On the moments of the aggregate discounted claims with dependence introduced by a FGM copula


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1 On th momnts of th aggrgat discountd claims with dpndnc introducd by a FGM copula  Mathiu BARGES Univrsité Lyon, Laboratoir SAF, Univrsité Laval  Hélèn COSSETTE Ecol Actuariat, Univrsité Laval, Québc, Canada  Stéphan LOISEL Univrsité Lyon, Laboratoir SAF  Etinn MARCEAU Ecol Actuariat, Univrsité Laval, Québc, Canada 29.3 WP 23 Laboratoir SAF 5 Avnu Tony Garnir Lyon cdx 7
2 On th Momnts of th Aggrgat Discountd Claims with Dpndnc Introducd by a FGM Copula Mathiu Bargès Hélèn Cosstt Stéphan Loisl Étinn Marcau Octobr 22, 29 Abstract In this papr, w invstigat th computation of th momnts of th compound Poisson sums with discountd claims whn introducing dpndnc btwn th intrclaim tim and th subsqunt claim siz. Th dpndnc structur btwn th two random variabls is dfind by a FarliGumblMorgnstrn copula. Assuming that th claim distribution has finit momnts, w giv xprssions for th first and th scond momnts and thn w obtain a gnral formula for any mth ordr momnt. Th rsults ar illustratd with applications to prmium calculation, momnt matching mthods, as wll as inflation strss scnarios in Solvncy II. Kywords : Compound Poisson procss, Discountd aggrgat claims, Momnts, FGM copula, Constant intrst rat. Introduction W considr a continuoustim compound rnwal risk modl for an insuranc portfolio and w dfin th compound procss of th discountd claims X i, i =, 2,... occurring at tim T i, i =, 2,... by = { t, t } with { Nt t = i= δt i X i, N t >, N t =, whr N = {N t, t } is an homognous Poisson counting procss and δ th constant nt intrst rat. In actuarial risk thory, it is assumd that th claim amounts X i, i =, 2,... ar indpndnt and idntically distributd i.i.d. random variabls r.v. s and th intrclaim tims W = T and W j = T j T j, j = 2, 3,... ar also i.i.d. r.v. s. Th r.v. s X i and W i, i =, 2,... ar Univrsité d Lyon, Univrsité Claud Brnard Lyon, Institut d Scinc Financièr t d Assurancs, 5 Avnu Tony Garnir, F697 Lyon, Franc Écol d Actuariat, Univrsité Laval, Québc, Canada
3 classically supposd indpndnt. This last assumption also implis that X i, i =, 2,... ar indpndnt from N. This risk procss has bn usd in ruin thory by many authors such as Taylor 979, Watrs 983, Dlban and Hazndonck 987, Willmot 989, Sundt and Tugls 995 and mor rcntly Kalashnikov and Konstantinids 2, Yang and hang 2 and Tang 25. Thy mainly focusd on th ruin probability and rlatd ruin masurs. Only a fw rcnt works dal with th distribution of th aggrgat discountd claims t. Lévillé and Garrido 2a provid th first two momnts of this procss. Ths first two momnts wr also obtaind in Jang 24 using martingal thory. This rsult has sinc bn gnralizd by rlaxing som of th classical assumptions prsntd abov. Lévillé and Garrido 2b and Lévillé t al. 29 drivd rcursiv formulas for all th momnts of th aggrgat discountd claims considring a compound rnwal procss whr N is not ncssarily a Poisson procss. In Jang 27, th Laplac transform of th distribution of a jump diffusion procss and its intgratd procss is drivd and usd to obtain th momnts of th compound Poisson procss t. Kim and Kim 27 and Rn 28 studid th discountd aggrgat claims in a Markovian nvironmnt which modulats th distributions of th intrclaim tims and claim sizs for th formr and th distribution of th intrclaim tims for th lattr. Thy both providd th Laplac transform of th distribution of th discountd aggrgat claims and thn gav xprssions for its first two momnts. Th aggrgation of discountd random variabls is also usd in many othr filds of application. For xampl, it can b usd in warranty cost modling, s Duchsn and Marri 29, or in rliability in civil nginring, s van Noortwijk and Frangopol 24 or Portr t al. 24. In this papr, w want to introduc som dpndnc btwn th intrclaim tims and th subsqunt claim amounts. In risk thory, this dpndnc has alrady bn xplord. For xampl, Albrchr and Boxma 24 supposd that if a claim amount xcds a crtain thrshold, thn th paramtrs of th distribution of th nxt intrclaim tim is modifid. In Albrchr and Tugls 26 th dpndnc is introducd with th us of an arbitrary copula. Convrsly to Albrchr and Boxma 24, Boudrault t al. 26 assumd that if an intrclaim tim is gratr than a crtain thrshold thn th paramtrs of th distribution of th nxt claim amount is modifid. In a similar dpndnc modl, but with mor frdom in th choic of th copula btwn ach intrclaim tim and th subsqunt claim amount, Asimit and Badscu 29 considr a constant forc of intrst and havytaild claim amounts. Dpndnc concpts usd in Boudrault t al. 26 wr thn xtndd in Biard t al. 29 whr thy suppos that th distribution of a claim amount has its paramtrs modifid whn svral prcding intrclaim tims ar all gratr or all lowr than a crtain thrshold. All ths paprs wr intrstd in finding xact xprssions or approximations for som ruin masurs such as th ruin probability or th GrbrShiu function. In our study, this assumption of indpndnc btwn th claim amount X j and th intrclaim tim W j is rlaxd to allow {X j, W j, j N } to form a squnc of i.i.d. random vctors distributd as th canonical random vctor X, W in which th componnts may b dpndnt. W follow th ida of Albrchr and Tugls 26 supposing that dpndnc is introducd by a copula btwn an intrclaim tim and its subsqunt claim amount. Mor spcifically, w us th FarliGumblMorgnstrn FGM copula which is dfind by C F GM θ u, v = uv θuv u v, 2
4 for u, v [, [, and whr th dpndnc paramtr θ taks valu in [,. Whil thr ar a larg numbr of copula familis, w choos th FGM copula bcaus it offrs th advantag of bing mathmatically tractabl as it is illustratd in Cosstt t al. 29. Evn if th FGM copula introducs only light dpndnc, it admits positiv as wll as ngativ dpndnc btwn a st of random variabls and includs th indpndnc copula whn θ =. It is also known that th FGM copula is a Taylor approximation of ordr on of th Frank copula s Nlsn 26, pag 33, AliMilkhailHaq copula and Placktt copula s Nlsn 26, pag. Th papr is structurd as follows. In th scond sction, w prsnt th modl of th continuous tim compound Poisson risk procss that w us and giv som notation. Th first momnt, th scond momnt and thn a gnralization to th mth momnt ar drivd in Sction 2. Applications to prmium calculation, momnt matching mthods and Solvncy II ar givn in th third sction. In particular, w show how our mthod may b usd to dtrmin Solvncy Capital Rquirmnts and to prform part of Own Risk and Solvncy Assssmnt ORSA analysis in Solvncy II for som cat risks and inflation risk. 2 Th modl As xplaind in th introduction, w considr th continuoustim compound Poisson procss = { t, t } of th discountd claims X,..., X Nt occurring at tims T,..., T Nt with t = whr E[Xi k < for i =, 2,... { Nt i= δt i X i, N t >, N t =, W introduc a spcific structur of dpndnc basd on th FarliGumblMorgnstrn copula btwn th ith claim amount and th ith intrclaim tim such that, using, th joint cumulativ distribution function c.d.f. for th canonical random vctor X, W is F X,W x, t = C F X x, F W t = F X x F W t θf X x F W t F X x F W t, for t, x R R and whr F X and F W ar th marginals of rspctivly X and W. This dpndnc rlation implis that X, X 2, X 3,... ar no mor indpndnt of N. Rcalling th dnsity of th FGM copula c F GM θ u, v = θ 2u 2v, for u, v [, [,, th joint probability dnsity function p.d.f. of X, W is f X,W x, t = c F θ GM F X x, F W t f X x f W t = f X x f W t θf X x f W t 2F X x 2F W t, whr f X and f W ar th p.d.f. s of rspctivly X and W. 3
5 µ m Th mth momnt of t is dnotd by µ m t = E [ m t and its Laplac transform by r with m R. W s in th nxt sction how to driv xplicit formulas for ths momnts. 3 Momnts of th aggrgat discountd claims 3. First momnt To driv th xprssion for th first momnt µ t of t, w condition on th arrival of th first claim whr Ltting 2 bcoms µ t = E [ t [ [ = E E δs X δs t s W = s = E [X W = s = E [ X = f W s δs E [X W = s ds = = E [X θ θ xf X W =s x dx f W s δs µ t s ds, x { θ 2F X x 2F W s} f X x dx x 2 2F X x 2F W s f X x dx x 2F W s f X x dx = E [X θ 2F W s θ 2F W s F X x 2 dx < F X x 2 dx. 2 F X x dx = E [X, E [X E [ X E [X θ 2F W s. 3 From 3, w can driv th following rmarks. If θ > θ < and s < F W.5 s > F W.5, rspctivly, thn E [X W = s < E [X. Convrsly, if θ > θ < and s > FW.5 s < F W.5, rspctivly, thn E [X W = s > E [X. W considr th cas whr th canonical r.v. W has an xponntial distribution with man 4
6 and f W t = t, 4 F W t = t, 5 f W s = E [ sw = s. For simplification purposs, w us th xprssions to driv th momnts of t. W obtain th following xprssion for µ t µ t = = = h s; γ = γ γs ĥ r; γ = γ γ r f W s δs E [X ds θ E [ X E [X f W s δs 2F W s ds f W s δs µ t s ds s δs E [X ds θ E [ X E [X s δs µ t s ds h s; δ E [X ds δ θ E [ X E [X θ E [ X E [X 2 h s; 2 δ ds 2 δ h s; δ ds δ s δs 2 s ds δ h s; δ µ t s ds. 6 W tak th Laplac transform on both sids of 6 and aftr som rarrangmnts, w obtain µ r = which is quivalnt to µ r = r ĥr;δ r δ δr δ E [X θ E [X E [X 2 ĥr;δ 2δ r δ ĥr;δ r δ ĥ r; δ 7 δ E [X θ E [X E [X 2 2δ δ δ δr r 2δ 2δr δ r δ δr. 8 5
7 Rarranging 8, w dduc µ r = E[X rδ r θ E[X E[X. 9 r2 δ r Invrting 9, w obtain µ t = E[X δt δ θ E[X E[X 2δt. 2 δ Notic that whn th r.v. s X and W ar indpndnt which corrsponds to θ =, th xpctd valu of th compound procss of th discountd claims, notd ind t, bcoms µ ind t = E[X δt. δ 3.2 Scond momnt As for th first momnt of th discountd total claim amount, w condition on th arrival of th first claim to obtain th scond momnt of t [ µ 2 [E t = E δs X δs t s 2 W = s = f W s 2δs E [ X 2 W = s ds 2 f W s 2δs µ 2 t s ds. f W s 2δs E [X W = s µ t s ds Similarly as in 2, w hav E [ X 2 W = s = E [ X 2 θ 2F W s θ 2F W s 2x F X x 2 dx = E [ X 2 [ E X 2 E [ X 2 θ 2F W s, whr E [ X 2 = 2x F X x 2 dx < 2x F X x dx = E [ X 2. 6
8 µ 2 W find th following xprssion for µ 2 t t = = 2 f W s 2δs E [ X 2 ds θ E f W s 2δs E [X µ t s dsds [ X 2 E [ X 2 f W s 2δs 2F W s ds 2θ E [ X E [X f W s 2δs 2F W s µ t s ds 2δ h s; 2δ E [ X 2 ds [ X θ E 2 E [ X h s; 2 2δ 2 2δ 2δ h s; 2δ E [X µ t s ds 2θ E [ X E [X 2δ 2 h s; 2 2δ 2 2δ h s; 2δ ds 2δ f W s 2δs µ 2 t s ds h s; 2δ µ t s ds 2δ h s; 2δ µ2 t s ds. W tak th Laplac transform on both sids of and aftr som rarrangmnts, w obtain µ 2 r = [ĥr; 2δ 2δ ĥr; 2δ r 2δ E[X2 θ E[X 2 E[X 2 2 ĥr; 2 2δ ĥr; 2δ 2 2δ r 2δ r 2E[X 2δ ĥr; 2δ µ r 2θ E [ X E[X 2 ĥr; 2 2δ 2 2δ ĥr; 2δ µ r, 2δ which bcoms µ 2 r = E[X2 r2δ r θ [ E X 2 E[X 2 2 E[X r2 2δ r 2δ r µ r 2θ E[X E[X µ r 2 2δ r = E[X2 r2δ r θ [ E X 2 E[X 2 2 E[X E[X r2 2δ r 2δ r rδ r θ E[X E[X r2 δ r 2θ E[X E[X E[X 2 2δ r rδ r θ E[X E[X r2 δ r = E[X2 r2δ r θ [ E X 2 E[X 2 2 E[X 2 2 r2 2δ r rδ r2δ r 2θ 2 E[X E [X E[X r2 δ r2δ r 2θ 2 E[X E [X E[X rδ r2 2δ r 2θ 2 2 E [X E[X 2 r2 δ r2 2δ r. 2 7
9 This last Laplac transform is a combination of trms of th form fr = rα rα 2 r...α n r, with f a function dfind for all nonngativ ral numbrs. As dscribd in th proof of Thorm. in Baumr 23, ach of ths trms can b xprssd as a combination of partial fractions such as whr γ = α...α n and, for i =,..., n, fr = γ r γ α r γ 2 α 2 r... γ n α n r, 3 γ i = α i n j=;j i α j α i. 4 Sinc th invrs Laplac transform of α i r is α it, it is asy to invrs f and obtain ft = γ γ α t γ 2 α 2t... γ n α 2t. 5 Using 5 in 2, it rsults that µ 2 t = E[X 2 2δ 2δt θ E [ X 2 E[X 2 2δ 2 2δ 22δt 2 2δ 2 2 E[X 2 2δ 2 δt δ 2 2δt 2δ 2 2θ 2 E[X E[X E[X 2δ2 δ 2δt 2 δ 2 δ 2δt 2δ 2 δ 2θ 2 E[X E[X E[X δ2 2δ δt δ2 δ 22δt 2 2δ2 δ 2θ 2 2 E[X E[X 2 2 δ2 2δ 2δt δ2 δ 22δ δ2 2δ. 6 8
10 3.3 mth momnt W now gnraliz th prvious rsults to th mth momnt of th discountd total claim amount. Conditioning on th arrival of th first claim lads to µ m t = f W s mδs E [X m W = s ds f W s mδs µ m t s ds. m j= m t j f W s mδs E [ X j W = s µ m j t s ds µ m For th Laplac transform of µ m t, w find mδ r = [ĥ r; mδ ĥ r; mδ r mδ E [Xm θ E [ X m E [X m 2 2 mδ m m E [ X j j mδ j= 2 ĥ r; 2 mδ 2 mδ which can also b xprssd as follows µ m m E[X m r = m rmδ r θ m j= ĥ r; 2 mδ r ĥ r; mδ µm j r θ ĥ r; mδ mδ m θ E [X m E[X m m r2 mδ r m E[X j E[X j j 2 mδ r µ m j r. ĥ r; mδ mδ r m m [ X E j E [ X j j j= µ m j r m j= m E[X j j mδ r µm j r 7 Noting for i =,..., m, j =,..., m and k =, ζi; j; k = w can rwrit µ r and µ 2 r as i θ k E[X j k E[X j E[X j k j k 2 iδ r µ r = r = [ ζ,, ζ,,, Λi; j; k k 2 iδ r, 8 9
11 r = [ ζ2, 2, ζ2, 2, [ζ2,, ζ2,, [ζ,, ζ,, r = [ ζ2, 2, ζ2, 2, ζ2,, ζ,, ζ2,, ζ,, ζ2,, ζ,, r ζ2,, ζ,,. µ 2 Th trm µ m r can also b xprssd using 8 µ m r = r m n= i,j,k,...,i n,j n,k n A mn ζi n, j n, k n... ζi, j, k, 9 { whr A mn = i, j, k,..., i n, j n, k n ; i = m, i...i n = m n, i >... > i n, j = m n, } j... j n = m, j... j n, k. {, }. To invrs 9, lt I ζi ; j ; k ;...; ζi n ; j n ; k n b th invrs Laplac transform of r ζi ; j ; k... ζi n ; j n ; k n, for n =,..., m. Using 3 and 5, w hav I ζi ; j ; k ;...; ζi n ; j n ; k n = Λi ; j ; k... Λi n ; j n ; k n γ γ αi ;k t... γ n αi n;k n t with, rfring to 4, γ = αi ;k...αi n ;k n and γ u =,,..., n. It finally rsults that αi u ;k u u v=;v u αi v ;k v αi u ;k u, u = µ m t = m n= i,j,k,...,i n,j n,k n A mn I ζi ; j ; k ;...; ζi n ; j n ; k n. 2 4 Applications As w hav alrady discussd in th introduction, svral scintific domains hav rcours to discountd aggrgations. W prsnt hr som applications of our rsults in actuarial scincs whr th claim distributions ar assumd to b positiv and continuous. 4. Prmium calculation Now that w ar abl to comput th momnts of t, it is possibl to comput th prmium rlatd to th risk of an insuranc portfolio rprsntd by t. W propos hr to study svral prmium calculation principls. Th loadd prmium Πt consists in th sum of th pur prmium
12 P t, which is th xpctd valu of th costs rlatd to th portfolio, and a loading for th risk Lt as Πt = P t Lt = E[t Lt. Th loading for th risk diffrs according to th prmium calculation principls. as Dnot by κ th safty loading. Th xpctd valu principl dfins th loadd prmium Πt = E[t κe[t, whr Lt = κe[t. Th varianc principl givs Πt = E[t κv art, whr Lt = κv art. And finally, w introduc th standard dviation principl which is dtrmind by whr Lt = κ V art. Πt = E[t κ V art, As w only nd th first two momnts for ths xmpls, w can us th quations and 6 to dtrmin th loading for th risk and thn th loadd prmium s.g. Rolski t al. 999 for dtails on prmium pincipls. 4.2 First thr momnts basd approximation for th distribution of t Hr, w suggst to us a momnt matching approximation for its distribution. As said in Tijms 994, th class of mixtur of Erlang distributions is dns in th spac of positiv contiuous distributions. So, w propos, as an illustration, to match th first thr momnts of t to a mixtur of two Erlang distributions of common ordr. This mthod coms from Johnson and Taaff 989 whr a momnt matching mthod with th first k momnts is fasibl for a mixtur of Erlang distributions of ordr n is prsntd. Th distribution function of a mixtur of two Erlang distributions with rspctiv rat paramtrs λ and λ 2 and common ordr n is givn by F Y y = p F y p 2 F 2 y, whr F and F 2 ar two Erlang c.d.f. s and p and p 2 thir rspctiv wight in th mixtur. Th p.d.f. Y is f Y y = p f y p 2 f 2 y,
13 whr f and f 2 ar two Erlang p.d.f. s. Th nth momnt of th mixtur of two Erlang distributions is E[Y n = p µ n p 2 µ n 2, whr µ n and µ n 2 ar th rspctiv nth momnt of two Erlang distributions. Undr som conditions, Thorm 3 of Johnson and Taaff 989 givs th paramtrs of th mixtur of two Erlang distributions with th sam ordr n as follows and λ i = B i B 2 4AC / 2A p = p 2 = whr A = nn 2µ y, B = x = µ µ 3 n2 n µ 2 2. µ n λ 2 / λ λ 2, nx nn2 n y2 n 2µ 2 y, C = µ x, y = µ 2 n µ 2 and For th numrical illustration, suppos that X Expλ = /, th intrclaim tim distribution paramtrs =, 5 and, th intrst rat δ = 4%. W us thr diffrnt valus for th copula paramtr θ =,, and fix th tim t = 5. Th mth momnt of X is E[X m = λ m m!. As E[X m = mx m F X x 2 dx, w hav that E[X m = 2λ m m!. Th first thr momnts of t and th matchd paramtrs for th mixtur of Erlang distributions ar prsntd in Tabls, 2 and 3. θ µ 5 µ 2 5 µ3 5 n λ λ 2 p p n Tabl : Momnts of 5 and paramtrs of th mixtur of Erlang distributions for =. For this last cas, Figurs, 2 and 3 in th appndix show th drawings of th simulatd c.d.f. of 5 vrsus th approximatd c.d.f. of 5 with a mixtur of Erlang distributions momnt matching. W s on our illustration that th fit of th approximations is satisfying. In Tabls 4, w compar th VaR obtaind from MontCarlo simulations of 5 against th VaR for th mixtur of Erlang distributions approximation for a confidnc lvl α = 99.5%. Onc 2
14 θ µ 5 µ 2 5 µ3 5 n λ λ 2 p p Tabl 2: Momnts of 5 and paramtrs of th mixtur of Erlang distributions for = 5. θ µ 5 µ 2 5 µ3 5 n λ λ 2 p p Tabl 3: Momnts of 5 and paramtrs of th mixtur of Erlang distributions for =. again, th approximatd VaR s ar satisfying. θ MC = MM = MC = 5 MM = 5 MC = MM = Tabl 4: VaR calculatd from th MontCarlo simulations and th momnt matching. Rmark Lt St = t whn th forc of intrst δ =. As, in gnral for δ, w hav E[φt E[φSt for vry nondcrasing function φ, w hav that t sd St whr sd dsignat th stochastic dominanc ordr. Furthrmor, this implis that V ar α t V ar α St for vry α [,. 4.3 Solvncy II intrnal modl Th Europan Solvncy II projct is going to lay down som nw rgulatory rquirmnts that vry insuranc company insid th Europan Union will hav to fulfill. In addition, svral othr countris outsid th Europan Union.g. Canada, Columbia or Mxico ar likly to us similar principls. Th dirctiv has bn adoptd in April 29 and th implmntation masurs ar in progrss in ordr to hav th nw systm in forc on Octobr 3st, 22. Dtrmination of Solvncy Capital Rquirmnt SCR is on of th main points of th quantitativ pillar of this rform: in addition to th bst stimat which is dfind as th xpctd prsnt valu of all potntial futur cash flows that would b incurrd in mting policyholdrs liabilitis of liabilitis and a risk margin, insuranc companis and rinsurrs will hav to own an xtra capital to cop with unfavorabl vnts. Th computation of th Solvncy II standard formula for SCR is basd on th yar 99.5%Valuatrisk VaR. Most oftn in th standard formula, it is assumd that th havinss of th tail of th distribution of random loss X is quit modrat, and so th SCR, dfind as th diffrnc V ar 99.5% X EX, is rplacd by a proxy qσ X, whr σ X dnots th 3
15 standard rror cofficint of X and q is a quantil factor which should b st at q = 3. Sldom, if appropriat, factor q = 3 may b rplacd by a largr valu, clos to 5 for xampl, to tak into account potntial havir tails. This is th cas in particular in th currnt vrsion of th Countrparty Risk modul s Consultation Papr 5 of CEIOPS. Although quantil factors may vary from on lin of businss to th othr, it has bcom classical to comput th SCR in th standard formula as a multipl of th standard rror cofficint of th random loss, or with strss scnarios. Evn if intrnal modls or partial intrnal modls ar bing ncouragd, companis will anyway hav to provid th SCR computations with th standard formula as complmnt. Som of thos partial intrnal modls ar basd on a diffrnt tim horizon, up to 5 or yars for som rinsurrs. Bsids, all insurrs hav to provid an Own Risk and Solvncy Assssmnt ORSA which aims to study risks that may affct th longtrm solvncy of th company. Eithr for ORSA or for SCR computations, it may b usful to dtrmin th first two momnts of th discountd aggrgat claim amount, both with constant intrst rat and inflation, and in a strss scnario whr inflation incrass. Inflation is vry low currntly, but thr is a clar risk that it incrass quit a bit whn th crisis nds. In an ORSA analysis, it would b intrsting to study th impact of inflation on Bst Estimat BE and on th SCR: what would b th BE and th SCR in thr yars from now if insuranc risk xposur was th sam as today, but inflation was much highr? This is what w invstigat in Tabls 6 and 7. Solvncy II standard formula oftn uss th indpndnc btwn claim amounts and th claim arrival procss. In practic, for risks lik arthquak risk or flood and drought risks, th nxt claim amount is not indpndnt from th tim lapsd bfor th prvious claim, and this must b takn into account in partial intrnal modls. Th advantag of our mthod is that it rmains valid for ngativ valus of δ as long as thy ar not too ngativ, which can b sn as th diffrnc btwn th intrst rat and th inflation rat. If inflation bcoms largr than th intrst rat, thn δ bcoms ngativ, and our mthod still applis for small nough valus of δ. Som othr approachs ar possibl as cat risk is somtims addrssd dirctly by th mans of xtrm scnarios. Hr w comput th SCR in th standard formula approach and in th intrnal modl approach for a 5yar horizon for xponntially distributd intrclaim tims and Exponntial and Parto claim amount distributions. For th intrnal modl approach, w us Equations?? and?? from th prvious xampl to comput th mth momnt of t whn th claim amounts ar xponntially distributd. If th claim amount r.v. X is Parto with c.d.f. γ κ F X x =, x >, γ x and mth momnt E[X m = γ m m! m i= κ i 2 for γ > and κ > m thn E[X m bcoms E[X m = m i= γ m m! 22 2κ i. Thus th mth momnt of t can b xplicitly xprssd using and 6 for th first and scond momnts, or using 2 for gratr momnts. Th SCR for th intrnal modl is obtaind 4
16 from th first momnt of t and a simulatd VaR with MontCarlo mthod. Lt th FGM dpndnc paramtr b, or, and δ = 3%. Th paramtr for th intrclaim tim distribution is = 2. Assum that th claim amount r.v. X Expλ = / for th Exponntial cas and that X P artoκ = 2.5, γ = 5 for th Parto cas with th sam xpctd valu but with variancs rspctivly qual to and 5. As discussd abov, w st th quatil factor q for th standard formula approach at 3 for th Exponntial cas and at 5 for th Parto cas. Th SCR s for th standard formula and th intrnal modl approachs ar prsntd in Tabl 5. Using th intrnal modl approach, w also comput th SCR and th Bst Estimat BE with inflation criss δ =.5%,.5% or 5% in comparison to δ = 3% for th Parto cas. Th rsults ar shown in Tabl 6. Exponntial cas Parto cas Copula paramtr Standard formula q = 3 Intrnal modl Standard formula q = 5 Intrnal modl θ = θ = θ = Tabl 5: Comparison btwn th standard formula and th intrnal modl approachs for th SCR, 5yar tim horizon. Copula paramtr δ = 3% δ =.5% δ =.5% δ = 5% θ = BE SCR θ = BE SCR O θ = BE SCR Tabl 6: Effct of inflation crisis for Parto claim amounts, 5yar tim horizon. W also provid som rsults for th sam valus for δ whn th tim horizon is qual to yars and th copula paramtr θ = in Tabl 7. Finally, w also provid in Tabl 8 a fw rsults with θ = and =.5 to s th influnc of paramtr and to illustrat th cas whr larg claims occur in avrag vry m yars, with m >. Rgarding dpndncy btwn intrclaim tims and claim amounts, both SCR and Bst Estimat ar incrasing with th dpndnc paramtr θ. This is logical as positiv dpndnc btwn intrclaim tims and claim amounts is a form of divrsification ffct. SCR ar largr for Parto claim amounts than for Exponntial claim amounts, as usual. Nvrthlss, Tabl 5 shows that th socalld intrnal modl approach lads to highr valus of SCR than th ons obtaind by th 5
17 δ = 3% δ =.5% δ =.5% δ = 5% θ = BE SCR Tabl 7: Effct of inflation crisis for Parto claim amounts, yar tim horizon. δ = 3% δ =.5% δ =.5% δ = 5% θ = BE SCR Tabl 8: Effct of inflation crisis for Parto claim amounts, yar tim horizon, =.5. standard formula for Exponntially distributd claim amounts, whil it is th opposit for Parto distributd claim amounts. Finally, th impact of inflation cannot b nglctd: in Tabl 8, th cas whr δ = 5% which corrsponds to scnarios whr th inflation rat bcoms 5% largr than th intrst rat lads to mor than a 5%incras in Bst Estimat and SCR, in th most favorabl cas whr θ =. 5 Acknowldgmnts Th rsarch was financially supportd by th Natural Scincs and Enginring Rsarch Council of Canada, th Chair d actuariat d l Univrsité Laval, and th ANR projct ANR8BLAN34. H. Cosstt and E. Marcau would lik to thank Prof. JanClaud Augros and th mmbrs of th ISFA for thir hospitality during th stays in which this papr was mad. 6
18 Rfrncs Albrchr, H. and Boxma, O. J. 24. A ruin modl with dpndnc btwn claim sizs and claim intrvals. Insuranc Math. Econom., 352: Albrchr, H. and Tugls, J. L. 26. Exponntial bhavior in th prsnc of dpndnc in risk thory. J. Appl. Probab., 43: Asimit, A. and Badscu, A. 29. Extrms on th discountd aggrgat claims in a tim dpndnt risk modl. Scand. Actuar. J. from Baumr, B. 23. On th invrsion of th convolution and Laplac transform. Trans. Amr. Math. Soc., 3553:2 22 lctronic. Biard, R., Lfèvr, C., Loisl, S., and Nagaraja, H. 29. Asymptotic finittim ruin probabilitis for a class of pathdpndnt claim amounts using Poisson spacings. Talk at 2b or not 2b Confrnc, Lausann, Switzrland. Boudrault, M., Cosstt, H., Landriault, D., and Marcau, E. 26. On a risk modl with dpndnc btwn intrclaim arrivals and claim sizs. Scand. Actuar. J., 5: Cosstt, H., Marcau, É., and Marri, F. 29. Analysis of ruin masurs for th classical compound poisson risk modl with dpndnc. To appar in Scandinavian Actuarial Journal. In prss. Dlban, F. and Hazndonck, J Classical risk thory in an conomic nvironmnt. Insuranc Math. Econom., 62:85 6. Duchsn, T. and Marri, F. 29. Gnral distributional proprtis of discountd warranty costs with risk adjustmnt undr minimal rpair. IEEE Transactions on Rliability, 58:43 5. Jang, J. 24. Martingal approach for momnts of discountd aggrgat claims. Journal of Risk and Insuranc, pags 2 2. Jang, J. 27. Jump diffusion procsss and thir applications in insuranc and financ. Insuranc Math. Econom., 4:62 7. Johnson, M. A. and Taaff, M. R Matching momnts to phas distributions: Mixturs of rlang distributions of common ordr. Stoch. Modls, 54: Kalashnikov, V. and Konstantinids, D. 2. Ruin undr intrst forc and subxponntial claims: a simpl tratmnt. Insuranc Math. Econom., 27: Kim, B. and Kim, H.S. 27. Momnts of claims in a Markovian nvironmnt. Insuranc Math. Econom., 43: Lévillé, G. and Garrido, J. 2a. Momnts of compound rnwal sums with discountd claims. Insuranc Math. Econom., 282: Lévillé, G. and Garrido, J. 2b. Rcursiv momnts of compound rnwal sums with discountd claims. Scand. Actuar. J., 2:98. 7
19 Lévillé, G., Garrido, J., and Wang, Y. 29. Momnt gnrating functions of compound rnwal sums with discountd claims. To appar in Scandinavian Actuarial Journal. Nlsn, R. B. 26. An introduction to copulas. Springr Sris in Statistics. Springr, Nw York, scond dition. Portr, K., Bck, J., Shaikhutdinov, R., Au, S., Mizukoshi, K., Miyamura, M., Ishida, H., Moroi, T., Tsukada, Y., and Masuda, M. 24. Effct of sismic risk on liftim proprty valu. Earthquak Spctra, 24: Rn, J. 28. On th Laplac Transform of th Aggrgat Discountd Claims with Markovian Arrivals. N. Am. Actuar. J., 22:98. Rolski, T., Schmidli, H., Schmidt, V., and Tugls, J Stochastic procsss for insuranc and financ. Wily Sris in Probability and Statistics. John Wily & Sons Ltd., Chichstr. Sundt, B. and Tugls, J. L Ruin stimats undr intrst forc. Insuranc Math. Econom., 6:7 22. Tang, Q. 25. Th finittim ruin probability of th compound Poisson modl with constant intrst forc. J. Appl. Probab., 423: Taylor, G. C Probability of ruin undr inflationary conditions or undr xprinc rating. Astin Bull., 2: Tijms, H Stochastic modls: an algorithmic approach. John Wily, Chistr. van Noortwijk, J. and Frangopol, D. 24. Two probabilistic lifcycl maintnanc modls for dtriorating civil infrastructurs. Probabilistic Enginring Mchanics, 94: Watrs, H Probability of ruin for a risk procss with claims cost inflation. Scand. Actuar. J., pags Willmot, G. E Th total claims distribution undr inflationary conditions. Scand. Actuar. J., : 2. Yang, H. and hang, L. 2. On th distribution of surplus immdiatly aftr ruin undr intrst forc. Insuranc Math. Econom., 292:
20 APPENDIX Th first two momnts of th distribution of t which ar givn by and 6 rspctivly. Th third momnt is µ 3 t = 3 n= i,j,k,...,i n,j n,k n A 3n I ζi ; j ; k ;...; ζi n ; j n ; k n, { whr A 3n = i, j, k,..., i n, j n, k n ; i = 3, i... i n = 3 n, i >... > i n, j = 3 n, } j... j n = 3, j... j n, k. {, }. It can b dvlopd as µ 3 t = 3 E[X 3 3 3δ 3δt 3 θ E[X 3 E[X 3 3δ 3 2 3δ 23δt 2 3δ 3 2 E[XE[X 2 2 3δ 2 δt 2δ 2 3δt 6δ 2 3 θ 2 E[X 2 E[X 2δt 3δt E[X 2 3δ 2 δ 2 δ 2 2δ 3δ 2 2δ 3 θ 2 E[X E[X 2 E[X 2 δt 23δt 2 δ 2 3δ δ 2 2δ 2 3δ 2 2δ 3 θ 2 2 E[X E[X E[X 2 E[X δ 2 3δ 2δt 2δ 2 δ 23δt 2δ 2 3δ E[XE[X 2 2 6δ 2 2δt 2δ 2 3δt 3δ θ 2 E[X E[X 2 E[X 2 22δt 3δt 2 3δ 2 2δ 2 2δ 2 δ 3δ 2 δ 3 2 θ 2 E[X 2 E[X E[X 2 2δ 2 3δ 2δt 23δt 2δ 2 δ 2 δ 2 3δ 3 2 θ 2 2 E[X E[X E[X 2 E[X δ 2 3δ 22δt δ 2 2δ 23δt δ 2 3δ E[X 3 6δ 3 δt 2δ 3 2δt 2δ 3 3δt 6δ θ 3 E[X 2 E[X 2δt 2δt 3δt E[X 3δ 2 2 δ 2 δ 2 δ 2 2δ 2δ 2 2 δ 3δ 2 2 2δ 3 2 θ 3 E[X 2 E[X δt 22δt 3δt E[X 3δ 2 2 2δ 2δ 2 2 δ 2 2δ 2 δ 2 δ 6δ 2 2 δ 3 2 θ 3 E[X 2 E[X δt 2δt 23δt E[X 2δ 2 2 3δ δ 2 2 2δ 2δ 2 2 δ 2 3δ 2 2δ 2 δ E[X 2 E[X 3 2 θ 2 3 E[X E[X 2 E[X 3δ 2 δ 2 2δ 2δt δ 2 δ 2 2δ 22δt δ 2 2δ 2 δ 3δt 3δ 2 2δ 2 δ 23δt 3 2 θ 2 3 2δt 2δt E[X 2δ 2 δ 2 3δ 2δ 2 δ 2 δ 2δ 2 δ 2 δ 2δ 2 3δ 2 δ 3 2 θ 2 3 E[X E[X δt 22δt 23δt 2 E[X δ 2 2δ 2 3δ δ 2 δ 2 2δ δ 2 2δ 2 δ δ 2 3δ 2 2δ 3 2 θ 3 3 E[X 3 E[X 2 δ 2 2δ 2 3δ 2δt 2δ 2 2 δ 22δt δ 2 2 2δ 23δt 2δ 2 2 3δ 9
21 Figur : Figur 2: 2
22 Figur 3: 2
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