Stable solutions for optimal reinsurance problems involving risk measures

Transcription

1 Ivative Applicatis f O.R. Stable slutis fr ptimal reisurace prblems ivlvig risk measures Alejadr Balbás, Beatriz Balbás, Ati Heras Uiversity Carls III f Madrid, CL. Madrid, 6, 93 Getafe, Madrid, Spai Uiversity f Castilla la Macha, Avda. Real Fábrica de Seda, s/, 456 Talavera, Tled, Spai Uiversity Cmplutese f Madrid, Smsaguas-Campus, 3 zuel de Alarcó Madrid, Spai article if abstract Article histry: Received 3 May Accepted 9 May Available lie 7 May Keywrds: Optimal reisurace Risk measure Sesitivity Stable ptimal reteti Stp-lss reisurace The ptimal reisurace prblem is a classic tpic i actuarial mathematics. Recet appraches csider a cheret r expectati buded risk measure ad miimize the glbal risk f the cedig cmpay uder adequate cstraits. Hwever, there is csesus abut the risk measure that the isurer must use, sice every risk measure presets advatages ad shrtcmigs whe cmpared with thers. This paper deals with a discrete prbability space ad aalyzes the stability f the ptimal reisurace with respect t the risk measure that the isurer uses. We will demstrate that there is a stable ptimal reteti that will shw sesitivity, isfar as it will slve the ptimal reisurace prblem fr may risk measures, thus prvidig a very rbust reisurace pla. This stable ptimal reteti is a stp lss ctract, ad it is easy t cmpute i practice. A fast liear time algrithm will be give ad a umerical example preseted.. Itrducti Sice the paper by Artzer et al. (999) itrduced the cheret measures f risk may authrs have further exteded the discus si, which shws the imprtace that this tpic is achievig i fiace ad isurace. Amg thers, Gvaerts et al. (4) itr duced the csistet risk measures, als studied i Burgert ad Rüschedrf (6), Frittelli ad Scadl (5) aalyzed risk measures fr stchastic prcesses, ad Rckafellar et al. (6) de fied the deviatis ad the expectati buded risk measures. Classical actuarial ad fiacial prblems have bee the revis ited usig risk measures beyd the variace. Amg thers, Naka (4) ad Balbás et al. () drew risk measures whe pricig i icmplete markets, Masii et al. (7) ad Schied (7) dealt with prtfli chice prblems, ad Aaert et al. (9) checked the efficiecy f the classical prtfli isurace prblem if the risk level is give by the value at risk (VaR) r the cditial value at risk (CVaR). The ptimal reisurace prblem is a mai issue i actuarial sciece. A cmm apprach attempts t miimize sme measure f the first isurer risk after reisurace. Semial papers by Brch (96) ad Arrw (963) used the variace as a risk measure ad prved that the stp lss reisurace miimizes the retaied risk if Crrespdig authr at: Uiversity Carls III f Madrid, CL. Madrid, 6, 93 Getafe, Madrid, Spai. addresses: alejadr.balbas@uc3m.es (A. Balbás), beatriz.balbas@uclm.es (B. Balbás), aheras@ccee.ucm.es (A. Heras). premiums are calculated fllwig the expected value premium priciple. The subsequet research fllwed similar ideas ad tried t take it accut mre geeral risk measures ad premium prici ples, which may give ptimal ctracts ther tha stp lss. Recetly, Gajec ad Zagrdy (4) csidered mre geeral symmetric ad eve asymmetric risk fuctis such as the abs lute deviati ad the trucated variace f the retaied lss, u der the stadard deviati premium priciple. Kaluszka (5) studied reisurace ctracts with may cvex premium prici ples (expetial, semi deviati ad semi variace, Dutch, dis trti, etc.). Other well kw fiacial risk measures such as the VaR r the tail value at risk (TVaR) are als beig csidered. Fr example, Kaluszka (5) uses the TVaR as a premium priciple ad Cai ad Ta (7) calculate the ptimal reteti fr a stp lss reisurace by csiderig the VaR ad the cditial tail expectati risk measures (CTE), uder the expected value pre mium priciple. The mst recet papers have fially icrprated cheret ad/ r expectati buded risk measures i the bjective fucti t be miimized by the cedig cmpay. Alg with the paper f Cai ad Ta (7) abve, ther iterestig examples are Cai et al. (), Balbás et al. (9) r Berard ad Tia (9). The differ eces amg their appraches are caused by the isurer behavir. Very cmplete ifrmati may be fud i the survey f Cete ad Simes (9). Despite the iterest f the prblem, as far as we kw there are aalyses fcusig the stability f the ptimal reisurace. This shuld be a imprtat tpic sice the ptimality f may

2 reisurace plas will critically deped the risk measure ad the pricig priciple. There is csesus abut the risk measure that the isurer must use, sice every risk measure presets advatages ad shrtcmigs whe cmpared with thers. This paper csiders that the reisurer s premium priciple is give by a cvex fucti ad deals with the ptimal reisurace prblem if risk is measured by cheret ad expectati buded risk measures. The fcus is the stability i the large f the pti mal reteti pla with respect t the chse risk measure. Stabil ity i the large is used i the sese f Samuels (947), i.e., we will aalyze whether the ptimal ctract remais cstat as the risk measure becmes mre ad mre risk adverse (the risk measure icreases). The paper s utlie is as fllws. Secti will preset the basic cditis ad prperties f the risk measure q t be used. Secti 3 prvides ur geeral ptimal reisurace prblem. We will pres et the prblem i a discrete prbability space. Actually, this sim plifies the mathematical expsiti, ad every prbability space admits a discrete apprximati which achieves as much accuracy as eeded. May actuarial ad fiacial aalyses deal with discrete prbability spaces (see Beati, 3 K et al., 5 Masii et al., 7, r Miller ad Ruszczyski,, amg may thers), sice this is t a restricti i practice. The prpsed ptimal reisurace prblem seems t be quite flexible ad geeral, sice it allws us t icrprate may particular situatis such as bud get cstraits, the maximizati f the isurer expected wealth, etc. The mst imprtat results i Secti 3 are Therem ad Crllary 4, sice they characterize the ptimal reteti by meas f Karush Kuh Tucker (KKT) like cditis ad permit us t itrduce the stable ptimal reteti, which will slve the prblem fr all f the risk measures with a subgradiet satisfyig adequate prperties. Therefre, the stable ptimal reteti may be uderstd as a rbust ptimal reisurace pla. Secti 4 is devted t cmputig i practice the stable ptimal reteti. Here we will assume that the reisurer uses a liear value priciple, ctaiig the expected value premium priciple as a particular case. Of curse it is t ecessary, sice practical ptimal ity cditis have bee give i a much mre geeral framewrk, but the specific sluti f the ptimizati prblem depeds the premium priciple we take, ad csiderig mre tha e wuld sigificatly elarge the paper. As already idicated, previ us literature measurig the isurer risk by a geeral risk measure is still limited, s it seems t be atural ad f iterest t aalyze ccrete prblems by takig the mst used premium priciple. The mst imprtat result f this secti is Therem, because it gives explicit expressis fr the stable ptimal reteti ad the KKT multipliers f the prblem. Accrdig t Therem, the stable ptimal reteti is a stp lss reisurace. Therem is used i Secti 5 s as t itrduce a fast algrithm that gives the stable ptimal reteti i umerical applicatis. The algrithm is t time csumig sice there is a liear relati ship betwee its cmputatial cmplexity ad the cmplexity f the prtfli f isurace plicies. A illustrative umerical exam ple is als prvided, which clarifies hw t use the algrithm i practice ad shws the rbustess f the give reisurace, i the sese that mst f the usual risk measures lead t this sluti. The last secti f the paper summarizes the mst imprtat cclusis.. relimiaries ad tatis As usual, csider the prbability space ðx F Þ cmpsed f the set f states f the wrld X, the r algebra F ad the prba bility measure. As said abve, we will be dealig with a discrete framewrk, s X will be cmpsed f a fiite umber f elemets, X fx x... x g: ðþ We will csider the prbability f every sigle evet p i ðx i Þ > i =,,...,. Dete by EðyÞ the mathematical expectati f every radm variable y, ad dete by L the Hilbert space f R valued radm variables y X edwed with the rm = kyk Eðjyj Þ fr every y L. 3 Let [,T] be a time iterval. Frm a ituitive pit f view, e ca iterpret that every y L may represet the wealth at T f a arbitrary isurer. Let q : L! R be the geeral risk fucti that a isurer uses i rder t ctrl the risk level f his fial wealth at T. Dete by D q z L EðyzÞ 6 qðyþ y L : ðþ We will assume that D q is cvex ad cmpact, ad qðyþ Max EðyzÞ : z D q ð3þ hlds fr every y L. Furthermre, we will als suppse that the cstat radm variable z = is i D q ad D q z L EðzÞ : ð4þ Summarizig, we have: Assumpti. The set D q give by () is cvex ad cmpact, (3) hlds fr every y L, z = is i D q, ad (4) hlds. h The assumpti abve is clsely related t the represetati therem f risk measures stated i Rckafellar et al. (6). Fl lwig their ideas, it is easy t prve that the fulfillmet f Assumpti hlds if ad ly if q satisfies: (a) qðy þ kþ qðyþ k ð5þ fr every y L ad k R. (b) qðayþ aqðyþ ð6þ fr every y L ad a >. (c) qðy þ y Þ 6 qðy Þþqðy Þ fr every y, y L. (d) qðyþ EðyÞ ðþ fr every y L. 4 ð7þ Isurace premiums are usually give by cvex fuctis, see, fr istace, Deprez ad Gerber (95). Actually, the give KKT like cditis are t exactly the same as the stadard KKT cditis f the prblem, ad that is the reas why we say KKT like. Nevertheless, they are ecessary ad sufficiet ptimality cditis, ad are geerated by the KKT cditis f a equivalet ptimizati prblem preseted i Balbás et al. (9). Further details may be fud i that paper. 3 Actually, X beig discrete ad ctaiig N elemets the dimesi f L is fiite ad equals. Thus, L = L p fr every p [,] ad the rm kk abve is equivalet t the rm kk p. Thugh we have chse p =, every p [,] may play the same rle. 4 Actually, the prperties abve are almst similar t thse used by Rckafellar et al. (6) i rder t itrduce their expectati buded risk measures.

3 It is easy t see that if q satisfies prperties (a) (d) the it is als cheret i the sese f Artzer et al. (999) if ad ly if D q L þ z L ðz Þ : ð9þ articular iterestig examples are the cditial value at risk (CVaR) f Rckafellar et al. (6), the weighted cditial value at risk (WCVaR) f Chery (6), the dual pwer trasfrm (DT) f Wag () ad the Wag measure (Wag, ), amg may thers. Furthermre, fllwig the rigial idea f Rckafellar et al. (6) t idetify their expectati buded risk measures ad their deviati measures, it is easy t see that qðyþ rðyþ EðyÞ ðþ satisfies (a) (d) if r : L! R is a deviati, that is, if r satisfies (b), (c), (e) rðy þ kþ rðyþ fr every y L ad k R, ad (f) rðyþ fr every y L. Amg may thers, a particular example is the classical p deviati fr every p [,), give by r p ðyþ E jeðyþ yj p =p r the dwside p semi deviati, give by r p ðyþ E jmaxfeðyþ y gj p =p : The classical separati therems allw us t prve that there is a e t e mappig q M D q betwee the risk measures satisfyig Assumpti that are cheret ad the set f cvex ad cmpact subsets f L such that z = is i D q, ad (4) ad (9) hld. Further mre (3) shws that this mappig is icreasig, i.e., q (y) 6 q (y) hlds fr every y L if ad ly if D q D q hlds. Accrdigly, the maximum cheret risk measure satisfyig Assumpti is that C assciated with the set D C z L þ : EðzÞ : ðþ It is easy t see that the risk measure C is CðyÞ Mifyðx i Þ : i... g ðþ fr every y L. Similarly, y! E (y) is the miimum risk measure satisfyig the cditis abve, sice D E fg. Thus CðyÞ qðyþ EðyÞ ð3þ hlds fr every y L ad every cheret q satisfyig Assumpti. Fially, ce agai the separati therems allw us t prve that every cvex cmbiati q Xm i w i q i f risk measures satisfyig (5) (9) als satisfies (5) (9), ad D q hlds. X m i w i D qi 3. Optimal reisurace: Geeral prblem ad ptimality cditis ð4þ Csider that the isurace cmpay receives the fixed amut S (premium) ad will have t pay the radm variable y L þ withi a give perid [, T] (claims). Withut lss f geerality we will assume that ðy > Þ, sice the absece f claims is a urealistic situati i practice. Suppse that a reisurace ctract is siged i such a way that the cmpay will ly pay y L, whereas the reisurer will pay y y. If the reisurer premium priciple is give by the cvex ad icreasig fucti, p : L! R such that p() =, ad S > is the largest amut that the isurer wuld like t pay fr the ctract, the the isurace cmpay will chse y (ptimal reteti) s as t slve the bi criteria ptimiza ti prblem Mi ðs y pðy yþþ >< Max EðS y pðy yþþ ð5þ pðy yþ 6 S 6 y 6 y beig a cheret risk measure that satisfies Assumpti. C ditis p() = ad S > imply that y = y satisfies the cstrait, s (5) is always feasible (Therem belw will shw that it is als buded ad slvable). Ntice that, if desired, cstrait p(y y) 6 S may be remved withut mdifyig (5), sice p is icreasig ad therefre it is sufficiet t chse S > p(y ). First f all let as see that the multibjective ptimizati prb lem (5) is cvex. Lemma. With the tatis f (5) we have that the three fuctis L 3 y! ðs y pðy yþþ R L 3 y! EðS y pðy yþþ R ad L 3 y! pðy are cvex. yþr ð6þ rf. Let us prve that (6) is cvex sice the remaiig cases are aalgus. Thus, suppse that y, y L ad 6 k 6. ðs ðky þð kþy Þ pðy ðky þð kþy ÞÞÞ ðkðs y Þþð kþðs y Þ pðkðy y Þþð kþðy y ÞÞÞ: Sice p is cvex we have that pðkðy y Þþð kþðy y ÞÞ kpðy y Þ ð kþpðy y Þ: is decreasig because it is cheret. 5 Hece ðs ðky S y pðy yþy Þ pðy ðky þð kþy ÞÞÞ 6 ðkðs y Þþð kþðs y Þ kpðy y Þ ð kþpðy y ÞÞ: Fially, sice is cvex, ðs ðky þð kþy Þ pðy ðky þð kþy ÞÞÞ 6 k ðs y pðy y ÞÞþð kþ ðs y pðy y ÞÞ: Sice the multibjective ptimizati prblem (5) is cvex, it may be slved by scalarizati methds, i.e., i rder t btai ar et slutis e ca miimize a cvex cmbiati f ad E. Accrdigly, take w ad w egative ad such that w + w =, let q w w E, ad slve 5 See Artzer et al. (999), r verify that is decreasig frm (3) ad (9). 3

4 >< MiqðS y pðy yþþ pðy yþ 6 S 6 y 6 y : ð7þ Bearig i mid the ideas f the previus secti, q satisfies Assumpti ad is cheret, sice it is a cvex cmbiati f ad E. It is wrth remarkig that the first (secd) bjective f (5) may be remved ad the prblem still fits i (7), because e ca take w = ad w =(w = ad w = ). Next we will give ecessary ad sufficiet Karush Kuh Tucker ptimality cditis. Therem. rblem (7) is buded ad slvable. Mrever, the existece f ðs z ÞRL satisfyig the fllwig Karush Kuh Tucker like cditis is ecessary ad sufficiet t guaratee the ptimality f y L. Eðy zþ 6 Eðy z Þ z D q >< s ðpðy y Þ S Þ pðy y Þ S 6 Eðy z Þþðþs Þpðy y Þ 6 Eðyz Þþðþs Þpðy yþ 6 y 6 y s R s 6 y 6 y z D q ðþ (s,z ) will be called KKT multiplier f (7). rf. The dimesi f L is fiite due t the assumptis (X is discrete ad fiite, see () ad Ftte 3). Thus, the fiite dime si f L ad the cvexity f p : L! R guaratees the ctiuity f p (Lueberger, 969). Similarly (6) ad (7) shw that q is cvex ad therefre ctiuus. Besides, the last cstrait f (7) shws that the feasible set is buded, ad therefre cmpact. Hece, the Weierstrass therem shws that (7) is buded ad slvable. Fially, we will t prve the Karush Kuh Tucker like cditis because a aalgus prf may be fud i Balbás et al. (9). h A first imprtat csequece is that e ca give cditis esurig that the sluti f (7) remais the same if q is replaced by a lwer e. 6 Hece we ca give the first result guarateeig the stability f the ptimal isurace (reteti) with respect t the risk measure. Crllary 3. Suppse that y L slves (7) ad (s,z ) is a KKT multiplier. Take the cheret risk measure q satisfyig Assumpti ad such that q 6 q. Ifz D ~q ad q replaces q the y L still slves (7) ad (s,z ) is still a KKT multiplier. rf. O the e had, y ad (s,z ) satisfy (). O the ther had, accrdig t that prperties give i the previus secti, D q D q because q 6 q. Thus, z D q implies that () still hlds if D q replaces D q. h Crllary 4. Suppse that y C L slves (7) ad ðs C z CÞ is a KKT multiplier fr the risk measure C f (). The y C still slves (7) ad ðs C z C Þ is still a KKT multiplier fr every q with z C D q. rf. It trivially fllws frm the previus crllary ad (3). 6 With the tatis f (5), tice that q decreases if s des, i.e., h Remark. With the tatis f Crllary 4,ifz C R D q e still ca lk fr a risk measure q q quite similar t q ad such that z C D q, ad therefre y C still slves (7) ad s C z C is still a KKT multiplier if e csiders q. Ideed, it is sufficiet t take the fllwig cvex ad cmpact set, 7 D ~q C D q [fz C g bviusly assciated with the risk measure ~qðyþ Max qðyþ Eðyz C Þ ð9þ fr every y L. Fr this reas hereafter the sluti y C L f (7) fr the risk measure C f () will be called stable ptimal reteti. h Remark. If the cedig cmpay is als iterested i maximizig the expected wealth ad deals with prblem (5), the C may be replaced by w C w E (with w i, i =,, ad w + w = ). Ideed, i such a case, () ad (4) shw that D w C w E z L EðzÞ ad z w : ðþ Obviusly, Crllary 3 prves that if y w C L slves (5) ad s w C z w C is a KKT multiplier fr the risk measure w C w E abve, the y w C still slves (5) ad s w C z w C is still a KKT mul tiplier fr every qsuch that z w C D w q w E. Furthermre, a ew cmmet similar t Remark applies. h 4. Characterizig ad cmputig the stable ptimal reteti Let us give prperties makig it easier t verify the fulfillmet f the iequalities f (). T this purpse, ad takig it accut Crllary 4, Remark ad the first cditi i (), let us give a istrumetal lemma. Lemma 5. Suppse that 6 y 6 y ad z D wcw E (see ()). Eðy zþ 6 Eðy z Þ hlds fr every z D wcw E if ad ly if y ðx j Þ Maxfy ðx i Þ : i... g hlds fr every j =,,..., such that z (x j ) > w. rf. The iequality abve hlds if ad ly if z slves the liear ptimizati prblem Max y ðx i Þzðx i Þp i >< i zðx i Þp i i w 6 zðx i Þ i... : Accrdig t the classical Karush Kuh Tucker cditis, this is equivalet t the existece f l l... l R such that y ðx i Þp i þ l p i l i i... >< z ðx i Þp i i ðz ðx i Þ w Þl i i... l i i... z ðx i Þ w i... : Hece, the result trivially fllws if e takes l Maxfy ðx i Þ : i... g ~ ) w w E w ~ w E: 7 As usual, C(A) detes the cvex hull f every set A L. 4

5 ad pi l i l y ðx i Þ i =,,...,. h Despite the fact that previus aalyses are quite geeral, the slutis f () will deped the specific assumptis e im pses. Hecefrth we will assume that the reisurer uses a liear premium priciple. Actually, as idicated i the itrducti, pre vius literature csiderig a geeral risk measure is scat, s it seems t be atural ad f iterest t aalyze ccrete prblems by takig the mst used premium priciple, which is the expected value premium priciple, i.e., there exists k > such that pðyþ keðyþ ðþ fr every y L. We will impse smethig strictly weaker, such as the existece f z p L such that ðz p > Þ ðþ Eðz p Þ > ð3þ ad pðyþ Eðyz p Þ ð4þ fr every y L. Assumpti. Hecefrth we will assume the existece f z p L such that () (4) hld. h Nevertheless, it is wrth pitig ut that the previus develp mets are mre geeral, ad therefre they als apply t altera tive premium priciples. Frm Assumpti the ecessary ad sufficiet ptimality c ditis () becme Eðy zþ 6 Eðy z Þ z D q >< s ðeððy y Þz p Þ S Þ Eððy y Þz p Þ S 6 Eðy ðz ð þ s Þz p ÞÞ 6 Eðyz ð ð þ s Þz p ÞÞ 6 y 6 y s R s 6 y 6 y z D q : ð5þ Next let us preset tw simple lemmas. The first e simplifies the furth cditi f (5). Lemma 6. Let z L,y L with 6 y 6 y, ad s R. The, Eðy ðz ð þ s Þz p ÞÞ 6 Eðyz ð ð þ s Þz p ÞÞ hlds fr every y L with 6 y 6 y if ad ly if there exists a partiti X X [ X [ X 3 such that >< z ðxþ > ð þ s Þz p y ðxþ if x X z ðxþ ð þ s Þz p if x X z ðxþ < ð þ s Þz p y ðxþ y ðxþ if x X 3 : rf. It is bvius if we realize that the sluti f MiEðyz ð ð þ s Þz p ÞÞ 6 y 6 y ð6þ must be as large as pssible (i.e., must equal y ) wheever z ( + s )z p < ad as small as pssible (i.e., zer) if z ( + s )z p >, whereas its value is t relevat at all if z ( + s )z p =. h Lemma 7. y = des t slve (7). rf. If y = slved (7) the (6) wuld lead t z ( + s )z p. Bearig i mid (4) ad (3), ad takig expectatis, e has the ctradicti ð þ s ÞEðz p Þ >. h As already said the stp lss reisurace is fte btaied as the ptimal reteti (Balbás et al., 9). Recall that y L ad ly ig betwee ad y is said t be a stp lss reisurace if there exists a such that y y 6 a y ð7þ a y > a: Hereafter the radm variable f (7) will be deted by y a. Crllary 4 ad Remark shw the imprtace f slvig (7) whe q w C w E, sice the sluti will geerate a very stable ptimal reisurace ctract. Therem. Csider prblem (7) with the risk measure w C w E. Suppse that ðz p > w Þ. 9 ad (a) There exists a > such that y a slves (7). (b) Suppse that y a slves (7), ðy a Þ ad s w C z w C is a KKT multiplier f (7). The ( z w C w if y < a ðþ ð þ s w C Þz p if y > a : (c) Suppse that y a slves (7), there is a uique x i X with y ðx i Þ a ad s w C z w C is a KKT multiplier f (7). The w y <a >< z w C ðxþ y ðxþ>a ðþs w C ÞzpðxÞ w y ðxþ<a ðþs w C Þ p i x x i ðþs w C Þz pðxþ y >a y ðxþ>a ð þ s w C Þz pðx i Þ w y ðxþ<a ð þ s w C Þ p i 6 ð þ s w C Þz pðx i Þ ð9þ ð3þ hld. (d) Suppse that y a slves (7) ad s w C z w C is a KKT multi plier f (7). Suppse that q is cheret ad satisfies Assumpti. If z w C D w q w E the y a slves (7) fr w q w E. rf (a) Take the sluti y f (7) whse existece is guarateed by Therem, ad defie a Maxfy ðx i Þ : i... g: Ntice that () is a particular case f (4) that arises if z p remais cstat ad equals k. 9 () implies the fulfillmet f this prperty wheever w =. Sice w 6, the prperty hlds if the reisurer draws the expected value premium priciple, sice the z p = k >. 5

6 Lemma 7 implies that a >. Let us see that y y a. Ideed, y beig (7) feasible we have that y 6 y, sa 6 y (x) wheever y (x)=a. Besides, if y (x)<a ad s w C z w C is a KKT multiplier (its existece fllws frm Therem ), the the first cditi i (5) ad Lemma 5 lead t z w C ðxþ w. Hece, the furth cditi i (5), Expressi (6) ad z p > w lead t y (x)=y (x), ad therefre y y a. (b) If y a ðxþ < a (r y (x)<a ) the the equality z w C ðxþ w may be prved with the same argumets. Suppse that y a ðxþ a. The, csider the partiti f Lemma 6 ad bviusly x X r x X 3, sice y a ðxþ. But x X 3 wuld imply y (x)=a, which cat hld. (c) As i the prf f (b), z w C ðxþ w wheever y a ðxþ < a. Suppse that y a ðxþ a. The, csider the partiti f Lemma 6 ad bviusly x X r x X 3, sice y a ðxþ. But x X 3 implies that y (x)=a, ad therefre x x i. Thus, takig it accut (4), we have (9). Fially, (3) cmes frm (6), because y a implies that z w C 6 þ s w C z p. (d) It trivially fllws frm Crllary 4 ad Remark. h Remark 3. Accrdig t the previus therem the stable ptimal reteti f Remark is a stp lss reisurace y a. Therem als prvides the multiplier z w C (see () r (9)), s the cditi z w C D w q w E is very easy t verify i practical examples. Actu ally, we will see i the ext secti that the assumptis f state mets b ad c are usually fulfilled i practice. h Remark 4. Rckafellar et al. (6) itrduced the risk measure CVaR l l ð Þ beig the level f cfidece. CVaR l is becm ig very imprtat ad ppular amg practitiers ad research ers fr its iterestig prperties. Ideed, it is cheret ad expectati buded (Rckafellar et al., 6), ad cmpatible with the secd rder stchastic dmiace ad the classical util ity fuctis (Ogryczak ad Ruszczyski, ). Rckafellar et al. (6) prved that D CVaRl z L 6 z 6 EðzÞ : ð3þ l Csider w = (the expected wealth is t ptimized by the cedig cmpay). Thus, if q CVaR l i prblem (7), the y a will slve the prblem (i.e., (3) will ctai the radm variable z C ) as lg as z l C ð3þ which clearly hlds fr l clse eugh t %. Aalgusly, if the isurace cmpay deals with prblem (5) ad CVaR l, the the sluti y a f (5) fr w C w E will be still the sluti fr the w CVaR l w E as lg as w þ w l z w C which is als bvius fr w > ad l large eugh. A illustrative umerical example will be give i Secti 5. h 5. Algrithm ad umerical experimet Next let us pit ut that the cditis f Therem usually hld i practice, ad the stable ptimal reteti y a ad the KKT multiplier s C z C may be easily calculated by drawig a apprpriate algrithm. The algrithm just tests the fulfillmet f Therem. First f all we will itrduce the algrithm ad the we will preset a umerical example. I rder t simplify the expsiti, i this secti we will assume that w = (the expected wealth is t maximized, ad ly the risk level is miimized), thugh the extesi fr w > is straightfrward. Ntice that, accrdig t Therem, y a ad s C z C will be kw ce we cmpute a ad s C, i.e., we ly have t estimate tw real umbers. I rder t itrduce the algrithm we will assume that X fx x... x g R ð33þ < x < x < < x ð34þ ad y is the idetity map, s ðy x i Þ p i ð35þ i =,,...,. Actually, this is a particular framewrk strictly mre restricted tha that i Therem, but this is a stadard simplifica ti i the literature abut the ptimal reisurace prblem. See, fr istace, Gajec ad Zagrdy, 4 Kaluszka, 5 Cai et al.,, ad may thers, where the authrs d t deal with the ri gial prbability space ðx F Þ, but with its image ðr B Þ by y, cmpsed f the real lie R, the Brel r algebra B f R, ad the prbability measure give by ðbþ ðx X y ðxþ BÞ fr every Brel subset B B. I such a particular case y is replaced by the idetity map. Besides, i practical situatis isurers usually deal with ðr B Þ ad the idetity map t, which meas that they d t distiguish differet evets leadig t the same cst f claims. Fially, thugh the ew settig (33) is much mre re stricted tha the rigial e, the simplificati des t mdify the cmputati f the sluti f (7). Ideed, Therem guara tees that we are lkig fr a stp lss reisurace, ad there bvi usly exists a e t e mappig betwee the stp lss ctracts f the iitial prbability space ðx F Þ ad the stp lss ctracts f its image ðr B Þ. Hece, assume (33) (35) ad defie a Max x : Obviusly, y a Max y is (7) feasible because S > ad p() =. Due t (), the premium priciple f (4) geerates a strictly icreasig fucti p. Csequetly, pðy y a Þ strictly decreases as a grws. Csider a first case (Case_) such that p(y ) 6 S, which implies that y a is (7) feasible fr every a ad therefre we will csider a Mi : If p(y )>S the the ctiuity f a! pðy tece f a uique a Mi (,a Max ) such that pðy y a Mi Þ S : y a Þ implies the exis Let us distiguish tw situatis. Case_ arises if a Mi R X, i which case we will chse i as the smallest subscript such that pðy x i Þ < S : Case_3 hlds if a Mi x i X fr sme i. Obviusly, fr the three cases y a is (7) feasible if ad ly if a Mi 6 a 6 a Max : Recall that the stadard deviati is t cmpatible with the secd rder stchastic dmiace if asymmetries are ivlved (Ogryczak ad Ruszczyski, 999), ad the stp-lss reisurace bviusly geerates asymmetric results. i.e., p (y )<p(y ) wheever y 6 y ad y y. Actually, Cstrait p(y y) 6 S is redudat i this case, ad may be remved i (7). 6

7 Algrithm. Suppse that Case_ hlds. Lemma 6 implies that y a Mi des t slve (7), s the stable ptimal reteti y a satisfies p y y a < S ad the secd cditi i () leads t s C. Hece, we ly have t estimate a. Step. Defie x a a x x a þ x 3 a 4 x... a x þ x a x : Step. Fr j =t check whether y a j ad if x < a z j j z p if x a j satisfy (5) ad (6). If these cditis are satisfied fr sme y a j the we will have the stable ptimal reteti ad the KKT multi plier. The algrithm ca stp sice the stable ptimal reteti has bee fud. Ntice that tw differet values f j cat satisfy (5) ad (6), sice () implies that Eðz j Þ strictly decreases with j ad there fre (4) cat hld tw times. Furthermre, if these cditis hld fr sme j the every a (a j,a j ) will geerate a stp lss stable ptimal reteti y a, sice the same KKT multipliers z j ad s j will still apply. Step 3. Suppse that Step did t lead t the stable ptimal reteti. Fr j =t check whether y a j ad x < a j >< z j ðxþ zpðxþ x>a j p j x a j ð36þ z p ðxþ x > a j satisfy (5) ad (6). Every time these cditis are satisfied we will have a sluti f (7) fr q = C. Ntice that (6) will imply x>a j z p ða j Þ 6 z p ða j Þ: ð37þ p j Algrithm. Suppse that Case_ hlds. The prceed as i Alg rithm with mir mdificatis i Steps 3. Nw, i Step we must defie a i a Mi þ x i a i x i a i þ x i þ x i þ... x a þ x a x : Obviusly, Steps ad 3 will start with j = i rather tha j =. Step 4. If Steps ad 3 did t lead t the stable ptimal rete ti the we must address Step 4 s as t check the pti mality f y a Mi. I this case s C > may hld ad we are i the cditis f Therem b. We must verify whether y a Mi s C ad ( if y z < a Mi C ð3þ þ s C zp if y > a Mi satisfy (5) ad (6) fr sme s C. Actually the ly cditi e must check is (4), i.e., X þ s C z p ðx i Þ x i >a Mi s the ptimality f y a Mi hlds if ad ly if s C x i >a Mi z p ðx i Þ : ð39þ Thus, Step 4 reduces t the verificati f the iequality i (39). If this iequality hlds the the equality i (39) prvides us with s C (3) prvides us with z C, ad ya Mi is the stable ptimal reteti. h Algrithm 3. Suppse that Case_3 hlds. The prceed as i Alg rithm with a mir mdificati i Step. Nw we must mdify a i accrdig t x a i þ x i x i a i x i a i þ x i þ i þ... a x þ x a x : Oce agai, Step ad Step 3 will start with j = i. Step 5. We still have t check the ptimality f y a Mi y x i. This reteti level is ptimal if ad ly if we ca fid s C such that y x i s C ad z C ðxþ x < x i >< þs x>x C i ð ÞzpðxÞ p i x x i þ s C zp ðxþ x > x i ð4þ satisfy (5) ad (6). The existece f s C is easy t verify, because, bearig i mid the fidigs f Sectis 3 ad 4, e ly eeds t check the cditis X 6 þ s C zp ðxþ ð4þ ad x>x i x>x þ s i C zp x i 6 þ s C zp x i X x>x i p i X þ s C zp ðxþþ þ s C x>x i ð4þ z p ðxþðxþ : ð43þ Equality (43) yields s C, ad the the iequalities (4) ad (4) are equivalet t s C 6 x>x z i pðxþ ad ð44þ s C ð45þ z p x i pi þ respectively. Thus, Step 4 reduces t the cmputati f s C by meas f (43) ad the the verificati f the iequalities s C (44) ad (45). If the three iequalities hld the (4) prvides us with the multiplier ad y a Mi is the stable ptimal reteti. h Remark 5. Ntice that the existece f sluti f (7) ad the fidigs f Sectis 3 ad 4 shw that at least e f the three algrithms must geerate a stable ptimal reteti. h Remark 6. As said abve, tice that the algrithm just tests the fulfillmet f Therem, ad csequetly it is t very time c sumig. Actually, it is a liear time algrithm, i the sese that there is a liear relatiship betwee its cmputatial cmplexity ad the umber f realizatis f the glbal cst y. h Next let us preset a simple umerical example. Our ly bjective is t illustrate hw t use the algrithm i practical situatis. 7

8 Example. Suppse that y ca reach the values,, 3, 4 ad 5 with a similar prbability.. Suppse that the reisurer uses the expected value premium priciple with a price % higher tha the expected claims, i.e., pðyþ :EðyÞ: Suppse fially that the cedig cmpay des t impse ay bud get cstrait, i.e., we are i Case_ abve. With the tatis f Algrithm, defie a 5 a a 3 5 a 4... a 9 45 a 5: I Step we have t check the ptimality f five stp lss c tracts. The first e is y 5. Csider if x < 5 z : if x 5: Obviusly, z remais cstat ad equals., s it is t i the set D C f (). The, y 5 is t a stable ptimal reteti. If e repeats the aalysis with the fur remaiig cadidates the similar re sults apply, s Step des t geerate ay stable ptimal reteti. I Step 3 we have t check the ptimality f the remaiig five stp lss ctracts. The first e is y, ad (36) gives >< if x < z : if x : if x > which d t belg t D C. Repeat the exercise with the remaiig values f a, ad fr a = we get >< if x < z 4 :4 if x : if x > which implies that y is t a stable ptimal reteti either. Aalgusly, fr a = 3 we get >< if x < 3 z 6 :4 if x 3 : if x > 3 ad y 3 is the stable ptimal reteti we are lkig fr. It is easy t check that y 4 ad y 5 are t stable ptimal retetis. I fact, fr y 4 e btais >< if x < 4 z 3: if x 4 : if x > 4 ad this multiplier is t feasible because (37) des t hld. A aalgus caveat arises fr y 5. Reisurace y 3 will be the ptimal reteti fr may risk measures. Fr istace, if e csiders q CVaR l, accrdig t (3) y 3 slves the prblem if l : which hlds fr l.45 (r l 45%), ad, i particular, fr the usual values f this parameter i the idustry, which are higher tha 9%. Fially, it is wrthwhile t pit ut that the rle f the CVaR l may be als played by may ther imprtat risk measures i actuarial scieces, such as, WCVaR, DT, Wag, etc. h 6. Cclusis The ptimal reisurace prblem is a classic tpic i actuarial thery. Sice cheret ad expectati buded risk measures are becmig very imprtat i Fiace ad Isurace, recet ap praches deal with them s as t address the ptimal reisurace prblem. Hwever, there is csesus abut the risk measure that e must use, sice every risk measure presets advatages ad shrtcmigs whe cmpared with thers. This article aalyzes the stability i the large f the ptimal reisurace with respect t the risk measure that the isurer uses. It has bee pited ut that there is a stable ptimal reteti that will shw sesitivity, isfar as it will slve the ptimal reisurace prblem fr may risk measures, prvidig a very r bust reisurace pla. Fr the expected value premium priciple this stable ptimal reteti is a stp lss ctract, ad it is easy t cmpute i practice. A fast liear time algrithm has bee give ad a umerical example preseted. The apprach is geeral e ugh. Actually, if desired, the aalysis permits us t icrprate bth budget cstraits ad the simultaeus maximizati f the cedig cmpay expected wealth. 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