1 On Valuing Equiy-Linked Insurance and Reinsurance Conracs Sebasian Jaimungal a and Suhas Nayak b a Deparmen of Saisics, Universiy of Torono, 100 S. George Sree, Torono, Canada M5S 3G3 b Deparmen of Mahemaics, Sanford Universiy, 450 Serra Mall, Sanford, CA USA Second Version : Sepember 21, 2006 Firs Version : Ocober 24, 2005 Through he issuance of equiy-linked insurance policies, insurance companies are increasingly facing losses ha have heavy exposure o capial marke risks. In his paper, we deermine he coninuous premium rae ha an insurer exposed o such risks charges via he principle of equivalen uiliy. Using exponenial uiliy, we obain he resuling premium rae in erms of an expecaion under he unique minimal maringale measure and perform a perurbaion expansion around a risk-neural invesor. Wihin he same consisen framework, we address he problem of pricing of a double-rigger reinsurance conrac, aking ino accoun couner-pary risk. The indifference price is found o saisfy a non-linear and nonlocal PDE. This price is furher expanded around he risk-neural price resuling in closed form soluions in he form of risk-neural expecaions. Finally, we recas he pricing PDE as a linear sochasic conrol problem and provide an implici-explici finie-difference scheme for solving he PDE numerically. 1. Inroducion Wih he S&P 500 index yielding reurns of 7.5% over he las year and 17.8% over he las wo years, i is no wonder ha individuals seeking insurance are more ofen oping for equiy-linked conracs raher han heir fixed paymen counerpars. Equiy-linked insurance conracs are highly popular opions for policyholders because hey also provide downside proecion in addiion o he upside equiy like growh poenial. From he insurer s perspecive, such conracs induce claim sizes ha are linked o he flucuaions in he value of he index and, as such, possess significan marke risk in addiion o he radiional moraliy risk. Deermining he premium rae for his class of conracs is a dauning ask which, due o he non-hedgable naure of he conracs, requires a delicae balancing of he insurer s risk preference, moraliy exposure, and The Naural Sciences and Engineering Research Council of Canada helped suppor his work. 1
2 On Valuing Equiy-Linked Insurance and Reinsurance Conracs 2 marke exposure. In his work, we adop he principle of equivalen uiliy, also known as uiliybased or indifference pricing o value such conracs. This pricing principle prescribes a premium rae a which he insurer is indifferen beween (i) no aking on he risk and receiving no premium or (ii) aking on he risk while receiving a premium. We review he mehodology in more deail a he end of 2. Uiliy mehods are pervasive in incomplee marke seings, and were firs used by Hodges and Neuberger (1989) o value opions subjec o ransacion coss. This early work was laer elaboraed by Davis, Panas, and Zariphopoulou (1993) and generalized by Barles and Soner (1998). We coninue along his rend bu apply i o a new and ineresing seing. Equiy-linked life insurance policies have been considered in a few recen works. Young and Zariphopoulou (2002, 2003) were he firs o use uiliy-based mehods o price insurance producs, albei wih uncorrelaed insurance and financial risks. Young (2003), on he oher hand, sudies equiy-linked life insurance policies wih a fixed premium and wih a deah benefi ha is linked o an index. She demonsraes, hrough uiliy indifference, ha he insurance premium saisfies a non-linear Black-Scholes-like PDE where he nonlineariy arises due o he presence of moraliy risk. I is well known ha insurance risks induce incompleeness ino he economy and in such incomplee markes, equivalen uiliy pricing mehods are boh useful and powerful. Even when he risky asse iself has non-hedgable jump risks, Jaimungal and Young (2005) demonsrae ha he indifference pricing mehodology yields racable and inuiively appealing resuls. Our work exends hese recen sudies in wo main direcions: firsly, by considering equiy-linked losses ha coninually arrive a Poisson imes; and secondly, by simulaneously considering he valuaion of a reinsurance produc wihin one consisen framework. Such claims process can arise from a large porfolio of equiy-linked erm life insurance policies, and he reinsurance conrac may be purchased on his large porfolio eiher a policy iniiaion, or a some fuure poin in ime. We assume ha he equiy-linked claims (losses) arrive a Poisson imes and ha he insurer may inves coninuously and updae her holdings in he equiy on which he claims are wrien. As wih all uiliy-based approaches, his requires a specificaion of he real world (as opposed o risk-neural) evoluion of equiy reurns and claim arrivals. We employ he usual geomeric Brownian moion assumpions on he index and presen our specific modeling assumpions in 2. In 3, we hen deermine he premium rae for his porfolio of insurance claims hrough he principle of equivalen uiliy, focusing exclusively on exponenial uiliy for several well-known reasons: Firsly, he opimal invesmen sraegies dicaed by exponenial uiliy is independen of he insurer s wealh. Secondly, he difference beween he equiy holdings, wih and wihou he insurance risk, reduces o he Black and Scholes (1973) hedge in he complee marke case. Thirdly, in he limi in which he invesor becomes risk-neural, he premium reduces o he risk-neural expeced losses over he insurer s invesmen ime horizon. Through exponenial uiliy, we derive he wo Hamilon-Jacobi-Bellman (HJB) equaions corresponding o he premium problem and solve hem explicily for any level of risk-aversion and any (reasonably well behaved) equiy-linked loss funcion. We find ha he resuling indifference premium q is proporional o he expecaion of an exponenially weighed average of he equiylinked loss funcion under he minimal maringale measure. Ineresingly, he premium rae can
3 On Valuing Equiy-Linked Insurance and Reinsurance Conracs 3 be reinerpreed as he scaled difference beween he expeced number of claims in a risk-adjused measure and he minimal maringale one. For consan losses, we are able o express he premium rae explicily in erms of he exponenial inegral funcion. In he opion heoreic framework and in he limi of zero risk-averse invesor leads, uiliy indifference produces prices ha are expecaions of he payoff under he minimal enropy measure as proved by Becherer (2001). This coincides wih he so-called fair price, inroduced by Davis (1998), associaed wih an infiniesimal posiion in he opion. Moivaed by hese works, we invesigae he behavior of a near risk-neural insurer by performing an asympoic expansion of he exac resul around he zero risk-aversion case. If he claim sizes are posiive and bounded by a power of he index level, he series converges uniformly. Ineresingly, all erms occurring in he series expansion are expressible as expecaions, under he minimal maringale measure, of various powers of he loss funcion. We also generalized he problem o include he arrival of several loss funcions, arising, for example, from claims induced in several groups of policies. Any insurer who akes on equiy-linked insurance risks is exposed o poenially large losses in he even of good marke condiions and/or poor underwriing; consequenly, in 4, we consider he relaed problem of pricing double-rigger reinsurance conracs once he insurer has fixed her premium rae. One can view he pricing problem ogeher wih he insurance premium problem, or a a laer dae once he policy rae has been fixed. In eiher case, he reinsurance conrac is assumed o pay a funcion of he oal observed losses, and he equiy value, o he insurer a he mauriy dae. The insurer is assumed o pay an upfron fee (single benefi premium) for his conrac he generalizaion o periodic paymens is no difficul. Uiliy indifference is once again invoked o deermined he value he insurer assigns o he conrac. We prove ha his price saisfies a Black-Scholes-like PDE wih non-linear and nonlocal correcion erms due o he presence of he non-hedgable moraliy risk. When he reinsurance payoff is independen of he loss level, he indifference price reduces o he Black-Scholes price of he corresponding equiy opion. This is an appealing resul since in ha limi he marke becomes complee. Purchasing a reinsurance conrac leaves he insurer exposed o poenially large couner-pary risk; consequenly, we model he defaul ime of he reinsurance company as he firs arrival ime of an inhomogenous Poisson process. This inroduces one more non-linear erm o he PDE arising in he absence of counerpary risk. Analogous o he expansion of he premium rae around a risk-neural insurer, we perform an asympoic expansion of he reinsurance indifference price around a zero risk-aversion parameer. We once again find ha he price can be wrien in erms of expecaions under he minimal maringale measure, now wih a defaul-adjused rae of discoun. In 4.2, we provide a probabilisic inerpreaion of he indifference price in erms of a dual opimizaion problem. Wihin his framework, he indifference price is he minimum of he riskneural expeced value of he reinsurance conrac wih a penaly erm, where he minimum is compued over he aciviy raes of he doubly sochasic Poisson processes driving he claim arrivals and couner-pary defaul. In 4.3, we provide a simple implici-explici numerical scheme for he reinsurance conrac price. We provide wo illusraive prooypical examples: (i) a sop-loss payoff; and (ii) wo relaed double-
4 On Valuing Equiy-Linked Insurance and Reinsurance Conracs 4 rigger sop-loss payoffs. The double rigger producs are riggered by he asse level rising above or dropping below a criical level. The sum of he wo double rigger payoffs resuls in a pure sop-loss reinsurance conrac; however, he sum of heir values do no equal o he value of he sop-loss conrac. This is indicaive of he non-linear pricing rules implied by uiliy indifference. 2. The modeling and pricing framework To model he problem for insurers exposed o equiy-linked losses, we assume ha here is a risky asse whose price process follows a Geomeric Brownian moion, and ha losses arrive a Poisson imes wih claim sizes depending on he price of he risky asse a he claim arrival ime. More specifically, le S()} 0 T denoe he price process for a risky asse; le L()} 0 T denoe he loss process for he insurer; le F S F S } 0 T denoe he naural filraion generaed by S(); le F L F L } 0 T denoe he naural filraion generaed by L(); le F F S F L denoe he produc filraion generaed by he pair S(), L()}; and le (Ω, P, F) represen he corresponding filered probabiliy space wih saisical probabiliy (or real-world) measure P. The insurer is assumed o inves coninuously in he risky asse S() and a risk-free money marke accoun wih consan yield of r 0. Furhermore, he risky asse s price process saisfies he SDE: ds() = S() µ d + σ dx()}, (1) where X()} 0 T is a sandard P-Brownian process, and µ > r. Equivalenly, S() = S(0)e (µ 1 2 σ2 ) +σ X(). (2) The claims arrive a he arrival imes i } of an inhomogenous Poisson process wih deerminisic hazard rae λ(), and each claim size is a funcion of he prevailing index level and possibly ime: g(s( i ), i ). Noice ha he loss size depends on he price of he risky asse prevailing a he ime he loss arrives. This is a defining feaure of equiy-linked insurance producs and inroduces a new dimension o he opimal sochasic conrol problem associaed wih pricing he premium sream. The loss process may be wrien in erms of an underlying Poisson couning process N()} 0 T as follows L() = N() n=1 g(s( i ), i ), We implicily assume ha he claims are posiive g(s, ) 0 and are bounded for every finie pair (S, ) [0, ) [0, T ]. Since our assumpions on he dynamics of he risky asse and he loss process have been addressed, we urn aenion o he dynamics of he wealh process for he insurer. There are wo separae siuaions of ineres: (i) he insurer does no ake on he insurance risk, however, he insurer does inves in he risky asse and he riskless money-marke accoun; and (ii) he insurer akes on he insurance risk in exchange for receiving a coninuous premium of q and simulaneously invess in he risky asse and he riskless money-marke accoun. Le W ()} 0 T and W L ()} 0 T denoe, respecively, he wealh process of he insurer who does no ake on he insurance risk (3)
5 On Valuing Equiy-Linked Insurance and Reinsurance Conracs 5 (as in case (i)) and he wealh process of he insurer who does ake on he insurance risk (as in case (ii)). The process π (π(), π 0 ())} 0 T denoes an F -adaped self-financing invesmen sraegy, where π() and π 0 () represen he amoun invesed in he risky asse and he amoun in he money-marke accoun, respecively. The wealh processes hen saisfy he following wo SDEs: dw (u) = [r W (u) + (µ r) π(u)] du + σ π(u) dx(u), (4) W () = w, dw L = [ r W L (u ) + (µ r) π(u ) + q ] du + σ π(u ) dx(u) dl(u), W L () = w, (5) where w represens he wealh of he insurer a he iniial ime ; and f(u ) represens he value of he process f prior o any jump a u. To complee he model seup, we suppose ha he insurer has preferences according o an exponenial uiliy of wealh u(w) = 1ˆα e ˆα w for some ˆα > 0. The parameer ˆα is he consan absolue risk-aversion rˆα (w) u (w)/u (w) = ˆα as defined by Pra (1964). Furhermore, he insurer seeks o maximize her expeced uiliy of erminal wealh a he invesmen ime horizon T. This resuls in wo separae sochasic opimal conrol problems. The value funcion of he insurer who does no accep he insurance risk is denoed by V (w, ), and he value funcion of he insurer who does accep he insurance risk is denoed by U(w, S, ; q). Explicily, he value funcions are defined as follows: V (w, ) = sup E [u(w (T )) W () = w], and (6) π A U(w, S, ; q) = sup E [ u(w L (T )) W L () = w, S() = S ]. (7) π A Here, A is he se of admissible, square inegrable, and self-financing, F -adaped rading sraegies for which T π 2 (s) ds < +. This resricion is necessary for he exisence of a srong soluion o he wealh process SDEs (4) and (5) (see Fleming and Soner, 1993). A priori, i is no obvious ha V should depend solely on he wealh process and ime; similarly, i is no obvious ha U should be independen of he loss L. However, hrough he explici soluions in he nex secion, we deermine ha his is indeed he case a familiar resul when working wih exponenial uiliy. Alhough he value funcions are found o depend on he insurer s wealh, he opimal invesmen sraegy is, in fac, independen of he wealh. This oo is a consequence of exponenial uiliy. Nex, he indifference premium is defined as he premium q such ha he wo value funcions are equal: V (w, ) = U(w, S, ; q). (8) Inuiively, his implies ha he insurer is equally willing eiher o accep he risk and receive a premium, or o decline he risk and receive no premium.
6 On Valuing Equiy-Linked Insurance and Reinsurance Conracs 6 Once he indifference premium is obained, he problem of pricing a reinsurance conrac is considered in 4. The conrac is assumed o pay an arbirary funcion h(l(t ), S(T )) of he oal observed losses and he risky asse s price a he ime horizon T. The associaed value funcion of an insurer who receives his paymen will be denoed U R (w, L, S, ; q) and is explicily expressed as U R (w, L, S, ; q) = sup E[u(W L (T ) + h(l(t ), S(T ))) W L () = w, S() = S, L() = L]. (9) π A Noice ha he reinsurance payoff is relevan only a he erminal ime, and is role is simply o increase he insurer s wealh by he conrac value. Alhough he conrac payoff plays an explicily role only a mauriy, i will induce a feed back ino he opimal invesmen sraegy, and consequenly, feed back ino he value funcion iself. The value funcion U R is well defined for any choice of he premium rae q, and in paricular, his rae may be differen from he indifference premium rae solving (8). In analogy wih he indifference premium of he insurance sream, he indifference price P (L, S, ) of he reinsurance conrac is he amoun of iniial wealh he insurer who receives he reinsurance paymen is willing o surrender such ha he value funcion wih he reinsurance paymen is equal o he value funcion wihou he reinsurance paymen. Tha is, he indifference price saisfies he equaion: U R (w P (L, S, ), L, S, ; q) = U(w, L, S, ; q). (10) A poseriori, he price is found o be independen of wealh for exponenial uiliy. Furhermore, even hough he reference value funcion U R conains exposure o he insurance risk, we find ha he indifference price is independen of he premium ha he insurer charges. Since U may be greaer, equal, or less han V, he insurer may or may no be in favor of holding he insurance risk in he firs place. On he oher hand, if he insurer is charing he indifference premium rae implicily defined by (8), hen he indifference price defined in (10) will render he insurer indifferen o aking on he insurance, receiving he premium rae q and simulaneously receiving he reinsurance payoff afer giving up iniial wealh P. In oher words, wih q chosen o be he indifference premium, here will be a riple equaliy: U R (w P, L, S, ; q) = U(w, L, S, ; q) = V (w, ). The indifference price P, for an arbirary premium rae q, may be hough of as a marginal or relaive indifference price. Mos noably, his price does no coincide wih he indifference price of he reinsurance conrac in oal absence of receiving he insurance premium. This price is similar, bu no idenical, o he relaive indifference price inroduced by Musiela and Zariphopoulou (2001, 2004) and employed by Soikov (2005) in he conex of pricing volailiy derivaives by adding a small posiion o an already exising porfolio of opions. To complee he pricing mehodology, we consider he effec ha couner-pary risk plays wihin his framework. Defaul of he reinsurer is modeled by he firs arrival ime τ of a second, sochasically independen, inhomogenous Poisson process M() wih rae of arrival κ(). The payoff h(l(t ), S(T )) is now uncerain and mus be replaced by ĥ(l(t ), S(T ), τ) h(l(t ), S(T )) I(τ > T ), and he relevan value funcion is U RC (w, L, S, ; q) = sup E[u(W L (T ) + ĥ(l(t ), S(T ), τ)) W L () = w, S() = S, L() = L, τ > ].(11) π A
7 On Valuing Equiy-Linked Insurance and Reinsurance Conracs 7 The indifference price is once again defined hrough he balancing equaion U RC (w P (L, S, ), L, S, ; q) = U(w, L, S, ; q). (12) We find ha couner-pary risk serves only o add one addiional non-linear, bu local, erm o he pricing PDE which vanishes as he defaul inensiy reduces o zero. 3. The indifference premium rae problem Now ha he sochasic model for he insurer has been described, and he pricing principle has been specified, we can focus on he deails of he pricing problem iself. In he nex subsecion, he value funcion wihou he insurance risk is reviewed. The resuls of ha secion are essenially hose of Meron (1969). Those resuls are hen used in 3.2 o solve he HJB equaion for he insurer exposed o he insurance risk. In 3.3, we deermine he indifference premium for a general loss funcion and provide specific examples. In 3.4, we address he issue of hedging he risk associaed wih his premium choice The value funcion wihou he insurance risk The value funcion of he insurer who does no ake on he insurance risk is defined in (6), and we now use he dynamic programming principle o deermine he opimal invesmen sraegy and he value funcion iself. Given a paricular invesmen sraegy π, we deermine ha V saisfies he following SDE: dv (W, s) = [ V + (r W + (µ r) π) V w σ2 π 2 V ww ] ds + πσ Vw dx. (13) The subscrips denoe he usual parial derivaives of V, and he ime dependence of he various processes are suppressed for breviy. Through he usual dynamic programming principle, V solves he HJB equaion: V + r w V w + max π [ (µ r) π Vw σ2 π 2 V ww ] = 0, V (w, T ) = u(w). We may assume ha he opimal invesmen is provided by he firs order condiion, and he Verificaion Theorem confirms he resul. To his end, he opimal invesmen sraegy is π () = µ r σ 2 V w V ww. On subsiuing π ino (14), V is found o saisfy he PDE V 1 ( ) µ r 2 Vw 2 + r w V w = 0. (16) 2 σ V ww Assuming ha (14) (15) V (w, ) = 1ˆα e α() w+β(), (17) wih β(t ) = 0 and α(t ) = ˆα, he HJB equaion reduces o ( ) µ r 2 = 0, (18) (α + rα) w + β 1 2 σ
8 On Valuing Equiy-Linked Insurance and Reinsurance Conracs 8 which mus hold for all w and. Therefore, α() = ˆα e r(t ) and (19) ( ) µ r 2 (T ), (20) β() = 1 2 σ resuling in he sandard opimal (Meron, 1969) invesmen of π () = µ r ˆα σ 2 e r(t ). Since he soluion saisfies he requiremens of he Verificaion Theorem, π corresponds o he opimal invesmen sraegy for (6), and V, given in (17), is he soluion of he original opimal sochasic problem The value funcion wih insurance risk While assuming he insurance company akes on he insurance risk and receives a premium rae of q, we mus solve for he opimal invesmen and value funcion U, given in (7). Through sraighforward mehods, we esablish he following HJB equaion for he value funcion U: 0 = U + (r w + q)u w + µ S U S σ2 S 2 U SS +λ() (U(w g(s, ), S, ) U(w, S, )) U(w, S, T ; q) = u(w). + max π 1 2 σ2 U ww π 2 + π [ (µ r)u w + σ 2 S()U ws ]}, The nonlocal erm appears due o he presence of he Poisson claims, and can be explained by observing ha a claim arrives in (, + d] wih probabiliy λ() d, causing he wealh o drop by g(s(), ). A firs sigh, he presence of his nonlocal erm appears o render he problem inracable. However, on closer inspecion, we find ha he HJB equaion can be solved explicily for arbirary claims. Theorem. 3.1 The soluion o he HJB sysem (22) is } U(w, S, ; q) = V (w, ) exp ˆα q er(t ) 1 + γ(s, ), (23) r (21) (22) where γ(s(), ) = E Q [ T λ(u) ( ) ] e α(u) g(s(u),u) 1 du, (24) and he process S() saisfies he following SDE in erms of he Q-Wiener process X()} 0 T, ds() = S() r d + S() σ dx(). (25) Furhermore, he opimal invesmen sraegy is independen of wealh and equals } π e r(t ) µ r (S, ) = ˆα σ 2 + S γ S. (26)
9 On Valuing Equiy-Linked Insurance and Reinsurance Conracs 9 Proof. By assuming U ww < 0, he firs order condiion supplies he opimal invesmen sraegy as π (S, ) = (µ r)u w + σ 2 S()U ws σ 2 U ww. (27) Subsiue his ino (22), and make he subsiuion U(w, S, ; q) = V (w, ; q) expγ(s, )}, (28) where V (w, ; q) denoes he value funcion of he insurer who receives a premium rae of q bu does no accep he insurance risk. If we denoe he wealh process for such an insurer as W q (), hen W q () = W () + q. Noice ha V (w, ; 0) = V (w, ), and ha V (w, ; q) saisfies he HJB equaion: V (w, T ; q) = 0 = V + (r w + q) V w + max π [ (µ r) π Vw σ2 π 2 V ww ], u(w). Through sraighforward calculaions } V (w, ; q) = V (w, ) exp ˆα q er(t ) 1 r is shown o saisfy (29). Making he subsiuion (28) ino (22), we discover, afer some edious calculaions, V (w, ; q) facors ou of he problem, and he funcion γ(s, ) saisfies he inhomogeneous linear parial differenial equaion: 0 = λ() (e α()g(s,) 1) + rsγ s σ2 S 2 γ SS + γ, γ(s, T ) = 0. The Feynman-Kac heorem direcly leads o soluion (39) from which we observe ha U ww < 0 so ha he maximizaion erm is indeed convex. For reasonably well-behaved loss funcions g(s(), ), he Verificaion Theorem implies ha (39) is he value funcion for he problem and sraegy (27) is opimal. Accordingly, subsiuing he ansäz (28) ino π leads o (26) The indifference premium Now ha boh value funcions V and U are found, an explici represenaion of he indifference premium is an easy consequence. Corollary 3.2 The insurer s indifference premium rae q is independen of wealh [ ] r T q(s(), ) = e r(t ) 1 EQ λ(u) eα(u) g(s(u),u) 1 du. (32) ˆα Proof. The indifference premium rae q is defined as he rae q such ha U(w, S, ; q) = V (w, ). Expression (32) hen immediaely follows from Theorem 3.1. Noice ha if u (, T ], g(s(u), u) > 0 Q a.s., i.e. (29) (30) (31) he claims are almos surely posiive, hen he indifference premium is posiive. Furhermore, he below lemma shows ha premium is monoonically increasing in ˆα, implying ha increasing risk-aversion induces an increase in he premium which renders he insurer indifferen o he insurance risk. This inuiively appealing resul should be expeced of any reasonable pricing rule which incorporaes risk-aversion.
10 On Valuing Equiy-Linked Insurance and Reinsurance Conracs 10 Lemma 3.3 The indifference premium rae q is monoone in he risk-aversion level ˆα if he claim size g(s(), ) is sricly posiive. Proof. I is easy o check ha p(y) := (e y x 1) /y is monoonically increasing in y for every x > 0. Then, under he posiiviy assumpion of g, for every even ω Ω, we have, T λ(u) eα(u) g(s(u),u; ω) 1 ˆα is increasing in ˆα and he resul follows. In analyzing he value funcion U, we assumed ha q was consan; however, on glancing a (32) i can be inferred ha q is no a consan, and herefore, our assumpions are false, discrediing he analysis. This iniial reacion is premaure. The siuaion is bes explained by appealing o he familiar case of a forward conrac. On signing of a forward conrac, he delivery price is se such ha he conrac has zero value. This delivery price is a funcion of he prevailing spo price of he asse and bond prices a he ime of signing. Alhough he conrac value on signing is zero, he forward price, a any fuure dae, will no equal o he delivery price, and he conrac s value is no longer zero. In he presen conex, he insurer is looking forward o a fuure ime horizon, and is deciding on a rae o charge so ha she is indifferen o aking he risk. Our analysis shows ha he rae (32), which depends on he prevailing price of he risky asse, should be charged. This rae is fixed unil he end of he ime horizon, and does indeed render he insurer indifferen o he insurance risk a he curren ime. (33) However, as ime evolves, he prevailing indifference premium a ha fuure poin in ime may be higher or lower han he rae he insurer iniially se. Consequenly, if he insurer ook on he insurance risk a ime in exchange for q(s(), ) unil he horizon end, hen a some fuure ime she may develop a preference eiher owards releasing he insurance risk or for holding ono i. Wih he forward conrac analogy, i is no surprise hen ha he premium rae depends on he risky asse s spo price. The indifference premium (32) has some addiional noeworhy properies. The risk-neural measure Q appearing in he premium calculaion is independen of he risk-aversion level of he insurer. Wihin his risk-neural measure, he disribuion of claim sizes has no been disored from is real world disribuion. Indeed, he Radon-Nikodym derivaive process which performs he measure change is ( ) dq η() = exp 1 ( ) } µ r 2 + µ r X(). (34) dp 2 σ σ This is he same measure change ha Meron (1976) uses in his jump-diffusion model and corresponds o risk-adjusing only he diffusion componen. Since he equiy risk and he losses are inerrelaed only hrough he loss size, and no hrough any insananeous correlaion, his risk-neural measure corresponds o he minimal maringale one. This is similar o he case of indifference pricing for financial opions, where only he radable asse s risk process are disored such ha heir drif is he risk-free one under he pricing measure, while orhogonal sochasic degrees of freedom are lef undisored. Even hough he risk-aversion level does no feed ino he
11 On Valuing Equiy-Linked Insurance and Reinsurance Conracs 11 probabiliy measure used for compuing expecaions, i does manifes iself in he disorion of he claim sizes hrough he exponenial erm. This exponenial disorion is, no surprisingly, inheried from he choice of uiliy funcion. A paricularly ineresing inerpreaion of he premium is supplied by spliing he expecaion erm appearing in (32) ino wo and wriing each erm as he expeced number of claims under wo differen measures. Specifically, [ T ( ) ] [ T ] [ T λ(u) e α(u) g(s(u),u) 1 du = E Q λ(u) e α(u) g(s(u),u) du E Q E Q = E e Q [N(T )] EQ ] λ(u) du [N(T )]. (35) Under he measure Q, he process N() is a doubly sochasic Poisson process wih aciviy rae λ() λ() e α() g(s(),). When ineres raes are zero and he losses hemselves are consan, i is easy o check ha he measure Q is he minimizer of he penalized enropy: min E e [ ( ) ] Q d Q ln ˆα g N(T ). (36) eq<<p dp T As a final poin of ineres, alhough he premium is a non-linear funcional of he claim sizes i is linear in he arrival rae of he claims λ(). This observaion suggess ha he generalizaion o muliple claims disribuions is sraighforward. In he heorem below, we provide he resuls for muliple claims disribuions. The proof is omied for breviy as i follows along he same lines as hose in he previous wo secions. Theorem. 3.4 Suppose ha he insurer is exposed o losses from m differen sources of risk. Explicily, he loss process is modeled as follows: L() = m j=1 N j () n=1 g j (S( n j ), n j ), (37) where N j () : j = 1,..., m} are independen inhomogenous Poisson processes wih arrival raes λ j () : j = 1,..., m}, g j (S, ) denoes he loss funcions for he j-h source of risk, and n j denoes he n-h arrival ime for he j-h process. Then, he value funcion of he insurer who akes on he insurance risk and receives a premium of q(w, S, ) is } U(w, S, ; q) = V (w, ) exp ˆα q er(t ) 1 + γ(s, ), (38) r where γ(s(), ) = m j=1 E Q [ T λ j (u) ( ) ] e α(u) gj(s(u),u) 1 du, (39) and he process S() saisfies he following sochasic differenial equaion in erms of he Q-Wiener process X()} 0 T : ds() = S() r d + S() σ dx(). (40)
12 On Valuing Equiy-Linked Insurance and Reinsurance Conracs 12 Furhermore, he insurer s indifference premium is independen of wealh and is explicily [ r m ] T q(w, S, ) = e r(t ) E Q λ j (u) eα(u) gj(s(u),u) 1 du. (41) 1 ˆα j=1 This lineariy is an ineresing consequence of he coninual arrival of claims a Poisson imes. If insead, we assumed ha here was a maximum finie number of claims, hen he problem would no be linear. We are currenly invesigaing how a large, bu finie, number of claims alers he resuls in his aricle Consan losses An ineresing esing ground for our resuls is he case when he losses hemselves are consan, i.e. g(s, ) = l. This case is a paricular example of he model in Young and Zariphopoulou (2003) where he claims disribuion is a Dirac dela disribuion; however, hey did no repor he resul for his simple consan claims case. We deermine he indifference premium rae as λ ( ) q = ˆα ( e r(t ) 1 ) Ei(ˆα l e r(t ) ) Ei(ˆα l) (T )r, (42) where Ei(x) denoes he so called exponenial inegral, defined as he following Cauchy principle value inegral: Ei(x) x e d. The exponenial inegral has he following asympoic expansion:ei(x) = γ + ln(x) + n=1 xn n! n ; where γ is Euler s consan. As expeced, he indifference premium is non-linear in he claim size l. There is good evidence ha insurers who have well diversified porfolios and large reserves exhibi near risk-neural behavior. I is herefore ineresing o invesigae he impac his has on he valuaion of he insurance sream. If he insurer is near risk-neural, hen an expansion in ˆα l can be carried ou, and we find he indifference premium rae o linear order is ( q = λ l ( ) ) e r(t ) + 1 ˆα l + o(ˆα l). (44) 4 As such, a risk-neural insurer exposed o fixed losses, will charge a rae equal o he expeced loss per uni ime λ l an inuiively sound resul. As expeced, he sign of he firs order correcion is posiive Near risk-neural insurer Exending he near risk-neural insurer analysis o general claims funcion is no difficul. For losses ha grow a mos power like, i.e. here exiss c > 0, b() > 0 and S () > 0 such ha for each and S > S (), g(s, ) b() S c, he rae has he following perurbaive expansion in erms of he risk-aversion parameer ˆα: λ r ˆα n 1 T q = e r(t ) e n r(t u) E Q 1 n! [ gn (S(u), u)] du. (45) n=1 (43)
13 On Valuing Equiy-Linked Insurance and Reinsurance Conracs 13 This series converges by appealing o he Lebesgue dominaed convergence heorem and noing ha E Q [Sn (u)] = S()e n(r+ 1 2 σ2 (n 1))(u ). This growh condiion can be weakened considerably; however, a his poin we are concerned wih aiding inuiion and as a resul, omi such deails from he analysis. he opions conex. Expansions similar o he one above have been explored in Davis (1998) in He demonsraes ha he zeroh order erm is equivalen o price of an infiniesimal posiion in he opion he so called marginal price. Based on (45), a risk-neural insurer would hen charge a premium rae of q = r e r(t ) 1 T λ(u) e r(t u) E Q [ g(s(u), u)] du. (46) Observe ha he facor in fron of he expecaion can be represened as (a) 1 wih a = T e r(t u) du. The expression a is precisely he accumulaed value of $1 per annum received coninuously over he ime span (, T ]. Furhermore, he expecaion can be inerpreed as he risk-neural expeced claims accumulaed o he mauriy dae T. Wih hese poins in mind, he risk-neural indifference premium (46) rae balances, in expecaion, beween coninually receiving he premium and paying ou he claims Floor, capped, and marke paricipaion claims In his secion we provide an explici example of he premium when he losses are funcions of he logarihm of he sock index. While sill mainaining he essenial properies of linear claim sizes, we use he logarihm of he sock price because i allows for parially closed form soluions. To his end, define A(u) as he expecaion appearing under he inegral in he indifference premium (32), i.e. A(u) E Q [ e α(u) g(s(u),u)]. Then, he indifference premium q can be wrien in erms of A(u) explicily as q = r ˆα ( e r(t ) 1 ) T (47) λ(u) A(u) 1} du. (48) Consider insurance claims which have a cap and a floor proecion in addiion o a paricipaion in he risky asse s reurn independen of ime ypical feaures found in equiy-linked insurance payoffs. In his case, he claim sizes are θ, S() < c 1, g(s(), ) = θ + β (log (S()) log(c 1 )), c 1 S() < c 2, θ + β (log(c 2 ) log(c 1 )), S() c 2. To mainain posiiviy of he claim sizes in all oucomes, we resric θ > 0, β > 0 and c 1 < c 2. Afer some edious calculaions, he inegrand A(u) reduces o ( ) β α(u) A(u) = e Φ(d α(u)θ 1 (c 1 )) + c2 c Φ( d1 1 (c 2 )) ( ) } β α(u) (50) + S() c 1 e β α(u)(r 1 2 σ2 (1 β α(u)))(u ) (Φ(d 1 (c 3 )) Φ(d 2 (c 1 ))). (49)
14 On Valuing Equiy-Linked Insurance and Reinsurance Conracs 14 Indifference Premium [ q ] $130 $110 α = 0 α = 0.1 α = 0.2 $90 $70 $80 $90 $100 $110 $120 $130 Indifference Premium [ q ] $130 $110 α = 0 α = 0.1 α = 0.2 $90 $70 $80 $90 $100 $110 $120 $130 Spo Price [ S() ] Spo Price [ S() ] Figure 1. The dependence of he indifference premia on he underlying equiy spo price for losses given in equaion (49). The model parameers are θ = 1, β = 1, c 1 = 90, c 2 = 110, r = 4%, σ = 15%, and λ = 100. The erms in he lef/righ panels are one and five years respecively. In Figure 1, he dependence of he premium on he underlying spo price is illusraed for hree choices of he risk-aversion parameer ˆα and for erms of one and five years respecively. The boxed line shows he pure loss funcion (49) scaled by he aciviy rae for comparison purposes. As he risk-aversion parameer increases, he premia increases illusraing he monooniciy propery. Noice ha when he erm increases he premium decreases for large spo prices, while i increases for small spo prices. This is analogous o he pricing for a sandard bull-spread opion in he Black-Scholes model Hedging he insurance risk Now ha we have deermined he indifference premium ha he insurer charges, i is ineresing o explore he hedging sraegy ha she would follow. In his incomplee marke seing, i is impossible o replicae he insurance claims; noneheless, he insurer holds differen unis of he risky asse when she is exposed o he insurance risk and when she is no exposed o he insurance risk. As a resul, we can define an analog of he Black-Scholes Dela hedging parameer. To his end, he Dela is defined as he excess unis of he risky asse ha he insurer holds when aking on he risk and receiving he premiums, and when here is an absence of insurance risk. Corollary 3.5 The Dela of he insurer s posiion is (S, ) 1 S (π U π V ) = e r(t ) ˆα γ S (S, ). (51) Proof. The opimal invesmen in he risky asse wihou he insurance risk appears in (15), and wih he insurance risk appears in (26). The resul is quie similar o he Black-Scholes Dela for an opion. Alhough i is possible o rewrie he Dela in erms of he indifference premium rae q, i is mos naurally represened in erms of he auxiliary funcion γ. Moreover, as T, he Dela vanishes; his is quie differen from he behavior of he Black-Scholes Dela of an opion wih payoff g(s(t ), T ). In he case of
15 On Valuing Equiy-Linked Insurance and Reinsurance Conracs 15 Dela α = 0 α = 0.1 α = 0.2 Dela α = 0 α = 0.1 α = $70 $80 $90 $100 $110 $120 $130 Spo Price 0.0 $70 $80 $90 $100 $110 $120 $130 Spo Price Figure 2. The dependence of he Dela on he underlying equiy spo price for losses given in equaion (49). The model parameers are θ = 1, β = 1, c 1 = 90, c 2 = 110, r = 4%, σ = 15%, and λ = 100. The erms in he lef/righ panels are one and five years respecively. a European opion, he Dela becomes equal o he derivaive of he payoff funcion wih respec o he spo price, and is zero only where he opion s payoff becomes fla. In he presen conex, he Dela vanishes as mauriy approaches because he probabiliy of a loss arriving in he nex small ime inerval close o mauriy is λ T. Therefore, probabilisically, here is no need o hold addiional shares of he risky-asse near mauriy. In Figure 2, we show how he Dela behaves as a funcion of he spo-level, risk-aversion parameer, and ime o mauriy for he example in The general shape of hese curves is expeced. The payoff is asympoically fla ouside of he paricipaion region (see Figure 1); implying a decaying Dela in he ails. The Dela is wider when mauriy is furher away and, conrary o an opion s dela, i is increases wih mauriy due o he larger number of poenial losses. 4. The indifference price for reinsurance Now ha we have deermined he indifference premium ha he insurers charges, we can address he dual problem of pricing a reinsurance conrac which makes paymens a he end of he ime horizon. In secion 2, we describe he value funcion associaed wih he insurer who akes on he insurance risk, receives he premium rae q, and receives a reinsurance paymen of h(l(t ), S(T )). The value funcion of such an insurer was denoed U R as defined in equaion (9). The associaed HJB equaion for his value funcion is essenially he same as he one for U (see equaion (22)); however, he boundary condiion is now alered o accoun for he presence of he reinsurance, and we mus also keep rack of he loss process explicily. Through he usual dynamic programming principle, we deermine ha U R saisfies he following HJB equaion: 0 = U R + (rw + q)uw R + µ S US R σ2 S 2 USS R +λ() (U R (w g(s, ), L + g(s, ), S, ) U R (w, L, S, )) U R (w, L, S, ; q) = u(w + h(l, S)). + max π 1 2 σ2 U R ww π 2 + π [ (µ r)u R w + σ 2 S()U R ws]}, (52)
16 On Valuing Equiy-Linked Insurance and Reinsurance Conracs 16 The nonlocal erm now conains wo ypes of shifing: he firs, due o he decrease in he wealh of he insurer; and he second, due o he increase in he loss process. However, boh shifs come from he same risk source. We can immediaely address he issue of couner-pary risk by modifying he payoff o include an indicaor of he even ha he reinsurer survives o he mauriy dae. As described a he end of secion 2, he reinsurer s defaul ime is modeled by a separae inhomogeneous Poisson process M() wih aciviy rae κ(). I is easy o see ha he value funcion in he presence of couner-pary risk, which we denoed by U RC, saisfies he HJB: 0 = U RC + (rw + q)uw RC + µ S US RC σ2 S 2 USS RC +λ() (U RC (w g(s, ), L + g(s, ), S, ) U RC (w, L, S, )) U R (w, L, S, ; q) = u(w + h(l, S)). +κ() (U(w, L, S, ) U RC (w, L, S, )) + max 1 π 2 σ2 Uww RC π 2 + π [ (µ r)uw RC + σ 2 S()Uws RC ]}, The only difference beween (52) and (53) is he presence of he nonlocal erm proporional o he defaul inensiy κ(). Is appearance can be undersood as follows: if a defaul occurs over he nex infiniesimal ime, he value funcion U RC revers o U since he insurer will no longer receive he reinsurance paymen a mauriy; however, she is sill exposed o he insurance risks and sill receives he premium rae q. Since we can recover he defaul-free case by seing κ() = 0, he remaining analysis focus only on solving equaion (53). Once again, exponenial uiliy allows us o obain a soluion of he HJB equaion in a semi-explici form. Theorem. 4.1 The soluion o he HJB sysem (53) can be wrien as U RC (w, L, S, ) = U(w, S, )φ(l, S, ), (54) (53) where φ saisfies he non-linear PDE ( ) 0 = φ + r Sφ S σ2 S 2 φ SS φ2 S φ + κ() (1 φ(l, S, )) +λ() e α()g(s,) (φ(l + g(s, ), S, ) φ(l, S, )), φ(l, S, T ) = e ˆα h(l,s). (55) Furhermore, he opimal invesmen in he risky-asse is [ π e r(t ) µ r (S, ) = ˆα σ 2 + S γ S + φ ]} S. (56) φ Proof. Assuming ha U RC ww < 0, he firs order condiions allow he opimal invesmen sraegy o be wrien, π () = (µ r)u RC w + σ 2 S()U RC ws σ 2 U RC ww. (57) On subsiuing he ansäz (54) and he opimal invesmen (57) ino he HJB equaion (52), we
17 On Valuing Equiy-Linked Insurance and Reinsurance Conracs 17 esablish 0 = φ U + (rw + q)u w + µsu S σ2 S 2 U SS 1 2 +U φ + } ((µ r)u w+σ 2 SU ws) 2 σ 2 U ww ( ) µ (µ r) U w 2 U U ww S φ S σ2 S (φ 2 SS U w 2 φ 2 S U U ww φ 2 +κ() U(w, S, ) [1 φ(l, S, )] +λ() [U(w g(s, ), S, ) φ(l + g(s, ), S, ) U(w, S, ) φ(l, S, )], ( UwS U w U U ww ) )} U S U φ S subjec o he boundary condiion U(w, S, T )φ(l, S, T ) = u(w + h(l, S)). From (22), he erms inside } in he firs line of he above expression equals λ() [U(w g(s, ), S, ) U(w, S, )]; collecing his wih he las line and making use of he ideniies U(w g(s, ), S, ) = U(w, S, ) e α()g(s,), (58) U 2 w U ww U = 1, and U wsu w U ww U = U S U = γ S, (59) we find, hen, equaion (58) disills o (55). I can hen be proven ha U R ww < 0. Using he ansäz (54), he opimal invesmen π can be rewrien as (56). For smooh g, he Verificaion Theorem allows us o confirm ha he consruced soluion is he value funcion for he original problem and ha he described sraegy is clearly opimal. Corollary 4.2 The insurer s indifference price P (L(), S(), ) for he reinsurance conrac saisfies he nonlinear nonlocal PDE: r P = P + r S P S σ2 S 2 P SS κ() α() P (L, S, T ) = h(l, S). ( 1 e α() P (L,S,) ) + λ() α() eα()g(s,) ( 1 e α()[p (L+g(S,),S,) P (L,S,)]), Proof. The indifference price P saisfies U RC (w P, L, S, ) = U(w, S, ). (60) The facorizaion (54) ogeher wih he ideniy U(w g(s, ), S, ) = U(w, S, ) e α()g(s,) implies ha P (L, S, ) = 1 α() ln φ(l, S, ). On subsiuing φ in erms of P in (55), we obain (60). Noice ha if he payoff funcion h(l, S) is independen of he loss level, i.e. h(l, S) = h(s), hen (60) reduces o r P = P + r S P S σ2 S 2 P SS κ() α() P (L, S, T ) = h(s). ( 1 e α() P ), When here is no couner-pary risk, he above pricing equaion is precisely ha of a European opion wih payoff h(s) in he Black and Scholes (1973) model. (61) This resul is expeced since he reinsurance conrac is hen exposed only o he hedgable risk he risky asse and no o he non-hedgable claims risk or couner-pary risk. Therefore, our resul should reduce o he no arbirage Black-Scholes price for an insurer of any degree of risk-aversion. When couner-pary risk is he only non-hedgable risk, i.e. κ() 0, bu h is a funcion only of S, hen he pricing equaion is idenical o he indifference pricing equaion for an equiy-linked pure endowmen paying h(s) condiional on survival o mauriy as sudied in Moore and Young (2003). Alernaively, his price may be viewed as he value of a defaulable opion wih payoff h(s).
18 On Valuing Equiy-Linked Insurance and Reinsurance Conracs Near risk-neural insurer Le he price of a risk-neural insurer, aken as he limi of a risk-averse insurer, be denoed P 0 (L, S, ) = limˆα 0 + P (L, S, ). Then, he pricing PDE for P 0 following from (60) is (r + κ()) P 0 = P 0 + r S PS σ2 S 2 PSS 0 + λ() P 0 P 0 (L, S, T ) = h(l, S), where P 0 denoes he increase in he price due o a loss arrival: P 0 (L, S, ) P 0 (L + g(s, ), S, ) P 0 (L, S, ). (63) Consequenly, hrough he Feynman-Kac Formula, a risk-neural insurer would be willing o pay P 0 (L, S, ) = E Q [e R ] T (r+κ(s))ds h(l(t ), S(T )) = E Q [e R ] T r ds h(l(t ), S(T )) I(τ > T ) (64) for he reinsurance conrac, where he Q-dynamics of S() appears in (40), while he loss arrival rae and couner-pary defaul rae are boh unalered from heir real world values. (62) Alhough his marke is incomplee, and herefore here exiss many risk-neural measures equivalen o he real world measure (Harrison and Pliska, 1981), he indifference pricing mehodology selecs a unique measure. This measure is he minimal maringale measure under which sochasic degrees of freedom ha are orhogonal o he driving diffusion of he radable asse s price process remain undisored. I is ineresing o invesigae he firs order correcion in he risk-aversion parameer ˆα o gain some undersanding of he perurbaions around he risk-neural price. This is similar o he work of Soikov (2005) where he invesigaes he linear correcions of he price of volailiy derivaives when he invesor has already aken posiions in a porfolio of derivaives. In our conex we are pricing he reinsurance opion in he presence of he insurance risk. The main difference here is ha our reinsurance payoff is no considered small relaive o he background porfolio as in Soikov (2005). If we assume ha he payoff funcion is bounded from above, and hence he price is also bounded, hen he price can be expanded in a power series in ˆα. Specifically, wrie P (L, S, ) = P 0 (L, S, ) + ˆαP 1 (L, S, ) + o(ˆα), (65) subjec o P 0 (L, S, T ) = h(l, S) and P 1 (L, S, T ) = 0. The linear correcion vanishes a mauriy since we have fully accouned for he payoff in he zeroh order erm. When insering his ansäz ino (60) and using (62), we deermine P 1 (L, S, ) saisfies he following PDE: (r + κ()) P 1 = P 1 + r S PS σ2 S 2 PSS 1 + λ() P 1 ( +e r(t ) 1 2 κ() ( P 0) 2 + λ() g 2 (S, ) [ P 0 (L, S, ) g(s, ) ] }) 2 + o(ˆα), P 1 (L, S, T ) = 0. Through Feynman-Kac, he firs order correcion can be represened as a risk-neural expecaion as well, and we find he following resul: [ T P 1 (L, S, ) = E Q e R [ u 1 κ(s) ds 2 κ(u) ( P 0 (L(u), S(u), u)) 2) (66)
19 On Valuing Equiy-Linked Insurance and Reinsurance Conracs 19 +λ(u) g 2 (S(u), u) [ P 0 (L(u), S(u), u) g(s(u), u) ] } ] ] 2 du.(67) Ineresingly, he payoff funcion h(l, S) does no explicily appear in P 1 ; raher, i feeds from he risk-neural price funcion P 0 which does explicily depend on he payoff. The sign of his correcion erm is difficul o discern on firs observaion due o he erm proporional o λ(u). However, we may deduce ha if (i) h is increasing in L, (ii) g is non-negaive, and (iii) h is Lipschiz-coninuous wih Lipschiz consan 2, hen he correcion erm is non-negaive Probabilisic inerpreaion of he indifference price Alhough explici soluions o he general pricing PDE (60) were no consruced, we follow Musiela and Zariphopoulou (2003) and show ha he price funcion solves a paricular sochasic opimal conrol problem. By using he convex dual of he non-linear erm, he PDE is linearized and resuls in a pricing resul similar o he American opion problem. However, in he curren conex, he opimizaion is no over sopping imes. Insead, we find ha he opimizaion is over he hazard raes of he driving Poisson processes. Theorem. 4.3 The soluion of he sysem (60) is given by he value funcion [ P (S, L, ) = sup inf EˆQ z Y y Y e R T (r+ˆκ(s)) ds h(l(t ), S(T )) + 1ˆα T R T e u ( ) }] ˆλ(u) ˆκ(s) ds y(u) ˆβ(y(u)) ˆκ(u) ˆβ(z(u)) du (68) where Y is he se of non-negaive F -adaped processes, he loss process L() = ˆN() n=1 g(s( i ), i ) (69) and i are he arrival imes of he doubly-sochasic Poisson process ˆN(). In he measure ˆQ, he F -adaped hazard rae process for ˆN() is ˆλ() = y() λ() e α()g(s(),), (70) and ˆκ() = z()κ(). Finally, S() saisfies he SDE: ds() = r S() d + σ S() d ˆX(), (71) where ˆX()} 0 T is a ˆQ-Wiener process. Proof. Le β(x) denoe he non-linear erm in (60), i.e. β(x) = 1 e x, (72) and le ˆβ(y) denoe is convex-dual so ha ˆβ(y) = max (β(x) x y) = 1 y + y ln y. (73) x
20 On Valuing Equiy-Linked Insurance and Reinsurance Conracs 20 Clearly, ˆβ(y) is defined on (0, ) and is non-negaive on is domain of definiion. Furhermore, ( ) β(x) = min ˆβ(y) + y x. (74) y 0 Rewriing he exponenial erms in (60) in erms of heir convex dual, we find ha he PDE becomes linear in P : r P = P + r S P S σ2 ( S 2 P SS + κ() α() max z() 0 ˆβ(z()) ) z() α() P (L, S, ))] ( ) + λ() α() eα()g(s,) min y() 0 ˆβ(y()) + y() α() P (L, S, ) P (L, S, T ) = h(l, S). Through he usual dynamic programming principle, we find ha he value funcion (68) saisfies he above HJB equaion. We have shown ha he pricing problem reduces o simulaneously finding he aciviy rae which minimizes and he ineres rae ha maximizes he Black-Scholes price of he reinsurance conrac, subjec o a penaly erm which is iself a funcion of he aciviy rae and ineres raes. I is useful o illusrae how he risk-neural resul of he previous subsecion is recovered. In he limi in which ˆα 0 +, he penaly erm increases o infiniy and he process y which minimizes (68) is clearly he one in which ˆβ(y(u)) = 0 for all u [, T ]. Similarly, he process z which minimizes (68) saisfies ˆβ(z(u)) = 0. This is achieved when y(u) = z(u) = 1. The opimal hazard rae is hen equal o is real world value ˆλ() = λ() and he rae ˆκ() = κ() implying an ineres rae of r + κ(). The price herefore reduces o (64) Numerical examples In he absence of explici soluions, we now demonsrae how he pricing PDE can be used, noneheless, o obain he value of reinsurance conracs hrough a simple implici-explici finiedifference scheme. Since we are no concerned wih proving ha he scheme converges in a wide class of scenarios, we ake a praciioner s viewpoin and apply he scheme o siuaions in which he loss funcion and reinsurance conrac iself are boh bounded and asympoically consan. To his end, i is convenien o rewrie he problem using he log of he forward-price process z() ln S() + r(t ). Also, i is appropriae o scale he price funcion by he risk-aversion parameer and he discoun facor by inroducing he funcion P (L, z, ) α() P (L, e z r(t ), ). (76), (75) Wih hese subsiuions, he pricing PDE (60) becomes 0 = P 1 2 σ2 P z σ2 P zz ) ) κ() (1 e P (L,z,) + λ(z, ) (1 e (P (L+g(z,),z,) P (L,z,)) P (L, z, T ) = ˆα h(l, e z ),, (77) where g(z, ) = g(e z r(t ), ) and λ(z, ) = λ() e α()g(z). Now, we inroduce a M L M z N grid for he (L, z, ) plane wih sep sizes of ( L, z, ) so ha L j = j L, z k = z min + k L, n = n. (78)