Scand. Acuarial J. 00; 4: 46 79 ORIGINAL ARTICLE Pricing Dynamic Insurance Risks Using he Principle of Equivalen Uiliy VIRGINIA R. YOUNG and THALEIA ZARIPHOPOULOU Young VR, Zariphopoulou T. Pricing dynamic insurance risks using he principle of equivalen uiliy. Scand. Acuarial J. 00; 4: 46 79. We inroduce an expeced uiliy approach o price insurance risks in a dynamic nancial marke seing. The valuaion mehod is based on comparing he maximal expeced uiliy funcions wih and wihou incorporaing he insurance produc, as in he classical principle of equivalen uiliy. The pricing mechanism relies heavily on risk preferences and yields wo reservaion prices one each for he underwrier and buyer of he conrac. The framework is raher general and applies o a number of applicaions ha we exensively analyze. Key words: Dynamic insurance risks, reservaion prices, incomplee markes, expeced uiliy, Hamilon -Jacobi -Bellman equaions. 1. INTRODUCTION The purpose of his work is o inroduce a coheren mehod for he valuaion of insurance risks in a dynamic marke seing. Generally, acuaries have analyzed and priced such risks by using mehods ha rely primarily on saic sraegies. Due o he ever-increasing complexiy of he producs inroduced daily in insurance markes, i is imperaive o nd mechanisms ha are more sophisicaed and able o accommodae he individual feaures of he inheren insurance risks. The valuaion of dynamic risks has been a fundamenal issue in nancial markes, primarily in he area of derivaive securiies. One successful pricing heory is based on a sraegy in which one creaes a porfolio ha accuraely replicaes he payoff of he produc. The risk associaed wih he nancial produc is hereby compleely eliminaed or hedged. Thus, one can argue ha he value of he produc mus be he cos of seing up he hedging porfolio. This is he key ingredien of he celebraed Black-Scholes mehod ha has been a landmark in derivaive asse pricing (Black & Scholes, 1973). Despie is success, which resuled in he grea growh of derivaive markes, he Black-Scholes approach breaks down enirely once he fundamenal assumpions of he characerisics of he marke are removed. These assumpions include compleeness of he marke, liquidiy, absence of ransacion coss and rading consrains, consan volailiy, and perfecly observable asses, o name a few. For an overview, see Wilmo e al. (1995). 00 Taylor & Francis. ISSN 0346-138 DOI: 03461301101063 7
Scand. Acuarial J. 4 Pricing dynamic insurance risks 47 In he case of incomplee markes, here is no universal heory o dae ha successfully addresses all aspecs of pricing, for example, numeraire properies, speci caion of hedging sraegies, and robusness of prices. Various alernaive pricing mechanisms have been developed ha are srongly oriened owards he speci c naure of each marke fricion. For example, he assumpion of consan volailiy can be relaxed, and a number of sochasic volailiy models have been proposed for valuaing and calibraing volailiy (Hull & Whie, 1987; Heson, 1993; Dupire, 1994; Renaul & Touzi, 1996). In he case of oher fricions, such as ransacion coss or rading consrains, imperfec replicaing or super-replicaing sraegies have been inroduced ha minimize he slippage error in a (model-relaed) appropriae sense (Leland, 1985; Jouini & Kallal, 1995; Cvianic & Karazas, 1996; Karazas & Kou, 1996; Cvianic e al., 1999). A differen approach is one ha is based on expeced uiliy argumens and produces he so-called reservaion prices. This mehodology is buil around he invesors preferences owards he risks ha canno be eliminaed due o marke fricions. The risk preferences are inroduced via uiliy funcionals for he buyer and he wrier of he nancial claim. To esablish he wrier s reservaion price, for example, one examines her maximal expeced uiliy wih and wihou wriing he claim. The compensaion a which he wrier is indifferen beween he wo alernaive invesmen opporuniies yields his reservaion price. The fundamenal idea for his approach sems from he basic economic principle of cerainy equivalen, bu modi ed and exended o accommodae he dynamic aspecs of he marke environmen. I was inroduced by Hodges & Neuberger (1989) for he valuaion of European calls in he presence of ransacion coss and laer exended by Davis e al. (1993). Since hen, a subsanial body of work has been produced by using eiher sochasic conrol mehods (Davis & Zariphopoulou, 1995; Consaninides & Zariphopoulou, 1999, 001; and Barles & Soner 1998) or by using maringale heory argumens (Davis, 1997; Karazas & Kou, 1996; and Rouge & El Karoui, 000). Our purpose herein is o exend an expeced uiliy mehod, he principle of equivalen uiliy, o price dynamic insurance risks. The moivaion o underake his ask comes from he fac ha insurance markes are de faco incomplee markes. In fac, he risks we wan o price are relaed o uncerainies ha do no correspond o ucuaions of a radable asse; herefore, we are no able o use he classical Black-Scholes analysis and hereby eliminae he relevan risk. This is a fundamenal dif culy, and we are going o inroduce pricing crieria based on uiliy argumens in order o overcome i and o consruc meaningful prices. Our approach exends he one applied in earlier work in acuarial science (for example, Borch, 1961; Bowers e al., 1997; and Gerber & Pafumi, 1998), in which acuaries use expeced uiliy of erminal wealh o calculae prices in a saic seing he so-called principle of equivalen uiliy. As will be apparen in subsequen secions, he speci caion of he reservaion price is a formidable ask. Indeed, one needs o solve wo sochasic opimizaion
48 V. R. Young & T. Zariphopoulou Scand. Acuarial J. 4 problems and o exrac he argumen ha makes heir value funcions equal. In general, explici soluions are no readily available. However, i urns ou ha using exponenial uiliy faciliaes he compuaions and, hus, he speci caion of he price. Because our purpose is primarily o inroduce he principle of equivalen uiliy in a dynamic seing, our examples use only his class of uiliy funcions. We sar our presenaion wih he sochasic opimizaion model of expeced uiliy of erminal wealh. This fundamenal model was inroduced by Meron (1969 and 1971; or Chapers 4 and 5 in 199) and subsequenly revisied, generalized, and exended by a number of auhors. In Secion 3, we inroduce he principle of expeced uiliy and de ne he reservaion prices of insurance claims. To illusrae he use of expeced uiliy argumens, we sar wih a claim of xed expiraion ime and derive is fair price in he simple case of a complee marke. In he absence of marke fricions, he price naurally coincides wih he Black-Scholes price of he claim and can be direcly calculaed by replicaion argumens. Even hough, under marke compleeness, he uiliy mehod appears redundan, we choose o presen he relevan argumens so ha he audience becomes accusomed wih he sochasic conrol framework and he underlying srucure of he reservaion prices. In Secion 4, we consider liabiliies ha are payable a a xed ime T and are independen of he underlying risky asse. We begin by regarding a single insured life and calculae he reservaion prices for erm life insurance; hen, we exend ha model o one ha includes more han one independen life. We nex consider pure endowmen insurance, and we end he secion by modeling insurance risks as diffusion and Poisson processes. In each case, he reservaion prices we obain are calculaed via he so-called value funcions (opimal expeced uiliy of erminal wealh) ha are shown o be soluions of cerain non-linear parial differenial equaions, known as he Hamilon-Jacobi-Bellman equaions. In Secion 5, we consider insurance payable a he ime of incurrence of he loss, and in Secion 6, we look a claims involving a random ime,, such as he ime of deah. In boh Secions 5 and 6, we parallel he opics from Secion 4. In he examples using exponenial uiliy, we nd in Secions 4 and 5 ha he prices are independen of he risky sock process. Thus, he prices are idenical o he ones obained when invesmen is limied o he riskless bond. For ha reason, our mehod may appear o be raher complicaed perhaps unnecessarily so. However, in Secion 6, we learn he raher ineresing fac ha when he horizon is random, he prices depend on he parameers of he risky sock process. Also, our mehod applies o any smooh (increasing and concave) uiliy funcion; herefore, our mehod can be applied o oher uiliy funcions, such as power or logarihmic uiliy. In hose cases, he prices will depend on he risky sock process, in conras wih he examples in Secions 4 and 5. In Secion 7, we conclude our paper wih a summary and suggesions for furher research.
Scand. Acuarial J. 4 Pricing dynamic insurance risks 49. BACKGROUND RESULTS ON STOCHASTIC OPTIMIZATION AND EXPECTED UTILITY In his secion, we review he fundamenal classical model of opimal porfolio managemen for expeced uiliy of erminal wealh. This model was inroduced by Meron in his seminal papers (1969 and 1971; or Chapers 4 and 5 in 199), and is exensions have araced grea ineres boh from academics and praciioners. Two main mehodologies have been used in he analysis of expeced uiliy models one relies on maringale echniques and he oher uses opimal sochasic conrol and non-linear parial differenial equaions. For an overview of he wo approaches, see he monograph of Karazas (1996) and he review papers by Zariphopoulou (1999b, 001). Meron s model examines he opimal invesmen sraegies of an individual who, endowed wih iniial wealh, seeks o maximize her expeced uiliy of erminal wealh, i.e., wealh a he end of a (prespeci ed) rading horizon. The invesor has he opporuniy o rade beween a riskless bond and a risky sock accoun. The price of he sock S s is modeled as a geomeric Brownian moion: ds s ¾S s (m ds s db s ), S ¾S\0. (.1) The process B s is a sandard Brownian moion on a probabiliy space (V, F, P), and he coef ciens m and s are given posiive consans, known, respecively, as he mean rae of reurn and he volailiy coef cien. I is assumed hroughou he analysis ha m\r\0, in which r is he rae of reurn of he riskless bond. The invesor is given, a ime \0, an iniial endowmen we0, and she rades dynamically beween he wo accouns; in oher words, he invesor chooses he 0 amouns p s and ps, 0 s 0T o inves in he bond and he sock accoun, respecively. The consan T\ 0 represens he end of he rading horizon. The oal curren wealh sais es he budge consrain W s ¾p s 0 ps and follows he sae dynamics dw s ¾rW s ds (m¼r)ps ds sps db s, 0s0T. (.) One can easily derive his equaion by using he de niion of W s and he dynamics in (.1); see Meron (1969). Noe ha he budge consrain, ogeher wih he log-normaliy assumpion on he sock dynamics, enables us o eliminae one of he conrol policies in he conrolled wealh diffusion process. In he absence of any addiional risk, for example, a random liabiliy or payoff, he invesor seeks o maximize he expeced uiliy of erminal wealh V(w, )¾ sup {p}ï A E[u(W T ) W ¾w]. (.3) The se A is he se of admissible policies, {ps}, ha are F s -progressively measurable (in which F s is he augmenaion of s(w u : 0u0s)) and ha saisfy
50 V. R. Young & T. Zariphopoulou Scand. Acuarial J. 4 he inegrabiliy condiion E T p s dsb Ä. The uiliy funcion u: R R is assumed o be increasing, concave, and smooh. The soluion of (.3) is known as he value funcion, and i sais es he Hamilon-Jacobi-Bellman (HJB) equaion V max p (m¼r)pv w 1 s p V ww rwv w ¾0, (.4) V(w, T) ¾u(w). (.5) The HJB equaion is he offspring of he principle of dynamic programming and of sochasic calculus. If i can be shown a priori ha he value funcion is smooh (C,1 (R½[0, T])), hen resuls, which are well known by now, yield ha he value funcion equals he unique smooh soluion of he HJB equaion. Addiionally, he opimal policies can be speci ed via he rs-order condiions arising in (.4). Indeed, he concaviy of he uiliy funcion u, ogeher wih he lineariy of he sae equaion (.) wih respec o he wealh and he porfolio process, implies ha he value funcion iself inheris his propery of concaviy. Therefore, he maximum in (.4) is well de ned and achieved a p*(w, )¾ ¼ ( m¼r) s V w (w, ) V w w (w, ). (.6) One can esablish ha he opimal policy is given via (.6) in a feedback law in he following sense: The opimal invesmen process in he sock accoun is P* s ¾p*(W* s, s)¾ ¼ ( m¼r) V w (W* s, s), 0s0T, (.7) s V w w (W* s, s) in which V solves (.4) and W* s is he opimal wealh process solving (.) wih P* s used for ps. The above classical opimaliy resuls are known as he Veri caion Theorem (Fleming & Soner, 1993, Chaper 6)..1 Remark. Because we assume ha he sock price follows a lognormal process, i does no appear as an exra sae variable. This is no he case if he dynamics are non-linear (Zariphopoulou, 1999a) or if he volailiy is modeled as a correlaed process (Zariphopoulou, 001).. Remark. If he HJB equaion does no have smooh soluions, as is usually he case in incomplee markes, one needs o work wih a relaxed class of soluions and o de ne opimaliy conceps herein. I urns ou ha a rich class of weak soluions, which are appropriae for he applicaions a hand, are he so-called viscosiy soluions. They were inroduced by Crandall & Lions (1983) for rs-order non-linear equaions and by Lions (1983) for second-order ones. In he conex of expeced uiliy models, Zariphopoulou (199, 1994) rs inroduced viscosiy soluions, which have by now become a sandard ool for he analysis of sochasic
Scand. Acuarial J. 4 Pricing dynamic insurance risks 51 opimizaion models in markes wih fricions. For an overview, see he review aricles by Zariphopoulou (1999b, 001)..3 Remark. Noe ha in our analysis, he expiraion is a xed ime; his is a naural feaure in models of asse pricing and porfolio managemen. However, when we analyze insurance models, his is no necessarily he case, and one may need o consider sochasic horizons, a naural propery of random evens ha affec he enire pricing mechanism. We do his in Secion 6 a he expense of increasing he complexiy of he model..4 EXAMPLE. Suppose u(w)¾ ¼(1:a) e ¼ aw, for some a\0. Then, a sraighforward, bu edious, calculaion shows ha V(w, )¾ ¼ 1 a exp ¼ aw e r(t ¼ ) ¼ ( m¼r) s (T¼). By using (.7), we nd ha he opimal invesmen in he risky asse is P* ¾ ( m¼r) e ¼ r(t ¼ ). s a Observe ha P* is no sochasic; in paricular, i is independen of wealh. In models of lognormal sock dynamics, such independence from wealh is generally observed in calculaions wih exponenial uiliy because he absolue risk aversion ¼u (w):uæ(w) (Pra, 1964) is consan (equal o a). Noe ha as he risk aversion of he decision maker increases, as measured by a, and as he ime unil expiry increases, he amoun of money invesed in he risky asse decreases. For arbirary uiliy funcions, a closed form soluion is no generally available, excep in he case of hyperbolic absolue risk aversion (HARA) uiliies ha are of he form u(w)¾(a Bw) g, for given consans A, B, and gb1. In he general case, he sandard way o proceed is o linearize he HJB equaion and hereby work wih he dual funcion VÑ (y, )¾sup w (V(w, )¼wy). Equivalenly, one may inroduce he ransformaion V w ( f(y, ), )¾y and deermine he funcion f. The laer urns ou o solve a linear parial differenial equaion ha is easy o analyze. For relaed argumens, see Karazas & Shreve (1998, Chaper 3). In he opimizaion models ha we will encouner for valuaing insurance risks, such ransformaion s do no linearize he HJB equaion, because he marke is incomplee. In fac, he echnically-oriene d reader will recognize ha linearizaion is always compaible wih compleeness. Because of he echnical dif culies ha arise for general uiliy funcions, we work examples using only exponenial uiliy whose scaling properies are convenien for he speci caion of he relaed value funcions. However, we presen he HJB equaions for general uiliy funcions.
5 V. R. Young & T. Zariphopoulou Scand. Acuarial J. 4 3. RESERVATION PRICES In his secion, we exend he principle of equivalen uiliy o price dynamic insurance risks. The main ingredien of he pricing mehodology is he use of individual risk preferences owards he risks ha canno be eliminaed hrough rading in he nancial marke. Boh paries involved in he insurance claim, namely, he insurer and he buyer, are endowed wih a uiliy funcion of erminal wealh. We denoe boh uiliy funcions by u; however, our framework allows for each pary o possess a differen uiliy funcion. For example, one expecs ha an insurer will be less risk averse han a buyer of insurance, so ha in he noaion of Example.4, ainsurer B abuyer. Boh paries have he opporuniy o inves in a riskless asse and a risky one wih he goal of maximizing heir expeced uiliy of erminal wealh. The relevan sochasic opimizaion problem is idenical o he one described in he previous secion. We inroduce an insurance claim ha, for he sake of exposiion only, is aken o be of European ype, namely, i is represened as he insurer s liabiliy or he (poenial) buyer s obligaion Y T a expiraion T. For he insurer, he uiliy-based pricing mechanism relies on considering and subsequenly comparing he following possibiliies: Eiher he insurer can choose o accep he risk, receive some premium, and inves in he nancial marke, or he insurer can choose no o insure he risk and simply inves his wealh in he marke wih resuling value funcion V, as in (.3). The reservaion price is de ned as he premium a which he insurer is indifferen beween hese wo opions; see De niion 3.1 below. Similarly, he buyer of insurance considers wo possibiliies: Eiher he buyer can purchase insurance for some premium and inves in he nancial marke wih resuling value funcion V, as in equaion (.3), or he buyer can reain he risk and inves in he marke. The reservaion price of he buyer is de ned as he premium a which he buyer is indifferen beween hese wo opions; see De niion 3.1 below. If he insurer (he poenial buyer of insurance) insures (reains) he risk, hen we need o de ne a value funcion similar o he one for V in Secion. As we menioned earlier, we assume ha he insurance liabiliy Y T is payable a ime T by eiher an individual who has no ye purchased insurance or by an insurance company ha has underwrien he liabiliy. We also assume ha he liabiliy canno be raded afer is ransfer from he buyer o he insurer and before is expiraion. In his ime horizon, only rading beween he wo available marke asses is permied. Then, he value funcion of he agen is de ned o be U(w, y, ) ¾ sup {p} ÏA E[u(W T ¼Y T ) W ¾w, Y ¾y], (3.1) in which A is he se of admissible policies for he agen. The process Y is he loss process, in he sense ha i models he cumulaive loss incurred up o ime. If he agen is he insurance company, hen U is he value funcion if he insurance company insures he risk Y T, while V in (.3) is he value funcion if he insurance company does no accep he risk. If he agen is he (poenial) buyer of insurance, hen U is he value funcion if he buyer does no buy insurance bu
Scand. Acuarial J. 4 Pricing dynamic insurance risks 53 insead reains he risk Y T, while V in (.3) is he value funcion if he buyer does buy insurance. A fundamenal assumpion is ha Y is independen of B, he Brownian moion ha derives he sock price. I is imporan o observe ha he loss process does no represen an asse on which one can rade and hereby creae a hedging porfolio. The fac ha Y is no radable resuls in a fundamenal dif culy for he speci caion of he price via classical arbirage -free argumens based on perfec replicaion. Anoher imporan issue is how one de nes admissible sraegies in A. If one insiss on having he consrain W T ¼Y T E0 a.s., in analogy wih he non-negaiviy consrain of wealh in Meron s problem, hen in nie prices may resul due o he non-perfec correlaion beween Y and S. Such admissibiliy consrains generae many echnical dif culies and, in general, aler he prices in a raher complex way; see he discussion in Consaninides & Zariphopoulou (1999). We do no assume ha such consrains are binding herein, bu we will deal wih his issue in fuure work; see Young & Zariphopoulou (001). This is, in fac, one reason we chose o work examples wih exponenial uiliy, which is well de ned for negaive argumens. 3.1 DEFINITION. The (sae dependen) reservaion price of he underwrier (or insurer), P I (w, y, ), is de ned as he compensaion P I such ha V(w, )¾U(w P I, y, ), (3.) for a given (w, y, ). Similarly, he (sae dependen) reservaion price of he buyer, P B (w, y, ), is de ned as he obligaion P B such ha V(w¼P B, )¾U(w, y, ), (3.3) for a given (w, y, ). Essenially, we equae he value of no insuring he risk wih he value of insuring he risk for a price P I received a ime. One can hink of P I as he minimum premium ha he insurer is willing o accep a ime, given he sae (w, y), o insure he liabiliy Y T a ime T. Respecively, we equae he value of buying complee insurance for a price of P B wih he value of no insuring he risk. One can hink of P B as he maximum premium ha he buyer is willing o pay a ime, given he sae (w, y), for insurance agains he liabiliy Y T a ime T. The saic analogues of he above prices are presened in Bowers e al. (1997, Equaions (1.3.6) and (1.3.1)) in he conex of insurance risks. They are he essence of he principle of equivalen uiliy for pricing insurance; see also Gerber & Pafumi (1998). Similar saic crieria are relaed o he noion of indifference and compensaing prices in classical economics. The speci caion of such prices is much more complex if inermediae rading in a nancial marke is allowed, and his is he ask underaken herein.
54 V. R. Young & T. Zariphopoulou Scand. Acuarial J. 4 In general, due o he non-lineariy of he crieria (3.) and (3.3) and he marke incompleeness, he prices P I and P B do no coincide. Moreover, because of he way P I and P B are de ned, i is ineviable ha hey depend no only on he liabiliy process bu also on he curren wealh. In his sense, P I and P B are no universal, as opposed o he Black-Scholes price ha is independen of he individual s porfolio holdings and depends only on he dynamics of he asse on which he claim is wrien (as well as he riskless ineres rae). Overall, universaliy is a highly desirable propery ha is no, in general, valid in incomplee markes. In fac, one may easily verify ha if he loss process Y and he sock price S are no perfecly correlaed, he pricing equaliies (3.) and (3.3) canno be valid for all levels of (y, ), if one insiss on price funcions P I and P B ha are wealh independen. Besides he dependence on wealh, P I and P B are also expeced o depend on risk preferences as represened by he uiliy funcion; afer all, hese prices are inroduced in order o price risks ha canno be hedged away. From he above discussion, we see ha here are wo pifalls of he pricing mechanism ha uses he principle of equivalen uiliy he dependence on wealh and on risk aversion. We will see ha he laer canno be eliminaed because i is he direc consequence of marke incompleeness and risks ha canno be hedged. However, he dependence on wealh can be addressed in wo ways one may use exponenial uiliy or work wih universal price bounds. Pra s measure of absolue risk aversion is independen of wealh for exponenial uiliy (as menioned in Example.4). This propery, in urn, yields invesmen sraegies ha are independen of wealh, which, ogeher wih cerain scaling properies, implies wealh-independen reservaion prices. We will observe his phenomenon in our examples. An alernaive approach is o seek price bounds ha are independen of wealh bu saisfy (3.) and (3.3) as inequaliies, insead of equaliies, for all w. 3. DEFINITION. The universal wrie price, PÉ I (y, ), is de ned as he minimum price ha sais es V(w, )0U(w PÉ I, y, ), (3.4) for all wealh levels w. Similarly, he universal buy price, P B (y, ), is de ned as he maximum price ha sais es V(w¼P B, )EU(w, y, ), (3.5) for all wealh levels w. According o he above de niion, he insurer (respecively, buyer) should no accep o wrie, or insure, (respecively, buy) he liabiliy a a price lower (respecively, higher) han PÉ I (respecively, P B ). Therefore, wo agens wih he same risk preferences mus rade he liabiliy a prices wihin he above spread. Consaninides & Zariphopoulou (1999) inroduced universal prices in markes wih rans-
Scand. Acuarial J. 4 Pricing dynamic insurance risks 55 acion coss, and subsequenly ohers used hem for differen marke imperfecions (see, for example, Consaninides & Zariphopoulou, 001; Munk, 000; and Mazaheri, 001). We do no pursue his approach in his paper. We conclude his secion by illusraing how he principle of equivalen uiliy can be used o price dynamic risks ha are perfecly correlaed wih he underlying risky securiy. This is he well known complee marke seing, and he risks can be priced by classical arbirage argumens which yield he Black-Scholes price. The laer is unique and independen of he curren wealh and he individual preferences (Black & Scholes, 1973). Clearly, one expecs he reservaion prices o be equal o each oher and o coincide wih he Black-Scholes price. Moreover, he same price sais es boh inequaliies (3.4) and (3.5), which reduce o a universal equaliy among he relevan value funcions. Even hough he noion of reservaion price is redundan in such a perfec marke seing, in he calculaions below we rederive he Black-Scholes price. We choose o do his in order o familiarize he audience wih he involved echnical seps and also o have a benchmark case wih which o compare when we analyze risks ha canno be priced wih radiional mehods. To his end, we assume ha he insurer can choose o underwrie a liabiliy Y T a expiraion ime T, such ha Y T ¾g(S T ) for some funcion g: R R and S s given by equaion (.1). We look for a funcion h I (S, ) such ha V(w, )¾U(w h I (S, ), S, ), (w, S, )ÏR½R ½[0, T], (3.6) in which U here is de ned somewha differenly han in equaion (3.1), namely, U(w, S, ) ¾ sup {p } Ï A E[u(W T ¼g(S T )) W ¾w, S ¾S]. We de ne U his way because he liabiliy in his case is a funcion of he underlying risky asse. Similarly, he buyer of insurance agains he liabiliy is willing o pay h B (S, ) ha solves V(w¼h B (S, ), )¾U(w, S, ), (w, S, )ÏR½R ½[0, T]. (3.7) Noe we posulae ha he price does no depend on he curren wealh level. This independence is no known a priori nor can be seen easily from he model in general. On he oher hand, one can argue if here exiss a funcion ha sais es boh (3.6) and (3.7), hen i coincides wih he unique Black-Scholes price. In fac, one can easily argue ha if he claim is wrien (respecively, bough) a a price higher (respecively, lower) han he Black-Scholes price, hen here exis sraegies ha creae arbirage opporuniies (Musiela & Rukowski, 1997). Therefore, in he calculaions below, i suf ces o nd a candidae ha is wealh independen and o jusify ha he corresponding pricing equaliies hold. As he calculaions below demonsrae, his candidae price solves he Cauchy problem (3.10), which has a unique soluion (Fleming & Soner, 1993, Chaper 6). Alernaively, one could sar wih prices ha are wealh dependen and follow he calculaions as done below. Afer more edious analysis, he candidae price
56 V. R. Young & T. Zariphopoulou Scand. Acuarial J. 4 would urn ou o saisfy a nonlinear parial differenial equaion. Using argumens from he heory of viscosiy soluions, one could show ha he parial differenial equaion has a unique viscosiy soluion (Ishii & Lions, 1990), say H(w, S, ). From he erminal daa, however, one sees ha H is wealh independen, because H(w, S, T)¾g(S T ). By he uniqueness of viscosiy soluion, one concludes ha he soluion is unique and coincides wih he wealh independen soluion of (3.10). In he following calculaion, we compue he buyer s price. To his end, he HJB equaion for U urns ou o be U max p (m¼r)pu w 1 s p U ww s pu w S 1 s S U S S msu S rwu w ¾0, U(w, S, T)¾u(w¼g(S)), or, equivalenly, U ¼ [ s SU ws (m¼r)u w ] s U ww 1 s S U S S msu S rwu w ¾0, U(w, S, T)¾u(w¼g(S)). (3.8) Differeniaing (3.7) yields ¼h V w V ¾U, V w ¾U w, V w w ¾U ww, ¼h S V w ¾U S, ¼h S S V w h S V w w ¾U S, ¼h S V w w ¾U ws, in which h¾h B. Afer insering hese expressions in (3.8) and rearranging erms, we obain V w ¼h rh¼ 1 s S h SS ¼rSh S V ¼ ( m¼r) s V w r(w¼h(s, ))V w ¾0, V w w (3.9) in which all he derivaives are evaluaed a (w ¼h(S, ), ). Observe ha he second brackeed erm in (3.9) vanishes because V solves he HJB equaion (.4). Therefore, h B sais es rh B ¾h B 1 s S h B SS rsh SB, h B (S, T)¾g(S), (3.10) he seminal Black-Scholes equaion. By using equaion (3.6) in place of (3.7), a similar argumen shows ha he insurer s price, h I, also sais es (3.10). The reason ha he insurer s and he buyer s prices are equal and independen of he uiliy funcion u is ha he marke is complee.
Scand. Acuarial J. 4 Pricing dynamic insurance risks 57 4. INSURANCE CLAIMS PAYABLE AT EXPIRATION TIME T In his secion, we apply he principle of equivalen uiliy o deermine he value of insurance producs when paymen occurs a a erminal, prespeci ed ime T. We will consider a variey of such producs. We will sar wih he simples class of insurance claims saic losses of a single life in Secion 4.1 and of a group of insured lives in Secion 4., as in erm life insurance payable a ime T. In ha case, T could be aken o be one year, so ha we are considering erm life insurance payable a he end of he year of deah. Nex, we consider he problem of pricing pure endowmen insurance on a single life in Secion 4.3. Finally, we consider losses modeled as sochasic processes, namely, diffusion and Poisson processes in Secions 4.4 and 4.5, respecively. We derive he reservaion prices and compare hem wih he benchmark premia derived hrough radiional mehods based on presenvalue argumens. To specify he reservaion prices for insurance, we need o solve sochasic opimizaion problems of expeced uiliy ha are generalizaions of Meron s model in a non-rivial way. In fac, he complexiy of he payoff resuls in HJB equaions wih non-local erms and forms ha canno be manipulaed in a sraighforward fashion. In our analysis, we derive he associaed HJB equaion for general uiliy funcions, bu we specify he reservaion prices only for exponenial uiliy. Generally speaking, under arbirary choice of preferences, one can esablish ha he value funcion of he agen (he buyer or seller of insurance) is a soluion o he HJB equaion, a leas in he viscosiy sense. As a maer of fac, one can show ha he value funcion is he unique soluion in he class of viscosiy soluions. The argumens used o show he viscosiy properies are rouine adapaions of by-now classical resuls in he area of sochasic opimizaion, and we only discuss formally he main seps of he analysis. For a deailed exposiion of he use of viscosiy soluions for HJB equaions arising in asse valuaion models, we refer he reader o he review aricles of Zariphopoulou (1999b, 001). 4.1. Single insured life erm life insurance We sar wih he case of a claim ha pays, a expiraion T, a random variable aking he values 0 or 1 wih probabiliies ha depend on he curren ime. To be concree, we consider an individual aged x, who is seeking o buy erm life insurance ha will pay 1 uni a ime T if he individual dies before ime T, and 0 oherwise. For he res of his secion, we wrie (x) o refer o his individual. To his end, denoe by T ¼ q x he probabiliy ha (x) will die before ime T given ha (x) is alive a ime. Then, T ¼ q x ¾ F x (T)¼F x (), 1¼F x () in which F x is he cumulaive disribuion funcion of he ime unil deah of (x). Oher life funcions can be obained similarly. By assuming enough differeniabiliy for he disribuion funcion, we will employ he hazard funcion, oherwise known
58 V. R. Young & T. Zariphopoulou Scand. Acuarial J. 4 as he force of moraliy, lx (), given by lx ()¾f x ():(1¼F x ()), in which f x is he probabiliy densiy funcion of he ime unil deah of (x). Nex, we consider he opimizaion problem of he seller of he insurance produc, as inroduced in equaion (3.1) and rewrien here for convenience, U(w, )¾ sup {p } ÏA E[u(W T ¼Y T ) W ¾w]. (4.1) Recall ha he curren wealh W s is de ned as in Secion and solves (.). The expecaion is wih respec o he produc measure P ½Q, in which (V Æ, G, Q) is a probabiliy space on which Y T is de ned. Observe ha he liabiliy, even hough i is indexed by T, is no a funcion of ime (compare, for example, wih Y T ¾g(S T ) a he end of Secion 3). In fac, he liabiliy is a saic risk, and his is he reason ha U depends explicily on (w, ), wih he dependence on y being absorbed in he measure according o which we calculae he expecaion a erminal ime. The classical principle of dynamic programming yields U(w, )EE[U(W h, h) W ¾w]; (4.) see Fleming & Soner (1993). We coninue wih a formal derivaion of he HJB equaion. Le us assume ha {p* s : 0s 0 h} is he opimal sraegy ha he insurer follows. Denoe by W* s he wealh under {p* s }. If he individual (x ) dies during [, h], in which ime is measured from age x, hen we are in he cerain siuaion wih respec o he insurance risk in he following sense: The insurer will pay 1 a ime T for his deah and will have o charge e ¼ r(t ¼ ) a ime o cover his payou. Therefore, for his problem, he insurer s value funcion equals E[V(W* h ¼e ¼ r(t ¼ ¼ h), h) W ¾w] muliplied by h q x, he probabiliy ha (x ) dies during [, h], in which ime is measured from age x. If he individual (x ) survives o ime h, an even ha happens wih probabiliy h p x, he insurer s value funcion is E[U(W* h, h) W ¾w W ¾w] muliplied by h p x. Therefore, U(w, )EE[U(W* h, h) W ¾w] h p x E[V(W* h ¼e ¼ r(t ¼ ¼ h), h) W ¾w] h q x. (4.3)
Scand. Acuarial J. 4 Pricing dynamic insurance risks 59 By assuming enough regulariy condiions and appropriae inegrabiliy on he value funcions and heir derivaives (Bjørk, 1998), we ge E[U(W* h, h) W ¾w]¾U(w, ) E E h h {U (W* s, s) (rw* s (m¼r)p* s )U w (W* s, s)} ds W ¾w 1 s p* s U ww (W* s, s) ds W ¾w. (4.4) We obain a similar expression for E[V(W* h ¼e ¼ r(t ¼ ¼ h), h) W ¾w] ha combined wih (4.3) and (4.4) yields U(w, )EU(w, ) h p x V(w¼e ¼ r(t ¼ ), ) h q x E h {U (W* s, s) (rw* s (m¼r)p* s )U w (W* s, s) 1 s p* s U ww (W* s, s)} ds W ¾w h p x E h {V (W* s ¼e ¼ r(t ¼ s), s) (rw* s (m¼r)p* s )V w (W* s ¼e ¼ r(t ¼ s), s)} ds W ¾w ½ h q x E h 1 s p* s V ww (W* s ¼e ¼ r(t ¼ s), s) ds W ¾w h q x. (4.5) By subracing U(w, ) h p x from boh sides and dividing by h, we obain q U(w, ) h x q EV(w¼e ¼ r(t ¼ ), ) h x h h E 1 h h U (W* s, s) (rw* s (m¼r)p* s )U w (W* s, s) 1 s p* s U w w (W* s, s) ds W ¾w h p x E 1 h h {V (W* s ¼e ¼ r(t ¼ s), s)
60 V. R. Young & T. Zariphopoulou Scand. Acuarial J. 4 (rw* s (m¼r)p* s )V w (W* s ¼e ¼ r(t ¼ s), s)} ds W ¾w h q x E 1 h h 1 s p* s V ww (W* s ¼e ¼ r(t ¼ s), s) ds W ¾w h q x. (4.6) By aking he limi as h goes o 0 from he righ, we ge 0E U (rw (m¼r)p)u w s p U w w lx ()[V(w¼e ¼ r(t ¼ ), )¼U(w, )], which in urn yields, along he opimum, he HJB equaion U max p (m¼r)pu w 1 s p U w w rwu w lx ()[V(w¼e ¼ r(t ¼ ), )¼U(w, )]¾0, (4.7) 1 U(w, T)¾u(w). By following rouine argumens, we can show ha he value funcion U is concave, which implies, as in (.6), ha he above maximum is well de ned and achieved a p*(w, s)¾ ¼ ( m¼r) s U w (w, s) U ww (w, s). Similarly o (.7), he opimal invesmen policies are given via he laer funcion in a feedback form. Speci cally, he opimal invesmen process in he sock accoun is P* s ¾p*(W* s, s)¾ ¼ ( m¼r) s U w (W* s, s) U w w (W* s, s), (4.8) in which U solves (4.7) and W* s is he opimal wealh solving (.) wih P* s used for ps. Therefore, we can rewrie (4.7) as U rwu w ¼ ( m¼r) U(w, T)¾u(w). s U w lx ()[V(w¼e ¼ r(t ¼ ), )¼U(w, )]¾0, U w w (4.9) 1 The reader familiar wih Meron s work will recognize ha equaion (4.7) resembles he equaion relaed o an expeced uiliy maximizaion problem wih discoun facor lx () and inermediae uiliy payoff lx ()V(w¼e ¼ r(t ¼ ), ).
Scand. Acuarial J. 4 Pricing dynamic insurance risks 61 One may derive equaion (4.7) rigorously by using argumens from he heory of viscosiy soluions. We refer he reader o he review aricle of Zariphopoulou (001) for a concise exposiion. Nex, we calculae he reservaion prices for he case of exponenial uiliy. 4.1 EXAMPLE. Suppose u(w)¾ ¼(1:a) e ¼ aw, in which a\0 is he risk aversion coef cien. Because of he uniqueness of soluions o he HJB equaion, i suf ces o consruc a candidae soluion. To his end, le U(w, ) ¾ V(w, )f(), wih f(t)¾1 and V as in Example.4. Then, (4.9) becomes V f VfÆ rwv wf¼ ( m¼r) s V w f l x () [V e a ¼Vf]¾0. V ww The rs, hird, and fourh erms cancel because V sais es he HJB equaion (.4), and we can cancel a facor of V from he remaining hree erms o obain he ordinary differenial equaion for f: 0¾fÆ lx ()[e a ¼f], wih boundary condiion f(t)¾1. The soluion o his equaion is given by f()¾e ¼ T lx (s) ds e a 1¼e ¼ T lx (s) ds he momen generaing funcion of Y T and P B, boh equal ¾ T ¼ p x e a T ¼ q x ¾M Y T (a), (4.10) evaluaed a a. The reservaion prices, P I P I (w, )¾P B (w, )¾ 1 a e¼ r(t ¼ ) ln M Y T (a), (4.11) in which he disribuion of Y T depends on he curren ime. Noe ha he price increases wih respec o he absolue risk aversion a. As a approaches 0, he price approaches e ¼ r(t ¼ ) T ¼ q x, he ne premium for his risk. Also, noe ha he price is independen of he risky asse; hus, his price is idenical o he one obained by allowing only invesmen in he riskless bond. In his example, he reservaion prices of he insurer and he buyer of insurance are equal. In realiy, one expecs ha he insurer s risk aversion will be less han he buyer s, from which i will follow ha P I BP B because he price increases wih respec o a. This spread allows for he rade of insurance. 4.. Group of insured lives erm life insurance Suppose he loss payable a ime T equals 1 for each of Y T people who have died from a group of n people alive a ime 0. Assume ha he n people are all aged x wih independen and idenically disribued imes unil deah. Then, he number who die in he inerval [, h] is disribued according o he Binomial(n¼Y,h q x ), in which Y is he number who have died by ime. One can follow
6 V. R. Young & T. Zariphopoulou Scand. Acuarial J. 4 he reasoning in Secion 4.1 o show ha he value funcion U(w, y, ; n) given by (3.1) solves he following recursive HJB equaion: U(w, n, ; n)¾v(w¼n e ¼ r(t ¼ ), ); For y¾0, 1,..., n¼1, (4.1) U rwu w ¼ ( m¼r) U w (n¼y)l s x ()[U(w, y 1, ; n)¼u(w, y, ; n)] ¾0, U ww U(w, y, T; n)¾u(w¼y). 4. EXAMPLE. Suppose u(w)¾ ¼(1:a) e ¼ aw, in which a\0 is he risk aversion coef cien, as in Examples.4 and 4.1. From Example 4.1, we know ha U(w, 0, ; 1)¾V(w, )f(), in which f is given by (4.10) and V is as in Example.4. As in Example 4.1, one can show ha U(w, y, ; n)¾v(w, ) e ay f() n ¼ y. I follows ha boh he reservaion prices equal P I (w, y, ; n)¾p B (w, y, ; n)¾y e ¼ r(t ¼ ) n¼y e ¼ r(t ¼ ) ln f(), (4.13) a an immediae generalizaion of he price given in (4.11) for a single life (y¾0, n¾1). Noe ha his price conains a riskless provision for he y people who have died by ime and a risk-loaded price for each of he remaining n ¼y people. The risk-loaded price for each person equals he one in equaion (4.11). 4.3. Single insured life pure endowmen insurance Pure endowmen insurance for he period [0, T] pays 1 uni a ime T if (x) survives o ime T and 0 oherwise. This insurance is he building-block of life annuiies; see Bowers e al. (1997). We nd he reservaion prices for pure endowmen insurance in his secion, hen laer use our resul when we consider pricing a emporary life annuiy in Secion 5.3. In his case, he HJB equaion for U reduces o U rwu w ¼ ( m¼r) s U(w, T)¾u(w¼1). U w lx ()[V(w, )¼U(w, )]¾0, U w w (4.14) 4.3 EXAMPLE. Suppose u(w)¾ ¼(1:a) e ¼ aw, in which a\0 is he risk aversion coef cien. Wrie U(w, )¾V(w, )h(), in which V is given in Example.4. Then, h solves he rs-order differenial equaion 0¾hÆ lx ()[1¼h],
Scand. Acuarial J. 4 Pricing dynamic insurance risks 63 wih boundary condiion h(t)¾e a. I follows ha h equals h()¾e a T ¼ p x T ¼ q x ¾M Y T (a), (4.15) and ha boh reservaion prices equal P I (w, )¾P B (w, )¾ 1 a e¼ r(t ¼ ) ln M Y T (a). (4.16) Again, if he insurer is less risk averse han he buyer, we will have P I BP B. Compare he expression for h in equaion (4.15) wih he one for f in equaion (4.10). Noe he parallel beween he wo equaions in ha p and q swich roles in he wo equaions, which re ecs he fac ha 1 uni is paid if he person dies under life insurance bu lives under pure endowmen insurance. Also, noe ha he sum of he premiums for life insurance and for pure endowmen insurance equals e ¼ r(t ¼ ) 1 1 a ln 1 e a: ¼e ¼ a: T ¼ p x ¼ T ¼ q x, which is greaer han e ¼ r(t ¼ ), he presen value a ime of 1 payable a ime T, a direc consequence of he non-linear pricing mechanism induced by marke incompleeness. 4.4. Losses modeled as a diffusion process In his secion, we allow losses o follow a diffusion process; ha is, Y follows he process dy s ¾u(Y s, s) ds z(y s, s) dbñ s, Y ¾yE0, (4.17) wih BÑ s a sandard Brownian moion independen of B s, he Brownian moion for he sock process (.1). Assume ha he drif and volailiy coef ciens u(y, ) and z(y, ) saisfy he usual growh and Lipschiz condiions u(y, )¼u(x, ) z(y, )¼z(x, ) 0K y¼x, u(y, ) z(y, ) 0K(1 y), for some posiive consan K. These condiions guaranee ha a unique soluion o equaion (4.17) exiss (Gihman & Skorohod, 197, Chaper 6). A diffusion equaion is ofen used o model insurance losses, especially from he poin of view of he insurer. See, among ohers, Grandell (1990), Asmussen & Taksar (1997) and Højgaard & Taksar (1997).
64 V. R. Young & T. Zariphopoulou Scand. Acuarial J. 4 The corresponding HJB equaion for he value funcion U(w, y, ) reduces U rwu w ¼ ( m¼r) s U(w, T, y)¾u(w¼y). U w u(y, )U y 1 U w w z (y, )U yy ¾0, (4.18) 4.4 EXAMPLE. Suppose u(w)¾ ¼(1:a) e ¼ aw, in which a\0 is he risk aversion coef cien. Also, suppose u and z are independen of y. Then, U(w, y, )¾ V(w, ) e ay c(), in which V is as in Example.4 and c solves he ordinary differenial equaion cæ() au() 1 a z ()¾0, c(t)¾0. Thus, boh reservaion prices equal T P I (w, y, )¾P B (w, y, )¾y e ¼ r(t ¼ ) ¼ r(t ¼ ) e u(s) 1 az (s) ds, (4.19) a erm for he loss already incurred and a erm for he discouned expeced loss plus a loading proporional o he variance of he loss during he period [, T]. Tha is, he exponenial premium principle in his case is a variance premium principle. 4.5 EXAMPLE. Suppose u(w)¾ ¼(1:a) e ¼ aw, in which a\0 is he risk aversion coef cien. Also, suppose u and z are given by u(y, )¾(n¼y)lx () and z(y, )¾ (n¼y)lx (), in which n is a xed posiive number, Y 0 ¾0, and lx is he hazard funcion for a person aged x. One can hink of his as a limiing case of he model presened in Secion 4.. Speci cally, as n¼y ges large, one can approximae he Binomial (n¼y,h q x ) by he Normal((n¼Y ) h q x, (n¼y ) h q x h p x ) ¾Normal (n¼y ) h q x h h, (n¼y ) h q x h p x h :Normal((n¼Y )lx ()h, (n¼y )lx ()h). Suppose ha lx ()Ål, a posiive consan. Afer a fair amoun of work, we obain ha (n¼y) U(w, y, ; n)¾v(w, ) exp a n¼ ( a) e l(t ¼ ) ¼a h.
Scand. Acuarial J. 4 Pricing dynamic insurance risks 65 I follows ha boh reservaion prices equal P I (w, y, )¾P B (w, y, )¾y e ¼ r(t ¼ ) (n¼y) e ¼ r(t ¼ ) ( a)(1 ¼e ¼ l(t ¼ ) ) ( a)¼a e ¼ l(t ¼ ), similar in form o he expression given in (4.13) wih (1:a) ln[(1¼e ¼ l(t ¼ ) ) e a e ¼ l(t ¼ ) ] replaced by ( a)(1 ¼e ¼ l(t ¼ ) ):[( a)¼a e ¼ l(t ¼ ) ]. The wo prices are nearly equal if a 3 :0 and e ¼ 3l(T ¼ ) :0. 4.6 EXAMPLE. Suppose u(w)¾ ¼(1:a) e ¼ aw, in which a\0 is he risk aversion coef cien. Also, suppose u(y, )¾uy and z(y, )¾zy, for some consans u and z, wih z\0. Se U(w, y, )¾V(w, )f(y, ), wih f(y, T)¾e ay. Then, by subsiuing his form of U ino (4.18), we learn ha f solves f uyfy z y f(y, T)¾e ay. fyy ¾0, By using sandard argumens relaed o he Feynman-Kac represenaion (Karazas & Shreve, 1991, Theorem 5.7.6), we can represen he soluion of he above linear parial differenial equaion in erms of he expecaion of he exponenial of a diffusion process wih generaor L ¾uy É Éy 1 z y É Éy. I hen follows ha f(y, )¾E[exp(k 1 e k Z )], T¼, and ZºN(0, 1). Thus, boh reserva- in which k 1 ¾ay e (u¼ (z :))(T ¼ ), k ¾z ion prices equal P I (w, y, )¾P B (w, y, )¾ 1 a e¼ r(t ¼ ) E[exp(k 1 e k Z )]¾ 1 a e¼ r(t ¼ ) M X (k 1 ), in which X¾e k Z is lognormally disribued and M X is he momen generaing funcion of X. 4.5. Losses modeled as a Poisson process In his secion, we allow losses o follow a Poisson process; ha is, Y follows he process dy s ¾L(W s, Y s, s) dn s, Y ¾y, (4.0)
66 V. R. Young & T. Zariphopoulou Scand. Acuarial J. 4 wih N s a non-homogeneous Poisson process wih deerminisic parameer f(s). We assume ha N s is independen of B s, he Brownian moion for he sock process (.1). Also, L is he (random) loss amoun a ime s, independen of N s. Noe ha we allow he loss o depend on he wealh a ime s and he losses o dae. The corresponding HJB equaion for he value funcion U(w, y, ) reduces o U rwu w ¼ ( m¼r) s U(w, y, T)¾u(w¼y). U w f()[eu(w, y L(w, y, ), )¼U(w, y, )]¾0, U w w (4.1) See Meron (199, Secion 5.8). 4.7 EXAMPLE. Suppose u(w)¾ ¼(1:a) e ¼ aw, in which a\0 is he risk aversion coef cien, and suppose ha he loss L(w, y, ) is independen of w and y. Se U(w, y, )¾V(w, ) e ay j() wih V given in Example.4. Then, j solves jæ f()[m L() (a)¼1]j¾0, j(t)¾1. Thus, j()¾exp T f(s)[m L(s) (a)¼1] ds, and boh reservaion prices equal P I (w, y, )¾P B (w, y, )¾y e ¼ r(t ¼ ) 1 T r(t ¼ ) e¼ a f(s)[m L(s) (a)¼1] ds. (4.) There is an ineresing relaionship beween he premia given by equaions (4.19) and (4.) when he expeced losses and he variances of he loss during [, T] are equal. In ha case, he Poisson premium in (4.) is greaer han he diffusion premium in (4.19). Indeed, if he expeced losses are equal, hen T u(s) ds¾ T f(s)e[l(s)] ds, and if he variances of he loss are equal, hen T z (s) ds¾ T f(s)e[l (s)] ds.
Scand. Acuarial J. 4 Pricing dynamic insurance risks 67 Thus, he premium in (4.19) becomes T y e ¼ r(t ¼ ) ¼ r(t ¼ ) e u(s) a z (s) ds T ¾y e ¼ r(t ¼ ) ¼ r(t ¼ ) e f(s) EL(s) a E[L (s)] ds T By e ¼ r(t ¼ ) ¼ r(t ¼ ) e f(s)(m L(s) (a)¼1) ds, where he laer is he premium in (4.). This inequaliy beween he premiums is wha one expecs because he Poisson process is a jump process and he diffusion process is a coninuous one. The laer is, hereby, less risky, and he reservaion prices re ec ha. 5. INSURANCE PAYABLE AT INCURRENCE MAXIMIZING EXPECTED UTILITY AT TIME T We assume, as before, ha he agen (wheher buyer or seller of insurance) seeks o maximize expeced uiliy of wealh a ime T. Unlike in Secion 4, he insurance now is payable when he loss is incurred. In Secions 5.1 hrough 5.6, we parallel Secions 4.1 hrough 4.5. 5.1. Single insured life erm life insurance Consider again he problem from Secion 4.1, bu his ime he insurance will be paid a he ime of deah of (x) if (x) dies before ime T. We sill seek o maximize he expeced uiliy of erminal wealh a ime T. By following reasoning similar o ha in Secion 4.1, he HJB equaion for U reduces o U rwu w ¼ ( m¼r) U w s lx ()[V(w¼1, )¼U(w, )] ¾0, U w w (5.1) U(w, T)¾u(w). Noe ha his equaion is essenially he same one given in (4.9) wih V(w¼e ¼ r(t ¼ ), ) replaced by V(w¼1, ) because he insurance is payable a he momen of deah of (x). 5.1 EXAMPLE. As in Example 4.1, we can derive he price for exponenial uiliy o be P I (w, )¾P B (w, )¾ 1 a e¼ r(t ¼ ) ln c(), in which c() is given by c()¾e ¼ T lx (s) ds T lx (s) e a er(t ¼ s) ¼ s lx (u) du ds ¾ T ¼ p x T e a er(t ¼ s) lx (s) s ¼ p x ds.
68 V. R. Young & T. Zariphopoulou Scand. Acuarial J. 4 As before, he price is independen of he risky asse, and i equals he price we ge if we resric invesing only o he riskless bond. The price is also he same wheher we are calculaing he reservaion price of he insurer or of he buyer of insurance, unless he insurer is less risk averse han he buyer. Boh of hese phenomena occur because we are using exponenial uiliy. Also, noe ha his price is greaer han he one from Example 4.1 because he insurance bene here is payable when he insured dies, insead of a ime T. 5.. Group of insured lives erm life insurance Consider again he problem from Secion 4., bu his ime he insurance will be paid a he ime of deah of (x) if (x) dies before ime T. We sill seek o maximize he expeced uiliy of erminal wealh a ime T. By following reasoning similar o ha in Secion 5.1, one can show ha he value funcion U(w, y, ; n) given by (3.1) solves he following recursive HJB: U(w, n, ; n)¾v(w, ); For y¾0, 1,..., n¼1, (5.) U rwu w ¼ ( m¼r) U w s (n¼y)lx ()[U(w¼1, y 1, ; n)¼u(w, y, ; n)]¾0, U ww U(w, y, T; n)¾u(w). Noe ha his equaion is parallel o he one given in (4.1) wih changes because he insurance bene is payable a he momen of deah. 5. EXAMPLE. As in Example 4., we can derive he price for exponenial uiliy o be P I (w, y, ; n)¾p B (w, y, ; n)¾ n¼y e ¼ r(t ¼ ) ln c(), a in which c() is given in Example 5.1. Noe ha he premium is simply (n¼y) imes he premium for erm insurance on a single life, as we saw in Example 4. for erm insurance payable a ime T. There is no provision for he y lives who have already died because hose bene s have been paid. 5.3. Insurance on a single life emporary life annuiy immediae In his secion, we build on he work in Secion 4.3 o nd he reservaion prices for a emporary life annuiy ha pays 1 uni o (x) a he end of each period for T periods as long as (x) is alive. We assume ha T is a posiive ineger. For Ï(T¼n, T¼n 1], wrie U(w, ; n) for U. Then, he HJB equaion for he value funcion U reduces o U rwu w ¼ ( m¼r) s U w lx ()[V(w, )¼U(w, )] ¾0, U ww U(w, T¼n; n 1)¾U(w¼1, T¼n; n), U(w, T; 1)¾u(w¼1). for n¾1,,..., T¼1, (5.3)
Scand. Acuarial J. 4 Pricing dynamic insurance risks 69 5.3 EXAMPLE. Suppose u(w)¾(1:a) e ¼ aw, in which a \0 is he risk aversion coef cien. Then, by using he work in Example 4.3 and inducion on n, we obain ha he reservaion prices a ime 0 equal P I (w, 0)¾P B (w, 0)¾ 1 a e¼ rt ln[q x e a e(t ¼ 1)r p x q x 1 e a(e(t ¼ 1)r e (T ¼ )r ) p x q x e a(e(t ¼ 1)r 1) T p x ]. 5.4. Insurance on a single life emporary coninuous life annuiy In his secion, we assume ha he emporary life annuiy is payable coninuously a a rae of 1 uni per period for as long as (x) is alive or unil ime T expires. In his case, we incorporae he loss of 1 uni per period ino he wealh equaion so ha wealh W s follows dw s ¾[rW s (m¼r)p s ¼1] ds sp s db s, W ¾w. The corresponding HJB equaion for U reduces o U (rw¼1)u w ¼ ( m¼r) U(w, T)¾u(w). s U w lx ()[V(w, )¼U(w, )]¾0, U ww (5.4) 5.4 EXAMPLE. Suppose u(w)¾(1:a) e ¼ aw, in which a \0 is he risk aversion coef cien. Le U(w, )¾V(w, )h(), in which h(t)¾1. Then, h solves he following rs-order differenial equaion hæ [lx ()¼a e r(t ¼ ) ]h()¼lx ()¾0, so h is given by h()¾ T lx (s) exp s [a e r(t ¼ u) ¼lx (u)] du ds exp T [a e r(t ¼ s) ¼lx (s)] ds ¾ T e s a e r(t ¼ u ) du lx (s) s ¼ p x ds e T a e r(t ¼ s) ds T ¼ p x. (5.5) I follows ha he reservaion prices equal P I (w, )¾P B (w, )¾ 1 a e¼ r(t ¼ ) ln h(), in which h is given in equaion (5.5). 5.5. Losses modeled as a diffusion process Since we assume ha he insurance claims are payable a he ime of loss, i makes sense o incorporae hose losses ino he wealh equaion, if possible, as in Secion