Captive insurance companies and the management of non-conventional corporate risks

Transcription

1 Captve nsurance companes and the management of non-conventonal corporate rsks Jean-Baptste Lesourd a and Steven Schlzz b* a EJCM, Unversté de la Médterranée, Marselle, France b School of Agrcultural and Resource Economcs, The Unversty of Western Australa, Crawley, WA 6009, Australa *E-mal address: Steven.Schlzz@uwa.edu.au 22 February 2011 Workng Paper 1105 School of Agrcultural and Resource Economcs Ctaton: Lesourd, J-B. and Schlzz, S. (2011) Captve nsurance companes and the management of non-conventonal corporate rsks, Workng Paper 1105, School of Agrcultural and Resource Economcs, Unversty of Western Australa, Crawley, Australa. Copyrght remans wth the authors of ths document. 1

2 ABSTRACT We examne under what condtons settng up a captve nsurance company wth rensurance s an optmal soluton for rsk-averse frms when the nsured frm, the nsurer and the rensurer do not know the probablty dstrbuton of some rsks, and have conflctng estmates of ths dstrbuton. Keywords : corporate nsurance, rensurance, uncertanty, ambguty, non-conventonal rsks, captve nsurance companes JEL Classfcaton N : D 81, G22, Q2 1. Introducton Ths paper nvestgates the corporate management of non-conventonal rsks such as (among others rsks) envronmental rsks and dsasters, medcal malpractce, ltgaton-related rsks, and terrorsm rsks (see e.g. Goller (2007), p.11). Envronmental and catastrophc rsks have been studed by a number of authors. Zagalsk [1991], Kronenberg [1995], Freeman and Kunreuther (1997), and Lesourd and Schlzz (2003: chapter 7) have been concerned wth the specfc features of envronmental rsks. Catastrophc rsks have been studed by Kleffner and Doherty (1996), Zeckhauser [1996], and, n the context of crop nsurance, by Mranda and Glauber [1996] and Duncan and Myers (2000). Many of these studes show that nsurance and rensurance companes can refuse to nsure non-conventonal rsks such as envronmental rsks. Informaton about the probablty dstrbutons of such non-conventonal rsks s most often ncomplete or unknown. Ths leads to behavours that can be ncompatble wth standard expected utlty theory, as was clearly demonstrated by Ellsberg [1961] n hs semnal paper. In ths paper, he defnes ambguty as uncertanty on the probablty dstrbutons. In ths case, decson makers wll consder a range of probablty dstrbutons and wll usually gve more weght to the more pessmstc ones. Ths s the standard nterpretaton of the so-called Ellsberg paradox. Uncertanty regardng probablty dstrbutons also means that dfferent agents wll typcally have dfferent and even conflctng estmates of the rsks nvolved. One may see ths as a consequence of Ellsberg s ambguty and wll be of mportance n the analyss that follows. In partcular, dfferent partes to an nsurance contract for non-conventonal rsks, the nsured 2

3 frm, the nsurer and the rensurer, can, f the rsks are ambguous, have dvergng estmates of the rsks nvolved. Cabantous (2007) has examned the effects on nsurablty when dfferent partes have conflctng probablty estmates. She concludes that nsurers wll set a hgher premum for a rsk wth ambguous probablty than for a rsk wth non ambguous probablty. When the partes to an nsurance polcy dsagree on the set of probablty dstrbutons, she adds that nsurers wll set a hgher premum for an ambguous rsk wth a conflctng probablty than for an ambguous rsk wth a consensual but mprecse probablty. Of the technques avalable for the management of unnsurable corporate rsks s the settng up of a captve nsurance company (Porat and Powers 1995, 1999; Scords, Barrese, and Yokohama, 2007). The partcular case of envronmental rsks has been dscussed by Lesourd and Schlzz (2001). A captve nsurance company (hereafter smply referred to as "a captve") s an nsurance company entrely owned by a parent company, provdng nsurance servces manly, or exclusvely, to ts parent company. A captve provdes drect access to rensurance; however, n some cases a rensurance captve can be establshed, whch specalzes n rensurng ts parent company. The potental advantages of a captve le, frst, n ts drect access to rensurance, savng transacton costs of ntermedaton and, secondly, n the fact that there no longer s the rsk of moral hazard, snce the parent company and the captve should n prncple share the same nformaton. For coverng non-conventonal rsks, settng up a captve can be cheaper than contractng conventonal nsurance: the captve s able to charge lower premums than conventonal nsurance contracts, for varous reasons, some of whch wll be examned n ths paper. Captves consttute a partcular case, and a partcular practcal soluton, of the more general self-nsurance problem (Chu, 2000; Goller, 2003). Our concern n ths paper dffers from most of the lterature on nsurance captves, whch has mostly been concerned wth ther fscal status (see e.g. La and Wtt, 1995; Porat and Powers, 1995, 1999). Here we are nterested n determnng the condtons under whch a captve wll be preferred to conventonal nsurance when rsks are ambguous and the partes have dfferent and possbly conflctng perceptons of the rsks nvolved. Among the few studes that dscuss non-fscal ssues concernng captves, one can menton Dallo and Km [1989], who are nterested n asymmetrc nformaton, Scords and Porat (1998), who address captve manager-owner conflcts, and Scords, Barrese, and Yokohama (2007), who study the value of captves for ther parent company. Though these topcs are somewhat related to ours, ther approach, whch s emprcal and econometrc, s dfferent. For ths purpose, a model derved from the maxmn expected utlty (MMEU) models of Glboa and Schmedler (1989) and Kelsey (1994) s developed. A necessary consequence of 3

4 our model s that the nsurance premum can be too hgh for a conventonal nsurance contract to fnd a market. Ths s n lne wth, and adds to, what authors lke Enhorn and Hogarth (1985, 1986), Goller (2005, 2007) and Cabantous (2007) fnd, that f probabltes are ambguous, the nsurer wll ask for a hgher premum, or even refuse altogether to provde nsurance coverage. We study ths problem wth the am of dervng the condtons under whch a captve s a better soluton than conventonal nsurance for addressng corporate rsks characterzed by ambguty, as dscussed above. Our model leads to as yet unpublshed condtons for settng up a captve wth rensurance. The rest of our paper s organsed as follows. The second secton develops our model for nsurng ambguous rsks, and the thrd presents and dscusses the condtons under whch a captve s economcally attractve. A fourth secton concludes. 2. Insurng rsks under ambguous dstrbutons: the model Under condtons of ambguty n Ellsberg s (1961) sense, experments have shown that the nsured frm, the nsurer, and the rensurer can all have dfferent estmates of the unknown probablty dstrbuton (Cabantous, 2007). Ths s n lne wth the ambguty hypothess of Enhorn and Hogarth [1985, 1986]. These authors assume that an agent mplctly calculates a judged probablty functon whch can be ether larger or smaller than what they term an anchor probablty. They descrbe an anchor probablty as a frst estmate of the unknown probablty. When losses are at stake and when the agent s ambguty averse, ths estmate s larger or more pessmstc 1, meanng hgher cumulatve probabltes of losses, so that a more pessmstc dstrbuton domnates, n the sense of frst-order stochastc domnance, a less pessmstc one. Glboa and Schmedler [1989] and Kelsey [1994] developed a dscrete maxmn expected utlty (MMEU) model n whch an economc agent takes nto account a pessmstc estmate of some poorly known probablty dstrbuton. Ozdenoren, Casadessus and Klbanoff (2000) sutably generalzed ther approach to contnuous probablty dstrbutons. Bose, Ozderenen and Pape (2006) show, n the context of auctons, that ambguty, and stuatons such as the one descrbed by Ellsberg (1961), can be equvalently descrbed by Glboa and Schmedler s MMEU approach. Glboa and Schmedler (1989) and Kelsey (1994) developed a dscrete maxmn expected utlty (MMEU) model n whch an economc agent takes nto account a pessmstc estmate of some poorly known probablty dstrbuton. Ozdenoren, Casadessus and Klbanoff [2000] sutably generalzed ther approach to contnuous probablty 4

5 dstrbutons. Bose, Ozderenen and Pape [2004] L n show, n the context of auctons, that ambguty, and stuatons such as the one descrbed by Ellsberg [1961], are compatble wth Kelsey s MMEU approach. Applyng ths MMEU approach n the case of nsurance, we descrbe the case where, under a dscrete dstrbuton of losses, the nsured, the nsurer, and the rensurer, have dfferent perceptons of the dstrbuton of losses. Ths s related to Enhorn and Hogarth s (1985, 1986) model. Moreover, ther conclusons, rather than those of the standard subjectve expected utlty (SEU) model, have been vndcated by experment (D Mauro and Maffolett, 2001). Therefore, our MMEU approach seems approprate. Let us now develop our model. Let L 0, L 1, L 2,, L, L n be the possble losses, wth L 0 =0 < L 1 < L 2 < < L < L +1 < < L n-1 < L n = L wth dscrete probablty dstrbutons P d {p 0, n p 1, p 2,, p, p n } ( p = p 0 ). Let probablty dstrbuton P d {p 0, p 1, p 2,, p, p n } be = 1 a dstrbuton that s possble accordng to the agent s nformaton, but s less pessmstc than P d, wth p 1 > p 1, p 2 > p 2, p > p, p n > p n, and at least one j (0 < j < n) such that p j > p j. P d domnates dstrbuton P n the sense of frst-order stochastc domnance and s the most pessmstc dstrbuton wthn the set of dstrbutons that the frm assumes possble and compatble wth the nformaton avalable. Let U be the nsured frm s utlty functon, whch we assume to be a strctly ncreasng, twce dfferentable and strctly concave functon. We also assume that an nsurer s able to offer an nsurance polcy provdng for an nsurance premum π wth a deductble D, calculated on the bass of the nsured frm s dstrbuton P d. Our results have been derved usng the followng assumptons: Assumpton 1: The utlty functon s strctly ncreasng and concave, contnuous and twce dfferentable. Assumpton 2: The subjectve set of dstrbutons accordng to whch the nsured party s assumed to use for ts decsons s a set of dstrbutons for the possble losses L 0 =0 < L 1 < L 2 < < L < L +1 < < L n-1 < L n = L defned as the set of dstrbutons S d {s 0, s 1, s 2,, s, s n } such that, f P d = {p 0, p 1, p 2,, p, p n } ( p = p 0 ), and P d = {p 0, p 1, p 2,, p, p n } one has: n = 1 1 Goller (2007) ponts out that pessmsm s another word for ambguty averson. 5

6 p 1 > s 1 > p 1, p 2 > s 2 > p 2, p > s > p, p n > s n > p n, wth at least one (0 < < n) such that s j > p j. Assumpton 3: The decson crteron s MEU on the bass of the lmtng dstrbuton P d = {p 0, p 1, p 2,, p, p n } (the most pessmstc dstrbuton). The frm s mnmax expected utlty under ths polcy s, under ts P d dstrbuton : n MMEU = p U[R L + sup(l D,0) π] = 0 (1) In ths equaton, π = E[sup( L D,0)](1 + a) = E[ L D](1 a), s the nsurance premum, + wth an nsurance load factor a >0, and W(D, a, P d ) = R L + sup(l D, 0) - π. Let n + = p j j= h+ 1 P., the frst-order condton for maxmzaton of MMEU s : h + + MMEU = P D (1 + a) p U (R L π) + [P (1 + a) 1] U' (R D π) = 0 (2) = 0 In ths equaton, h s a state of nature such that, for all < h, L < D; for = h, L < D, and for j > h, L > D. For any gven a > 0, the second-order condton ensures a unque soluton n D snce : 2 MMEU = 2 D n = 0 2 W p D U"(W ) < 0 (3) Snce U (W ) always strctly negatve, ths s always negatve, so that there always exsts one, and only one optmal soluton D that obeys the frst-order condton (2). Let E(L D) be the mean of L D; we also have : 2 MMEU = P D a + (1 + a)e(l D) n = 1 p U"(W ) + P + (1 + a) n = 0 p U' (W ) > 0 (4) Ths quantty s always strctly postve, whence, defnng the optmum nsurance coverage, L n - D though equaton (2) as an mplct functon C(a) of the load factor a (the demand for C coverage), one can see that < 0, so that ths demand functon s strctly decreasng. a 6

7 We now dscuss the behavour of both the nsurer and the rensurer f the dstrbuton s not well known. It s reasonable to assume that both the rensurer and the nsurer, not knowng the actual dstrbuton, wll assume a more pessmstc probablty dstrbuton of the losses L ( > 0) than the nsured frm. Why n practce an nsurance company wll have less nformaton than the nsured frm about the loss dstrbuton P and wll therefore resort to a more pessmstc dstrbuton n order to prce ts premum, s open to dscusson. Several general reasons have recently been stated by several authors. Goller (2007) nvokes a number of reasons, among whch the most mportant are adverse selecton (some agents are more rsky than the average populaton, but the reasons for ths cannot be observed fully), so that nsurers wll ncrease ther premum rates for all ther clents, ex ante moral hazard (the nsured wll not dsclose some pertnent nformaton, such as ts efforts to decrease the nsured rsks), and ex post moral hazard (the rsk of fraudulent clams). Cabantous (2007) dscusses the general problem of ambguty averson, whch she defnes as uncertanty about the probablty, especally n the case of rsks wth a lack of large, relable hstorcal data base ; even f there s a consensual but mprecse probablty, ambguty wll nduce nsurers to set hgher premums, and, n the case n whch there are conflctng estmates of probablty ranges, hgher premums than n the consensual case. It seems reasonable to assume that the nsured frm wll not dsclose part of the nformaton avalable to t to any external party. Such dsclosure can be costly or techncally dffcult, and t could reveal secret detals of the frm s technology. Propretary nformaton and prvate nformaton gve the frm a compettve edge, and n many cases s unartculated, and hardly even artculable (Hayek, 1988) meanng that t cannot even be clearly stated. Even f the nsured frm wll dsclose such nformaton, t s reasonable to assume that the nsurer wll not wthout hgh costs have the ablty to gather such hghly techncal nformaton; n addton, such nformaton mght not appear suffcently trustworthy. In the case of a rensurer, thngs can be qute dfferent. Rensurers usually have larger captal reserves, and overall operate at a much larger scale than ordnary nsurers; thus they can afford techncal and engneerng expertse and acqure better nformaton on rsk dstrbutons. They also have better statstcs on the probabltes of large losses than ordnary nsurers. Fnally, the establshment of a captve whch can drectly negotate wth a rensurance company can be more effectve n protectng prvate techncal nformaton held by the frm. Let us specfy more precsely these assumptons. It seems reasonable to assume that the nsurer, n ts assessment of the dstrbuton P, due to the above nformaton problems under possbly conflctng estmates of the dstrbutons, wll ncrease ts premum above the optmal 7

8 premum calculated on the bass of the nsured frm s estmate of the dstrbuton. However, n the lmtng case of consensual but stll ambguous estmates, the nsurer could accept the already pessmstc dstrbuton P d. As far as the rensurer s concerned, one can reasonably assume that t s concerned only wth the hgher losses. Let us assume for smplfcaton that the rensurer s coverng only large losses, for nstance, losses above the loss L m wth m < n The rensurance market especally addresses the fact that nsurers are constraned by ther sze not to nsure major rsks, but n many cases they can acheve a lower prce by rensurng, so that nsurers can also be drven by market forces to rensure the rsk of the hgher losses. Thus, under our hypotheses, the rensurer wll cover only the rsk of the hgher loss L j (L j > L m > 0 for some m < n) n terms of non-proportonal rensurance. Although ths s a smplfyng assumpton, we also assume that the load factor a s the same for the nsurer and the rensurer, as argued by Blazenko (1986). Thus, the nsurer wll cover only the rsk of the lower losses L, 0 < < m, for whch t has less nformaton than the nsured frm, but as much nformaton as the rensurer. As far as the largest rsks are concerned, the rensurer can be assumed to possbly know better the probablty dstrbutons than the nsurer, but less than the nsured snce he s oblged to rely on the nsurer s apprecaton of the nsured s partcular stuaton. All ths results n the followng hypotheses : And : p < p d < p r = p R (0 < < m) (5) p < p d < p R < p r (m < < n) (6) We assume that an nsurer s able to offer an nsurance polcy wth an nsurance premum π d and a deductble D. The queston now s whether ths nsurance coverage s marketable or not, that s, acceptable or not for the nsured frm. Ths nsurance premum, wth a > 0 beng the load factor, s assumed to be exogenous. Ths premum s calculated under the dstrbuton of losses assumed by the nsurer (wth rensurance of the rsk of the hgher losses L j ). If the nsurer rensures the rsk of the hgher losses, L j, ths leads to a rensurance premum calculated accordng to the rensurer s estmate of the dstrbuton pertnent for the hgher rsk (P R ), More precsely : 8

9 π r = E r (L D)(1+ a) + E R (L j D)(1+ a) (0 < < m < j < n) (7) Let us defne the dfferences between the probablty dstrbutons p as assumed by the nsurer (p r for 1< < m), and by the rensurer (p jr for m < j < n), and ther respectve counterparts n the nsured frm s estmates as : q r = p r p d (0 < < m) (8) And : q jr = p jr p jd (m < j < n) (9) Clearly, accordng to our hypotheses, q r > 0 (0 < < m), and q jr > 0 (m < j < n). It s evdent that, f we consder the MEU functon of the nsured frm takng nto account the premum MEU prced accordng to (7), < 0 q r MEU (0 < < m); < 0 q jr (L n D) (m < j < n), and < 0 q r (0 (L n D) < < m); < 0 q jr (m < j < n), so that MEU and the deductble S are strctly decreasng wth the q r (0 < < m), and the q jr (m < j < n). 3. Condtons underlyng the attractveness of a captve Equaton (2) gves as an mplct functon the demand for nsurance, whch s a strctly decreasng functon of a. However, f one assumes that both the nsurer and the rensurer are rsk-neutral, the supply curve wll be a vertcal lne a = constant. Ths also means the load factor a, nterpreted as the prce of nsurance coverage, s exogenous. It reflects the costs of coverage wth a sutable mark-up for proft; t s unque f the market s perfectly compettve. Thus, as shown on Fgure 1, for each value of a (for example a 1 ) the equlbrum demand for nsurance s the ntercept of the demand lne and of a vertcal lne wth a = constant, provded such an ntercept exsts for a postve and meanngful value of coverage (.e. to the left of a 0 ). If the nsurer, as s assumed here, covers the rsks for the smaller loss L 1, the deductble D (besdes beng nonnegatve) has to be smaller than L m. 9

10 Thus, gven our assumptons, a necessary but not suffcent condton for feasblty of nsurance coverage s: 0 < D < L n (9) Proposton 2 leads us to dscuss several possbltes that can occur n terms of the optmal nsurance decson for the nsured frm. Frstly, n the lmtng case where p 1r = p 1d and p 2rr = p 2d, and f 0 < D < L 1, nsurance wth rensurance, or establshng a captve wth rensurance, s ndfferent. Secondly, stll under p 1r = p 1d, f p 2d < p 2rr, and f 0 < D < L 1, establshng a captve wth rensurance s the optmal soluton. Ths means that cooperaton between the captve and the rensurer wll provde some addtonal nformaton about the dstrbuton of the larger loss, resultng n the economcally more attractve soluton of a captve wth rensurance. Ths s emprcally observed as dscussed n Lesourd and Schlzz [2003: chapter 7] and others. Thrdly, f p 1d < p 1r, and f p 2d = p 2rr, and f 0 < D < L 1, establshng a captve wth rensurance s agan the optmal soluton. Ths means that the captve has some addtonal nformaton about the dstrbuton of the smaller loss, but that drect access to rensurance provdes no addtonal nformaton about ts dstrbuton; ths agan leads to the economcally more attractve soluton of a captve wth rensurance. Fourthly, f p 1d < p 1r, and f p 2d < p 2rr, and f 0 < D < L 1, establshng a captve wth rensurance s a thrd case where t s an optmal soluton. Ths means that the captve has some addtonal nformaton about the dstrbuton of the smaller loss, and ts drect access to rensurance also provdes addtonal nformaton about the dstrbuton of the larger loss. It can also happen that the necessary condton for nsurance (0 < D < L 1 ) s not met, but that demand for rensurance s stll postve (L 1 < D < L 2 ). In ths case, nsurance coverage would be too expensve at current market prce and under the nsurer s estmate of the rsk. Then, nether the external nsurance company, nor an nsurance captve can provde any nsurance coverage. A rensurance captve can, n ths case, be useful to rensure aganst the hgher rsk. Table 1 hereafter summarses all these condtons. 4. Concluson 10

11 Ths paper addresses the problem of nsurng non-conventonal rsks such as envronmental rsks, product rsks, medcal malpractce, ltgaton-related rsks and terrorsm. Conventonal rsks, such as, for nstance, rsks related to road accdents, theft and fre, can be charactersed by well known probablty dstrbutons whch can be assumed to be avalable precsely and at low cost. On the contrary, non-conventonal rsks are charactersed by the fact that ther probablty dstrbuton s not well known. In ths case, the nsured frm, the nsurer and the rensurer all dffer n ther estmates of the probablty dstrbuton, whch leads to dfferent prces on the nsurance or rensurance markets. Ths s a stuaton frst descrbed n terms of ambguty by Ellsberg n 1961 and later by Enhorn and Hogarth [1985, 1986]. We address ths problem by developng a maxmn expected utlty model as n Glboa and Schmedler [1989], and as n Kelsey (1994), whch, as shown by Bose, Ozdenoren and Pape [2004], s equvalent to a descrpton n terms of ambguty. More precsely, we assume on reasonable grounds that (1) rensurance companes specalse n hgher rsks for whch they have a compettve advantage leadng to a lower estmate of the probablty of these hgher rsks, that (2) nsurers can be less nformed than the nsured frm about the lower rsks and that (3) there can be a gan for an nsured frm from drect access to rensurance. Under these assumptons, we are led to specfy condtons under whch the operaton of a captve wth rensurance of hgher rsks s optmal for the frm. Ths wll occur whenever the nsured frm, and the captve t establshes, has more nformaton about the lower rsks than a standard nsurance company, and whenever drect access to rensurance through a captve s proftable. Wthn a smplfed framework, we provde as yet unpublshed condtons on the perceved probabltes of both the lower and the hgher rsks. If the demand for nsurance coverage s postve, under the above condtons, the operaton of a captve wth rensurance s optmal. In the future, t mght be nterestng to extend our approach to more general dstrbutons, and to ntroduce costly nformaton as well as moral hazard nto our model. Followng D Mauro and Maffolett [2001] and other earler work such as Hogarth and Kunreuther [1985], Camerer [1987], and Camerer and Kunreuther [1989], an expermental nvestgaton of the predctons of our model mght allow us to probe deeper nto ths ncreasngly mportant ssue. 11

12 Condtons P jd = p jr P jd < p jr P d = p r P d < p r P d = p r P d < p r P d = p r P d < p r 0 < D < L m Captve wth rensurance, or nsurance wth rensurance ndfferent Captve wth rensurance optmal, nsurance wth rensurance suboptmal L m < D < L n Rensurance captve, or nsurance wth rensurance ndfferent Rensurance captve optmal, nsurance wth rensurance suboptmal L n < D No nsurance and rensurance feasble No nsurance and rensurance feasble Captve wth rensurance optmal, nsurance wth rensurance suboptmal Captve wth rensurance optmal, nsurance wth rensurance suboptmal Rensurance captve optmal, nsurance wth rensurance suboptmal Rensurance captve optmal, nsurance wth rensurance suboptmal No nsurance and rensurance feasble No nsurance and rensurance feasble Table 1 12

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