis fr lss sizes. Secdly, her depedece srucures, i.e. depedece bewee lss arrivals r depedece bewee lsses ad heir arrivals ca be icluded i lss prcesses. Lasly, bu challegig eugh, we ca csider exedig mulidimesial risk frm bivariae risk. Refereces Bäuerle, N. ad Grübel, R. (25) : Mulivariae cuig prcesses: cpulas ad beyd, accceped ASTIN BULLTIN. Caslema, K. R. (1996) : Digial Image Prcessig, Preice Hall, glewd Cliffs, NJ. Chavez-Demuli, V., mbrechs, P. ad Nešlehvá, J. (25) :Quaiaive Mdels fr Operaial Risk: xremes, Depedece ad Aggregai, Dep. f Mahemaics, TH Zürich. C, R. ad Takv, P. (24) : Fiacial Mdellig Wih Jump Prcesses, Champa & Hall, Bca Ra, FL. Dassis, A. ad mbrechs, P. (1989) : Marigales ad isurace risk, Cmmu. Sa.-Schasic Mdels, 5(2), Dassis, A. ad Jag, J. (23) : Pricig f caasrphe reisurace & derivaives usig he Cx prcess wih sh ise iesiy, Fiace & Schasics, 7/1,

13 Davis, M. H. A. (1984) : Piecewise deermiisic Markv prcesses: A geeral class f diffusi schasic mdels, J. R. Sa. Sc. B, 46, mbrechs, P., Lidskg, F., McNeil, A.(23) : Mdellig Depedece wih Cpulas ad Applicais Risk Maageme I: Hadbk f Heavy Tailed Disribuis i Fiace, ed. S. Rachev, lsevier, Chaper 8, Gzalez, R. C. ad Wds, R.. (22) : Digial Image Prcessig, 2d dii, Preice Hall, Upper Saddle River, NJ. Gzalez, R. C., Wds, R.. ad ddis, S. L. (24) : Digial Image Prcessig Usig MATLAB, Preice Hall, Upper Saddle River, NJ. Lidskg, F. ad McNeil, A. J. (23): Cmm Piss Shck Mdels: Applicais Isurace ad Credi Risk Mdellig, ASTIN BULLTIN, Vl. 33, N. 2, Nelse, R. B. (1999) : A Irduci Cpulas, Spriger-Verlag, New Yrk. Pfeifer, D. ad Nešlehvá, J. (24) : Mdelig ad Geeraig Depede Risk Prcesses fr IRM ad DFA, ASTIN BULLTIN, Vl. 34, N. 2, Rlski, T., Schmidli, H., Schmid, V. ad Teugels, J. L. (1998) : Schasic Prcesses fr Isurace ad Fiace, Jh Wiley & Ss, UK. 13