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1 econstor Der Open-Access-Pblikationsserver der ZBW Leibniz-Informationszentrm Wirtschaft The Open Access Pblication Server of the ZBW Leibniz Information Centre for Economics MacMinn, Richard; Richter, Andreas Working Paper Hedging Brevity Risk with Mortality-based Secrities Discssion Papers in Bsiness Administration, No Provided in Cooperation with: University of Mnich, Mnich School of Management Sggested Citation: MacMinn, Richard; Richter, Andreas (26) : Hedging Brevity Risk with Mortality-based Secrities, Discssion Papers in Bsiness Administration, No. 26-3, nbn-resolving.de/rn:nbn:de:bvb:9-epb-29-6 This Version is available at: Ntzngsbedingngen: Die ZBW rämt Ihnen als Ntzerin/Ntzer das nentgeltliche, rämlich nbeschränkte nd zeitlich af die Daer des Schtzrechts beschränkte einfache Recht ein, das asgewählte Werk im Rahmen der nter nachzlesenden vollständigen Ntzngsbedingngen z vervielfältigen, mit denen die Ntzerin/der Ntzer sich drch die erste Ntzng einverstanden erklärt. Terms of se: The ZBW grants yo, the ser, the non-exclsive right to se the selected work free of charge, territorially nrestricted and within the time limit of the term of the property rights according to the terms specified at By the first se of the selected work the ser agrees and declares to comply with these terms of se. zbw Leibniz-Informationszentrm Wirtschaft Leibniz Information Centre for Economics

2 with Mortality-based Secrities Richard MacMinn, Andreas Richter Discssion Paper 26-3 October 26 LMU LUDWIG-MAXIMILIANS-UNIVERSITÄT MÜNCHEN MUNICH SCHOOL OF MANAGEMENT Online at

3 with Mortality-based Secrities By Richard MacMinn Edmondson-Miller Chair College of Bsiness Illinois State University Normal, Illinois Andreas Richter Professor, Chair in Risk and Insrance Ldwig-Maximilians-University Mnich, Germany First draft September 24 Latest draft October 26 Abstract In 23, Swiss Re introdced a mortality-based secrity designed to hedge excessive mortality changes for its life book of bsiness. The concern was apparently brevity risk, i.e., the risk of prematre death. The brevity risk de to a pandemic is similar to the property risk associated with catastrophic events sch as earthqakes and hrricanes and the secrity sed to hedge the risk is similar to a CAT bond. This work looks at the incentives associated with insrance-linked secrities. It considers the trade-offs an insrer or reinsrer faces in selecting a hedging strategy. We compare index and indemnity-based hedging as alternative design choices and ask which is capable of creating the greater vale for shareholders. Additionally, we model an insrer or reinsrer that is sbject to insolvency risk, which creates an incentive problem known as the jdgment proof problem. The corporate manager is assmed to act in the interests of shareholders and so the jdgment proof problem yields a conflict of interest between shareholders and other stakeholders. Given the fact that hedging may improve the sitation, the analysis addresses what type of hedging tool wold be best to se. We show that an indemnity-based secrity tends to worsen the sitation, as it introdces an additional incentive problem. Index-based hedging, on the other hand, nder certain conditions trns ot to be beneficial and therefore clearly dominates indemnity-based strategies. This reslt is frther spported by showing that for the same strike prices the crrent shareholder vale is greater with the index-based secrity than the indemnitybased secrity. Keywords: alternative risk transfer, insrance, defalt risk JEL classification: G22, G32, D82

4 Introdction As poplations grow and change, markets integrate and medical science advances, the character and strctre of the risks faced by governments, corporations and individals also change. The threat of SARS in 23 and avian fl in 24 have provided reminders that governments and life insrers face correlated risks on a large scale in events sch as pandemics. In December 23, Swiss Re introdced a mortality-based secrity designed to hedge excessive mortality changes for its life book of bsiness. The concern was apparently brevity risk, i.e., the risk of prematre death. Brevity risk can be managed with the standard tools as long as there are no correlated mortality srprises. Sch wold be the case with a recrrence of the Spanish fl or more generally with the occrrence of a new avian fl. The potential for pandemics introdces correlated risks on a large scale and so the potential for mortality srprises. The brevity risk de to a pandemic is similar to the property risk associated with catastrophic events sch as earthqakes and hrricanes and the secrity sed to hedge the risk is similar to a CAT bond (Dbinsky and Laster 23). The model constrcted here is designed to analyze the potential seflness of mortalitybased secrities in hedging risk. A pblicly held and traded corporation with books of life bsiness is constrcted. The corporation may be an insrer or reinsrer, althogh it will be referred to as a reinsrer throghot most of this article. The organization is strctred so that it faces brevity risk in addition to other standard risks sch as credit and interest rate risk. Under these conditions, a (re)insrer facing a capital constraint may find a mortality-based secrity to be a natral risk management tool and therefore trn to the capital markets to hedge these risks. It may also traditionally reinsre its book of bsiness. The model employed here is sfficiently general to allow for both types of instrments to be considered. The focs, however, is aimed at highlighting a design choice that is particlarly important in catastrophe bond isses; the qestion is whether an index or indemnity trigger is sed as the nderlying for sch a transaction. The literatre on Alternative Risk Transfer (ART) explains how the secritization of catastrophic exposres can create vale. Some articles have identified the trade-offs involved in the design of optimal risk management programs integrating traditional (re)insrance and ART instrments (Doherty 997; Froot 997; Croson and Knrether 2). On the one hand, secritization of insrance risk offers advantages over traditional reinsrance arrangements, sch as the potential to sbstantially redce moral hazard, credit risk and transaction costs. On the other hand, possible improvements typically come at a cost of the basis risk incrred by an index-linked transaction; this is tre since an index cannot perfectly represent the individal risk and wold therefore only provide an imperfect hedge. This recent literatre focses on transactions based pon index triggers. This approach seems jstified in light of empirical observations in the CAT bond market: While earlier CAT bond isses were mainly based pon indemnity triggers (which have also traditionally been sed in insrance and reinsrance coverage), the last few years indicate a greater market share for indexed instrments (McGhee, Fast and Clarke 25). 2 As index-based soltions create the problem of basis risk, their recent poplarity A similar mortality-based instrment was introdced by Swiss Re in April 25. See Artemis the alternative risk transfer portal. 2 A recent stdy by Gy Carpenter & Company (Gy Carpenter 25), for instance identifies new risk capital in the amont of $95.3 million ($.47 billion) that was provided throgh index-linked CAT bonds in 24 (23),

5 natrally raises the qestion of why the indstry prefers index over indemnity triggers. The straightforward answer is that, besides potentially redcing transaction cost, an appropriately constrcted index redces or eliminates moral hazard. The introdction of a catastrophe index in a CAT bond isse or the se of a poplation s average life expectancy in a mortality-based secrity solve the moral hazard problem inherent in almost any insrance or reinsrance transaction. An index trigger is a new device for addressing moral hazard. If compensation from a reinsrance contract or any other hedging instrment is based pon an index beyond the hedging party s control, this party will still reap the entire benefit of loss control in addition to the hedge. The other party, e.g. a reinsrer or the investors in an insrancelinked secrity, do not need to be concerned anymore abot monitoring the cedent s, respectively the isser s, risk selection or loss-handling practices. A trade-off reslts between these benefits and the basis risk incrred by the index instrment. A few papers have addressed the trade-offs analytically: Cmmins and Mahl (Cmmins and Mahl 2) consider an insrance prodct that is sbject to credit risk as well as basis risk, 3 as the insrer s payment is tied to an exogenos index. The interaction between these two factors is also analysed by Richter (Richter 23) albeit with two different instrments: Insrance, on the one hand, is sbject to credit risk bt can be sed to generate a perfect hedge. Risk secritization, on the other hand, comes withot credit risk bt incrs basis risk. The analysis shows that nder these conditions the indexed secrity is beneficial whenever the credit risk on the reinsrance exists. As a tool that mainly conteracts reinsrance credit risk, secritization primarily replaces reinsrance for high levels of the loss. The latter reslt is confirmed by Nell and Richter (Nell and Richter 24) who stdy the trade-off between the implicit transaction cost incrred by a reinsrer s risk aversion and the basis risk of a CAT bond. The trade-off between moral hazard and basis risk has been discssed analytically by Doherty and Mahl (Doherty and Mahl 2) and Doherty and Richter (Doherty and Richter 22), who investigate the interaction of these two problems, when insrance can be sed to cover the basis risk of an index-linked transaction. It is shown that combining the two hedging tools might extend the possibility set and therefore lead to efficiency gains. This work will consider the trade-offs an insrer or reinsrer faces in selecting a hedging strategy to maximize crrent shareholder vale. Like Doherty and Mahl and Doherty and Richter we consider index and indemnity triggers; the focs here, however, is on a pblicly held and traded corporation acting in the interests of shareholders rather than a manager maximizing expected tility. Rather than considering a mix of hedging instrments, we compare index and indemnity-based hedging as alternative design choices and ask which is capable of creating the greater vale for shareholders. Additionally, and very importantly, we model an insrer or reinsrer that is sbject to insolvency risk; this risk of insolvency creates an additional incentive problem known as the jdgment proof problem. The corporate manager is assmed to act in the interests of shareholders and so the jdgment proof problem yields a conflict of interest between shareholders and other stakeholders. The jdgment proof problem then yields a sitation in which management does not have an incentive to select the socially optimal level of care. while new indemnity-based transactions only amonted to (26) million. Contrasting this, indemnitybased transactions in 998 (997) amonted to $846. ($43) million while index-based CAT bonds generated risk capital in the amont of $ ($22 million). 3 We refer to one of the risks as credit rather than defalt risk since the organization that is the object of analysis is not sbject to defalt bt rather owns a contract that is sbject to defalt. The recently pblished version of (Cmmins and Mahl 24), however, does not inclde the basis risk. 2

6 A soltion for the nderinvestment problem sggested in the risk management literatre is that potential creditors demand that the corporation hedges insolvency risk, e.g., (Jensen and Meckling 976; Smith and Stlz 985; Mayers and Smith 987; Froot, Scharfstein and Stein 993; Garven and MacMinn 993; MacMinn 25). This reqirement can be enforced, for instance, by adding a covenant to the debt that reqires the company to hedge. Given the fact that hedging improves the sitation, the following analysis will address, in light of the new financial instrments described above, what type of hedging tool wold be best to se. We ask whether one of the two types of hedging that were discssed earlier is better than the other as a soltion for the incentive distortions created by insolvency risk. Ths, the primary interest here is in the incentives associated with index verss indemnity-based insrance-linked secrities (or other forms of hedging) in a framework in which the isser faces the risk of insolvency. We consider the impact sch secrities have on the company s actions, e.g., regarding nderwriting. The analysis here extends previos work by incorporating basis risk and moral hazard in modeling an insrer or reinsrer facing insolvency risk on its books of bsiness. In a financial market setting, we show that an indemnity-based secrity tends to worsen the sitation, as it introdces the additional incentive problem. Index-based hedging, on the other hand, nder certain conditions trns ot to be beneficial and therefore clearly dominates indemnity-based strategies. This reslt is frther spported by showing that for the same strike prices the crrent shareholder vale is greater with the index-based secrity than the indemnity-based secrity. The financial market model with a life reinsrer s brevity risk is introdced in the next section. The socially efficient operating decisions are also derived here. The following section on triggers and incentives introdces the indemnity and index instrments; there the incentive effects of each are analyzed. The penltimate section compares the crrent shareholder vales for the indemnity and index triggers given the same strike prices on those options. A Financial Market Model with Brevity and Insolvency Risk Consider an organization in a competitive economy operating between the dates t = and (MacMinn 987). The dates t = and are sbseqently referred to as now and then, respectively. Decisions are made now and payoffs on those decisions are received then. The economy is composed of organizations and risk averse investors. Investors make portfolio decisions on personal accont to maximize expected tility sbject to a bdget constraint. The organization will initially take the corporate form and will be assmed to act on behalf of its principals, i.e., the investors who are shareholders. 4 In a standard reinsrance transaction the profitability of a contract depends on the cedent s loss control effort or eqivalently the level of care. A (re)insrer selects a portfolio of risks and negotiates the contract terms with the insred. This incldes reqired safety and loss redction operations as well as aspects of prodct design sch as dedctibles, retention levels or coinsrance arrangements. When claims arise, a primary settles those claims with its policyholders. Each of these activities and considerations is costly, yet each activity can affect the freqency and severity of claims. If a primary is heavily reinsred or if a reinsrer is covered by a significant retrocession, it still bears the cost of loss redction, bt the other contracting party reaps the benefit. To address 4 The assmption is only for convenience. The corporate objective fnction can be derived; for example, see MacMinn (25). 3

7 this incentive conflict, reinsrance has contractal controls. Contracts may be experience rated or retrospectively priced. Additionally, long term and brokered relationships are common in reinsrance which provide frther incentive to ndertake loss control. We abstract from the plethora of methods available to the reinsrer to manage its risks and focs on a single activity we call the level of care and sppose that increasing the level of care redces risk. The insrance company considered will face the standard capital market risks sch as interest rate and insolvency risks bt will also face life risk. The premim income will be generated now and invested in a capital market portfolio. The composition of the organization s portfolio will be determined endogenosly and will be shown to depend on the risk management choices made. The losses on the books of bsiness occr then and depend on the state of natre revealed. The following partially smmarizes the notation sed in the development of the model: ω state of natre Ω=, set of states p(ω) P(ω) basis stock price now sm of basis stock prices ε ω; ω P( ω ) = p( ε)d ε Γ(ω) premim income then on the book of bsiness; Γ > 5 a effort spent in composing the book of bsiness measred in dollars La, ( ω) loss on book of bsiness Π ( a, ω) payoff on book of bsiness, i.e., ( ) ( ) Π a, ω = Γ( ω) L a, ω a I(ω) mortality index i exercise price for mortality secrity S stock vale 5 As the economy improves in state so does the premim income then since the premim income is invested. 4

8 Sppose the financial markets are competitive. In the absence of any insrance-linked secrity and any insolvency risk, the stock market vale of the (re)insrer may be expressed as the vale of its books of bsiness as follows: { } S(a) = max, Π dp = ΠdP Ω () Consider the reinsrer. The reinsrer has the payoff max{, Π} in the absence of the mortality-based secrity. The reinsrer may create a mortality-based secrity for its life book by forming a special prpose entity (SPE) similar to that for a CAT bond; the essence of the secrity from the perspective of the insrer, however, is the creation of an option that yields a payoff of L(a, ω) dollars in state ω for losses on its life book in excess of a trigger amont i; eqivalently, the secrity pays {, L( a, ) i} max ω. This is the case if the reinsrer ses an indemnity trigger. Alternatively, if the reinsrer ses an index trigger then the essence of the SPE is the creation of a secrity that yields an indexed payoff of I(ω) in state ω for losses on its life book in excess of a trigger amont i; max, I ω i eqivalently, the index secrity pays { ( ) } In an economy sch as this the 958 Modigliani-Miller theorem (Modigliani and Miller 958) will hold and so mortality-based secrities will not, ceteris paribs, increase the stock vale of the (re)insrer. Once the ceteris paribs assmption is relaxed we will investigate the incentive effects of the mortality-based secrities and ltimately the impact on vale. Additionally, by adding longevity risk and a similar mortality-based secrity for life annity books of bsiness, the natral hedge between the (re)insrer's life and annity books of bsiness might be investigated. From the reinsrer s perspective the life book of bsiness exposes the corporation to the risk that an insred s life is briefer than expected and so we refer to it as brevity risk. The reinsrer may also be exposed to insolvency risk and credit risk in the financial markets; the insolvency risk introdces the jdgment proof problem with the associated incentive problems. The ability to secritize some risk in capital markets introdces a moral hazard problem in the nderwriting operation. In this section we concentrate on the life book of bsiness and the associated risks. First, consider the vale of the reinsrer with and withot the mortality-based secrity; we will refer to the mortality-based secrity more generally as an insrance-linked secrity (ILS). The nhedged reinsrer has a stock vale S. If there is insolvency risk in the event of a pandemic then let the state δ be implicitly defined by Π(a, δ ) =. The nhedged stock vale may then be expressed as { } S(a) = max, Π(a, ω)dp( ω) Ω = Π(a, ω)dp( ω) δ (2) Observe that the reinsrer selects the nderwriting effort to maximize the crrent shareholder vale. The first order condition is 5

9 ds da ( ) = D Π a, ω dp( ω) δ ( ) = D L(a, ω) dp( ω) 6 (3) δ = Eqation (3) implicitly defines the optimal nderwriting activity a. The nderwriting effort by the reinsrer is assmed to redce the brevity risk. This assmption is formalized in the following: Assmption One. The reinsrer s payoff L(a, ω ) satisfies the principle of decreasing ncertainty (PDU) and the PDU is defined by the following derivative properties: D2L < and D2L >. 7 After compensating for the change in the mean, the PDU provides a decrease in the risk of the payoff in the Rothschild-Stiglitz sense (Rothschild and Stiglitz 97; MacMinn and Holtmann 983). Next, consider the second order condition. Observe that ds ( ) ( ) 2 = D Π ω ω Π δ δ < 2 δ a, dp( ) D a, p( ) (4) da da The concavity of Π or eqivalently the convexity of L sffices to make the first term on the right hand side of (4) negative. The PDU sffices to show that D Π(a, δ ) > and dδ D Π(a, δ) 2 dδ = da D Π(a, δ ) < (5) Hence, the second order condition holds if the second term on the RHS of (4) is less than the first. It may also be noted that the second order condition redces to jst the first term in the absence of insolvency risk and so the concavity of the payoff sffices to show that the condition holds. We will assme that the second order condition is satisfied in the remaining analysis. It is sefl to compare the nderwriting decisions of the firms with and withot insolvency risk. Recall that Shavell (Shavell 986) has described the sitation in which an insrer does not possess the resorces to cover all losses with certainty as the jdgment proof problem. The insrer facing the jdgment proof problem does not have the incentive to select the socially efficient level of care as noted in the following claim e and proof. Let a denote the level of care or eqivalently the nderwriting choice made by the reinsrer with no insolvency risk. 8 6 DΠ is standard notation for the partial derivative of the fnction Π with respect to its first argment. Similarly D2Π is standard notation for the partial derivative of the fnction D2Π with respect to its first argment. 7 See MacMinn and Holtmann 983 for a description of this principle. It is a mirror image of the principle of increasing ncertainty introdced by Leland, i.e., see Leland, H. (972). "Theory of the Firm Facing Uncertain Demand." American Economic Review 62: The socially efficient care is that level that maximizes the vale for all stakeholders in the enterprise and so can also be described in sitations with insolvency risk as well. Eqation (6) wold still apply. 6

10 Claim: The level of care selected by the reinsrer is greater in the absence of insolvency e risk, i.e., a > a Proof. In the absence of insolvency risk the vale is S e where e S(a) = Π(a, ω)dp( ω) (6) and the first order condition for a socially optimal level of care is e ds da e D (a, )dp( ) = Π ω ω = Hence, the claim follows by noting that ( ) e ( ) D L(a, ) dp( ) (7) = Γω ω ω e ds da ds = D Π(a, ω)dp( ω) D Π ω ω δ (a, )dp( ) da a= a δ = D Π(a, ω)dp( ω) (8) > follows by the PDU since ω δ. QED D Π(a, δ) is positive and D Π < 2 yields D Π(a, ω ) > for all It may also be noted that the social optimm noted in eqation (7) is, with appropriate disconting, eqivalent to the optimm noted in the literatre by (Shavell 986; Kahan 989; MacMinn 22). The social optimm is the level of care sch that the present vale of the marginal benefit eqals that of the marginal cost, as seen in the following rewrite of eqation (7) e ds da e ( ) = DL(a, ω) dp( ω) e = DL(a, ω)dp( ω) dp( ω) (9) = The first term on the RHS of the second eqality is the marginal benefit or eqivalently the present vale of the marginal loss redction and the second term is the marginal cost or eqivalently the present vale of the last dollar spent on care. 7

11 Triggers and Incentives Next, consider the introdction of insrance-linked secrities. The first case considered here is that of a mortality-based bond issed by an SPE. The instrment is designed to pay the reinsrer in the event of mortality srprise, e.g., if the mortality rate is 3% or more of what had been projected. Sch an instrment may be constrcted with an indemnity, index or parametric trigger. In the indemnity case the payoff from the perspective of the reinsrer wold be an option payoff like max{, L(a, ω) i} where i is the strike price. In the second case of an index trigger the payoff from the perspective of the reinsrer wold be max{, I(ω) - i} where I(ω) is the loss index. Indemnity trigger The indemnity trigger case of an insrance-linked secrity costs C m dollars now where C m is the call option price for the coverage. Then Ω { } m C (a,i) = max,l(a, ω) i dp γ ( L(a, ) ) = ω i dp () where γ is the bondary of the option s in the money event as shown in figre one. 9 The stock vale now of the corporation with this ILS is Ω { { }} m S (a,i) = max, Π(a, ω ) + max,l(a, ω) i dp ( { }) = Π(a, ω ) + max,l(a, ω) i dp () where is the bondary of the insolvency event as shown in figre (b). Finally the crrent shareholder vale is S mo 9 The payoffs in the figres will be represented as linear only de to the athors limited drawing ability; the analysis does not depend on the linear fnctions represented in the figres. 8

12 mo m m S (a,i) = C (a,i) + S (a,i) γ ( ) ( { }) = L(a, ω) i dp + Π(a, ω ) + max,l(a, ω) i dp (2) ( ) ( ( )) = L(a, ω) i dp + Γ( ω) a L a, ω dp o Let S denote the crrent shareholder vale in the nhedged case with no insolvency risk. It follows that if the probability of insolvency is zero so that is zero then (2) becomes mo m m S (a,i) = C (a,i) + S (a,i) γ ( ) ( { }) = L(a, ω) i dp + Π(a, ω ) + max,l(a, ω) i dp = Π(a, ω)dp (3) = S o (a) and (3) demonstrates that no vale is added by the ILS. This is confirmation of the generalized version of the 958 Modigliani-Miller theorem. It is achieved with the sal ceteris paribs assmption; in this case no change in nderwriting care de to the ILS is allowed in making the comparison and that assmption will be changed in the mo sbseqent analysis. It may also be noted that the crrent shareholder vale S o diminishes relative to the nhedged crrent shareholder vale S given insolvency risk; this is not a real dimintion of vale bt rather a redistribtion of vale from shareholders to policyholders or other stakeholders. ( ) δ mo o S S = L(a, ω) i dp + Π(a, ω)dp Π(a, ω)dp( ω) ( ) = L(a, ω) i dp + Π(a, ω)dp δ (4) < Incentive effects of the indemnity trigger The ILS with an indemnity trigger will have an impact on the incentive to take care in nderwriting. The indemnity trigger has the effect of fll loss coverage in some states and that in trn impacts the nderwriting choice; eqivalently, a well known moral hazard problem (Shavell 979) occrs with this form of the ILS. The relationship between the option coverage and the nderwriting care will be specified in the fnction m a (i) where i is the exercise price of the option. The fnction a m is implicitly defined by the first order condition for the stock vale expressed in eqation (). 9

13 m S Π γ L = dp + dp a a a L γ L = dp + dp a a (5) = Sppose the second order condition for a maximm holds. Then the fnction and its derivative is m a (i) exists da di m 2 m S a i = 2 m S 2 a (6) if the nmerator is non-negative. To see that the nmerator in (6) is non-negative observe that S L L dp dp a i i a a 2 m γ = + L L L γ = p( ) p( ) + p( γ) a i a i a i (7) L γ = p( ) + p( γ) i a i > The ineqality in (7) follows becase is increasing in i, γ is decreasing in i and L is decreasing in a. Therefore, the ineqality in (6) is a strict ineqality and a m (i) is increasing in the strike price i. Figre one and eqation () are appropriate for some exercise prices bt there are two more generic cases that mst also be inclded. First, there is an exercise price î sfficiently large that eqals δ; for larger i the bondary of the insolvency event remains at δ. For i iˆ the stock vale is that provided in eqation (7) and so a m (i) is a Note that is implicitly defined by the condition Π(a, ) + ( L(a, ) i) = or eqivalently by Γ ( ) a i = and so = >. i Γ ( ) Similarly, γ is implicitly defined by the condition L(a, γ) i = and so γ = <. i L γ

14 constant on the interval i iˆ. Second, there is an exercise price i* sch that eqals zero and so i* eliminates insolvency risk; for i i* the stock vale is ( ) γ m S = Π+ L i dp+ ΠdP γ γ ( ) ( ) = Γ a i dp + Γ L a dp γ (8) and again it is apparent that a m (i) is a constant on the interval i i*; i* may bt need not be as small as zero bt at i = the care decision clearly goes to zero as shown in figre two. The reslts are collected in figre two. Note that a m (i) is non-decreasing and this is confirmation of a moral hazard problem since an increase in the strike price i is eqivalent to less loss coverage and that generates more care in nderwriting bt more loss coverage generates less care. The figre also sggests that the indemnity trigger ILS cannot provide the incentive for adeqate care in nderwriting since the maximm care is that for the nhedged case. Clearly, rather than improving or solving the incentive problem the introdction of the indemnity hedge aggravates the problem. Index trigger The index trigger case of an insrance-linked secrity costs C b dollars now where C b is the call option price for the coverage. Then ( ) { } b C(i) = max,i( ω) idp Ω = I( ω) i dp η (9) where η is the bondary of the in the money or eqivalently the exercise event for this option as shown in figre three. This claim may be jstified by direct calclation since γ increases withot limit.

15 The stock vale of the corporation with this ILS is { { }} b S (a,i) = max, Π(a, ω ) + max,i( ω) i dp Ω ( { }) = Π(a, ω ) + max,i( ω) i dp β (2) where β is the bondary of the insolvency event as shown in figre 3(b). The crrent shareholder vale in this case is bo b b S (a,i) = C (i) + S (a,i) η ( ) ( { }) = I( ω) i dp + Π(a, ω ) + max,i( ω) i dp = I( ω) i dp + Γ( ω) a+ I( ω) L(a, ω) i dp β η η ( ) ( ( ) ) β (2) ( ) + Γ( ω) a L(a, ω) dp η Note that the second term on the RHS of the third eqality incldes the basis risk (I L). The connection with the generalized 958 Modigliani-Miller theorem can be made here as in the last case when the insolvency risk is zero. Incentive effects of the index trigger The ILS with an index trigger will have an impact on incentive to take care in nderwriting. Unlike the indemnity trigger, this instrment does not generate a moral hazard problem bt it does generate basis risk. The relationship between the option coverage and the nderwriting care will be specified in the fnction a b (i) where i is the exercise price of the option. The fnction a b is implicitly defined by the first order condition for eqation (2) 2

16 b S a = β Π dp a L = dp β a (22) = Sppose the second order condition for a maximm holds. Then the fnction a b (i) exists and its derivative is da di b 2 b S a i = 2 b S 2 a (23) if the nmerator non-positive. Sppose that ( I ω L ω ) > so that the corporate loss distribtion has more weight in its tails. Then observe that 2 b S L = dp a i iβ a L β = p( β ) a i (24) < The ineqality follows becase β is an increasing fnction of i and by the PDU the term in parentheses is positive at β. 2 The ineqality in (24) yields a care fnction a b (i) that is decreasing in i. Figre three and eqation (2) are appropriate for some exercise prices bt again there are two more generic cases that mst also be inclded. First, there is an exercise price i sfficiently large that β eqals δ; for larger i the bondary of the insolvency event b remains at δ. For i i the stock vale is that provided in eqation (2) and so a(i) is a constant on the interval i i. Second, if there is an exercise price i sch that β eqals zero then i eliminates insolvency risk; for i i the stock vale is 2 Note that is implicitly defined by the condition Π(a, β ) + ( I( β) i) = or eqivalently by Γβ ( ) a ( L(a, β) I( β) ) i = and so β = >. i I L β β 3

17 ( ) γ b S = Π+ I i dp+ ΠdP γ γ ( ) ( = Γ a + (I L) i dp + Γ L a dp γ ) (25) and again it is apparent that a b (i) is a constant on the interval i i if an i exists which makes β =. If sch an i exists then below that threshold the care becomes the socially efficient choice as the following derivative shows b S γ L L = dp+ dp a a γ a L = a dp (26) = b since (26) is eqivalent to (7). The reslts are collected in figre for. Note that a(i) is non-increasing and this is confirmation that the moral hazard problem can be eliminated and the incentive distortions de to insolvency risk can be mitigated or eliminated. Figre 4 a am (i) i - ^ i* i i i The analysis shows that the index trigger dominates the indemnity trigger in the sense that it redces the insolvency withot creating an incentive problem; the indemnity trigger, on the other hand, redces the insolvency risk bt engenders an incentive that tends to increase the insolvency risk. The dominance is investigated in the next section by comparing crrent shareholder vales. Comparison of Crrent Shareholder Vales The analysis shows that the insrance-linked secrity with an index trigger can nder certain conditions provide the corporate manager, ceteris paribs, with the incentive to take additional care as the level of protection is increased. The secrity with an indemnity trigger, however, does not align incentives and in fact an additional protection provides the corporate manager, ceteris paribs, with an incentive to redce rather than increase care. Indeed, in the case of the indemnity trigger the care level taken by an nhedged firm provides an pper bond on the care that the manager with this 4

18 instrment will take. Still, the two triggers provide different crrent shareholder vales first becase of the difference in the cost of the protection and second becase of the difference in incentive effects. The goal of this section, therefore, is to compare the crrent shareholder vales with the index verss the indemnity trigger, taking into accont the incentives associated with bo and S evalated mo m these instrments. Ths we will compare S evalated at ( a (i),i) at ( ). b a(i),i First note that from (2), the crrent shareholder vale in the case of index trigger for a given (a, i) may be rewritten as follows: ( ) ( ) β bo S (a,i) = I( ω) i dp + Γ( ω) a L(a, ω) dp β (27) Second from (2), the crrent shareholder vale in the case of the indemnity trigger for a given (a, i) may be expressed as follows: ( ) ( ( )) mo S (a,i) = L(a, ω) i dp + Γω ( ) a L a, ω dp (28) Consider the difference in crrent shareholder vale at the same care and protection level for the standard case in which β >. β ( ) ( ) bo mo S (a,i) S (a,i) = L(a, ω) i dp I( ω) i dp ( ( )) ( ( )) + Γ( ω) a L a, ω dp Γ( ω) a L a, ω dp β β β ( ) ( ) ( ) = L I dp I i dp Γ a L dp (29) ( β ( ) ( ( ) ) = L I dp Γ a L I i dp > The ineqality in (29) follows since L > I for states in the insolvency event ω and becase the second term on the right hand side of the third eqality is negative for all ω, β ; to see that the second term on the right hand side of the third eqality is ) negative recall that the bondary β of the insolvency event is implicitly defined by the ( ) a L(a, ) I( ) i Γ( ω ) a L(a, ω ) I( ω) i increasing in condition Γβ ( β β) =. Hence, ( ) ω and zero at β sffices to show that this integral is negative. Next, observe that from the ineqality in (29) it follows that ( ) ( bo b mo m S a (i),i S a (i),i > ) mo b S decreases as a is redced from a m mo to a ; eqivalently, this is the case if S is increasing in a. Hence, observe that if 5

19 S a mo m a= a = DLdP ( L(a, ) i) p( ) + ( DL ) dp a ( Γ( ) a L(a, ) ) p( ) a ( ) ( ) = DLdP + DL dp Γ ( ) a i p( ) a (3) ( ) = dp + D L dp > The third eqality follows by noting that is implicitly defined by the condition m e Γ ( ) a i =. The ineqality follows by noting that a < a and so the second term on the right hand side of the third eqality is positive. Therefore, we have the crrent shareholder vale larger for the index trigger than the indemnity trigger, i.e., bo b mo m > S a (i),i. S ( a (i),i) ( ) Conclding Remarks The analysis begins by noting that insolvency risk in conjnction with limited liability creates an incentive problem known as the jdgment proof problem. The manager of a pblicly held and traded (re)insrance corporate represents the stockholders interest and the jdgment proof problem pts those interests in conflict with those of other stakeholders. Not srprisingly we show that sch a manager selects a level of care less than the socially efficient level. The conflict of interest described here also generates an nderinvestment problem; while not or focs, that problem does motivate the analysis of some new capital market instrments that have been designed to manage (re)insrer insolvency risk. The new capital market instrments considered here are similar to the CAT bonds discssed in the literatre in the sense that they may be designed with triggers that are either indemnity or index based so that the options attached to the bonds are in the money if the reinsrer sffers a sfficiently large loss or if the index of losses, i.e., mortality, is sfficiently large. We stdy the incentive effects associated with each instrment and show that the index based instrment dominates the indemnity based instrment in the sense that it redces insolvency risk and provides the corporate manager with the incentive to take more rather than less care. We go on to show that, given the same strike price, the crrent shareholder vale of the index based instrment exceeds that of the indemnity based instrment. There is a growing literatre that is concerned with hedging longevity risk, i.e., the risk of ot living ones wealth, e.g., see (MacMinn, Brockett and Blake 26). For frther research we note that to date there has been no similar comparison of mortality-based secrities to hedge excessive mortality changes for annity books of bsiness; sch mortality-based secrities wold, of corse, be designed to cover excessive mortality changes in the opposite direction. The concern here wold be longevity risk, i.e., the risk of living too long. Mortality improvements are being reported; there has been acceleration in the mortality improvements at older ages in Sweden (Wilmoth, Deegan, Lndstrom and Horichi 2). There has also been some evidence that genetics plays a 6

20 major role in the ability to srvive to extremely old ages and hence that genetic research may yield insights into how to slow the aging process (Strass 2). The mortality improvements do yield correlated risks for insrers with life annity books. To the extent that the improvements can be predicted accrately over the horizon of the life annities, the longevity risk can be managed by insrers with the standard tools. Life annities, however, have tails that are qite long and so althogh the mortality improvements may seem less srprising, the correlated risks are jst as problematic. Srvivor bonds (Blake and Brrows 2) have been sggested as an effective means of managing longevity risk. The srvivor bond is essentially a reverse tontine; the bond pays a copon that is proportional to the nmber of srvivors in a cohort. A basis risk problem might remain depending on how the cohort is strctred. An instrment similar to that issed by Swiss Re for brevity risk cold also be strctred for longevity risk. The Swiss Re instrment is in the money if the mortality rate becomes too large bt one cold also write a secrity that wold be in the money if the mortality rate became too small. References Blake, D. and W. Brrows (2). "Srvivor Bonds: Helping to Hedge Mortality Risk." Jornal of Risk and Insrance 68(2): Croson, D. C. and H. C. Knrether (2). "Cstomizing Indemnity Contracts and Indexed Cat Bonds for Natral Hazard Risks." Jornal of Risk Finance (3): Cmmins, J. D. and O. Mahl (2). Managing Catastrophic Risk with Insrance Contracts Sbject to Defalt Risk. Working Paper. Wharton. Doherty, N. (997). "Corporate Insrance: Competition from Capital Markets and Financial Instittions." Assrances 65(): Doherty, N. A. and O. Mahl (2). Mickey Mose and Moral Hazard: Uninformative bt Correlated Triggers. Working Paper. Wharton School. Doherty, N. A. and A. Richter (22). "Moral Hazard, Basis Risk and Gap Insrance." Jornal of Risk and Insrance 69(): Dbinsky, W. and D. Laster (23). Insrance-linked secrities, Swiss Re Capital Markets Corporation. Froot, K. (997). The limited financing of catastrophe risk : an overview. Cambridge, MA, National Brea of Economic Research. Froot, K. A., D. S. Scharfstein and J. C. Stein (993). "Risk Management: Coordinating Corporate Investment and Financing Policies." Jornal of Finance 48(5): Garven, J. R. and R. D. MacMinn (993). "The Underinvestment Problem, Bond Covenants and Insrance." Jornal of Risk and Insrance. Jensen, M. and W. Meckling (976). "Theory of the Firm: Managerial Behavior, Agency Costs and Ownership Strctre." Jornal of Financial Economics 3: Kahan, M. (989). "Casation and Incentives to Take Care nder the Negligence Rle." Jornal of Legal Stdies 8:

21 Leland, H. (972). "Theory of the Firm Facing Uncertain Demand." American Economic Review 62: MacMinn, R. D. (987). "Insrance and Corporate Risk Management." Jornal of Risk and Insrance 54(4): MacMinn, R. D. (22). "On the Jdgment Proof Problem." Geneva Papers on Risk and Insrance Theory 27: MacMinn, R. D. (25). The Fisher Model and Financial Markets. Singapore, World Scientific Pblishing. MacMinn, R. D., P. Brockett and D. Blake (26). "Longevity Risk and Capital Markets." Jornal of Risk and Insrance 73(4): MacMinn, R. D. and A. Holtmann (983). "Technological Uncertainty and the Theory of the Firm." Sothern Economic Jornal 5: Mayers, D. and C. W. J. Smith (987). "Corporate Insrance and the Underinvestment Problem." Jornal of Risk and Insrance: McGhee, C., J. Fast and R. Clarke (25). The Growing Appetite for Catastrophe Risk: The Catastrophe Bond Market at Year-End 24, Gy Carpenter & Company, Inc. Modigliani, F. and M. H. Miller (958). "The Cost of Capital, Corporation Finance and the Theory of Investment." American Economic Review 48(2): Nell, M. and A. Richter (24). "Improving Risk Allocation Throgh CAT Bonds." Geneva Papers on Risk and Insrance 29(2): Richter, A. (23). Catastrophe Risk Management - Implications of Defalt Risk and Basis Risk. Working Paper. Illinois State University. Rothschild, M. and J. E. Stiglitz (97). "Increasing Risk: I. A Definition." Jornal of Economic Theory 2: Shavell, S. (979). "On Moral Hazard and Insrance." Qarterly Jornal of Economics 93(4): Shavell, S. (986). "The Jdgement Proof Problem." International Review of Law and Economics 6: Smith, C. W. and R. M. Stlz (985). "The Determinants of Firms' Hedging Policies." Jornal of Financial and Qantitative Analysis 2(4): Strass, E. (2). "Longevity: Hints of a 'Master Gene' for Extreme Old Age." Science 293(5535): 57b-57. Wilmoth, J. R., L. J. Deegan, H. Lndstrom and S. Horichi (2). "Increase of Maximm Life-Span in Sweden, " Science 289(5488):