On Valuing Equity-Linked Insurance and Reinsurance Contracts

Transcription

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 Insurance companies are increasingly facing losses ha have heavy exposure o capial marke risks hrough he issuance of equiy-linked insurance policies. In his paper, we deermine he coninuous premium rae ha an insurer charges via he principle of equivalen uiliy. Using exponenial uiliy, we obain he resuling premium rae in erms of a risk-neural expecaion. We also consider he relaed problem of pricing double-rigger reinsurance conracs, paying a funcion of he risky asse and losses, once he insurer has fixed her premium rae. We solve he Hamilon-Jacobi-Bellman equaion arising in he indifference pricing problem and show ha he price saisfies a PDE wih a non-linear shif erm. Alhough a closed form soluion is no, generally, aainable, we obain analyical resuls in some special cases. Finally, we recas he pricing PDE as a linear sochasic conrol problem and provide an explici finie-difference scheme for solving he PDE numerically. 1. Inroducion Wih he S&P 500 index yielding reurns of 10% over he las year and 16% over he las wo years, i is no wonder ha individuals seeking insurance are more ofen oping for equiy-linked insurance conracs raher han fixed paymen conracs. Equiy-linked insurance conracs are highly popular opions for policyholders because hey also provide downside proecion. From he insurer s perspecive, such conracs induce claim sizes ha are correlaed o he flucuaions in he value of he S&P 500 index and, as such, posses 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 marke exposure. In his work, we adop he principle of equivalen uiliy, also known as uiliy-based pricing o value such conracs (see e.g. Bowers, e.al. (1997)). This pricing principle prescribes a premium rae a which he insurer is indifferen beween (i) 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. Equiy-linked life insurance policies have been considered in many previous works. Young (2003), for example, sudied equiy-linked life insurance policies wih a fixed premium and wih a deah benefi ha was linked o an index. She demonsraes ha he insurance premium saisfies a non-linear, Black-Scholeslike PDE, where he nonlineariy arises due o he presence of moraliy risk. Young and Zariphopoulou (2002, 2003) also use uiliy-based mehods o price insurance producs wih uncorrelaed insurance and financial risks. Insurance risks ofen resul in economies ha are incomplee, 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) sudied, he indifference pricing mehodology The Naural Sciences and Engineering Research Council of Canada helped suppor his work. 1

2 On Valuing Equiy-Linked Insurance and Reinsurance Conracs 2 yields racable and inuiively appealing resuls. Our work here exends hese earlier sudies in wo main direcions: firsly, by considering equiy-linked losses ha arrive a Poisson imes; and secondly, by simulaneously considering he valuaion of a reinsurance produc wihin one consisen framework. We assume ha he equiy-linked claims (losses) arrive a Poisson imes and ha he insurer may boh 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 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. We focus exclusively on exponenial uiliy for several reasons: firsly, we find ha he opimal invesmen sraegies dicaed by exponenial uiliy is independen of he insurer s wealh. Secondly, he difference beween he holdings, wih and wihou he insurance risk, in he equiy, reduces o he hedge in Black and Scholes (1973). Thirdly, in he limi a which he invesor becomes risk-neural, he premium reduces o he risk-neural expeced losses over he insurer s invesmen ime horizon. From his, 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 equiy-linked loss funcion. We find ha he resuling premium q is proporional o risk-neural expecaion of an exponenially weighed average of he equiy-linked loss funcion. The premium for an insurer who is almos risk-neural is also invesigaed via an asympoic expansion of he exac resul. 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. A mauriy, he reinsurance conrac pays a funcion of he oal observed losses and he equiy value o he insurer. The insurer pays an upfron single benefi premium for his conrac. We prove ha he price saisfies a Black-Scholes-like PDE wih a non-linear shif erm due o he presence of he non-hedgable moraliy risk. If he reinsurance payoff does no depend on he loss level, we show ha he indifference price reduces o he Black-Scholes price of he corresponding equiy opion. We also invesigae he price ha a near risk-neural insurer would be willing o pay, and find ha he price can be wrien in erms of an ieraed risk-neural expecaion. In 4.2, we provide a probabilisic inerpreaion of he indifference price in erms of a dual opimizaion problem. Wihinn his framework, he indifference price is he minimum of he risk-neural expeced value of he reinsurance conrac wih a penaly erm, where he minimum is compued over he aciviy rae of a doubly sochasic Poisson process driving he claim arrivals. Subsequenly, 4.3 provides numerical examples for he reinsurance conrac price in wo special cases: (i) a sop-loss payoff and (ii) a double-rigger sop-loss payoff. 2. The Model 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 follow a compound Poisson process wih claim sizes depending on he price of he risky asse a he loss 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 measure P. We assume ha he insurer is able 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

3 On Valuing Equiy-Linked Insurance and Reinsurance Conracs 3 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 loss process is assumed o following a compound Poisson process wih deerminisic hazard rae λ(), and loss sizes of size g(s(), ), where is he arrival ime of a loss. 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 ), where i are he arrival imes of he Poisson process. We implicily assume ha g(s, ) 0 and is 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 (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 process dynamics 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, where w represens he wealh of he insurer a he iniial ime ; and for each process f, f(u ) represens he value of he process 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 absolue risk-aversion rˆα (w) u (w)/u (w) = ˆα as defined by Pra (1964). We furher assume 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. We denoe he value funcion of he insurer who does no accep he insurance risk by V (w, ), and denoe he value funcion of he insurer who does accep he insurance risk 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 (3) (5)

4 On Valuing Equiy-Linked Insurance and Reinsurance Conracs 4 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, which is a familiar resul when using 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. 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 can be 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 conrac is relevan only a he erminal ime, and is role is simply o increase he insurer s wealh by he conrac value. Alhough he effec is explicily fel a mauriy, i will feed back ino he opimal invesmen sraegy which he insurer follows, and consequenly, i will feed back ino he value funcion iself. The indifference price P (L, S, ) of he conrac is he amoun of wealh he insurer who receives he reinsurance paymen is willing o surrender so 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 funcion is found o be independen of wealh for exponenial uiliy. Furhermore, he indifference price is independen of he premium ha he insurer charges. Raher, i equaes he uiliy of he insurer who receives some premium rae q and is exposed o he equiy-linked losses of an insurer who, in addiion, receives a reinsurance conrac paymen and pays upfron for ha reinsurance. 3. The HJB Equaion For The Indifference Premium 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 his secion are essenially hose of Meron (1969). These 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 The 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

5 On Valuing Equiy-Linked Insurance and Reinsurance Conracs 5 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. (11) 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, (12) V (w, T ) = u(w). We may assume ha he opimal invesmen is provided by he firs order condiion, and he Verificaion Theorem confirms ha he resul holds. To his end, he opimal invesmen sraegy is π () = µ r σ 2 V w V ww. On subsiuing π ino (12), we find ha V saisfies he PDE V 1 ( ) 2 µ r Vw 2 + r w V w = 0. (14) 2 σ V ww Assuming ha (13) V (w, ) = 1ˆα e α() w+β(), (15) wih β(t ) = 0 and α(t ) = ˆα, we deermine ha he HJB equaion reduces o (α + rα) w + β 1 ( ) 2 µ r = 0. (16) 2 σ The above mus hold for all w and ; herefore, α() = ˆα e r(t ) and (17) β() = 1 2 ( ) 2 µ r (T ), (18) σ resuling in he sandard Meron opimal invesmen of (Meron, 1969) π () = µ r ˆα σ 2 e r(t ). (19) The above soluion saisfies he requiremens of he Verificaion Theorem and, herefore π corresponds o he opimal invesmen sraegy for (6); and V, given in (15), 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 + (rw + q)u w + µ S U S σ2 S 2 U SS + λ() (U(w g(s, ), S, ) U(w, S, )) + max π 1 2 σ2 U ww π 2 + π [ (µ r)u w + σ 2 S()U ws ]}, U(w, S, T ; q) = u(w). The shif erm appears due o he presence of he claim arrivals, 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 (20)

6 On Valuing Equiy-Linked Insurance and Reinsurance Conracs 6 sigh, he presence of his non-linear shif erm appears o render he problem inracable. However, on closer inspecion, we find ha, since uiliy is exponenial, he HJB equaion can be solved explicily for arbirary claims funcion g. Theorem. 3.1 The soluion o he HJB sysem (20) is U(w, S, ; q) = V (w, ) exp q ˆα ( ) } e r(t ) 1 + γ(s, ), (21) r where [ T γ(s(), ) = E Q λ(u) ( ) e α(u) g(s(u),u) 1 ] du F, (22) 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(). (23) Furhermore, he opimal invesmen sraegy is independen of wealh and equals } π ) e r(t µ r (S, ) = ˆα σ 2 + S γ S. (24) 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. (25) Subsiue his ino (20), and make he subsiuion U(w, S, ; q) = V (w, ; q) expγ(s, )}, (26) 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: [ ] 0 = V + (r w + q) V w + max π (µ r) π Vw σ2 π 2 V ww, (27) V (w, T ; q) = u(w). Sraighforward calculaions yield he soluion: V (w, ; q) = V (w, ) exp q ˆα ( e r(t ) 1) }. (28) r Making he subsiuion (26) ino (20), 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, (29) γ(s, T ) = 0. The Feynman-Kac heorem direcly leads o soluion (34) from which we observe ha U ww < 0 so ha he maximizaion erm is indeed convex. For smooh loss funcions g(s(), ), he Verificaion Theorem implies, indeed, ha (34) is he value funcion for he problem and sraegy (25) is opimal. Accordingly, subsiuing he ansäz (26) ino π leads o (24). Noice ha if u (, T ], g(s(u), u) > 0 Q a.s. hen γ(s, ) is sricly posiive for every S. In he nex secion, under hese condiions he indifference premium is show o also be posiive.

7 On Valuing Equiy-Linked Insurance and Reinsurance Conracs The Indifference Premium Now given boh value funcions, V and U, i is possible o obain an explici represenaion of he indifference premium. Corollary 3.2 The insurer s indifference premium rae q is independen of wealh and is given by [ ] r T q(s(), ) = e r(t ) 1 EQ λ(u) eα(u) g(s(u),u) 1 du ˆα F. (30) Proof. The indifference premium rae q is defined as he rae q such ha U(w, S, ; q) = V (w, ). Expression (30) hen immediaely follows from Theorem 3.1. Since q is essenially proporional o γ, based on he remarks a he end of he previous secion, and if he claim sizes are posiive Q-a.s. over he ime horizon (, T ], hen he indifference premium is posiive. I is imporan o discuss he dependence of q on he risky asse s price and ime. In analyzing he value funcion U, we assumed ha q was consan; however, on glancing a (30) 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 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 amoun q(s(), ), 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. 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 depended on he risky asse s price. The indifference premium (30) also has a few very appealing properies which warran discussing. Regardless of he risk-aversion level of he insurer, he expecaion appearing in he premium calculaion is always compued in a risk-neural measure Q. Furhermore, he risk-neural disribuion of claim sizes has no been disored from he real world disribuion. Indeed, he Radon-Nikodym derivaive process which performs he measure change is η() ( dq dp ) = exp 1 2 ( µ r σ ) 2 + µ r σ X() }. (31) This is he same measure change ha Meron (1976) uses in his jump-diffusion model and corresponds o risk adjusing only he diffusion componen. Alhough he risk-aversion level does no feed ino he probabiliy measure used for compuing expecaions, i does manifes iself in he disorion of he claim sizes hrough he exponenial erm. The exponenial erm is inheried, indeed, from he uiliy funcion. Furhermore, noice ha he facor in fron of he expecaion can be represened as 1/ T e r(t u) du. The denominaor of his expression is simply he coninuous premium rae of $1 per annum which is accumulaed o end of he ime horizon T. Consequenly, he facor in fron of he expecaion can be viewed as a normalizaion consan. Finally, alhough he premium is a non-linear funcional of he claim sizes g(s(), ), i is linear in he arrival rae of he claims λ(). This observaion suggess ha he

8 On Valuing Equiy-Linked Insurance and Reinsurance Conracs 8 generalizaion o muliple claims disribuions is sraighforward. In he heorem below, we provide he resuls for muliple claims disribuions. The proof follows along he same lines as hose in he previous wo secions and we omi i for breviy. Theorem. 3.3 Suppose ha he insurer is exposed o losses from m differen sources of risk. Explicily, he loss process is modeled as follows: L() = m N j() j=1 n=1 g j (S( i j), i j), where N j () : j = 1,..., m} are independen Poisson processes wih arrival raes λ j () : j = 1,..., m} and g j (S, ) denoe he loss funcions for he i-h source of risk. 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 q ˆα ( ) } e r(t ) 1 + γ(s, ), (33) r where γ(s(), ) = [ m T E Q λ j (u) j=1 ( ) e α(u) gj(s(u),u) 1 (32) ] du F, (34) 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(). (35) Furhermore, he insurer s indifference premium is independen of wealh and is explicily [ r m ] T q(w, S, ) = E Q λ e r(t ) j (u) eα(u) gj(s(u),u) 1 du 1 ˆα F. (36) j= Consan Losses And Risk Neural Insurers In his subsecion, we consider claims which have consan losses g(s, ) = l and a consan arrival rae λ() = λ. In his case, we deermine he premium rae is λ ( ) q = ˆα ( e r(t ) 1 ) Ei(ˆα l e r(t ) ) Ei(ˆα l) (T )r, (37) where Ei(x) denoes he so called exponenial inegral, defined as he following Cauchy principle value inegral: x Ei(x) e d. If he insurer is near risk-neural, hen a Taylor 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). (39) 4 As such, a risk-neural insurer who is exposed o fixed losses, will charge a rae equal o he expeced loss per uni ime λ l - an inuiively appealing resul. As expeced, he sign of he firs order correcion is posiive. For losses ha grow a mos linearly, i.e. here exiss b() > 0 and S () > 0 such ha for each and S > S (), g(s, ) b() S, he rae has he following perurbaive expansion in erms of he risk-aversion parameer ˆα: q = λ r e r(t ) 1 n=1 ˆα n 1 n! T (38) e n r(t u) E Q [ g n (S(u), u) F ] du. (40)

9 On Valuing Equiy-Linked Insurance and Reinsurance Conracs 9 Indifference Premium [ q ] $150 $130 $110 α = 0 α = 0.1 α = 0.2 $90 $70 $90 $110 $130 Indifference Premium [ q ] $150 $130 $110 α = 0 α = 0.1 α = 0.2 $90 $70 $90 $110 $130 Spo Price [ S() ] Spo Price [ S() ] Figure 1. The dependence of he indifference premia on he underlying equiy spo price for losses described in The model parameers are F = 1, K = 100, β = 1, r = 4%, σ = 15%, and λ = 100. The erms in he lef/righ panels are one and five years respecively. This series is shown o converge by appealing o he Lebesgue dominaed convergence heorem and noing ha E Q [S n (u) F ] = S()e n(r+ 1 2 σ2 (n 1))(u ). In fac, he lineariy 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. 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) F ] du. (41) The above premium can be inerpreed as he average risk-neural expeced loss per uni ime, where he expeced losses have been accumulaed o mauriy and hen normalized (raher han discouned) back o ime. In he nex wo subsecions, we provide wo explici examples 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, we define A(u) as he expecaion appearing under he inegral in he indifference premium (30), i.e. A(u) E Q [ e α(u) g(s(u),u) F ]. (42) Then, he indifference premium q can be wrien in erms of A(u) explicily as q = r ˆα ( e r(t ) 1 ) T λ(u) A(u) 1} du. (43) Floor and Marke Paricipaion Claims In our firs explici example, we consider insurance claims which pay a minimum of F and hen grows proporionaely o he logarihm of he excess spo price above he srike level K; ha is, g(s(), ) = F + β (log (S()) log(k)) +. (44) Sraighforward, bu edious, calculaions lead o he resul: [ (S() ) ] βα(u) A(u) = e F α(u) Φ( d 2 (K)) e βα(u)(r 1 2 σ2 (1 βα(u)))(u ) + Φ(d 1 (K)), (45) K

10 On Valuing Equiy-Linked Insurance and Reinsurance Conracs 10 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 2. The dependence of he indifference premia on he underlying equiy spo price for losses described in 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. where d 1 (K) = ( log K S() ) (r 1 2 σ2 )(u ) σ, (46) u d 2 (K) = d 1 (K) β α(u) σ u, (47) and Φ( ) is he cumulaive disribuion funcion for he sandard normal disribuion. In Figure 1, we illusrae how he premium depends on he underlying spo price 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 (44) scaled by he aciviy rae for comparison purposes. Naurally, as he risk-aversion parameer increases, he premia increases. Furhermore, since he loss is increasing as he spo grows, he premia increases as he mauriy increases. This resul is analogous o he pricing behavior of a call opion in he Black-Scholes model Floor, Capped, and Marke Paricipaion Claims In our second explici example, we consider insurance claims which have a cap and a floor proecion in addiion o a paricipaion in he risky asse s reurn. 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, (48) θ + β (log(c 2 ) log(c 1 )), S() c 2. Once again, afer some edious calculaions, we find ha he inegrand A(u) reduces o ( S() ) β α(u) A(u) = e Φ(d α(u)θ 1 (c 1 )) + 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 ))) ( ) } β α(u) + c2 c Φ( d1 1 (c 2 )). (49) In Figure 2, we illusrae how he premium depends on he underlying spo price 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 (48) scaled by he aciviy rae for comparison purposes. Once again, as he risk-aversion parameer increases, he premia increases. In his case, he loss is bounded from above and below; herefore, increasing mauriy does no aler he premia as significanly as he uncapped case

11 On Valuing Equiy-Linked Insurance and Reinsurance Conracs 11 explored in he previous example. In fac, he premium acually decreases for larger spo prices when he erm increases. This resul is analogous wih he pricing for a sandard bull-spread opion in he Black-Scholes model Hedging The 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 exacly replicae he insurance claims; noneheless, he insurer sill holds differen unis of he risky asse when she is exposed o he insurance risk or 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, we define he Dela 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 3.4 The Dela of he insurer s posiion is (S, ) 1 S (π U π V ) = e r(t ) ˆα γ S (S, ). (50) Proof. The opimal invesmen in he risky asse wihou he insurance risk appears in (13), and wih he insurance risk appears in (24). Noice ha his resul is quie similar o he Black-Scholes Dela for an opion. However, here is a suble difference since he funcion γ(s(), ) appears in he resul, raher han in he premium rae iself. Moreover, as T, he Dela vanishes, his behavior conras wih he behavior of he Dela of an opion. In he case of a European opion, he Dela becomes equal o he derivaive of he payoff funcion, and is zero only when he opion has a consan payoff, namely, when he opion is acually a bond. To undersand why he Dela vanishes as mauriy approaches in our case, suppose ha he ime horizon ends in T 1 from now; hen, he probabiliy of a loss arriving is λ T and herefore here is no need o hold addiional shares of he risky-asse. In Figure 3, we show how he Dela behaves as a funcion of he spo-level, risk-aversion parameer, and ime o mauriy for he examples in and The general shape of hese curves is expeced. In he firs example, alhough he payoff grows logarihmically, i appears o grow linearly over he scale shown in diagram (see Figure 1), and herefore he Dela flaens ou. While in he second example, he payoff is asympoically fla ouside of he paricipaion region (see Figure 1), and herefore, he Dela decays in he ails. 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 and 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 (20)); 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

12 On Valuing Equiy-Linked Insurance and Reinsurance Conracs 12 Dela [ Δ(S(),) ] α = 0 α = 0.1 α = 0.2 Term = 5 years Term = 1 year 0.0 $70 $90 $110 $130 Dela [ Δ(S(),) ] Term = 5 years α = 0 α = 0.1 α = Term = 1 year 0.0 $70 $90 $110 $130 Spo Price [ S() ] Spo Price [ S() ] Figure 3. The dependence of he Dela on he underlying equiy spo price for losses described in and (lef/righ panel respecively). The model parameers are he described in Figures 1 and 2. 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, )) + max 1 π 2 σ2 Uww R π 2 + π [ (µ r)uw R + σ 2 S()Uws]} R, U R (w, L, S, ; q) = u(w + h(l, S)). (51) The shif 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. 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 (51) can be wrien as U R (w, L, S, ) = U(w, S, )φ(l, S, ), (52) where φ saisfies he non-linear PDE ( ) 0 = φ + r Sφ S σ2 S 2 φ SS φ2 S φ + λ() e α()g(s,) (φ(l + g(s, ), S, ) φ(l, S, )), φ(l, S, T ) = e ˆα h(l,s). (53) Furhermore, he opimal invesmen in he risky-asse is [ π ) e r(t µ r (S, ) = ˆα σ 2 + S γ S + φ ]} S. (54) φ Proof. Assuming ha U R ww < 0, we find he firs order condiions allow he opimal invesmen sraegy o be wrien, π () = (µ r)u R w + σ 2 S()U R ws σ 2 U R ww. (55) On subsiuing he ansäz (52) and he opimal invesmen (55) ino he HJB equaion (51), we esablish } 0 = φ U + (rw + q)u w + µsu S σ2 S 2 U SS 1 ((µ r)u w+σ 2 SU ws) 2 2 σ 2 U ww ( ) ( ) )} +U φ + µ (µ r) U 2 w U U ww S φ S σ2 S (φ 2 SS U 2 w φ 2 S U U ww φ 2 UwS U w U U ww U S U φ (56) S +λ() [U(w g(s, ), S, ) φ(l + g(s, ), S, ) U(w, S, ) φ(l, S, )],

13 On Valuing Equiy-Linked Insurance and Reinsurance Conracs 13 subjec o he boundary condiion U(w, S, T )φ(l, S, T ) = u(w + h(l, S)). From (20), 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,), U 2 w U ww U = 1, and U ws U w U ww U = U S U = γ S, (57) we find, hen, equaion (56) disills o (53). I can hen be proven ha Uww R < 0. Using he ansäz (52), he opimal invesmen π can be rewrien as (54). 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 shifed PDE: r P = P + r S P S σ2 S 2 P SS + λ() ( α() eα()g(s,) 1 e α()[p (L+g(S,),S,) P (L,S,)]), (58) P (L, S, T ) = h(l, S). Proof. The indifference price is defined as he amoun of iniial wealh P he insurer is willing o surrender so ha her value funcion wih he reinsurance paymen is equal o her value funcion wihou he reinsurance paymen. Specifically, he price P saisfies U R (w P, L, S, ) = U(w, S, ). (59) Hence, P (L, S, ) = 1 α() ln φ(l, S, ), and on subsiuing φ in erms of P in (53), we obain (58). Noice ha if he payoff funcion h(l, S) is independen of he loss level, i.e. h(l, S) = h(s), hen (58) reduces o r P = P + r S P S σ2 S 2 P SS, (60) P (L, S, T ) = h(s). The above price can be recognized as he price of a European opion wih payoff h(s) in he Black-Scholes model (Black and Scholes, 1973). This resul is expeced since he reinsurance conrac is now exposed only o he hedgable risk - he risky asse - and no o he non-hedgable claims risk. Therefore, our resul should reduce o he no arbirage Black-Scholes price for an insurer of any degree of risk-aversion 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 follows from (58) and reduces o r P 0 = P 0 + r S PS σ2 S 2 PSS 0 + λ() P 0 (61) P 0 (L, S, T ) = h(l, S), where P 0 denoes he increase in he price due o a loss arrival, i.e. P 0 P 0 (L + g(s, ), S, ) P 0 (L, S, ). Consequenly, hrough he Feynman-Kac Formula, a risk-neural insurer would be willing o pay [ ] P 0 (L, S, ) = E Q e r(t ) h(l(t ), S(T )) F (62) for he reinsurance conrac, where he Q-dynamics of S() appears in (35). Furhermore, he risk-neural dynamics of L() is unalered from is real world dynamics, and in paricular, he aciviy rae of he driving Poisson process remains a λ() under he measure Q. Alhough his marke is incomplee, and

14 On Valuing Equiy-Linked Insurance and Reinsurance Conracs 14 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. 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. 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), (63) subjec o P 0 (L, S, T ) = h(l, S) and P 1 (L, S, T ) = 0. When insering his ansäz ino (58) and using (61), we deermine P 1 (L, S, ) saisfies he following PDE: r P 1 = P 1 + r S PS σ2 S 2 PSS 1 + λ() P 1 +λ() e g r(t ) 2 (S, ) [ P 0 (L, S, ) g(s, ) ] } 2 + o(ˆα), (64) 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 λ(u) g 2 (S(u), u) [ P 0 (L(u), S(u), u) g(s(u), u) ] } ] 2 du F. (65) Ineresingly, he payoff funcion h(l, S) does no explicily appear in P 1 ; raher, i feeds from he riskneural price funcion P 0 which does explicily depend on he payoff. The sign of his correcion erm is difficul o discern on firs observaion. 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 The Indifference Price Alhough explici soluions o he general pricing PDE (58) 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 rae of he driving Poisson process. Theorem. 4.3 The soluion of he sysem (58) is given by he value funcion [ ] T P (S, L, ) = e r(t ) inf EˆQ r(t u) ˆλ(u) h(l(t ), S(T )) + e β(y(u)) du y Y y(u)α(u) F (66) where Y is he se of non-negaive F -adaped processes, he loss process L() = ˆN() n=1 g(s( i ), i ) (67) and i are he arrival imes of he doubly-sochasic Poisson process ˆN() where he F -adaped hazard rae process is ˆλ() = y() λ() e α()g(s(),) (68)

15 On Valuing Equiy-Linked Insurance and Reinsurance Conracs 15 in he measure ˆQ. Finally, S() saisfies he SDE: ds() = r S() d + σ S() d ˆX(), (69) where ˆX()} 0 T is a ˆQ-Wiener process. Proof. Le β(x) denoe he non-linear erm in (58), i.e. β(x) = 1 e x, (70) and le ˆβ(y) denoe is convex-dual so ha ˆβ(y) = max (β(x) x y) = 1 y + y ln y. (71) x Clearly, ˆβ(y) is defined on (0, ) and is non-negaive on is domain of definiion. Furhermore, ( ) β(x) = min ˆβ(y) + y x. y 0 Rewriing he exponenial erm in (58) in erms of ˆβ(y), we find ha he PDE becomes linear in P : r P = P + r S P S σ2 S 2 P SS ( ) + λ() α() eα()g(s,) min y() 0 ˆβ(y()) + y() α()[p (L + g(s, ), S) P (L, S)], (73) P (L, S, T ) = h(l, S). Through he usual dynamic programming principle, we find ha he value funcion (66) saisfies he above HJB equaion. The Verificaion Theorem, herefore, hen implies ha he soluions of (58) can be represened by (66). The pricing problem reduces o finding he aciviy rae ha minimizes he Black-Scholes price of he reinsurance conrac, subjec o a penaly erm, which is a funcion of he aciviy rae iself. 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 ha minimizes (66) is clearly he one in which ˆβ(y(s)) = 0 for all s [, T ]. This is achieved when y(s) = 1. The opimal hazard rae hen becomes equal o is real world value ˆλ() = λ(), and he price reduces o (62) Numerical Examples In he absence of explici soluions, we demonsrae how he pricing PDE can be used, noneheless, o obain he value of reinsurance conracs hrough a simple finie-difference 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. 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 ), ). (74) (72) Wih hese subsiuions, he pricing PDE (58) becomes ) 0 = P 1 2 σ2 P z σ2 P zz + λ(z, ) (1 e (P (L+g(z,),z,) P (L,z,)), P (L, z, T ) = ˆα h(l, e z ), (75) where g(z, ) = g(e z r(t ), ) and λ(z, ) = λ() e α()g(z). Here, we inroduce a 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. (76)

16 On Valuing Equiy-Linked Insurance and Reinsurance Conracs 16 We hen obain he following explici finie difference scheme for solving (75): P (L j, z k, n 1 ) = 1 2 σ2 ( z)p (L j, z k 1, n ) + (1 σ)p (L ( j, z k, n ) ) σ(1 1 2 z)p (L j, z k+1, n ) + λ(z k, n ) 1 e P (Lj,z k, n), where P (L j, z k, T ) = h(l j, e z k ), σ 2 = z 2 σ2 and (78) P (L j, z k, n ) = P (L j + g(z k, n ), z k, n ) P (L j, z k, n ). (79) However, o complee he descripion of he finie difference scheme, we need o impose appropriae boundary condiions. For he numerical examples in his secion, we assume ha claim sizes are generae losses as described in Since such claims are asympoically consan as S approaches zero and infiniy, he price of he reinsurance mus be asympoically consan as well. Accordingly, we impose he boundary condiions P S (L, z min, ) = 0 and P S (L, z max, ) = 0 for all loss levels L and imes. To illusrae he effecs of he insurer s risk aversion level on he pricing resuls, we focus on wo prooypical reinsurance payoff funcions: h 1 (L, S) = min(m, max(0, L m)) and (80) h 2 (L, S) = I(S > S ) min(m, max(0, L m)). (81) The firs reinsurance payoff funcion h 1 corresponds o a sop-loss reinsurance conrac wih paymens saring a losses of m and aaining a maximum of M. This reinsurance conrac makes paymens ha are independen of he risky asse s price a mauriy; however, because he loss sizes are linked o he equiy value, he value of he conrac a iniiaion will depend on he spo price of he risky asse. The second reinsurance payoff funcion h 2 corresponds o a double-rigger reinsurance conrac in which a sop-loss paymen is made if he risky asse s price rises above a criical value S. In Figure 4, we show how he price of he wo reinsurance conracs depend on he prevailing spo price for several levels of risk aversion ˆα. As expeced, for boh reinsurance conracs, he price increases as he insurer becomes more risk averse. Furhermore, for any given risk-aversion level, he price of he double-rigger sop-loss conrac is lower han he pure sop-loss conrac. This oo is expeced since he double-rigger conrac pays nohing if he risky asse s price is below he rigger level a mauriy. Finally, in he region of large risky asse prices, he wo conracs asympoically approach he same values. 5. Conclusions In his paper, we obained he premium an insurer requires if she akes on he risk of equiy-linked losses. To do so, we employed he principle of equivalen uiliy wih consan absolue risk-aversion, i.e. exponenial uiliy, o value he conrac, and alhough he insurer is risk-averse, we demonsraed ha he premium is obainable by compuing a risk-neural expecaion of an exponenially weighed average of he claim sizes. In he limi in which he insurer becomes risk-neural his expecaion reduces o he expeced loss per uni ime. Furhermore, we examined he indifference price for and an insurer who ook on he risk of he equiy-linked insurance conrac, bu he general non-linear PDE ha arises from he associaed HJB equaions was no solved. However, we were able o rewrie he non-linear PDE in erms of a dual linear opimizaion problem. This allowed us o provide a probabilisic inerpreaion of he pricing problem for he reinsurance conrac: The price in he dual represenaion, is a minimum of (77)

17 On Valuing Equiy-Linked Insurance and Reinsurance Conracs 17 Sop-Loss Price $150 $100 $50 α = 0.1 α = 0.2 α = 0 Double-Trigger Sop-Loss Price $150 $100 $50 α = 0.1 α = 0.2 α = 0 $0 $0 $50 $100 $150 $200 Spo Price $0 $0 $50 $100 $150 $200 Spo Price $250 $250 $1,000 Loss $500 $0 $0 $50 $100 $150 $200 $200 $150 $100 $50 $0 Spo Price Sop-Loss Price $1,000 Loss $500 $0 $0 $50 $100 $150 $200 $200 $150 $100 $50 $0 Spo Price Double-Trigger Sop-Loss Price Figure 4. The indifference price for he reinsurance conracs (80) and (81) wih losses described in The model parameers are hose used in Figure 2. In he boom panels, he risk-aversion parameer is se o α = 0.2. In all experimens, we used 1000 ime seps and a grid of size wih z min = 10, z max = 10 and L max = 2000.

18 On Valuing Equiy-Linked Insurance and Reinsurance Conracs 18 he risk-neural price plus a penaly erm, where he opimizaion is over a sochasic aciviy rae for he Poisson processes driving he arrival imes. In he limi of a risk-neural insurer, we demonsraed ha he price reduces o an expecaion of he reinsurance payoff over a risk-neural measure in which he disribuion of losses are idenical o he real-world disribuion. There are several avenues ha are open for furher exploraion. For example, i would be ineresing o obain he disribuion of he ruin imes for insurers facing hese equiy-linked risks. A closed form resul for general loss funcions g(s, ) is no likely. We suspec, however, ha in cases when g is a piecewise linear funcion of he log-sock price, a semi-explici form migh be available. Anoher exploraion could involve ruin-relaed problems: such as he opimal consumpion problem for he insurer where he value funcion runcaes a he ime of ruin. This is similar o he quesions Young and Zariphopoulou (2002) addressed in he conex of fixed loss sizes. Exending heir resuls o he case of equiy-linked losses would be quie ineresing. The Gerber and Shiu (1998) penaly funcion is anoher problem relaed o he ime of ruin, and alhough i is no explicily conneced o quesion of indifference pricing, i would also be ineresing o invesigae is equiy-linked exensions. References Black, F., and M. Scholes, 1973, The Pricing of Opions and Corporae Liabiliies, The Journal of Poliical Economy, 81, Fleming, W., and H. Soner, 1993, Conrolled Markov Processes and Viscosiy Soluions, Springer, New York. Gerber, H., and E. Shiu, 1998, On he ime vlaue of ruin, Norh American Acuarial Journal, 2, Harrison, J., and S. Pliska, 1981, Maringales and Sochasic Inegrals in he Theory of Coninuous Trading, Sochasic Processes and Their Applicaions, 11, Jaimungal, S., and V. Young, 2005, Pricing equiy-linked pure endowmens wih risky asses ha follow Lévy processes, Insurance: Mahemaics and Economics, 36, Meron, R., 1976, Opion pricing when underlying sock reurns are disconinuous, Journal of Financial Economics, 3, Meron, R. C., 1969, Lifeime porfolio selecion under uncerainy, Rev. Econom. Sais., 51, Musiela, M., and T. Zariphopoulou, 2003, Indifference prices and relaed measures, Preprin. Pra, J., 1964, Risk aversion in he small and in he large, Economerica, 32, Young, V., 2003, Equiy-indexed life insurance: pricing and reserving using he principle of equivalen uiliy, Norh American Acuarial Journal, 17, Young, V., and T. Zariphopoulou, 2002, Pricing dynamic insurance risks using he principle of equivalen uiliy, Scandinavian Acuarial Journal, 4, Young, V., and T. Zariphopoulou, 2003, Pricing insurance via sochasic conrol: opimal consumpion and wealh, preprin.