Dynamic Contracting: An Irrelevance Result


 Madlyn Wells
 1 years ago
 Views:
Transcription
1 Dynamic Conracing: An Irrelevance Resul Péer Eső and Balázs Szenes Sepember 5, 2013 Absrac his paper considers a general, dynamic conracing problem wih adverse selecion and moral hazard, in which he agen s ype sochasically evolves over ime. he agen s final payoff depends on he enire hisory of privae and public informaion, conracible decisions and he agen s hidden acions, and i is linear in he ransfer beween her and he principal. We ransform he model ino an equivalen one where he agen s subsequen informaion is independen in each period. Our main resul is ha for any fixed decisionacion rule implemened by a mechanism, he maximal expeced revenue ha he principal can obain is he same as if he principal could observe he agen s orhogonalized ypes afer he iniial period. In his sense, he dynamic naure of he relaionship is irrelevan: heagenonlyreceivesinformaionrensforheriniialprivae informaion. We also show ha any monoonic decisionacion rule can be implemened in a Markov environmen saisfying cerain regulariy condiions. Keywords: asymmeric informaion, dynamic conracing, mechanism design 1 Inroducion We analyze muliperiod principalagen problems wih adverse selecion and moral hazard. he principal s perperiod decisions and he moneary ransfers are governed by a conrac signed a he beginning of he relaionship, in he presence of some iniial informaional asymmery, and he agen s privae informaion sochasically evolves over ime. he agen s final payoff can depend, quie generally, on he enire hisory of privae and public informaion, conracible decisions and he agen s hidden acions, and i is linear in he ransfer beween her and he principal. We discuss he wideranging applicaions of such models in micro and macroeconomic modeling below. Deparmen of Economics, Oxford Universiy, Oxford, UK. Deparmen of Economics, London School of Economics, London, UK. 1
2 Our main resul is an irrelevance heorem: In a mechanism implemening a given aciondecision rule, he maximal expeced revenue ha he principal can obain (and his maximal payoff if i is linear in he ransfers) is he same as if he could conrac on whaever new (orhogonal) informaion is observed by he agen in any fuure period. Noe ha in he hypoheical benchmark case where he agen s fuure orhogonalized ypes are observable and conracible he paries need no inerac beyond he iniial period, and he agen has no access o dynamic deviaion reporing sraegies. In his sense he dynamic naure of he adverse selecion problem is irrelevan. his irrelevance resul holds in a rich environmen, wih very lile assumed abou he agen s uiliy funcion (no singlecrossing or monooniciy assumpions are made), he informaion srucure, and so on. We also show ha monoonic decision rules can be implemened in Markov environmens wih imeseparable payoffs, subjec o addiional regulariy condiions. he regulariy assumpions include familiar singlecrossing condiions for he agen s uiliy funcion, and also assumpions concerning he availabiliy of a conracible public signal abou he agen s acion and ype. If he signal is informaive (however imperfecly) abou a summary saisic of he agen s hidden acion and ype, and is disribuion is generic, hen any monoonic decision rule coupled wih any monoonic acion rule is approximaely implemenable. 1 If he conracible signal is uninformaive abou he agen s acion (bu he oher regulariy condiions hold), hen monoonic decision rules coupled wih agenopimal acions can be implemened. he significance of he implemenaion resuls is ha when hey apply, he dynamic conracing problem can indeed be reaed as a saic one and solved as follows. Consider he benchmark case in which he agen s only privae informaion is her iniial ype, and he principal can observe her orhogonalized fuure ypes. Solve his relaxed case, eiher by opimizing he acion rule as well or aking i o be he agenopimal one, depending on wheher or no a public summary signal abou he agen s ype and acion is available. If he resuling rule is monoonic in he agen s ype profile hen i can be implemened in he original problem wih he same expeced paymens as in he benchmark, hence i is opimal in he original problem as well. 2 I is imporan o noe ha despie he validiy of his soluion mehod he original and he benchmark problems are no equivalen: he monooniciy requiremen on he decision rule is more sringen, hence he se of implemenable decision rules is smaller, in he benchmark. However, under he regulariy condiions he opimal soluion in he relaxed (benchmark) problem is implemenable in he more resricive original problem. Models in he class of dynamic conracing problems ha we analyze can be, and indeed 1 he genericiy condiion and he noion of approximae implemenabiliy will be precisely defined in Secion 4. 2 However, Baaglini and Lamba (2012) poin ou ha he regulariy condiions guaraneeing he monooniciy of he poinwiseopimal decision rule are quie srong. 2
3 have been, applied o a wide range of economic problems. 3 he roos of his lieraure reach back o Baron and Besanko (1984) who used a muliperiod screening model o address he issue of regulaing a monopoly over ime. Coury and Li (2000) sudied opimal advance icke sales, Eso and Szenes (2007a) he opimal disclosure of privae informaion in aucions, Eso and Szenes (2007b) he sale of advice as an experience good. Farhi and Werning (2012), Golosov, roshkin and sivinsky (2011) and Kapička (2013) apply a similar approach o opimal axaion and fiscal policy design, respecively. Pavan, Segal and oikka (2012) apply heir (o dae, mos general) resuls on he muliperiod pure adverse selecion problem o he aucion of experience goods (bandi aucions). Garre and Pavan (2012) use a dynamic conracing model wih boh adverse selecion and moral hazard o sudy opimal CEO compensaion. Such mixed, hidden acion hidden informaion models could also be applied in insurance problems. In his paper we also develop hree applicaions in order o illusrae our echniques and new resuls. he firs wo examples are dynamic monopoly problems in which he buyer s valuaion for he good (her ype) sochasically evolves over ime. In he second example he valuaion also depends on he buyer s hidden, cosly acion: e.g., she may privaely inves in learning how o beer enjoy he good. he monopolis canno observe any signal abou he buyer s ype and acion; all he can do is o offer a dynamic screening conrac. We derive he opimal conrac and show ha our dynamic irrelevance heorem holds: all disorions are due o he buyer s iniial privae informaion. he hird applicaion is a dynamic principalagen problem wih adverse selecion and moral hazard, where he principal is an invesor and he agen an invesmen advisor. he conracible decision is he amoun of money invesed by he principal wih he agen. he agen s ype is her abiliy o generae higher expeced reurns, whereas her cosly acion is aimed a picking socks ha conform wih he principal s oher (e.g., ehical) consideraions. Here he principal (invesor) observes a summary signal abou he agen s (advisor s) ype and acion, in he form of he principal s flow payoff. We fully solve his problem as well and show ha he dynamic irrelevance heorem applies. In order o formulae he main, irrelevance resul of he paper we rely on an idea inroduced in our previous work (Eso and Szenes (2007a)): We ransform he model ino an equivalen orhogonal represenaion, in which he agen s privae informaion in each period is independen of ha obained in earlier periods. he irrelevance heorem obains by showing ha in he original problem (where he agen s orhogonalized fuure ypes and acions are no observable), in any incenive compaible mechanism, he agen s expeced payoff condiional on her iniial ype are fully deermined by her onpah (in he fuure, ruhful) behavior. 3 Our review of applicaions is deliberaely incomplee; for a more indeph survey his lieraure see Krähmer and Srausz (2012) or Pavan, Segal and oikka (2012). 3
4 herefore, he agen s expeced payoff (and paymens) coincide wih hose in he benchmark case, where he orhogonalized fuure ypes are publicly observable. he resuls on he implemenabiliy of monoonic decision rules are obained in Markovian environmens subjec o addiional regulariy condiions. Here, he key sep in he derivaion (also used in Eso and Szenes (2007a) in a simpler model) is o show ha if he agen is unruhful in a given period in an oherwise incenive compaible mechanism, she immediaely undoes her lie in he following period o make he principal s inference regarding her ype correc in all fuure periods. he explici characerizaion of ouofequilibrium behavior in regular, Markovian environmens enables us o pin down he ransfers ha implemen a given monoonic decision rule in a model wih adverse selecion. he resuls for models wih boh moral hazard and adverse selecion are obained by appropriaely reducing he general model o ones wih only adverse selecion, he exac way depending on he assumpions made regarding he observabiliy of a public signal on he agen s ype and acion. he echnical conribuions nowihsanding, we believe he mos imporan message of he paper is he dynamic irrelevance resul. he insigh ha he principal need no pay his agen rens for posconracual hidden informaion in a dynamic adverse selecion problem has been expressed in previous work (going back o Baron and Besanko (1984)). Our paper highlighs boh he deph and he limiaions of his insigh: Indeed he principal ha conracs he agen prior o her discovery of new informaion can limi he agen s rens o he same level as if he could observe he agen s orhogonalized fuure ypes; however, we also poin ou ha he wo problems are no equivalen. he paper is organized as follows. In Secion 2 we inroduce he model and describe he orhogonal ransformaion of he agen s informaion. In Secion 3 we derive necessary condiions of he implemenabiliy of a decision rule and our main, dynamic irrelevance resul. Secion 4 presens sufficien condiions for implemenaion in Markov environmens. Secion 5 presens he applicaions; Secion 6 concludes. Omied proofs are in he Appendix. 2 Model Environmen. here is a single principal and a single agen. ime is discree, indexed by =0, 1,...,. he agen s privae informaion in period is θ Θ, where Θ = θ, θ R. In period, he agen akes acion a A which is no observed by he principal. he se A is an open inerval of R. hen a conracible public signal is drawn, s S R. Afer he public signal is observed in period, a conracible decision is made, denoed by x X R n, which is observed by boh paries. Since x is conracible, i does no maer wheher i is aken by he agen or by he principal. he conrac beween he principal and he agen is 4
5 signed a =0, righ afer he agen has learned her iniial ype, θ 0. We denoe he hisory of a variable hrough period by superscrip ; for example x = (x 0,...,x ), and x 1 = { }. he random variable θ is disribued according o a c.d.f. G θ 1,a 1,x 1 suppored on Θ. he funcion G is coninuously differeniable in all of is argumen, and he densiy is denoed by g θ 1,a 1,x 1. he public signal s is disribued according o a coninuous c.d.f. H ( f (θ,a )), where f : Θ A R is coninuously differeniable. We may inerpre s as an imperfec public summary signal abou he agen s curren ype and acion; for example, in Applicaion 3 of Secion 5 i will be s = θ + a + ξ, where ξ is noise wih a known disribuion. In he general model we assume ha for all θ, θ and a here is a unique a such ha f (θ,a )=f ( θ, a ). 4 he agen s payoff is quasilinear in money, and is defined by u θ,a,s,x p, where p R denoes he agen s paymen o he principal, and u :Θ A S X R is coninuously differeniable in θ and a for all =0,...,. We do no specify he principal s payoff. In some applicaions (e.g., where he principal is a monopoly and he agen is cusomer) i could be he paymen iself, in ohers (e.g., where he principal is a social planner and he agen he represenaive consumer) i could be he agen s expeced payoff; in ye oher applicaions i could be somehing differen. A noaional convenion: We denoe parial derivaives wih a subscrip referring o he variable of differeniaion, e.g., ũ θ ũ/ θ, f θ f / θ, ec. Orhogonalizaion of Informaion. he model can be ransformed ino an equivalen one where he agen s privae informaion is represened by serially independen random variables. Suppose ha a each =0,...,, he agen observes ε = G θ θ 1,a 1,x 1 insead of θ. Clearly, ε can be inferred from θ,a 1,x 1. Conversely, θ can be compued from (ε,a 1,x 1 ), ha is, for all =0,..., here is ψ :[0, 1] A 1 X 1 Θ such ha ε = G ψ (ε,a 1,x 1 ) ψ 1 (ε 1,a 2,x 2 ),a 1,x 1, (1) where ψ (ε,a 1,x 1 ) denoes ψ 0 (ε 0 ),...,ψ (ε,a 1,x 1 ). In oher words, if he agen observes ε,a 1,x 1 a ime in he orhogonalized model, she can infer he ype hisory ψ (ε,a 1,x 1 ) in he original model. Of course, a model where he agen observes ε for all is sraegically equivalen o he one where she observes θ for all (provided ha in boh cases she observes x 1 and 4 his assumpion ensures ha he principal canno resolve he adverse selecion problem by requiring he agen o ake a cerain acion and using he public signal o deec he agen s ype. 5
6 recalls a 1 a ). By definiion, ε is uniformly disribued on he uni inerval 5 for all and all realizaions of θ 1, a 1 and x 1, hence he random variables {ε } 0 are independen across ime. here are many oher orhogonalized informaion srucures (e.g., ones obained by sricly monoonic ransformaions). In wha follows, o simplify noaion, we fix he orhogonalized informaion srucure as he one where ε is uniform on E =[0, 1]. he agen s gross payoff in he orhogonalized model, u : E A S X R, becomes u ε,a,s,x = u ψ ε,a 1,x 1,a,s,x. Revelaion Principle. A deerminisic mechanism is a fouruple Z, x, a,p, where Z is he agen s message space a ime, x : Z S X is he conracible decision rule a ime, a : Z S 1 A is a recommended acion a, andp : Z S R is he paymen rule. he agen s reporing sraegy a is a mapping from previous repors and informaion o a message. We refer o a sraegy ha maximizes he agen s payoff as an equilibrium sraegy and he payoff generaed by such a sraegy as equilibrium payoff. he sandard Revelaion Principle applies in his seing, so i is wihou loss of generaliy o assume ha Z = E for all, ando resric aenion o mechanisms where elling he ruh and aking he recommended acion (obedience) is an equilibrium sraegy. A direc mechanism is defined by a riple x, a, p, where x : E S X, a : E S 1 A and p : E S R. Direc mechanisms in which elling he ruh and obeying he principal s recommendaion is an equilibrium sraegy are called incenive compaible mechanisms. We call a decisionacion rule x, a implemenable if here exiss a paymen rule, p : E R such ha he direc mechanism x, a, p is incenive compaible. echnical Assumpions. We make hree echnical assumpions o ensure ha he equilibrium payoff funcion of he agen is Lipshiz coninuous in he orhogonalized model. Assumpion 0. (i) here exiss a K N such ha for all =1,..., and for all θ,a,s,x, u θ θ,a,s,x, u a θ,a,s,x <K. (ii) here exiss a K N such ha for all =1,...,, τ<, and for all θ,a 1,x 1, G θ θ θ 1,a 1,x 1, G θτ θ θ 1,a 1,x 1 <K. 5 o see his, noe ha since ε = G `θ θ 1,a 1,x 1, he probabiliy ha ε ε is Pr `G `θ θ 1,a 1,x 1 ε =Pr`θ G 1 ` ε θ 1,a 1,x 1 = G `G 1 ` ε θ 1,a 1,x 1 = ε. 6
7 (iii) here exiss a K N such ha for all =1,..., and for all θ,a, f θ (θ,a ) f a (θ,a ) <K. 3 he main resul We refer o he model in which he principal never observes he agen s ypes as he original model, whereas we call he model where ε 1,...,ε are observed by he principal he benchmark case. he conracing problem in he benchmark is a saic one in he sense ha he principal only ineracs wih he agen a =0, and he agen has no access o dynamic deviaion reporing sraegies. Our dynamic irrelevance resul is ha in any mechanism ha implemens a given decisionacion rule in he original model he principal s maximal expeced revenue is he same as i would be he benchmark case. Specifically, wha we show below is ha he expeced ransfer paymen of an agen wih a given iniial ype when he principal implemens decisionacion rule x, a in he original problem is he same (up o a ypeinvarian consan) as i would be in he benchmark. his implies ha he principal s maximal expeced revenue (and his payoff, in case i is linear in he revenue) when implemening a decisionacion rule in he original problem is jus as high as i would be in he benchmark. his does no imply, however, ha he wo problems are equivalen: sufficien condiions of implemenabiliy (of a decision rule) are sronger in he original problem han hey are in he benchmark. We will urn o he quesion of implemenabiliy in Secion 4. In he nex subsecion we consider a decisionacion rule x, a and derive a necesarry condiion for he paymen rule p such ha x, a, p is incenive compaible. his condiion urns ou o be he same in he benchmark case and in he original model. We hen use his condiion o prove our main, irrelevance resul. 3.1 Paymen rules We fix an incenive compaible mechanism x, a, p and analyze he consequences of ime0 incenive compaibiliy on he paymen rule, p, in boh he original model and he benchmark. We consider a paricular se of deviaion sraegies and explore he consequence of he nonprofiabiliy of hese deviaions in each case. o his end, le us define his se as follows: If he agen wih iniial ype ε 0 repors ε 0 hen (i) she mus repor ε 1,...,ε ruhfully, and (ii) for all =0,...,, afer hisory ε,s 1, she mus ake acion a (ε, ε 0,s 1 ) such ha he disribuion of s is he same as if he hisory were ε 0,ε 0,s 1 and acion a (ε 0,ε 0,s 1 ) were aken, where ε 0 =(ε 1,...,ε ). Since he disribuion of s only depends on f (θ,a ), 7
8 he acion a (ε, ε 0,s 1 ) is defined by where f θ, a (ε 0,ε 0,s 1 ) = f θ, a (ε, ε 0,s 1 ), (2) θ = ψ ε0,ε 0, a 1 ε 0,ε 1 0,s 2, x 1 ε 0,ε 1 0,s 1, θ = ψ ε, a 1 ε 1, ε 0,s 2, x 1 ε 0,ε 1 0,s 1. In oher words, he deviaion sraegies we consider require he agen (i) o be ruhful in he fuure abou her orhogonalized ypes, and (ii) o ake acions ha mask her earlier lie so ha he principal could no deec her iniial deviaion based on he public signals, even in a saisical sense. 6 Noe ha in he benchmark case we only need o impose resricion (ii) since he principal observes ε 1,...,ε by assumpion. Also noe ha he sraegies saisfying resricions (i) and (ii) include he equilibrium sraegy in he original model because if ε 0 = ε 0 he wo resricions imply ruhelling and obedience (adherence o he acion rule). We emphasize ha we do no claim by any means ha afer reporing ε 0 i is opimal for he agen o follow a coninuaion sraegy defined by resricions (i) and (ii). Neverheless, since he mechanism x, a, p is incenive compaible, none of hese deviaions are profiable for he agen. We show ha his observaion enables us o characerize he expeced paymen of he agen condiional on ε 0 up o a ypeinvarian consan. Le Π 0 (ε 0 ) denoe he agen s expeced equilibrium payoff condiional on her iniial ype ε 0 in he incenive compaible mechanism x, a, p. ha is, Π 0 (ε 0 )=E u ε, a ε,s 1,s, x ε,s p ε ε 0, (3) where E denoes expecaion over ε and s. Proposiion 1 If he mechanism x, a, p is incenive compaible eiher in he original model or in he benchmark case, hen for all ε 0 E 0 : ˆ ε0 Π 0 (ε 0 )=Π 0 (0) + E u ε0 y, ε 0, a y, ε 0,s 1,s, x y, ε 0,s dy ε 0 (4) 0 ˆ ε0 +E u a y, ε 0, a y, ε 0,s 1,s, x y, ε 0,s a ε0 y, ε 0,y,s 1 dy ε 0, 0 =0 6 Similar ideas are used by Pavan, Segal and oikka (2012) in a dynamic conracing model wihou moral hazard and by Garre and Pavan (2012) in a more resricive environmen wih moral hazard. 8
9 where y, ε 0 =(y, ε1,...,ε ). Proposiion 1 esablishes ha in an incenive compaible mechanism ha implemens a paricular decisionacion rule he expeced payoff of he agen wih a given (iniial) ype does no depend on he ransfers. Analogous o he necessiy par of he Spence Mirrlees Lemma in saic mechanism design (or Myerson s Revenue Equivalence heorem), necessary condiions similar o (4) have been derived in dynamic environmens by Baron and Besanko (1984), Coury and Li (2000), Eso and Szenes (2007a), Pavan, Segal and oikka (2012), Garre and Pavan (2012), and ohers. In our environmen, which is no only dynamic bu incorporaes boh adverse selecion and moral hazard as well, he real significance of he resul is ha he same formula applies in he original problem and in he benchmark case. I may be insrucive o consider he special case where he principal has no access o a public signal, or equivalenly, he disribuion of s is independen of (θ,a ). Since he choice of a has no impac on x and p, he agen chooses a o maximize her uiliy. A necessary condiion of his maximizaion is E u a ε, a (ε,s 1 ),s, x (ε,s ) ε,s 1 =0for all. As a consequence he las erm of Π 0 (ε 0 ), i.e. he second line of equaion (4), vanishes. Proof of Proposiion 1 Firs we express he agen s reporing problem a =0in he benchmark case as well as in he original problem subjec o resricions (i) and (ii) discussed a he beginning of his subsecion. In order o do his define U (ε 0, ε 0 )=E u ε, a ε, ε 0,s 1,s, x ε 0,ε 0,s ε 0 and P (ε 0 )=E p ε 0,ε 0,s ε 0,a = a ε, ε 0,s 1,x = x ε 0,ε 0,s, (5) where a is defined by (2). Recall ha he acion a ε, ε 0,s 1 generaes he same disribuion of s as if he agen s rue ype hisory was ε 0,ε 0 and he agen had aken a ε0,ε 0,s 1. he significance of his is ha E p ε 0,ε 0,s ε 0,a = a ε, ε 0,s 1,x = x ε 0,ε 0,s = E p ε 0,ε 0,s ε 0,a = a ε 0,ε 0,s 1,x = x ε 0,ε 0,s, so he righhand side of (5) is indeed only a funcion of ε 0 bu no ha of ε 0. In he benchmark case, he payoff of he agen wih ε 0 who repors ε 0 and akes acion a ε, ε 0,s 1 a every is W (ε 0, ε 0 ) = U (ε 0, ε 0 ) P (ε 0 ).NoehaW (ε 0, ε 0 ) is also 9
10 he payoff of he agen in he original model if her ype is ε 0 a =0, repors ε 0 and her coninuaion sraegy is defined by resricions (i) and (ii) above, ha is, she repors ruhfully afererwards and akes acion a ε, ε 0,s 1 afer he hisory ε,s 1. he incenive compaibiliy of x, a, p implies ha ε 0 arg max bε0 E 0 W (ε 0, ε 0 ) boh in he benchmark case and in he original model. In addiion, Π 0 (ε 0 )=W (ε 0,ε 0 ) and, by Lemma 6 of he Appendix, Π 0 is Lipshiz coninuous. herefore, heorem 1 in Milgrom and Segal (2002) implies ha almos everywhere. Noe ha =0 dπ 0 (ε 0 ) = U (ε 0, ε 0 ), dε 0 ε 0 bε 0 =ε 0 U (ε 0, ε 0 ) = E u ε0 ε, a ε,s 1,s, x ε,s ε 0 ε 0 bε 0 =ε 0 +E u a ε, a ε,s 1,s, x ε,s a ε0 ε, ε 0,s 1 bε 0 =ε 0 ε 0. Since Π 0 is Lipshiz coninuous, i can be recovered from is derivaive, so he saemen of he proposiion follows. By Proposiion 1, for a given decisionacion rule, incenive compaibiliy consrains pin down he expeced paymens condiional on ε 0 uniquely up o a consan in boh he benchmark case and he original model. paymen condiional on ε 0 can be expressed as o see his, noe ha from (3) and (4) he expeced E p ε,s ε 0, a, x = E u ε, a ε,s 1,s, x ε ε 0 Π0 (0) ˆ ε0 E u ε0 y, ε 0, a y, ε 0,s 1,s, x y, ε 0,s dy ε 0 0 ˆ ε0 E u a y, ε 0, a y, ε 0,s 1,s, x y, ε 0,s a ε0 y, ε 0,y,s 1 ε 0. 0 =0 An immediae consequence of his observaion and Proposiion 1 is he following Remark 1 Suppose ha x, a, p and x, a, p are incenive compaible mechanisms in he original and in he benchmark case, respecively. hen E p ε,s ε 0, a, x E p ε,s ε 0, a, x = c, where c R. 10
11 3.2 Irrelevance of dynamic adverse selecion Now we show ha he principal s maximal revenue of implemening a decision rule is he same as if he were able o observe he orhogonalized ypes of he agen afer =0. ha is, whenever a decision rule is implemenable, he agen only receives informaion ren for her iniial privae informaion. his is our irrelevance resul. o sae his resul formally, suppose ha he agen has an ouside opion, which we normalize o be zero. his means ha any mechanism mus saisfy Π 0 (ε 0 ) 0 (6) for all ε 0 E 0. We call he maximum (supremum) of he expeced paymen of he agen across all he mechanisms ha implemen x, a and saisfy (6) he principal s maximal revenue from implemening his rule. 7 heorem 1 Suppose ha he decision rule x, a is implemenable in he original model. hen he principal s maximal revenue from implemening his rule is he same as in he benchmark case. Proof. Consider firs he benchmark case where he principal observes ε 1,...,ε and le p denoe he paymen rule in a revenuemaximizing mechanism. hen he revenue of implemening x, a is jus E p ε,s a, x. Of course, he principal s revenue in he benchmark case is an upper bound on his revenue in he original model. herefore, i is enough o show ha he principal can achieve E p ε,s a, x from implemening x even if he he does no observe ε 1,...,ε.Suppose ha he direc mechanism x, a, p is incenive compaible. hen, by Remark 1, E p ε,s ε0, a, x = E p ε,s ε0, a, x + c for some c R. Define p ε,s o be p ε,s c. Since adding a consan has no effec on incenives, he mechanism x, a, p is incenive compaible. In addiion, E p ε,s ε 0, a, x = E p ε,s ε 0, a, x, ha is, he principal s revenue is he same as in he benchmark case. Finally, noice ha he paricipaion consrain of he agen, (6), is also saisfied because 7 Requiring (6) for all ε 0 implies ha we resric aenion o mechanisms where he agen paricipaes irrespecive of her ype. his is wihou he loss of generaliy in many applicaions where here is a decision which generaes a uiliy of zero for boh he principal and he agen. Alernaively, we could have saed our heorem for problems where he paricipaing ypes in he opimal conrac of he benchmark case is an inerval. 11
12 he agen s expeced payoff condiional on her iniial ype, ε 0,ishesameashainhe benchmark case. he saemen of heorem 1 is abou he revenue of he principal. Noe ha if he payoff of he principal is also quasilinear (affine in he paymen), hen he decision rule and he expeced paymen fully deermines his payoff. Hence, a consequence of heorem 1 is, Remark 2 Suppose ha he decisionacion rule x, a is implemenable in he original model and he principal s payoff is affine in he paymen. hen he principal s maximum (supremum) payoff from implemening x, a is he same as in he benchmark case. I is imporan o poin ou ha our dynamic irrelevance resul does no imply ha he original problem (unobservable ε 1,...,ε ) and he benchmark case (observable ε 1,...,ε ) are equivalen. heorem 1 only saes ha if an decisionacion rule is implemenable in he original model, hen i can be done so wihou revenue loss as compared o he benchmark case. his resul was obained under very mild condiions regarding he sochasic process governing he agen s ype, her payoff funcion and he srucure of he public signals. he obvious, nex quesion is wha ype of decisionacion rules can be implemened (under wha condiions) in he original problem. We find some answers o his quesion in he nex secion. 4 Implemenaion his secion esablishes resuls regarding he implemenabiliy of cerain decision rules. 8 We resric aenion o a Markov environmen wih imeseparable, regular (monoonic and singlecrossing) payoff funcions, formally saed in Assumpions 1 and 2 below. Firs, we show ha in he pure adverse selecion model (where here are neiher unobservable acions nor public signals) any monoonic decision rule is implemenable. hen we urn our aenion o he general model wih moral hazard. here he se of implemenable decision rules depends on he informaion conen of he public signal. If he public signal has no informaional conen, ha is, he disribuion of s is independen of f (θ,a ), hen naurally he agen canno be given incenives o choose any acion oher han he one ha maximizes her flow uiliy in each period. In his case, we show ha any decisionacion rule can be implemened if x is monoonic and a is deermined by he agen s per period maximizaion problem. he mos ineresing (and permissive) implemenaion resul is obained in he general model wih adverse selecion and moral hazard in case he signal is informaive and is dis 8 hroughou his secion we require a ypeinvarian paricipaion consrain for he agen wih her ouside opion normalized o zero payoff, ha is, (6) o hold. 12
13 ribuion saisfies a genericiy condiion due o McAfee and Reny (1992). his condiion requires ha he disribuion of s condiional on any given y = f (θ,a ) is no he average of signal disribuions condiional on oher y = y s such ha y = f ( θ, a ). In his case, we show ha any monoonic decisionacion rule x, a can be approximaely implemened (o be formally defined below). 9 he resul is based on argumens similar o he Full Surplus Exracion heorem of McAfee and Reny (1992) and explois he propery of he model ha f is approximaely conracible and he agen is risk neural wih respec o moneary ransfers. he main resul of his secion is ha in our general model, in a regular Markovian environmen wih ransferable uiliy and generic public signals, he principal is able o implemen any monoonic decisionacion rule while no incurring any agency cos apar from he informaion ren due o he agen s iniial privae informaion. In order o sae he regulariy assumpions made hroughou he secion, we reurn o he model wihou orhogonalizaion. hroughou his secion, we assume ha he public signal does no direcly affec he agen s payoff direcly and we remove s from he argumens of u, ha is, u :Θ A X R. We make wo ses of assumpions regarding he environmen. he firs se concerns he ype disribuion, he second one he agen s payoff funcion. Assumpion 1. (ype Disribuion) (i) For all {0,...,}, herandomvariableθ is disribued according o a coninuous c.d.f. G ( θ 1 ) suppored on an inerval Θ = θ, θ. (ii) For all {1,...,}, G ( θ 1 ) G ( θ 1 ) whenever θ 1 θ 1. Par (i) of Assumpion 1 saes ha he agen s ype follows a Markov process, ha is, he ype disribuion a ime only depends on he ype a 1. Inaddiion,hesupporofθ only depends on, soanyypeonθ can be realized irrespecive of θ 1. Par (ii) saes ha he ype disribuions a ime are ordered according o firsorder socahasic dominance. he larger he agen s ype a ime 1, he more likely i is o be large a ime. Assumpion 2. (Payoff Funcion) (i) here exis {u } =0, u :Θ A X R coninuously differeniable, such ha u θ,a,x = u θ,a,x. =0 (ii) For all {0,...,}, u is sricly increasing in θ. (iii) For all {0,...,}, θ Θ,a A : u θ θ,a,x u θ θ,a, x whenever x x. 9 he approximaion can be dispensed wih if he public signal is he summary saisic f (θ,a ) iself. 13
14 Par (i) of Assumpion 2 says ha he agen s uiliy is addiively separable over ime, such ha her flow uiliy a ime only depends on θ and a (and no on any prior informaion and acion) besides all decisions aken a or before. Par (ii) requires he flow uiliy o be monoonic in he agen s ype. Par (iii) is he sandard singlecrossing propery for he agen s ype and he conracible decision. We refer o he model as he one wih pure adverse selecion if u a 0 for all and he disribuion of s is independen of f. Nex, we sae our implemenaion resul for his case (Proposiion 2). hen, in Secions 4.1 and 4.2 we reurn o he general model wih moral hazard. In boh scenarios regarding he informaional conen of he public signal discussed above we reduce he problem of implemenaion o ha in an appropriaelydefined pure adverse selecion problem. Proposiion 2 Suppose ha Assumpions 0,1 and 2 hold in a pure adverse selecion model. hen a decision rule, x, x :Θ X,isimplemenableifx is increasing for all. By Corollary 2 of Pavan, Segal and oikka (2012), Assumpions 12 imply heir inegral monooniciy condiion; sligh differences in heir and our echnical assumpions nowihsanding, our Proposiion 2 appears o be an implicaion of heir heorem 2. We presen a proof of his resul in Secion 4.3 relying on he echniques used in Eso and Szenes (2007a). 4.1 Uninformaive public signal Suppose ha he public signal is uninformaive (i.e. s is independen of f ). We mainain he assumpion ha he payoff funcion of he agen is imeseparable and saisfies Assumpion 2, bu now he flow uiliy a ime is allowed o vary wih a. Recall ha he acion space of he agen a ime, A, was assumed o be an open inerval of R in Secion 2. We needed his assumpion because we posied ha for all θ, θ and a here is a unique a such ha f (θ,a )=f ( θ, a ). 10 Since here is no public signal in he case considered here, we can relax he requiremen ha A is open. In fac, in order o discuss he implemenabiliy of allocaion rules which may involve boundary acions, we assume ha A =[a, a ] is a compac inerval hroughou his subsecion. Assumpion 3. For all {0,...,}, for all θ Θ,a, a A,x, x X (i) u a 2 θ,a,x 0, (ii) u θ θ,a,x u θ θ, a, x whenever a a,and (iii) u a θ,a,x u a θ,a, x whenever x x. 10 If A was compac hen here would be a pair, (θ,a ), which maximizes f. herefore, if b θ / arg max θ [max a f (θ,a )], henherewasnoba such ha f (θ,a )=f ( b θ, ba ). 14
15 Par (i) of he assumpion saes ha he agen s payoff is a concave funcion of her acion. his is saisfied in applicaions where he acion of he agen is inerpreed as an effor, and he cos of exering effor is a convex funcion of he effor. Par (ii) saes ha he singlecrossing assumpion is also saisfied for he acion. In he previous applicaion, his means ha he marginal cos of effor is decreasing in he agen s ype. Par (iii) requires he singlecrossing propery o hold wih respec o acions and decisions. In wha follows, we urn he problem of implemenaion in his environmen wih adverse selecion and moral hazard ino one of pure adverse selecion. Since here is no publicly available informaion abou he agen s acion, her acion maximizes her payoff in each period and afer each hisory. ha is, if he agen has ype θ and he hisory of decisions is x, hen she akes an acion which maximizes u θ,a,x. Moivaed by his observaion, le us define he agen s new flow uiliy funcion a ime, v :Θ X R, obe v θ,x = max a u θ,a,x. We will apply our implemenaion resul for he pure adverse selecion case (Proposiion 2) o he seing where he flow uiliies of he agen are {v } =0 while keeping in mind ha he acion of he agen in each period maximizes u. o his end, le a θ,x denoe he generically unique arg max a u θ,a,x for all θ Θ and x X. By par (i) of Assumpion 3, if a θ,x is inerior, i is defined by he firsorder condiion u a θ, a (θ,x ),x =0. (7) he nex lemma saes ha he flow uiliies, {v } 0, saisfy he hypohesis of Proposiion 2. saisfy Assumpions 2 and 3. hen he func Lemma 1 Suppose ha he funcions {u } =0 ions {v } =0 saisfy Assumpion 2. Suppose ha he decisionacion rule x, a is implemenable. hen, since he agen s acion maximizes her payoff in each period, a θ = a θ, x θ. In addiion, he decision rule x mus be implemenable in he pure adverse selecion model, where he agen s flow uiliy funcions are {v } =1. Hence, he following resul is a consequence of Proposiion 2 and Lemma 1. Proposiion 3 Suppose ha Assumpions 03 hold. hen a decision rule, x, a, x : Θ X and a :Θ A,isimplemenableifx is increasing and a θ = a θ, x θ for all {0,...,}. 15
16 Of course, he saemen of his proposiion is valid even if he public signal is informaive (s depends on f ) bu he principal ignores i and designs a mechanism which does no condiion on s. However, if s is informaive abou f (θ,a ) he principal can implemen more decision rules which is he subjec of he nex subsecion. 4.2 Informaive public signal We urn our aenion o he case where public signal is informaive. he nex condiion is due o McAfee and Reny (1992); i requires ha he disribuion of he public signal condiional of any given value of y 0 = f (θ,a ) is no he average of he disribuion of s condiional on oher values of f. his condiion is generic. Assumpion 4. Suppose ha for all θ Θ and a A, f (θ,a ) Y =[y, y ]. hen, for all µ [y, y ] and y 0 [y, y ], µ ({y 0 }) = 1implies h ( y 0 ) = 1 0 h ( y) µ (dy). Nex, we make furher assumpions on he agen s flow uiliy, u,andonheshapeofhe funcion f. Assumpion 5. For all {0,...,}, for all θ Θ,a A,x X (i) u a θ,a,x < 0, (ii) here exiss a K N such ha f a (θ,a ),f θ (θ,a ) > 1/K, (iii) f a 2 (θ,a ) f θ (θ,a ) f a (θ,a ) f aθ (θ,a ), and (iv) u θx τ θ,a,x f a (θ,a ) u ax τ θ,a,x f θ (θ,a ). Par (i) requires he agen s flow uiliy o be decreasing in her acion. his is saisfied in applicaions where, for example, he agen s unobservable acion is a cosly effor from which she does no benefi direcly. Par (ii) says ha he funcion f is increasing in boh he agen s acion and ype. In many applicaions, he disribuion of he public signal can be ordered according o firsorder sochasic dominance. In hese applicaions, par (ii) implies ha an increase in eiher he acion or he ype improves he disribuion of s in he sense of firsorder sochasic dominance. Par (iii) is a subsiuion assumpion regarding he agen s ype and hidden acion in he value of f. I means ha an increase in a,holdinghevalueof f consan, weakly decreases he marginal impac of a on f. 11 his assumpion is saisfied, for example, if f (θ,a )=θ + a, bu i is clearly more general. As will be explained laer par (iv) is a sreghening of he single crossing propery posied in par (iii) of Assumpion 2. I requires he marginal uiliy in ype o be increasing in he conracible decision while 11 o see his inerpreaion, noe ha he oal differenial of f a (he change in he marginal impac of a ) is f a 2 da + f a θ dθ. Keeping f consan (moving along an isovalue curve) means dθ =( f a /f θ )da. Subsiuing his ino he oal differenial of f a yields (f a 2 f a θ f a /f θ )da. his expression is nonposiive for da > 0 if par (iii) is saisfied. 16
17 holding he value of f fixed. his assumpion is saisfied, for example, if he effor cos of he agen is addiively separable in her flow uiliy. he key observaion is ha due o Assumpion 4, he value of f becomes an approximaely conracible objec in he following sense. For each value of f, y, he principal can design a ransfer scheme depending only on s ha punishes he agen for aking an acion which resuls a value of f which is differen from y. Perhaps more imporanly, he punishmen can be arbirarily large as a funcion of he disance beween y and he realized value of f.we use his observaion o esablish our implemenaion resul in wo seps. Firs, we rea f (for all ) as a conracible objec, ha is, we add anoher dimension o he conracible decisions in each period. Since, condiional on θ,hevalueoff is deermined by a, we can express he agen s flow uiliy as a funcion of f insead of a. hese new flow uiliies depend only on ypes and decisions, so we have a pure adverse selecion model. We hen show ha he new flow uiliies saisfy he requiremens of Proposiion 2 and hence, every monoonic rule is implemenable. he second sep is o consruc punishmenransfers menioned above and show ha even if f is no conracible, any monoonic decision rule can be approximaely implemenable. For each y {f (θ,a ):θ Θ,a A } and θ Θ, le a (θ,y ) denoe he soluion o f (θ,a )=y in a. For each =0,...,, we define he agen s flow uiliy as a funcion of y as follows: w θ,y,x = u θ, a (θ,y ),x. Nex, we show ha he funcions {w } =0 saisfy he hypohesis of Proposiion 2. Lemma 2 Suppose ha Assumpions 25 are saisfied. hen he funcions {w } =0 saisfy Assumpion 2. By his lemma and Proposiion 2 if he value of f was conracible for all, any increasing decision rule was implemenable. However, f is no conracible; neverheless we can sill implemen increasing decisions rules approximiaely in he sense ha by following he principal s recommendaion he agen s expeced uiliy is arbirarily close o her equilibrium payoff. Formally: Definiion 1 he decision rule x, a is approximaely implemenable if for all δ>0 here exiss a paymen rule p: Θ S R such ha for all θ 0 Θ 0, E s =0 u θ, a θ, x θ p θ,s θ 0 Π 0 (θ 0 ) δ, (8) where Π 0 (θ 0 ) denoes he agen s equilibrium payoff wih iniial ype θ 0. 17
18 We are ready o sae he implemenaion resul of his subsecion. Proposiion 4 Suppose ha Assumpions 05 are saisfied. hen a decision rule, x, a, x :Θ X and a :Θ A,isapproximaelyimplemenableifx and a are increasing for all {0,...,}. 4.3 he proof of Proposiion 2 Since in a pure adverse selecion model u a 0 for all, hroughou his secion, we remove a from he argumens of u,hais,u :Θ X R. We firs inspec he consequences of Assumpions 1 and 2 on he orhogonalized model. Noe ha since θ does no depend on x 1, he inference funcions defined in equaion (1) do no depend on he decisions eiher, so ψ : E Θ. he imeseparabiliy of he agen s payoff (par (i) of Assumpion 2) is preserved in he orhogonalized model, excep ha he flow uiliy a, u : E X R, now depends on he hisory of ypes up o and including ime : u ε,x = u ψ ε,x. (9) Par (iii) of Assumpion 1 implies ha he larger he ype hisory in he orhogonalized model up o ime, he larger is he corresponding period ype in he original model. his, coupled wih par (ii) of Assumpion 2 implies ha u is weakly increasing in ε 1 and sricly in ε. Monoonicy in x as well as singlecrossing (par (iii) of Assumpion 2) are also preserved in he orhogonalized model. We sae hese properies formally in he following Lemma (see he proof in he Appendix). Lemma 3 (i) For all {0,...,} and ε,ε E, ε ε ψ ε ψ ε, (10) and he inequaliy is sric whenever ε <ε. (ii) he flow uiliy, u defined by (9), is weakly increasing in ε 1 and x 1,andsricly increasing in x and ε. (iii) For all {0,...,}, u ε ε,x u ε ε, x whenever x x. Anoher imporan consequence of par (i) of Assumpion 1 is ha for all ε +1 and ε, here exiss a ype σ +1 (ε +1, ε ) E such ha, fixing he principal s pas and fuure decisions as well as he realizaions of he agen s ypes beyond period +1, he agen s uiliy flow from period +1on is he same wih ype hisory ε +1 as i is wih ε 1, ε,σ +1 (ε +1, ε ). We will show below ha σ +1, inerpreed in Eso and Szenes (2007) as he agen s correcion of 18
19 a lie, defines an opimal sraegy for he agen a ime +1 afer a deviaion from ruhelling in an incenive compaible direc mechanism a. his is formally saed in he following Lemma 4 For all {0,..., 1}, ε +1 E +1 and ε E, here exiss a unique σ +1 ε +1, ε E+1 such ha for all k = +1,...,,allε k E k and x k X k, u k (ε 1,ε,ε +1, ε +2,...,ε k, x k )=u k (ε 1, ε,σ +1, ε +2,...,ε k, x k ). (11) he funcion σ +1 is increasing in ε,sriclyincreasinginε +1 and decreasing in ε. he saemen of he lemma migh appear somewha complicaed a firs glance, bu is meaning and is inuiive proof are quie sraighforward. Par (i) of Assumpion 1 requires he suppor of θ o be independen of θ 1. herefore, if he ype of he agen is ψ ε 1, ε a ime, here is a chance ha he period( + 1) ype will be ψ +1 ε +1. he ype σ +1 ε +1, ε denoes he orhogonalized informaion of he agen a +1 which induces he ransiion from ψ ε 1, ε o ψ+1 ε +1,hais, ψ +1 ε 1, ε,σ +1 ε +1, ε = ψ+1 ε +1. his means ha he inferred ype in he original model is he same afer he hisories ε 1, ε,σ +1 ε +1, ε and ε +1. Par (i) of Assumpion 1 and par (ii) of Assumpion 2 imply ha, given he decisions, he flow uiliies in he fuure only depend on curren ype which, in urn, imply (11). he decision rule in he orhogonalized model, x : E X, which corresponds o =0 {x } =0, is defined by x ε = x ψ ε for all and ε. Noe ha, by (10), if {x } =0 is increasing in ype (x is increasing in θ for all ) hen he corresponding decision rule {x } =0 in he orhogonalized model is also increasing in ype. 12 In fac, he monooniciy of {x } =0 implies a sronger monooniciy condiion on {x } =0. Consider he following wo ype hisories, ε k and ε 1,...,ε 1, ε,σ +1 ε +1, ε,ε+2,...,ε k. Noe ha he inferred ypes in he original model are exacly he same along hese hisories excep a ime. A ime, he inferred ype is smaller afer ε if and only if ε ε. Since x k is increasing in θ, he decision is smaller afer ε k if and only if ε ε.formally, Remark 3 If {x } =0 is increasing hen for all k =1,...,, <k, εk E k : x k (ε k ) x k ε 1, ε,σ +1 ε +1, ε,ε+2,...,ε k ε ε. (12) 12 o see his, noe ha if v bv hen x `bv = ex `ψ `bv ex `ψ `v = x `v, where he inequaliy follows from he monooniciy of {ex } 0 and (10). 19
20 o simplify he exposiion, we inroduce he following noaion for =0,...,, k : ζ k (ε k,y)= ε 1,y,ε +1,...,ε k, ρ k (ε k,y,ε )= ε 1, ε,σ +1 ε 1,y,ε +1, ε,ε+2,...,ε k. he vecors ζ k (ε k,y) and ρ k (ε k,y,ε ) are ype hisories up o period k, rue or repored, which are differen from ε k only a or a and +1.Fork = hese are appropriaely runcaed, e.g., ρ (ε,y,ε )= ε 1, ε. As we explained, he monooniciy of {x } =0 implies boh he monooniciy of {x } =0 and (12). herefore, in order o prove Proposiion 2, i is sufficien o show ha any increasing decision rule in he orhogonalized model which saisfies (12) can be implemened. In wha follows, fix a direc mechanism wih an increasing decision rule {x } =0 ha saisfies (12). Le Π (ε ε 1 ) denoe a ruhful agen s expeced payoff a condiional on ε. ha is, Π (ε ε 1 )=E k=0 u k ε k,x k (ε k ) p(ε ) Define he paymen funcion, p, such ha for all =0,..., and ε E, Π ε ε 1 =Π 0 ε 1 + E ˆ ε 0 k= ε u kε ζ k (ε k,y),x k ζ k (ε k,y) dy. (13) ε. (14) I is no hard o show ha he inegral on he righhand side of (14) exiss and is finie because of par (ii) of Assumpion 1, par (i) of Assumpion 2 and he monooniciy of x k.i should be clear ha i is possible o define p such ha (14) holds. In his mechanism, le π ε, ε ε 1 denoe he expeced payoff of he agen a ime whose ype hisory is ε and has repored ε 1, ε. his is he maximum payoff she can achieve from using any reporing sraegy from +1 condiional on he ype hisory ε and on he repors ε 1, ε. Π (ε ε 1 )=π (ε,ε ε 1 ). If he mechanism is incenive compaible hen, clearly, We call a mechanism IC afer ime if, condiional on elling he ruh before and a ime 1, i is an equilibrium sraegy for he agen o ell he ruh aferwards, ha is, from period on. By Lemma 4, he coninuaion uiliies of he agen wih ype ε +1 are he same as hose of he agen wih ype ε 1, ε,σ +1 ε +1, ε condiional on he repors and he realizaion of ypes afer +1. herefore, if a mechanism is IC afer +1, he agen whose ype hisory is ε +1 and repored ε 1, ε up o ime maximizes her coninuaion payoff by reporing σ +1 ε +1, ε a ime +1 and reporing ruhfully aferwards. If his were no 20
The Transport Equation
The Transpor Equaion Consider a fluid, flowing wih velociy, V, in a hin sraigh ube whose cross secion will be denoed by A. Suppose he fluid conains a conaminan whose concenraion a posiion a ime will be
More informationNiche Market or Mass Market?
Niche Marke or Mass Marke? Maxim Ivanov y McMaser Universiy July 2009 Absrac The de niion of a niche or a mass marke is based on he ranking of wo variables: he monopoly price and he produc mean value.
More informationANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS
ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS R. Caballero, E. Cerdá, M. M. Muñoz and L. Rey () Deparmen of Applied Economics (Mahemaics), Universiy of Málaga,
More informationPROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE
Profi Tes Modelling in Life Assurance Using Spreadshees PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Erik Alm Peer Millingon 2004 Profi Tes Modelling in Life Assurance Using Spreadshees
More informationEfficient Risk Sharing with Limited Commitment and Hidden Storage
Efficien Risk Sharing wih Limied Commimen and Hidden Sorage Árpád Ábrahám and Sarola Laczó March 30, 2012 Absrac We exend he model of risk sharing wih limied commimen e.g. Kocherlakoa, 1996) by inroducing
More informationEconomics Honors Exam 2008 Solutions Question 5
Economics Honors Exam 2008 Soluions Quesion 5 (a) (2 poins) Oupu can be decomposed as Y = C + I + G. And we can solve for i by subsiuing in equaions given in he quesion, Y = C + I + G = c 0 + c Y D + I
More informationStochastic Optimal Control Problem for Life Insurance
Sochasic Opimal Conrol Problem for Life Insurance s. Basukh 1, D. Nyamsuren 2 1 Deparmen of Economics and Economerics, Insiue of Finance and Economics, Ulaanbaaar, Mongolia 2 School of Mahemaics, Mongolian
More informationOptimal Investment and Consumption Decision of Family with Life Insurance
Opimal Invesmen and Consumpion Decision of Family wih Life Insurance Minsuk Kwak 1 2 Yong Hyun Shin 3 U Jin Choi 4 6h World Congress of he Bachelier Finance Sociey Torono, Canada June 25, 2010 1 Speaker
More information17 Laplace transform. Solving linear ODE with piecewise continuous right hand sides
7 Laplace ransform. Solving linear ODE wih piecewise coninuous righ hand sides In his lecure I will show how o apply he Laplace ransform o he ODE Ly = f wih piecewise coninuous f. Definiion. A funcion
More informationChapter 8: Regression with Lagged Explanatory Variables
Chaper 8: Regression wih Lagged Explanaory Variables Time series daa: Y for =1,..,T End goal: Regression model relaing a dependen variable o explanaory variables. Wih ime series new issues arise: 1. One
More informationDuration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is $613.
Graduae School of Business Adminisraion Universiy of Virginia UVAF38 Duraion and Convexiy he price of a bond is a funcion of he promised paymens and he marke required rae of reurn. Since he promised
More informationTrading on ShortTerm Information
428 Trading on ShorTerm Informaion by ALEXANDER GÜMBEL This paper shows ha invesors may wan fund managers o acquire and rade on shorerm insead of more profiable longerm informaion. This improves learning
More informationCooperation with Network Monitoring
Cooperaion wih Nework Monioring Alexander Wolizky Microsof Research and Sanford Universiy November 2011 Absrac This paper sudies he maximum level of cooperaion ha can be susained in perfec Bayesian equilibrium
More informationOption PutCall Parity Relations When the Underlying Security Pays Dividends
Inernaional Journal of Business and conomics, 26, Vol. 5, No. 3, 22523 Opion Puall Pariy Relaions When he Underlying Securiy Pays Dividends Weiyu Guo Deparmen of Finance, Universiy of Nebraska Omaha,
More informationThe Grantor Retained Annuity Trust (GRAT)
WEALTH ADVISORY Esae Planning Sraegies for closelyheld, family businesses The Granor Reained Annuiy Trus (GRAT) An efficien wealh ransfer sraegy, paricularly in a low ineres rae environmen Family business
More informationABSTRACT KEYWORDS. Markov chain, Regulation of payments, Linear regulator, Bellman equations, Constraints. 1. INTRODUCTION
QUADRATIC OPTIMIZATION OF LIFE AND PENSION INSURANCE PAYMENTS BY MOGENS STEFFENSEN ABSTRACT Quadraic opimizaion is he classical approach o opimal conrol of pension funds. Usually he paymen sream is approximaed
More informationPricing FixedIncome Derivaives wih he ForwardRisk Adjused Measure Jesper Lund Deparmen of Finance he Aarhus School of Business DK8 Aarhus V, Denmark Email: jel@hha.dk Homepage: www.hha.dk/~jel/ Firs
More informationJournal Of Business & Economics Research September 2005 Volume 3, Number 9
Opion Pricing And Mone Carlo Simulaions George M. Jabbour, (Email: jabbour@gwu.edu), George Washingon Universiy YiKang Liu, (yikang@gwu.edu), George Washingon Universiy ABSTRACT The advanage of Mone Carlo
More informationGraduate Macro Theory II: Notes on Neoclassical Growth Model
Graduae Macro Theory II: Noes on Neoclassical Growh Model Eric Sims Universiy of Nore Dame Spring 2011 1 Basic Neoclassical Growh Model The economy is populaed by a large number of infiniely lived agens.
More informationON THE PRICING OF EQUITYLINKED LIFE INSURANCE CONTRACTS IN GAUSSIAN FINANCIAL ENVIRONMENT
Teor Imov r.amaem.sais. Theor. Probabiliy and Mah. Sais. Vip. 7, 24 No. 7, 25, Pages 15 111 S 949(5)6344 Aricle elecronically published on Augus 12, 25 ON THE PRICING OF EQUITYLINKED LIFE INSURANCE
More informationOptimalCompensationwithHiddenAction and LumpSum Payment in a ContinuousTime Model
Appl Mah Opim (9) 59: 99 46 DOI.7/s45895 OpimalCompensaionwihHiddenAcion and LumpSum Paymen in a ConinuousTime Model Jakša Cvianić Xuhu Wan Jianfeng Zhang Published online: 6 June 8 Springer Science+Business
More informationChapter 7. Response of FirstOrder RL and RC Circuits
Chaper 7. esponse of FirsOrder L and C Circuis 7.1. The Naural esponse of an L Circui 7.2. The Naural esponse of an C Circui 7.3. The ep esponse of L and C Circuis 7.4. A General oluion for ep and Naural
More informationBALANCE OF PAYMENTS. First quarter 2008. Balance of payments
BALANCE OF PAYMENTS DATE: 20080530 PUBLISHER: Balance of Paymens and Financial Markes (BFM) Lena Finn + 46 8 506 944 09, lena.finn@scb.se Camilla Bergeling +46 8 506 942 06, camilla.bergeling@scb.se
More informationDYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS
DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS Hong Mao, Shanghai Second Polyechnic Universiy Krzyszof M. Osaszewski, Illinois Sae Universiy Youyu Zhang, Fudan Universiy ABSTRACT Liigaion, exper
More informationCooperation with Network Monitoring
Cooperaion wih Nework Monioring Alexander Wolizky Microsof Research and Sanford Universiy February 2012 Absrac This paper sudies he maximum level of cooperaion ha can be susained in perfec Bayesian equilibrium
More informationMorningstar Investor Return
Morningsar Invesor Reurn Morningsar Mehodology Paper Augus 31, 2010 2010 Morningsar, Inc. All righs reserved. The informaion in his documen is he propery of Morningsar, Inc. Reproducion or ranscripion
More informationINTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES
INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES OPENGAMMA QUANTITATIVE RESEARCH Absrac. Exchangeraded ineres rae fuures and heir opions are described. The fuure opions include hose paying
More informationChapter 6: Business Valuation (Income Approach)
Chaper 6: Business Valuaion (Income Approach) Cash flow deerminaion is one of he mos criical elemens o a business valuaion. Everyhing may be secondary. If cash flow is high, hen he value is high; if he
More informationAnalysis of Pricing and Efficiency Control Strategy between Internet Retailer and Conventional Retailer
Recen Advances in Business Managemen and Markeing Analysis of Pricing and Efficiency Conrol Sraegy beween Inerne Reailer and Convenional Reailer HYUG RAE CHO 1, SUG MOO BAE and JOG HU PARK 3 Deparmen of
More informationTerm Structure of Prices of Asian Options
Term Srucure of Prices of Asian Opions Jirô Akahori, Tsuomu Mikami, Kenji Yasuomi and Teruo Yokoa Dep. of Mahemaical Sciences, Risumeikan Universiy 111 Nojihigashi, Kusasu, Shiga 5258577, Japan Email:
More information4. International Parity Conditions
4. Inernaional ariy ondiions 4.1 urchasing ower ariy he urchasing ower ariy ( heory is one of he early heories of exchange rae deerminaion. his heory is based on he concep ha he demand for a counry's currency
More informationDependent Interest and Transition Rates in Life Insurance
Dependen Ineres and ransiion Raes in Life Insurance Krisian Buchard Universiy of Copenhagen and PFA Pension January 28, 2013 Absrac In order o find marke consisen bes esimaes of life insurance liabiliies
More informationMTH6121 Introduction to Mathematical Finance Lesson 5
26 MTH6121 Inroducion o Mahemaical Finance Lesson 5 Conens 2.3 Brownian moion wih drif........................... 27 2.4 Geomeric Brownian moion........................... 28 2.5 Convergence of random
More informationTowards Optimal Capacity Segmentation with Hybrid Cloud Pricing
Towards Opimal Capaciy Segmenaion wih Hybrid Cloud Pricing Wei Wang, Baochun Li, and Ben Liang Deparmen of Elecrical and Compuer Engineering Universiy of Torono Absrac Cloud resources are usually priced
More informationLongevity 11 Lyon 79 September 2015
Longeviy 11 Lyon 79 Sepember 2015 RISK SHARING IN LIFE INSURANCE AND PENSIONS wihin and across generaions Ragnar Norberg ISFA Universié Lyon 1/London School of Economics Email: ragnar.norberg@univlyon1.fr
More informationJumpDiffusion Option Valuation Without a Representative Investor: a Stochastic Dominance Approach
umpdiffusion Opion Valuaion Wihou a Represenaive Invesor: a Sochasic Doance Approach By Ioan Mihai Oancea and Sylianos Perrakis This version February 00 Naional Bank of Canada, 30 King Sree Wes, Torono,
More informationI. Basic Concepts (Ch. 14)
(Ch. 14) A. Real vs. Financial Asses (Ch 1.2) Real asses (buildings, machinery, ec.) appear on he asse side of he balance shee. Financial asses (bonds, socks) appear on boh sides of he balance shee. Creaing
More informationRandom Walk in 1D. 3 possible paths x vs n. 5 For our random walk, we assume the probabilities p,q do not depend on time (n)  stationary
Random Walk in D Random walks appear in many cones: diffusion is a random walk process undersanding buffering, waiing imes, queuing more generally he heory of sochasic processes gambling choosing he bes
More information11/6/2013. Chapter 14: Dynamic ADAS. Introduction. Introduction. Keeping track of time. The model s elements
Inroducion Chaper 14: Dynamic DS dynamic model of aggregae and aggregae supply gives us more insigh ino how he economy works in he shor run. I is a simplified version of a DSGE model, used in cuingedge
More informationOptimal Stock Selling/Buying Strategy with reference to the Ultimate Average
Opimal Sock Selling/Buying Sraegy wih reference o he Ulimae Average Min Dai Dep of Mah, Naional Universiy of Singapore, Singapore Yifei Zhong Dep of Mah, Naional Universiy of Singapore, Singapore July
More informationPATHWISE PROPERTIES AND PERFORMANCE BOUNDS FOR A PERISHABLE INVENTORY SYSTEM
PATHWISE PROPERTIES AND PERFORMANCE BOUNDS FOR A PERISHABLE INVENTORY SYSTEM WILLIAM L. COOPER Deparmen of Mechanical Engineering, Universiy of Minnesoa, 111 Church Sree S.E., Minneapolis, MN 55455 billcoop@me.umn.edu
More informationVerification Theorems for Models of Optimal Consumption and Investment with Retirement and Constrained Borrowing
MATHEMATICS OF OPERATIONS RESEARCH Vol. 36, No. 4, November 2, pp. 62 635 issn 364765X eissn 526547 364 62 hp://dx.doi.org/.287/moor..57 2 INFORMS Verificaion Theorems for Models of Opimal Consumpion
More informationNetwork Effects, Pricing Strategies, and Optimal Upgrade Time in Software Provision.
Nework Effecs, Pricing Sraegies, and Opimal Upgrade Time in Sofware Provision. YiNung Yang* Deparmen of Economics Uah Sae Universiy Logan, UT 84322353 April 3, 995 (curren version Feb, 996) JEL codes:
More informationTEMPORAL PATTERN IDENTIFICATION OF TIME SERIES DATA USING PATTERN WAVELETS AND GENETIC ALGORITHMS
TEMPORAL PATTERN IDENTIFICATION OF TIME SERIES DATA USING PATTERN WAVELETS AND GENETIC ALGORITHMS RICHARD J. POVINELLI AND XIN FENG Deparmen of Elecrical and Compuer Engineering Marquee Universiy, P.O.
More informationResearch on Inventory Sharing and Pricing Strategy of Multichannel Retailer with Channel Preference in Internet Environment
Vol. 7, No. 6 (04), pp. 365374 hp://dx.doi.org/0.457/ijhi.04.7.6.3 Research on Invenory Sharing and Pricing Sraegy of Mulichannel Reailer wih Channel Preference in Inerne Environmen Hanzong Li College
More informationOn the degrees of irreducible factors of higher order Bernoulli polynomials
ACTA ARITHMETICA LXII.4 (1992 On he degrees of irreducible facors of higher order Bernoulli polynomials by Arnold Adelberg (Grinnell, Ia. 1. Inroducion. In his paper, we generalize he curren resuls on
More informationInductance and Transient Circuits
Chaper H Inducance and Transien Circuis Blinn College  Physics 2426  Terry Honan As a consequence of Faraday's law a changing curren hrough one coil induces an EMF in anoher coil; his is known as muual
More informationTHE FIRM'S INVESTMENT DECISION UNDER CERTAINTY: CAPITAL BUDGETING AND RANKING OF NEW INVESTMENT PROJECTS
VII. THE FIRM'S INVESTMENT DECISION UNDER CERTAINTY: CAPITAL BUDGETING AND RANKING OF NEW INVESTMENT PROJECTS The mos imporan decisions for a firm's managemen are is invesmen decisions. While i is surely
More informationMathematics in Pharmacokinetics What and Why (A second attempt to make it clearer)
Mahemaics in Pharmacokineics Wha and Why (A second aemp o make i clearer) We have used equaions for concenraion () as a funcion of ime (). We will coninue o use hese equaions since he plasma concenraions
More informationChapter 1.6 Financial Management
Chaper 1.6 Financial Managemen Par I: Objecive ype quesions and answers 1. Simple pay back period is equal o: a) Raio of Firs cos/ne yearly savings b) Raio of Annual gross cash flow/capial cos n c) = (1
More informationWorking Paper No. 482. Net Intergenerational Transfers from an Increase in Social Security Benefits
Working Paper No. 482 Ne Inergeneraional Transfers from an Increase in Social Securiy Benefis By Li Gan Texas A&M and NBER Guan Gong Shanghai Universiy of Finance and Economics Michael Hurd RAND Corporaion
More informationTable of contents Chapter 1 Interest rates and factors Chapter 2 Level annuities Chapter 3 Varying annuities
Table of conens Chaper 1 Ineres raes and facors 1 1.1 Ineres 2 1.2 Simple ineres 4 1.3 Compound ineres 6 1.4 Accumulaed value 10 1.5 Presen value 11 1.6 Rae of discoun 13 1.7 Consan force of ineres 17
More informationSinglemachine Scheduling with Periodic Maintenance and both Preemptive and. Nonpreemptive jobs in Remanufacturing System 1
Absrac number: 050407 Singlemachine Scheduling wih Periodic Mainenance and boh Preempive and Nonpreempive jobs in Remanufacuring Sysem Liu Biyu hen Weida (School of Economics and Managemen Souheas Universiy
More informationDOES TRADING VOLUME INFLUENCE GARCH EFFECTS? SOME EVIDENCE FROM THE GREEK MARKET WITH SPECIAL REFERENCE TO BANKING SECTOR
Invesmen Managemen and Financial Innovaions, Volume 4, Issue 3, 7 33 DOES TRADING VOLUME INFLUENCE GARCH EFFECTS? SOME EVIDENCE FROM THE GREEK MARKET WITH SPECIAL REFERENCE TO BANKING SECTOR Ahanasios
More informationImprovement in Information, Income Inequality, and Growth
Improvemen in Informaion, Income Inequaliy, and Growh by Bernhard Eckwer Deparmen of Economics Universiy of Bielefeld Germany Izhak Zilcha The Eian Berglas School of Economics Tel Aviv Universiy Israel
More informationDifferential Equations in Finance and Life Insurance
Differenial Equaions in Finance and Life Insurance Mogens Seffensen 1 Inroducion The mahemaics of finance and he mahemaics of life insurance were always inersecing. Life insurance conracs specify an exchange
More informationForecasting and Information Sharing in Supply Chains Under QuasiARMA Demand
Forecasing and Informaion Sharing in Supply Chains Under QuasiARMA Demand Avi Giloni, Clifford Hurvich, Sridhar Seshadri July 9, 2009 Absrac In his paper, we revisi he problem of demand propagaion in
More informationWhy Did the Demand for Cash Decrease Recently in Korea?
Why Did he Demand for Cash Decrease Recenly in Korea? Byoung Hark Yoo Bank of Korea 26. 5 Absrac We explores why cash demand have decreased recenly in Korea. The raio of cash o consumpion fell o 4.7% in
More informationCan Individual Investors Use Technical Trading Rules to Beat the Asian Markets?
Can Individual Invesors Use Technical Trading Rules o Bea he Asian Markes? INTRODUCTION In radiional ess of he weakform of he Efficien Markes Hypohesis, price reurn differences are found o be insufficien
More informationTowards Optimal Capacity Segmentation with Hybrid Cloud Pricing
Towards Opimal Capaciy Segmenaion wih Hybrid Cloud Pricing Wei Wang, Baochun Li, and Ben Liang Deparmen of Elecrical and Compuer Engineering Universiy of Torono Torono, ON M5S 3G4, Canada weiwang@eecg.orono.edu,
More informationPresent Value Methodology
Presen Value Mehodology Econ 422 Invesmen, Capial & Finance Universiy of Washingon Eric Zivo Las updaed: April 11, 2010 Presen Value Concep Wealh in Fisher Model: W = Y 0 + Y 1 /(1+r) The consumer/producer
More informationA ProductionInventory System with Markovian Capacity and Outsourcing Option
OPERATIONS RESEARCH Vol. 53, No. 2, March April 2005, pp. 328 349 issn 0030364X eissn 15265463 05 5302 0328 informs doi 10.1287/opre.1040.0165 2005 INFORMS A ProducionInvenory Sysem wih Markovian Capaciy
More informationMarkit Excess Return Credit Indices Guide for price based indices
Marki Excess Reurn Credi Indices Guide for price based indices Sepember 2011 Marki Excess Reurn Credi Indices Guide for price based indices Conens Inroducion...3 Index Calculaion Mehodology...4 Semiannual
More informationA UNIFIED APPROACH TO MATHEMATICAL OPTIMIZATION AND LAGRANGE MULTIPLIER THEORY FOR SCIENTISTS AND ENGINEERS
A UNIFIED APPROACH TO MATHEMATICAL OPTIMIZATION AND LAGRANGE MULTIPLIER THEORY FOR SCIENTISTS AND ENGINEERS RICHARD A. TAPIA Appendix E: Differeniaion in Absrac Spaces I should be no surprise ha he differeniaion
More informationARCH 2013.1 Proceedings
Aricle from: ARCH 213.1 Proceedings Augus 14, 212 Ghislain Leveille, Emmanuel Hamel A renewal model for medical malpracice Ghislain Léveillé École d acuaria Universié Laval, Québec, Canada 47h ARC Conference
More informationA Note on Using the Svensson procedure to estimate the risk free rate in corporate valuation
A Noe on Using he Svensson procedure o esimae he risk free rae in corporae valuaion By Sven Arnold, Alexander Lahmann and Bernhard Schwezler Ocober 2011 1. The risk free ineres rae in corporae valuaion
More informationWHAT ARE OPTION CONTRACTS?
WHAT ARE OTION CONTRACTS? By rof. Ashok anekar An oion conrac is a derivaive which gives he righ o he holder of he conrac o do 'Somehing' bu wihou he obligaion o do ha 'Somehing'. The 'Somehing' can be
More informationOptimal Life Insurance Purchase, Consumption and Investment
Opimal Life Insurance Purchase, Consumpion and Invesmen Jinchun Ye a, Sanley R. Pliska b, a Dep. of Mahemaics, Saisics and Compuer Science, Universiy of Illinois a Chicago, Chicago, IL 667, USA b Dep.
More informationMeasuring macroeconomic volatility Applications to export revenue data, 19702005
FONDATION POUR LES ETUDES ET RERS LE DEVELOPPEMENT INTERNATIONAL Measuring macroeconomic volailiy Applicaions o expor revenue daa, 1970005 by Joël Cariolle Policy brief no. 47 March 01 The FERDI is a
More informationStock Trading with Recurrent Reinforcement Learning (RRL) CS229 Application Project Gabriel Molina, SUID 5055783
Sock raing wih Recurren Reinforcemen Learning (RRL) CS9 Applicaion Projec Gabriel Molina, SUID 555783 I. INRODUCION One relaively new approach o financial raing is o use machine learning algorihms o preic
More informationTowards Optimal Capacity Segmentation with Hybrid Cloud Pricing
Towards Opimal Capaciy Segmenaion wih Hybrid Cloud Pricing Wei Wang, Baochun Li, and Ben Liang Deparmen of Elecrical and Compuer Engineering Universiy of Torono Torono, ON M5S 3G4, Canada weiwang@eecg.orono.edu,
More informationOption Pricing Under Stochastic Interest Rates
I.J. Engineering and Manufacuring, 0,3, 889 ublished Online June 0 in MECS (hp://www.mecspress.ne) DOI: 0.585/ijem.0.03. Available online a hp://www.mecspress.ne/ijem Opion ricing Under Sochasic Ineres
More informationThe Interaction of Guarantees, Surplus Distribution, and Asset Allocation in With Profit Life Insurance Policies
1 The Ineracion of Guaranees, Surplus Disribuion, and Asse Allocaion in Wih Profi Life Insurance Policies Alexander Kling * Insiu für Finanz und Akuarwissenschafen, Helmholzsr. 22, 89081 Ulm, Germany
More informationDistributing Human Resources among Software Development Projects 1
Disribuing Human Resources among Sofware Developmen Proecs Macario Polo, María Dolores Maeos, Mario Piaini and rancisco Ruiz Summary This paper presens a mehod for esimaing he disribuion of human resources
More informationUNDERSTANDING THE DEATH BENEFIT SWITCH OPTION IN UNIVERSAL LIFE POLICIES. Nadine Gatzert
UNDERSTANDING THE DEATH BENEFIT SWITCH OPTION IN UNIVERSAL LIFE POLICIES Nadine Gazer Conac (has changed since iniial submission): Chair for Insurance Managemen Universiy of ErlangenNuremberg Lange Gasse
More informationA Generalized Bivariate OrnsteinUhlenbeck Model for Financial Assets
A Generalized Bivariae OrnseinUhlenbeck Model for Financial Asses Romy Krämer, Mahias Richer Technische Universiä Chemniz, Fakulä für Mahemaik, 917 Chemniz, Germany Absrac In his paper, we sudy mahemaical
More informationInventory Planning with Forecast Updates: Approximate Solutions and Cost Error Bounds
OPERATIONS RESEARCH Vol. 54, No. 6, November December 2006, pp. 1079 1097 issn 0030364X eissn 15265463 06 5406 1079 informs doi 10.1287/opre.1060.0338 2006 INFORMS Invenory Planning wih Forecas Updaes:
More informationMarket Liquidity and the Impacts of the Computerized Trading System: Evidence from the Stock Exchange of Thailand
36 Invesmen Managemen and Financial Innovaions, 4/4 Marke Liquidiy and he Impacs of he Compuerized Trading Sysem: Evidence from he Sock Exchange of Thailand Sorasar Sukcharoensin 1, Pariyada Srisopisawa,
More informationLife insurance cash flows with policyholder behaviour
Life insurance cash flows wih policyholder behaviour Krisian Buchard,,1 & Thomas Møller, Deparmen of Mahemaical Sciences, Universiy of Copenhagen Universiesparken 5, DK2100 Copenhagen Ø, Denmark PFA Pension,
More informationMultiprocessor SystemsonChips
Par of: Muliprocessor SysemsonChips Edied by: Ahmed Amine Jerraya and Wayne Wolf Morgan Kaufmann Publishers, 2005 2 Modeling Shared Resources Conex swiching implies overhead. On a processing elemen,
More informationThe option pricing framework
Chaper 2 The opion pricing framework The opion markes based on swap raes or he LIBOR have become he larges fixed income markes, and caps (floors) and swapions are he mos imporan derivaives wihin hese markes.
More informationWorking Paper On the timing option in a futures contract. SSE/EFI Working Paper Series in Economics and Finance, No. 619
econsor www.econsor.eu Der OpenAccessPublikaionsserver der ZBW LeibnizInformaionszenrum Wirschaf The Open Access Publicaion Server of he ZBW Leibniz Informaion Cenre for Economics Biagini, Francesca;
More informationTo Sponsor or Not to Sponsor: Sponsored Search Auctions with Organic Links and Firm Dependent ClickThrough Rates
To Sponsor or No o Sponsor: Sponsored Search Aucions wih Organic Links and Firm Dependen ClickThrough Raes Michael Arnold, Eric Darmon and Thierry Penard June 5, 00 Draf: Preliminary and Incomplee Absrac
More informationDesigning Optimal Disability Insurance: A Case for Asset Testing
Designing Opimal Disabiliy Insurance: A Case for Asse Tesing Mikhail Golosov Massachuses Insiue of Technology and Naional Bureau of Economic Research Aleh Tsyvinski Harvard Universiy and Naional Bureau
More informationAsymmetric Information, Debt Capacity, And Capital Structure *
Asymmeric Informaion, Deb Capaciy, And Capial Srucure * Michael. emmon Universiy of Uah Jaime F. Zender Universiy of Colorado a Boulder Curren Draf: July 20, 2011 Very Preliminary and Incomplee Do No Quoe
More informationThe Impact of Surplus Distribution on the Risk Exposure of With Profit Life Insurance Policies Including Interest Rate Guarantees.
The Impac of Surplus Disribuion on he Risk Exposure of Wih Profi Life Insurance Policies Including Ineres Rae Guaranees Alexander Kling 1 Insiu für Finanz und Akuarwissenschafen, Helmholzsraße 22, 89081
More informationTrading on ShortTerm Information
Trading on ShorTerm Informaion Preliminary version. Commens welcome Alexander Gümbel * Deparmen of Economics European Universiy Insiue Badia Fiesolana 5006 San Domenico di Fiesole (FI) Ialy email: guembel@daacomm.iue.i
More informationDynamic programming models and algorithms for the mutual fund cash balance problem
Submied o Managemen Science manuscrip Dynamic programming models and algorihms for he muual fund cash balance problem Juliana Nascimeno Deparmen of Operaions Research and Financial Engineering, Princeon
More informationGAMING PERFORMANCE FEES BY PORTFOLIO MANAGERS* Dean P. Foster and H. Peyton Young
GAMING PERFORMANCE FEES BY PORTFOLIO MANAGERS* Dean P. Foser and H. Peyon Young We show ha i is very difficul o devise performance based compensaion conracs ha reward porfolio managers who generae excess
More informationCredit Index Options: the noarmageddon pricing measure and the role of correlation after the subprime crisis
Second Conference on The Mahemaics of Credi Risk, Princeon May 2324, 2008 Credi Index Opions: he noarmageddon pricing measure and he role of correlaion afer he subprime crisis Damiano Brigo  Join work
More informationHedging with Forwards and Futures
Hedging wih orwards and uures Hedging in mos cases is sraighforward. You plan o buy 10,000 barrels of oil in six monhs and you wish o eliminae he price risk. If you ake he buyside of a forward/fuures
More informationSEMIMARTINGALE STOCHASTIC APPROXIMATION PROCEDURE AND RECURSIVE ESTIMATION. Chavchavadze Ave. 17 a, Tbilisi, Georgia, Email: toronj333@yahoo.
SEMIMARTINGALE STOCHASTIC APPROXIMATION PROCEDURE AND RECURSIVE ESTIMATION N. LAZRIEVA, 2, T. SHARIA 3, 2 AND T. TORONJADZE Georgian American Universiy, Business School, 3, Alleyway II, Chavchavadze Ave.
More informationUnemployment Insurance Fraud and Optimal Monitoring
Unemploymen Insurance Fraud and Opimal Monioring David L. Fuller, B. Ravikumar, and Yuzhe Zhang June 2014 Absrac An imporan incenive problem for he design of unemploymen insurance is he fraudulen collecion
More informationIndividual Health Insurance April 30, 2008 Pages 167170
Individual Healh Insurance April 30, 2008 Pages 167170 We have received feedback ha his secion of he e is confusing because some of he defined noaion is inconsisen wih comparable life insurance reserve
More informationThe Impact of Surplus Distribution on the Risk Exposure of With Profit Life Insurance Policies Including Interest Rate Guarantees
1 The Impac of Surplus Disribuion on he Risk Exposure of Wih Profi Life Insurance Policies Including Ineres Rae Guaranees Alexander Kling Insiu für Finanz und Akuarwissenschafen, Helmholzsraße 22, 89081
More informationA general decomposition formula for derivative prices in stochastic volatility models
A general decomposiion formula for derivaive prices in sochasic volailiy models Elisa Alòs Universia Pompeu Fabra C/ Ramón rias Fargas, 57 85 Barcelona Absrac We see ha he price of an european call opion
More informationThe Application of Multi Shifts and Break Windows in Employees Scheduling
The Applicaion of Muli Shifs and Brea Windows in Employees Scheduling Evy Herowai Indusrial Engineering Deparmen, Universiy of Surabaya, Indonesia Absrac. One mehod for increasing company s performance
More informationLEASING VERSUSBUYING
LEASNG VERSUSBUYNG Conribued by James D. Blum and LeRoy D. Brooks Assisan Professors of Business Adminisraion Deparmen of Business Adminisraion Universiy of Delaware Newark, Delaware The auhors discuss
More informationMarkov Chain Modeling of Policy Holder Behavior in Life Insurance and Pension
Markov Chain Modeling of Policy Holder Behavior in Life Insurance and Pension Lars Frederik Brand Henriksen 1, Jeppe Woemann Nielsen 2, Mogens Seffensen 1, and Chrisian Svensson 2 1 Deparmen of Mahemaical
More information