Optimal VaR-based Risk Management with Reinsurance

Transcription

1 manuscrpt No. (wll be nserted by the edtor) Optmal VaR-based Rsk Management wth Rensurance Janfa Cong Ken Seng Tan Receved: date / Accepted: date Abstract It s well-known that rensurance can be an effectve rsk management soluton for fnancal nsttutons such as the nsurance companes. The optmal rensurance soluton depends on a number of factors ncludng the crteron of optmzaton and the premum prncple adopted by the rensurer. In ths paper, we analyze the Value-at-Rsk (VaR) based optmal rsk management soluton usng rensurance under a class of premum prncples that s monotonc and pecewse. The monotonc pecewse premum prncples nclude not only those whch preserve stop-loss orderng, but also the pecewse premum prncples whch are monotonc and constructed by concatenatng a seres of premum prncples. By adoptng the monotonc pecewse premum prncple, our proposed optmal rensurance model has a number of advantages. In partcular, our model has the flexblty of allowng the rensurer to use dfferent rsk loadng factors for a gven premum prncple or use entrely dfferent premum prncples dependng on the layers of rsk. Our proposed model can also analyze the optmal rensurance strategy n the context of multple rensurers that may use dfferent premum prncples (as attrbuted to the dfference n rsk atttude and/or mperfect nformaton). Furthermore, by artfully mposng certan constrants on the ceded loss functons, the resultng model can be used to capture the rensurer s wllngness and/or capacty to accept rsk or to control counterparty rsk from the perspectve of the nsurer. Under some techncal assumptons, we derve explctly the optmal form of the rensurance strateges n all the above cases. In partcular, we show that a truncated stop-loss rensurance treaty or a lmted stop-loss rensurance treaty can be optmal dependng on the constrant mposed on the retaned and/or ceded loss functons. Some numercal examples are provded to further compare and contrast our proposed models to the exstng models. Keywords Rsk Management Rensurance Optmal strategy Value-at-Rsk (VaR) Monotonc pecewse premum prncple Multple rensurers Counterparty rsk 1 Introducton Rensurance s one of the most tradtonal and long standng rsk management solutons, partcularly from an nsurer s pont of vew. Its strategc use not only leads to an effectve rsk mtgaton, but t also enhances an nsurer s stablty and proftablty. Examples of the rensurance contracts (or treates) for whch an nsurer can transfer ts rsk to a rensurer nclude quota-share rensurance, stop-loss rensurance, excess-of-loss rensurance, surplus rensurance, and so on. Because of the varety of these rensurance J. Cong Department of Statstcs and Actuaral Scence, Unversty of Waterloo, 200 Unversty Avenue West, Waterloo, N2L 3G1, ON, Canada Tel.: ext E-mal: jcong@uwaterloo.ca K.S. Tan Department of Statstcs and Actuaral Scence, Unversty of Waterloo, 200 Unversty Avenue West, Waterloo, N2L 3G1, ON, Canada. Central Unversty of Fnance and Economcs, Chna Insttute for Actuaral Scence, Bejng, Chna Tel.: ext E-mal: kstan@uwaterloo.ca

2 treates that exst n the marketplace, the nsurers are therefore constantly seekng for better and more effectve rensurance strateges. The quest for optmal rsk management soluton usng rensurance s an actve area of research among academcs, actuares, and rsk managers. In a typcal formulaton of an optmal rensurance model, t nvolves at least the followng three components. Frst s the crteron (.e. objectve) that determnes the optmalty of the rensurance contracts. Second s the premum prncple that specfes how the rensurance premum s calculated. The thrd s the constrants, f any, that are mposed on the model. Examples of some typcal constrants nclude the restrcton on the structure of the rensurance contracts and the rensurance premum budget that an nsurer could spend on rensurng hs rsk va rensurance. In ths paper, we wll also demonstrate that an ngenous specfcaton of constrants could lead to an optmal rensurance model wth some desrable features, ncludng controllng the credt rsk of the rensurer and the counterparty rsk of the nsurer. The poneerng work on optmal rensurance s attrbuted to Borch (1960), Kahn (1961) and Arrow (1963). In partcular, Borch (1960) showed that the stop-loss rensurance s optmal n the sense of mnmzng the varance of the nsurer s retaned loss under the assumpton of expected rensurance premum prncple. Confnng to the expected rensurance premum prncple and the crteron of maxmzng the expected utlty of a rsk-averse nsurer s termnal wealth, Arrow (1963) also establshed that stop-loss rensurance s optmal. The classcal optmal rensurance models have been generalzed n a number of nterestng drectons, wth partcular emphass on the three aspects of the optmal rensurance models dscussed above,.e. more sophstcated crteron, more complex premum prncples, and more nvolved constrants. Just to name a few, Young (1999), Kaluszka (2001, 2005), Kaluszka and Okolewsk (2008) addressed the optmal rensurance strategy by consderng other premum prncples such as Wang s premum prncple, meanvarance premum prncples, maxmal possble clams prncple, convex premum prncples, etc. Ca and Tan (2007), and Ca et al. (2008), Balbas et al. (2009), Ch and Tan (2011), Tan et al. (2011) demonstrated that modern rsk measures such as the Value-at-Rsk (VaR) and the Condtonal Valueat-Rsk (CVaR) can be exploted n a rensurance model for a vable rsk management soluton. More recently, Ch and Tan (2013) broadened the optmal rensurance model by nvestgatng the VaR and CVaR rensurance models under a more general premum prncple. They mposed some constrants on the ceded loss functons and assumed that the premum prncple satsfes three basc axoms, namely dstrbuton nvarance, rsk loadng and stop-loss order preservng. See also Cheung et al. (2011) and Ch and Weng (2013). Whle the exstng results have studed the optmal rensurance solutons under a standard premum premum or a partcular class of premum prncples, n ths paper we propose a new class of premum prncple whch we denote as the monotonc pecewse premum prncple and show that the resultng optmal rensurance model nvolvng ths new class of premum prncple s stll tractable. By pecewse premum prncple we mean a premum prncple that can be constructed by ether concatenatng a seres of dfferent premum prncples or usng the same premum prncple but wth dfferent parameter values. There are many advantages to nvestgate the optmal rensurance model under ths new class of premum prncple. Frst and foremost s that the proposed monotonc pecewse premum prncple s able to capture the rsk atttude of the rensurer easly and ntutvely. If rsk were segmented nto dfferent layers so that a hgher layer of rsk refers to a greater rsk exposure wth a larger potental catastrophc loss, then a rensurer typcally has a dfferent level of rsk atttude on each of these layers. Ths mples that dfferent layers of rsk may be prced dfferently. More specfcally, a rensurer, n general, demands a hgher rsk premum (.e. hgher rsk loadng) on a rsk n hgher layers than a rsk n lower layers. The proposed monotonc pecewse premum prncple provdes an elegant way of addressng the dfferentate n rsk atttude. For example, f a rensurer prefers to consstently usng an expected premum prncple to prce all layers of rsk, then the dfferentate n rsk atttude can be reflected by attachng a hgher rsk loadng parameter of the expected premum prncple when prcng a hgher layer of rsk. The pecewse nature of the premum prncple also provdes a greater flexblty n modelng a rensurer s rsk atttude by allowng the rensurer to adopt dfferent premum prncples dependng on the layers. For nstance, the rensurer may use the expected value premum prncple when the clam s less than a certan threshold level, and Wang s premum prncple when the clam exceeds that threshold. Smlarly, f the rensurer uses prncple of equvalent utlty to prce the contracts, the rensurer may choose dfferent parameters or even dfferent utlty functons on dfferent layers and ths agan leads to premum prncple that s pecewse. 2

3 A second advantage to nvestgate the optmal rensurance under the proposed monotonc pecewse premum prncple s that t can be used to analyze the optmal rensurance n the context of multple rensurers. Ths s facltated by the fact that the pecewse nature of prcng layers of rsk can be vewed as beng rensured by dfferent rensurers. Each rensurer s rensurng one or more layers of rsk usng ts premum prncple. A thrd advantage s that t s a much wder class of premum prncples n that t encompasses the stop-loss preservng class of premum prncple consdered n Ch and Tan (2013). The stop-loss preservng premum prncple ncludes the followng eght classcal premum prncples: net, expected value, exponental, proportonal hazard, prncple of equvalent utlty, Wang s, Swss, and Dutch. Moreover, the class we consder here also ncludes the premum prncples whch are monotonc and constructed by concatenatng some combnatons of the above eght premum prncple. Another contrbuton of the paper s to demonstrate that by metculously mposng an approprate constrant on an optmal rensurance model, optmal rensurance strategy wth a certan desrable property can be obtaned analytcally. More specfcally, we propose two varants of the optmal rensurance models. The frst model takes nto consderaton the rensurers wllngness to rensure when desgnng the rensurance contract. Many of the studes on optmal rensurance mplctly assume that the rensurers wll accept any rensurance contracts proposed by the nsurance companes. Ths, however, may not be the case n practce. It s possble that the rensurers are not wllng to, or not allowed to due to concern wth credt rsk or constrant on rsk captal requrement. Ths ssue can be addressed by mposng a lmt on the losses that can be ceded to the rensurer. The second model s motvated by the presence of the counterparty rsk that the nsurer s concerned wth. In an deal arrangement, losses that are ceded to the rensurer become the oblgaton of the rensurer and wll be ndemnfed to the nsurer. However there are stuatons where the rensurer mght be facng cash flow straned or fnancal dstress that mpact ts ablty to meet ts oblgaton. When ths arses, the nsurer s responsble for the losses that are supposedly to have transferred to the rensurer and hence ultmately bearng the counterparty rsk. Ths suggests that when desgnng an optmal rensurance strategy, nsurer needs to take nto consderaton the counterparty rsk. In ths paper, we propose a new optmal rensurance model that reflects counterparty rsk. The basc setup of our optmal rensurance model s to seek an optmal rensurance strategy that mnmzes the VaR of the total exposed rsk of an nsurer gven some budget constrant and under the monotonc pecewse rensurance premum prncples. The model descrpton, ncludng the defnton of monotonc pecewse premum prncple and the constrants on the ceded loss functons are descrbed n Secton 2. The use of VaR as a relevant measure of rsk for the nsurer s prompted by ts popularty among banks and fnancal nsttutons for quantfyng rsk. Analytcal rensurance strateges for the basc rensurance model as well as ts varants are derved n Secton 3. In partcular, by requrng the retaned loss functon to be nondecreasng, ths secton demonstrates that the ceded loss functon of the followng form f(x) = (x d) + 1 {x v} (1) where 0 d v, (x) + = max{x, 0}, and 1 D denotes the ndcator functon of an event D, can be optmal. Ths ceded loss functon s commonly known as the truncated stop-loss rensurance treaty. The same type of ceded loss functon s also shown to be optmal n the rensurance models analyzed by Gajek and Zagrodny (2004b), Kaluszka (2005), Kaluszka and Okolewsk (2008), Bernard and Tan (2009), Ch and Tan (2011). The truncated stop-loss rensurance treaty has the pecular property that once the loss amount exceeds a certan threshold v, the rensurer wll have zero oblgaton to the nsurer. In other words, there s no ndemnfcaton from the rensurer to the nsurer for any loss exceedng v. From an nsurer s pont of vew, ths rsk management strategy seems counterntutve and not desrable snce there s no protecton to the nsurer when there s a catastrophc loss (or when loss exceeds v). From the rensurer s pont of vew, ceded loss functon of ths knd s also not desrable as t nduces an nsurer s moral hazard. There s an ncentve for the nsurer to underreport ts loss when the actual loss exceeds v. One smple soluton of preventng the truncated stop-loss functon beng optmal s to mpose the constrant that the ceded loss functon s also nondecreasng, n addton to nondecreasng constrant on the retaned loss functon. Ths s exactly the motvaton for Secton 4 whch demonstrates that wth the added constrant, a lmted stop-loss rensurance treaty wth the followng structure f(x) = mn{(x a) +, b} (2) 3

4 for some a 0, b > 0, can be optmal. Note that the lmted stop-loss treaty s smlar to the standard stop-loss rensurance except that t mposes an upper lmt on the loss that a rensurer s lable. Rsk management wth ths type of property s more reasonable as t does not expose the rensurer to unlmted rsk exposure. There are other studes that have smlarly shown that the above ceded loss functon can be optmal. These nclude the works of Kaluszka and Okolewsk (2008) and Gajek and Zagrodny (2004a). Usng the the crtera of maxmzng ether the expected utlty or the stablty of the cedent, the former authors establshed that (2) can be optmal for a fxed rensurance premum calculated accordng to the maxmal possble clams prncple. Smlarly the latter authors consdered more general symmetrc and even asymmetrc rsk measures and showed the optmalty of (2). By consderng a partcular form of a pecewse premum prncple, Secton 5 provdes a detaled llustraton on how the optmal form of the rensurance treates can be evaluated. Numercal examples are further gven to compare and contrast our proposed models to the exstng results. Secton 6 concludes the paper. 2 Rsk Measure Based Rensurance Model 2.1 Model descrpton Let X be the clam random varable that an nsurer s oblgated to pay. Wthout any loss of generalty, we assume that X s a non-negatve random varable wth cumulatve dstrbuton functon (c.d.f.) F X (x) = P(X x) and E(X) <. In the absence of rensurance the nsurer s rsk exposure s X. Let us now assume that the nsurer s usng rensurance to cede part of hs rsk to a rensurer. In ths case, the clam X s dvded nto two parts,.e. the ceded loss part, f(x), and the retaned loss part, R f (X). Ths means that X = f(x) + R f (X) and that a rensurance contract (or treaty) s unquely determned by ether the ceded loss functon f( ) or the retaned rsk functon R f ( ). Here we focus on the ceded loss functon f( ) to dentfy the rensurance treaty. Under the rensurance treaty f, the rensurer s oblgated to pay f(x) to the nsurer when a clam X arses. By transferrng part of the rsk to a rensurer, the nsurer ncurs an addtonal cost n the form of rensurance premum Π(f(X)) that s payable to the rensurer. Note that the rensurance premum s a functon of the ceded loss functon f( ) and the adopted premum prncple. In the presence of rensurance, the total rsk exposure of the nsurer s transformed from X to T f (X) where T f (X) = R f (X) + Π(f(X)). The transformed random varable T f (X) captures the tradeoff between rsk retanng and rsk transferrng. If the nsurer s conservatve and wshes to transfer most of the rsk to a rensurer, then the retaned rsk R f (X) can be made small but at the expense of hgher rensurance premum Π(f(X)). On the other hand, f the nsurer has a much hgher rsk tolerance, then the cost of rensurance Π(f(X)) wll be small but at the expense of hgher retaned rsk R f (X). Consequently the dea underlyng the optmal rensurance s to seek an optmal ceded functon f(x) that balances the tradeoff between rsk retanng and rsk transferrng. A plausble rsk measure based optmal rensurance model (see for example Ca and Tan, 2007 and Ch and Tan, 2013) can be formulated as { mn f L ρ(t f (X)) s.t. Π(f(X)) π 0, (3) where ρ( ) represents the rsk measure that s adopted by the nsurer, Π( ) s the rensurance premum prncple, π 0 s the maxmum budget an nsurer could spend on rensurance premum, and L s the admssble set of ceded loss functons. In ths paper, we analyze the optmal rensurance model by settng the rsk measure ρ to the value-at-rsk (VaR). Despte ts shortcomngs such as lackng coherence property (see Artzner et al., 1999), VaR remans promnent among fnancal nsttutons for quantfyng rsk (see Joron, 2006). Formally, VaR s defned as follows: Defnton 1 The VaR of a non-negatve varable X at the confdence level (1 α), where 0 < α < 1, s defned as VaR α (X) = nf{x 0 : P(X > x) α}. 4

5 The constant α, whch s typcally a small value such as 1% or 5%, reflects the the desred confdence level of the nsurer. The optmal rensurance model (3) under VaR crteron smplfes to { mn VaR α(t f (X)) f L (4) s.t. Π(f(X)) π 0. We now defne the admssble set L. Here we consder the followng two classes of L, whch are labeled as L 1 and L 2, respectvely: L 1 = {0 f(x) x : R f (x) x f(x) s a nondecreasng and left contnuous functon}, (5) L 2 = {0 f(x) x : both R f (x) and f(x) are nondecreasng functons, R f (x) s left contnuous}. (6) There are some common characterstcs among the above admssble sets L 1 and L 2. Frst, the loss that s ceded to a rensurer s nonnegatve and unformly bounded by the rsk tself. The latter restrcton ensures that the clam amount pad by the rensurer does not exceed the orgnal clam. Second, the retaned loss functon s at least a nondecreasng functon so that the nsurer needs to bear a correspondngly hgher clam for larger clam. Thrd, wthout loss of too much generalty, for any rensurance treaty f, we assume that the retaned loss functon R f (x) s a left contnuous functon wth respect to x. Fourth, the admssble set L 1 encompasses L 2 ;.e. L 2 L 1. Some argue that the ceded loss functon should be nondecreasng, smlar to the retaned loss functon. Ensurng both ceded loss functon and the retaned loss functon to be nondecreasng has the advantage of reducng the nsurer s moral hazard. It s for ths reason that we also nvestgate the optmal rensurance under the admssble class L 2. Ch and Tan (2013) smlarly analyzed the optmal rensurance under L 2 and stop-loss preservng class of premum prncple. The above two admssble sets L 1 and L 2 represent the two basc constrants we mpose on the ceded loss functons and the retaned loss functons, as we wll dscuss n Sectons 3 and Pecewse Premum Prncple Ths subsecton begns by frst descrbng the well-known stop-loss order preservng premum prncple. Then we formally defne the proposed class of premum prncple that s monotonc and pecewse. We conclude the subsecton by presentng an example n order to contrast the dfference between the proposed class of premum prncple and the class of stop-loss order preservng premum prncple. In order to understand what we meant by a class of premum prncple that s stop-loss order preservng, t s essental to frst ntroduce the defnton of stop-loss order between two rsks. Whle there are several dfferent but equvalent defntons of stop-loss order (see Hurlmann, 1998), here we just state the one whch s based on Theorem n Rolsk et al. (1999). Defnton 2 Suppose X 1 and X 2 are two random varables wth fnte means. If E [(X 1 d) + E [(X 2 d) +, d R, then we say that the random varable X 1 s smaller than the random varable X 2 n stop-loss order and we use the notaton X 1 sl X 2 to denote such orderng. Usng the above defnton, the stop-loss order preservng property of the nsurance premum prncple s defned as follows: Defnton 3 Suppose Π( ) s an nsurance premum prncple. If Π(X 1 ) Π(X 2 ) for any random varables X 1 and X 2 as long as they satsfy X 1 sl X 2, then we say that the nsurance premum prncple Π( ) s stop-loss order preservng. Now we ntroduce what we meant by a class of premum prncple that s monotonc. Defnton 4 Gven any two rsks X and Y such that X(ω) Y (ω) for all possble outcomes ω, then Π( ) s sad to be a premum prncple preservng monotoncty f Π(X) Π(Y ). 5

6 It should be emphaszed that monotoncty s a mld condton on the premum prncple. In partcular, the class of monotonc premum prncples ncludes the premum prncples whch preserve stop-loss orderng. The class of premum prncples whch preserves stop-loss orderng ncludes the followng eght classcal premum prncples: net, expected value, exponental, proportonal hazard, prncple of equvalent utlty, Wang s, Swss, and Dutch. It s also worth mentonng that monotoncty allows a premum prncple to have a very flexble pecewse structure. The pecewse premum prncple s defned as follows: Defnton 5 If there exst 0 = a 0 < a 1 < < a n 1 < a n =, a R, = 0, 1,..., n such that for any random varable X, Π(X) = n =1 Π (X 1 X [a 1,a )), where 1 denotes the ndcator functon and each Π ( ) s a specfc premum prncple, then we say that the premum prncple Π( ) s a pecewse premum prncple. If addtonally the pecewse premum prncple satsfes the monotoncty property, then the resultng premum prncple s both monotonc and pecewse. Note that any arbtrary classcal premum prncple s a specal case of the above pecewse premum prncple. Ths follows by settng n = 1 n the above defnton. For ths reason we wll manly focus our analyss on the pecewse premum prncple (.e. n > 1) nstead of the ordnary premum prncple. Furthermore, the monotonc pecewse premum prncple encompasses the stop-loss order preservng premum prncple so that the former premum prncple s more general than the latter premum prncple. In fact, the followng example confrms that a premum prncple can be monotonc and pecewse and yet does not preserve the stop-loss orderng. Example 1 Usng the notaton n Defnton 5, ths example consders a monotonc pecewse premum prncple wth n = 2, a 1 = 10, and Π, = 1, 2 are expectaton premum prncples wth rsk loadng factors ρ 1 = 0.1 and ρ 2 = 0.5, respectvely. Ths mples that Π 1 apples to the frst layer of rsk wth loss amount less than 10 whle Π 2 apples to the remanng layer wth loss amount exceedng or equal to 10. Hence the pecewse premum prncple s constructed by concatenatng two expectaton premum prncples wth the followng representaton: Π(X) = 1.1 E [ X 1 X [0,10) E [ X 1X [10, ). (7) Note that the expectaton premum prncple s monotonc and preserves stop-loss order and that the premum prncple (7) s a monotonc pecewse premum prncple. Let us now consder the followng two loss random varables X 1 and X 2 such that X 1 represents a determnstc loss of 10 n any scenaro whle X 2 equals to 5 wth probablty of 80% and unformly dstrbuted between 5 and 55 wth probablty of 20%. It s easy to verfy that both rsks have the same expectatons;.e. E [X 1 = E [X 2 = 10. Furthermore, the followng analyss confrms that X 1 sl X 2. (). If d 5, we have (). If 5 < d 10, we have E [(X 1 d) + = E [X 1 d = E [X 2 d = E [(X 2 d) +. E [(X 2 d) + E [(X 1 d) + = d d 50 = d d 3.95, 0.2 (10 d) whch s ncreasng wth respect to d when 5 < d 10. Therefore, t acheves ts mnmum when d = 5,.e., E [(X 2 d) + E [(X 1 d) + 1 > 0. (). If d 10, t s clear that E [(X 1 d) + = 0 E [(X 2 d) +. Hence accordng to Defnton 2, we have X 1 sl X 2. On the other hand, the premum prncple (7) s not stop-loss order preservng premum prncple though t s a monotonc pecewse premum prncple snce Π(X 1 ) = 15 and Π(X 2 ) =

7 3 Optmalty of truncated stop-loss rensurance treates By assumng the premum prncple s monotonc (see Defnton 4) and the ceded loss functons need not be non-decreasng (.e. the admssble set of ceded loss functons s gven by L 1 as defned n (5)), Subsecton 3.1 shows that the truncated stop-loss rensurance treaty (1) s optmal to the rensurance model (4). The same subsecton also demonstrates that the basc rensurance model can be extended to analyzng the optmal rensurance treates under the multple rensurers when the premum prncple s of the form pecewse as defned n Defnton 5. Two nterestng extensons of the optmal rensurance models are dscussed n Subsectons 3.2 and 3.3. In partcular, Subsecton 3.2 nvestgates the rensurance model (4) under the addtonal constrant that a lmt s mposed on the rensurance treaty whle Subsecton 3.3 examnes a generalzaton of the rensurance model (4) that ncorporates counterparty rsk. Interestngly, both varants of the optmal rensurance models stll confrm the optmalty of the truncated stop-loss rensurance treates. 3.1 Wthout nondecreasng assumpton on the ceded loss functons In ths subsecton, we show that for the rensurance model (4), the truncated stop-loss rensurance strategy s optmal among all the strateges n L 1. To proceed, for any ceded loss functon f from the set L 1, t s useful to consder the followng functon: g f (x) = { [x + f(v) v+, f 0 x v, 0, f x > v, (8) where v = VaR α (X). Note that by constructon, g f s also an element n L 1. Clearly, f the ceded loss functon of a rensurance treaty takes the form g f, then the rensurance treaty s a truncated stoploss rensurance treaty. The followng theorem shows that f the rensurance premum s monotonc, the truncated stop-loss rensurance treaty s the optmal form among all the admssble treates n L 1. Theorem 1 Consder the rensurance model (4) wth admssble ceded loss functons L 1. Assume further that the rensurance premum prncple Π( ) s a monotonc pecewse premum prncple. Then, for any ceded loss functon f L 1, we can construct the ceded loss functon g f L 1 usng (8) such that g f satsfes the followng propertes: (a) Π(f(X)) π 0 mples Π(g f (X)) π 0. Equvalently, f the ceded loss functon f satsfes the budget constrant, then the ceded loss functon g f wll also satsfy the budget constrant; (b) VaR α (T gf (X)) VaR α (T f (X)). Proof: (a) We frst clam that g f (x) f(x) for all x 0. In fact, for 0 x v, snce f L 1, the retaned loss functon correspondng to f s nondecreasng, whch mples that Therefore, and v f(v) x f(x). g f (x) = [x + f(v) v + f(x), g f (x) = 0 f(x), x > v. Thus, g f (x) f(x), x 0. As a result, the monotoncty of the premum prncple Π ( ) mmedately mples that Π(g f (X)) Π(f(X)), whch s the desred result. (b) The translaton nvarance property of VaR yelds V ar α (T f (X)) = V ar α (R f (X)) + Π(f(X)) = R f (V ar α (X)) + Π(f(X)) = V ar α (X) f(v ar α (X)) + Π(f(X)) V ar α (X) g f (V ar α (X)) + Π(g f (X)) = V ar α (T gf (X)), 7

8 where the second equalty s due to an applcaton of Theorem 1 n Dhaene et al. (2002) along wth the left contnuty and nondecreasng propertes of R f (x). Ths completes the proof. Remark 1 (a) The above theorem ndcates that the optmalty of the truncated stop-loss rensurance strategy s ndependent of the rensurance premum prncple. The truncated stop-loss rensurance strategy s optmal among all the strateges n L 1 as long as the premum prncple s monotonc. The actual specfcaton of the parameter values of the optmal ceded loss functon then depends on the premum prncple. (b) If we denote d = v f(v), then the truncated stop-loss functon g f defned n (8) can be succnctly represented as g f (x) = (x d) + 1 {x v}. Furthermore, t follows from Theorem 1 that the VaR-based optmal rensurance problem (4), wth admssble set of ceded loss functons L 1, can equvalently be rewrtten as ( mn X (X d)+ 1 {X v} + Π [g f (X) ) 0 d v VaR α s.t. Π[g f (X) Π [ (X d) + 1 {X v} π0. The above optmzaton problem reduces to mn d + Π [g f (X) 0 d v s.t. Π[g f (X) Π [ (X d) + 1 {X v} π0, (9) whch s smply an optmzaton problem nvolvng only one varable. Hence once the rensurance premum prncple s gven, t s techncally much easer to solve, as shown n the numercal examples n Secton 5. If there exst several rensurers whch adopt dfferent premum prncple n the market, then the nsurance company wll naturally take advantage of ths when cedng ts rsk to the rensurers. When determnng the optmal rensurance strategy, the nsurance company wll consder the exstence of multple rensurers, and the premum prncple s not so explct as that n the case of sngle rensurer. The followng theorem deals wth the case of multple rensurers. Theorem 2 Assume that there are n rensurers n the market and that rensurer adopts premum prncple, Π ( ), for = 1, 2,..., n. Each premum prncple Π ( ) s a monotonc pecewse premum prncple. We further assume that the nsurance company wll always seek the optmal way to cede hs rsk to the rensurers n order to mnmze the cost of rensurance. Under the above assumptons, the premum that the nsurance company pays assocated wth the ceded loss functon f s gven by Π(f(X)) = = mn {A } n =1 =1 n =1 n ( ) Π f(x) 1f(X) A Π (f(x) 1 f(x) A f ) (10) where n =1 A = n =1 Af = [0, + ). And {Af }n =1 s the optmal partton assocated wth the ceded loss functon f n the sense that t mnmzes the premum pad by the nsurance company. Then, for any ceded loss functon f L 1, we can construct the ceded loss functon g f accordng to (8), and g f satsfes the followng propertes: (a) Π(f(X)) π 0 mples Π(g f (X)) π 0. Equvalently, f the ceded loss functon f satsfes the budget constrant, then the ceded loss functon g f wll also satsfy the budget constrant; (b) VaR α (T gf (X)) VaR α (T f (X)). 8

9 Proof: (a) From the proof of Theorem 1, we know that g f (x) f(x), x 0. Therefore, for any set B, we have g f (X 1 X B ) f(x 1 X B ). The monotoncty of the premum prncple Π ( ) mmedately mples that Π (g f (X 1 X B )) Π (f(x 1 X B )), = 1,..., n. We assume that {A f }n =1 s the optmal partton assocated wth the ceded loss functon f n the sense that t mnmzes the premum pad by the nsurance company gven a ceded loss f. Ths means that the rensurance premum s calculated by Π(f(X)) = n =1 Π (f(x) 1 f(x) A f We further denote that B f = f 1 (A f ) as the nverse mage of Af under f, then the premum assocated wth the ceded loss functon f expressed n the above expresson can be rewrtten as Analogously we have B g f loss functon g f and B g f Π(f(X)) = = n =1 n =1 = g 1 f (Ag f ), where {A g f s the nverse mage of A g f Π(g f (X)) = = n =1 n =1 n =1 n =1 Π (f(x) 1 f(x) A f ) Π (f(x 1 X B f ). } n =1 ). ) s the optmal partton assocated wth the ceded under g f. Then Π (g f (X) 1 gf (X) A g f ) Π (g f (X 1 g X B f ) ) Π (g f (X 1 X B f ) ) Π (f(x 1 X B f ) = Π(f(X)), whch s the requred result. The proof of part (b) s the same as that of Theorem 1 and hence s omtted. Remark 2 (a) The above theorem dentfes the optmal form of the ceded loss functon when there are multple rensurers n the market. The optmal form of the rensurance strateges s also a truncated stop-loss type contract. (b) In ths theorem, we do not need to assume that the premum prncple Π( ) to be monotonc, though t s a weak and reasonable assumpton on the premum prncple. We just need to mpose the monotonc assumpton on each Π ( ), whch s the premum prncple adopted by the -th rensurer. (c) It s worth mentonng that the overall retaned loss functon s nondecreasng. Therefore, the rensurers wll accept ths treaty f they only requre the retaned loss functons to be nondecreasng for the concern of moral hazard. (d) It s mperatve to dstngush the works of Asmt and Badescu (2013) and Ch and Meng (2012) from ours as they have smlarly nvestgated the optmal rensurance n the context of multple rensurers. The key dfference les n how the ceded losses are dstrbuted to the rensurers. Ther formulatons assume that the ceded losses are dvsble n such a way that any loss s shared among the rensurers whle n our setup, the ceded losses are frst dvded nto layers and then each rensurer s responsble (entrely) for each layer of rsk. Because the potental clam s assumed to be segmentable, ) 9

10 ther optmal rensurance strateges and the correspondng mnmal exposed rsk may depend on the number of rensurance companes n the market. Even f all the rensurers are usng the same premum prncple, the number of rensurers n the market may stll affect nsurer s optmal strategy and the correspondng optmal exposed rsk level. Ths phenomenon appears to be counterntutve. In contrast, our proposed optmal strategy and the correspondng mnmal exposed rsk only depend on the premum prncples adopted by the rensurers and not on the number of rensurers n the market. (e) Remark 1 for Theorem 1 s smlarly appled to Theorem Exertng lmt on the rensurance treates In general, rensurers do not wsh to rensure catastrophc clams unless they are approprately compensated. Some rensurers may rase the rsk loadng factor on hgher layers of coverage, whch has been dealt wth by consderng the monotonc pecewse premum prncples n the last subsecton. Some rensurers may choose to mpose a lmt on the rensurance treates. Another reason for rensurer to mpose a lmt on the rensurance treates may be due to regulatory constrant. In ths subsecton we wll nvestgate an optmal rensurance strategy under ths motvaton. We suppose that the rensurers are only wllng to accept the rensurance treates subject to a lmt. Ths mples that the maxmal values of the ceded loss functons are bounded by a specfed constant c 1. Hence the admssble set of the ceded loss functons s revsed to L 1 = {0 f(x) mn{x, c 1 } :R f (x) x f(x) s a nondecreasng and left contnuous functon}. (11) Though the admssble set s dfferent, we can stll use the method smlar to the last subsecton to derve the optmal rensurance strateges. Ths s summarzed n the followng corollary. Corollary 1 Consder the rensurance model (4) wth admssble ceded loss functons L 1 and monotonc pecewse rensurance premum prncple Π( ). Then, for any ceded loss functon f L 1, we can construct the ceded loss functon g f usng (8) wth g f satsfes the followng propertes: (a) g f L 1. (b) Π(f(X)) π 0 mples Π(g f (X)) π 0. Equvalently, f the ceded loss functon f satsfes the budget constrant, then the ceded loss functon g f wll also satsfy the budget constrant; (c) VaR α (T gf (X)) VaR α (T f (X)). Proof: From the proof of Theorem 1 (a), we know that g f (x) f(x), x 0. Therefore, f(x) mn{x, c 1 } mples that g f (x) mn{x, c 1 }. It s easy to verfy that the retaned loss functon R gf, whch s assocated wth the ceded loss functon g f, s nondecreasng and left contnuous. Accordngly, we can conclude that g f L 1. Snce the constructon of g f s the same as that n Theorem 1, the proof of part (b) and (c) s exactly the same as that of Theorem 1 and hence s omtted. Remark 3 (a) The above corollary dentfes an optmal form of the ceded loss functon when there s a lmt mposed on the rensurance treates. The optmal form of the rensurance strateges s also a truncated stop-loss type contract. (b) Usng the notaton from Remark 1, we can express the ceded loss functon g f as follows g f (x) = (x d) + 1 {x v}. It s clear that max x 0 {g f (x)} = v d. Therefore, we need the condton d v c 1 to ensure that g f (x) s bounded by the constant c 1. Ths mples that we can smplfy the optmzaton problem as follows mn d + Π [g f (X) max{0,v c 1} d v s.t. Π[g f (X) Π [ (X d) + 1 {X v} π0. (12) 10

11 3.3 In the presence of counterparty rsk In an deal rensurance arrangement, the rensurer s lable for any clam as stpulated n the rensurance treaty and hence any clam that s ceded wll be rembursed by the rensurer. The nsurer s only concerned wth the resdual part of the rsk. Whle ths s true n theory, n practce the use of rensurance exposes the nsurer to another type of rsk known as the counterparty rsk. The counterparty rsk arses when the rensurer s not able to meet ts oblgaton for reasons such as the company s havng cash flow straned or facng nsolvency/bankruptcy. When ths occurs, the nsurer s ultmately responsble for the part of the rsk that s supposedly ceded to the rensurer. Ths suggests that n the desgn of optmal rensurance strategy, the credtworthness of the rensurer s one of the crtcal factors that cannot be gnored. Yet the counterparty rsk s often neglected n most formulatons of the optmal rensurance models. The objectve of ths subsecton s to demonstrate that by artfully modfyng some of the constrants of the rensurance models, the counterparty rsk could be ntegrated to the optmal rensurance models that we have dscussed so far. We frst assume that the actual clam that s ceded to the rensurer s so large that when t exceeds a certan threshold, then the rensurer s n fnancal stress and mght not be able to meet ts contractual oblgaton. In ths case, the loss that s supposedly ndemnfed to the nsurer wll be defaulted. We propose to reduce the counterparty rsk by ensurng that the probablty of the rensurer not meetng ts oblgaton does not exceed a certan acceptable tolerance level of the nsurer. If c 1 represents the threshold of the above rensurer and 0 β 1 denotes the desred tolerance level of the nsurer, then the above condton s translated to the probablstc constrant P(f(X) > c 1 ) β. The parameter β s predetermned by the nsurer and reflects the nsurer s rsk tolerance towards counterparty rsk. Clearly, the smaller the β, the less exposure the nsurer s to counterparty rsk. In the extreme case where β = 0, the counterparty rsk s completely elmnated snce the ceded clam can never exceed the threshold c 1 and hence the counterparty rsk wll never be trggered. The optmal rensurance model (4) can easly be modfed to reflect the above approach of controllng the counterparty rsk. Ths s acheved by seekng an optmal rensurance to the rensurance model (4) wth the admssble set of the ceded loss functon revses to L 1 ={0 f(x) x : P(f(X) > c 1 ) β, R f (x) x f(x) s a nondecreasng and left contnuous functon}. As n the last subsecton, we can stll use the same technque to derve the optmal rensurance strateges even though the admssble set s dfferent. The results are summarzed n the followng corollary. Corollary 2 Consder the rensurance model (4) wth admssble ceded loss functons L 1 and monotonc pecewse rensurance premum prncple Π( ). Then, for any ceded loss functon f L 1, we can construct the ceded loss functon g f usng (8) and that g f has the followng propertes: (a) g f L 1. (b) Π(f(X)) π 0 mples Π(g f (X)) π 0. Equvalently, f the ceded loss functon f satsfes the budget constrant, then the ceded loss functon g f wll satsfy the budget constrant as well; (c) VaR α (T gf (X)) VaR α (T f (X)). Proof: From the proof of Theorem 1 (a), we know that g f (x) f(x), x 0. Therefore, P(f(X) > c 1 ) β mples that P(g f (X) > c 1 ) β. It s easy to verfy that the retaned loss functon R gf, whch s assocated wth the ceded loss functon g f, s nondecreasng and left contnuous. Accordngly, we conclude that g f L 1. Snce the constructon of g f s the same as that n Theorem 1, the proofs of part (b) and (c) are exactly the same as that of Theorem 1, hence are omtted here. (13) Remark 4 (a) The above corollary dentfes the optmal form of the ceded loss functon whch takes nto consderaton counterparty rsk. The optmal form of the rensurance strateges s also a truncated stop-loss type contract. (b) Usng the notaton n Remark 1, we can express the ceded loss functon g f as follows g f (x) = (x d) + 1 {x v}. 11

12 Therefore, P(g f (X) > c 1 ) β s equvalent to d > Q 1 c 1 where Q 1 = nf{q 0 : P(q < X v) β}. Ths mples that we can smplfy the optmzaton problem as follows: mn d + Π [g f (X) max{0,q 1 c 1} d v s.t. Π[g f (X) Π [ (14) (X d) + 1 {X v} π0. (c) Note that the model we consdered n the last subsecton s a specal case of the model n ths subsecton. Ths can be seen by settng c 1 = c 1 and β = 0. Therefore, the optmal rensurance model proposed n ths subsecton s more general and that t gves the nsurer the addtonal flexblty of specfyng ts atttude on courterparty rsk. The nsurer s atttude on counterparty rsk s reflected by c 1 and β. Note also that f we let c 1 = + or β = 1, then the model n ths subsecton recovers the one n Subsecton Optmalty of lmted stop-loss rensurance treates In the last secton, we study a few varatons of the optmal rensurance model (4). All these varants share the same constrant that the ceded loss functons do not need to be nondecreasng and that the truncated stop-loss rensurance treates are optmal. These results mply that the losses that are ceded to the rensurer do not need to ncrease wth losses. In fact when the losses ncrease to a crtcal level, the losses ceded wll reduce drastcally to zero and reman at zero thereafter. Ths rases a concern to the rensurer as rensurance treaty of ths type potentally trggers nsurer s moral hazard. On the other hand, the truncated loss functon s also not a desrable rsk management strategy for the nsurer snce there s no protecton when the loss exceeds a certan threshold level. For these reasons, both nsurers and rensurers often prefer rensurance treates wth the property that the ceded losses are at least nondecreasng wth losses. As a result, the objectve of ths secton s to nvestgate the optmal ceded loss functon f to the optmzaton problem (4) when the premum prncple s monotonc and there s a monotonc assumpton mposed on the ceded loss functons. In ths case, the admssble set of the ceded loss functon corresponds to L 2. Smlarly, we wll extend our results to the case of multple rensurers and nvestgate the optmal strateges f there s a lmt on the rensurance treates or there exsts counterparty rsk. Our analyss reveals that the lmted stop-loss treaty (2) can be optmal. 4.1 Wth nondecreasng assumpton on the ceded loss functons In ths subsecton, we assume the admssble set s L 2 as defned n (6). We wll show that the so-called lmted stop-loss rensurance strategy (2) s optmal among all the strateges n L 2. We wll employ the same technque used n the prevous secton to derve the optmal solutons over L 2. Analogously, for any ceded loss functon f from set L 2 we construct the followng functon h f whch s also an element n L 2 : h f (x) = mn { [x (v f(v)) +, f(v) }, (15) where as defned prevously v = VaR α (X). It follows from the above representaton that the rensurance treaty wth the ceded loss functon h f (X) s commonly known as a lmted stop-loss rensurance treaty. The followng theorem shows that f the rensurance premum prncple s monotonc, then the lmted stop-loss rensurance treaty s the optmal form among all the admssble treates n L 2. Theorem 3 Consder the rensurance model (4) wth admssble ceded loss functons L 2. Assume the rensurance premum prncple Π( ) s a monotonc pecewse premum prncple. Then, for any ceded loss functon f L 2, we can construct the ceded loss functon h f L 2 accordng to (15), and h f satsfes the followng propertes: (a) Π(f(X)) π 0 mples Π(h f (X)) π 0. Equvalently, f the ceded loss functon f satsfes the budget constrant, then the ceded loss functon h f also satsfes the budget constrant; (b) VaR α (T hf (X)) VaR α (T f (X)). 12

13 Proof: The proof s smlar to that of Theorem 1. (a). We frst clam that g f (x) f(x) for all x 0. In fact, for 0 x v, snce f L 2, the retaned loss functon correspondng to f s nondecreasng, whch mples that Therefore, Snce f L 2 s a nondecreasng functon, we have v f(v) x f(x). h f (x) = [x + f(v) v + f(x). h f (x) = v f(x), x > v. Thus, h f (x) f(x), x 0. As a result, the monotoncty of the premum prncple Π ( ) mmedately mples that Π(h f (X)) Π(f(X)), whch s the requred result. (b). The proof s parallel to provng part (b) of Theorem 1 and hence s omtted. Remark 5 All the comments n Remark 1 for Theorem 1 are analogously applcable to the present case. In partcular, we make the followng remarks: (a) The above theorem ndcates that the optmalty of the lmted stop-loss rensurance strategy s ndependent of the rensurance premum prncple. The lmted stop-loss rensurance strategy s optmal among all the strateges n L 2 as long as the premum prncple s monotonc. (b) By denotng d = v f(v), the functon h f defned above can be rewrtten as h f (x) = (x d) + (x v) +. Based on the results from Theorem 3, t s easy to see that the VaR-based rensurance model (4) can be equvalently cast as { mn 0 d v VaR α {X (X d) + + (X v) + + Π [h f (X)} s.t. Π [h f (X) Π [(X d) + (X v) + π 0. The above optmzaton problem can be smplfed as follows { mn d + Π [h f (X) 0 d v (16) s.t. Π [h f (X) Π [(X d) + (X v) + π 0. The optmal rensurance problem agan reduces to an optmzaton problem of just a sngle varable. Smlar to the dscusson n the last secton, f there exst several rensurers whch adopt dfferent premum prncple n the market, then the nsurance company wll naturally take advantage of ths when cedng ts rsk to the rensurers. The followng theorem, as a counterpart of Theorem 2, deals wth the case of multple rensurance companes. Theorem 4 Consder the rensurance model (4) wth admssble ceded loss functons L 2. Assume that there are n rensurers n the market and each rensurer adopts premum prncples Π ( ) for = 1, 2,..., n. Every premum prncple s a monotonc pecewse premum prncple. We further assume that after the nsurance company determned hs ceded part, he wll seek the optmal way to cede hs rsk to the rensurers n order to mnmze the cost of rensurance. Under the above assumptons, the premum that the nsurance company pays assocated wth the ceded loss functon f s gven by (10). Then, for any ceded loss functon f L 2, we can construct the ceded loss functon h f accordng to (15), and h f satsfes the followng propertes: (a) Π(f(X)) π 0 mples Π(h f (X)) π 0. Equvalently, f the ceded loss functon f satsfes the budget constrant, then the ceded loss functon g f wll also satsfy the budget constrant; (b) VaR α (T hf (X)) VaR α (T f (X)). Proof: The proof s parallel to Theorem 2 and hence s omtted. Remark 6 All the comments n Remark 2 for Theorem 2 are analogously applcable here. We emphasze that the overall ceded loss functon and retaned loss functon are both nondecreasng, though the ceded loss functon wth respect to the -th rensurer mght not be. Therefore, the rensurers wll accept ths treaty snce there t reduces moral hazard. 13

14 4.2 Exertng lmt on the rensurance treates Smlar to Subsecton 3.2, here we study the optmal rensurance strateges f there s a lmt mposed on the rensurance treates. We suppose that the maxmal values of the ceded loss functons are bounded by a specfed constant c 2 so that the admssble set of the ceded loss functons changes to L 2 ={0 f(x) mn{x, c 2 } : both R f (x) and f(x) are nondecreasng functons, R f (x) s left contnuous}. (17) Usng the technque smlar to the last secton, we obtan the followng corollary. The proof s also smlar and hence s omtted. Corollary 3 Consder the rensurance model (4) wth admssble ceded loss functons L 2 and monotonc pecewse rensurance premum prncple Π( ). Then, for any ceded loss functon f L 2, we can construct the ceded loss functon h f usng (15) and h f satsfes the followng propertes: (a) h f L 2. (b) Π(f(X)) π 0 mples Π(h f (X)) π 0. Equvalently, f the ceded loss functon f satsfes the budget constrant, then the ceded loss functon h f also satsfes the budget constrant; (c) VaR α (T hf (X)) VaR α (T f (X)). Remark 7 All the comments n Remark 3 of Corollary 1 are analogously applcable here. In partcular, usng the notaton n Remark 5, the ceded loss functon h f can be expressed as h f (x) = (x d) + (x v) +. Snce max x 0 {h f (x)} = v d, h f (x) s bounded by the constant c 2 s equvalent to d v c 2. Therefore, the optmzaton problem can be reformulated as follows { mn d + Π [h f (X) max{0,v c 2} d v s.t. Π [h f (X) = Π [(X d) + (X v) + π 0. (18) 4.3 In the presence of courterparty rsk As n Subsecton 3.3, we model the counterpary rsk by seekng an optmal ceded loss functon such that the probablty that the ceded part exceeds the threshold c 2, whch s P(f(X) > c 2 ), s bounded by a predetermned parameter β. In ths case, the admssble set of the ceded loss functon s gven by L 2 ={0 f(x) x : P(f(X) > c 2 ) β, both R f (x) and f(x) are nondecreasng functons, R f (x) s left contnuous}. (19) where 0 β 1 s a predetermned parameter chosen by the nsurance company. Usng the same technque, the followng corollary gves the optmal rensurance strategy that reflects the counterparty rsk. The proof s agan omtted due to the smlarty. Corollary 4 Consder the rensurance model (4) wth admssble ceded loss functons L 2 and monotonc pecewse rensurance premum prncple Π( ). Then, for any ceded loss functon f L 2, we can construct the ceded loss functon g f accordng to (15), and g f satsfes the followng propertes: (a) g f L 2. (b) Π(f(X)) π 0 mples Π(g f (X)) π 0. Equvalently, f the ceded loss functon f satsfes the budget constrant, then the ceded loss functon g f also satsfes the budget constrant; (c) VaR α (T gf (X)) VaR α (T f (X)). 14

15 Remark 8 All the comments n Remark 4 for Corollary 2 are analogously applcable here. In partcular, usng the notaton n Remark 5, the ceded loss functon h f can be expressed as h f (x) = (x d) + (x v) +. Therefore, P(h f (X) > c 2 ) β s equvalent to d > Q 2 c 1 where Q 2 = VaR max{α,β} (X). Hence we can smplfy the optmzaton problem as follows { mn d + Π [h f (X) max{0,q 2 c 2} d v (20) s.t. Π [h f (X) = Π [(X d) + (X v) + π 0. 5 Illustratons The objectve of ths secton s to llustrate how the results obtaned n the last two sectons can be used to determne the optmal ceded loss functons by assumng the monotonc pecewse expected value premum prncple wth the followng representaton: Π(X) = (1 + ρ 1 ) E(X 1 X [0,a) ) + (1 + ρ 2 ) E(X 1 X [a,+ ) ) (21) where X s any random varable, a, ρ 1 and ρ 2 are fxed constants wth ρ 2 ρ 1. We note that the expected value premum prncple s the smplest premum prncple and t has been wdely studed due to ts tractablty. The drawback of ths premum prncple s that the rsk atttude of the rensurer s assumed to be nvarant to rsk. Ths s nconsstent wth practce snce rensurer often demands a hgher level of compensaton for larger rsk. Ths ssue s allevated by usng an expected value premum prncple that s monotonc and pecewse snce n ths case, the hgher layer of rsk s penalzed wth a larger loadng factor. Usng the monotonc pecewse expected value premum prncple (21), Subsecton 5.1 frst derves the generate expressons of the optmal ceded loss functons n term of parameters a, ρ 1 and ρ 2. By consderng a specfed set of numercal values, Subsecton 5.2 then calculates explctly the optmal ceded loss functon. The optmal ceded loss functons are compared and contrast to some exstng results. We emphasze that whle we have consstently used the pecewse expected value premum prncple n our llustratons, the optmal rensurance strateges under other pecewse premum prncples, such as prncple of equvalent utlty but wth pecewse parameter values, pecewse wth expected premum prncple and Wang s premum prncple, pecewse wth Dutch premum prncple and Wang s premum prncple, and so forth, can be calculated n a smlar fashon. 5.1 Pecewse Expected Value Premum Prncple The general optmal ceded loss functons, n term of parameters a, ρ 1 and ρ 2, are derved n the followng two subsectons for the optmal rensurance models that we have analyzed n the last two sectons. The frst subsecton assumes that the ceded loss functons need not be nondecreasng whle the second subsecton mposes the monotonc constrant on the ceded loss functons VaR-mnmzaton among L 1 Accordng to Theorem 1, the optmal ceded loss functon s of the followng form f 1 (x) = (x d 1 ) + 1 {x v}, where 0 d 1 < v and d 1 s yet to be determned. Recall that v = VaR α (X). It follows from (9) that the VaR of the nsurer s total exposed rsk correspondng to the ceded loss functon f 1 can be expressed as VaR α (T f1 (X)) = d 1 + Π [ (X d 1 ) + 1 {X v}. Now we wll determne the optmal retenton level d 1 under the assumed premum prncple (21). Snce the calculaton of the rensurance premum depends on the relatonshp between the ceded loss functon and the constant a, we need to consder the followng two dfferent cases: 15