risks Aricle Maximum Marke Price of Longeviy Risk under Solvency Regimes: The Case of Solvency II Susanna Levanesi 1, *, and Massimiliano Menziei 2, 1 Deparmen of Saisics, Sapienza Universiy of Rome, Viale Regina Elena, 295/G, 00161 Rome, Ialy 2 Deparmen of Economics, Saisics and Finance, Universiy of Calabria, Via P. Bucci, 87036 Rende CS), Ialy; massimiliano.menziei@unical.i * Correspondence: susanna.levanesi@uniroma1.i; Tel.: +39-06-4925-5303 These auhors conribued equally o his work. Academic Edior: Mogens Seffensen Received: 25 February 2017; Acceped: 4 May 2017; Published: 10 May 2017 Absrac: Longeviy risk consiues an imporan risk facor for life insurance companies, and i can be managed hrough longeviy-linked securiies. The marke of longeviy-linked securiies is a presen far from being complee and does no allow finding a unique pricing measure. We propose a mehod o esimae he maximum marke price of longeviy risk depending on he risk margin implici wihin he calculaion of he echnical provisions as defined by Solvency II. The maximum price of longeviy risk is deermined for a survivor forward S-forward), an agreemen beween wo counerparies o exchange a mauriy a fixed survival-dependen paymen for a paymen depending on he realized survival of a given cohor of individuals. The maximum prices deermined for he S-forwards can be used o price oher longeviy-linked securiies, such as q-forwards. The Cairns Blake Dowd model is used o represen he evoluion of moraliy over ime ha combined wih he informaion on he risk margin, enables us o calculae upper limis for he risk-adjused survival probabiliies, he marke price of longeviy risk and he S-forward prices. Numerical resuls can be exended for he pricing of oher longeviy-linked securiies. Keywords: longeviy risk; S-forwards; pricing; risk margin; Solvency II 1. Inroducion Currenly, one of he main risks affecing pension funds and insurance companies selling annuiy producs is longeviy risk, i.e., he risk o experience unexpeced improvemens in lifeime duraion. Longeviy risk implicaions are represened by he exension of he average period of he annuiy paymen, corresponding o unexpecedly higher acuarial liabiliies. In he echnical specificaions released by EIOPA 2014a) Longeviy risk is associaed wih he risk of loss, or of adverse change in he value of insurance liabiliies, resuling from changes in he level, rend, or volailiy of moraliy raes, where a decrease in he moraliy rae leads o an increase in he value of insurance liabiliies. According o he repor on fifh Quaniaive Impac Sudy EIOPA 2011), in he Solvency II life underwriing risk module, longeviy risk is ogeher wih he lapse risk) he main risk for life insurance companies. Longeviy risk can be hedged in several ways. One of he mos appealing soluions is o ransfer longeviy risk o capial markes hrough he so-called longeviy-linked securiies, financial conracs ha have payoffs linked o he level of a survivor index of a reference populaion. Over he pas few years, considerable aenion was paid o longeviy risk securiizaion on behalf of major invesmen banks, pension funds and reinsurers. Several longeviy-linked securiies have been proposed in he lieraure: e.g., longeviy bonds, q -forwards, survivor swaps and longeviy caps and floors Risks 2017, 5, 29; doi:10.3390/risks5020029 www.mdpi.com/journal/risks
Risks 2017, 5, 29 2 of 21 see, e.g., Blake e al. 2006) for a deailed descripion), bu only some of hem were issued in he marke. One of he main obsacles ha slows down he developmen of a worldwide longeviy marke is deermining he price of hese insrumens. The longeviy marke is currenly incomplee and makes i impossible o esimae a unique marke price of longeviy risk, i.e., he premium ha a life insurer or pension plan migh be willing o pay o release such a risk. Many auhors have addressed he quesion of how o price longeviy-linked securiies, bu he calibraion of he marke price of longeviy risk remains an open quesion see Bauer e al. 2010) for a review and comparison of differen pricing approaches). Some auhors explored he possibiliy o link he price of longeviy-linked securiies o he amoun ha he insurer should hold o cover unexpeced losses. In paricular, Börger 2010) assers ha he risk margin under Solvency II can be considered as he maximum price ha a life insurer would be willing o pay o ransfer longeviy risk via securiizaion. In he same conex, Sevens e al. 2009) deermine he maximum price of a porfolio of survivor swaps as he difference beween he buffer asse value in excess of he bes esimae liabiliies) sufficien o cover all fuure liabiliies wih a cerain probabiliy wih and wihou a survivor swap. The work in Meyricke and Sherris 2014) analyzes he rade off beween he coss and benefis of hedging longeviy risk, where he benefis arise from reducion in he Solvency Capial Requiremen ha would oherwise have o be held agains longeviy risk under Solvency II. Under his approach, we propose a mehod o esimae he maximum price of longeviy risk from he risk margin for longeviy implici in he marke-consisen evaluaion of echnical provisions under Solvency II legislaion. According o Solvency II, insurance liabiliies ha canno be hedged, e.g., he ones deriving from longeviy risk, are calculaed as he sum of a bes esimae plus a risk margin RM), which is he marke value of he uncerainy on insurance obligaions. The RM is deermined by he cos of capial approach based on fuure solvency capial requiremens SCR). The overall SCR is hen calculaed by aggregaing capial requiremens from individual risk modules longeviy risk and oher risk facors), using he relevan correlaion marices. As poined ou by Börger 2010), a shorcoming of he approach o esimaing a marke price of longeviy risk is ha he maximum price an insurer is willing o pay may be differen due o diversificaion effecs wih oher risks han longeviy. Diversificaion can be achieved by naural hedging, i.e., by combining producs ha have negaive correlaion, in order o lower he insurer s overall risk exposure. However, he insurer has is own specific porfolio and ailored naural hedging sraegies ha would clearly differ from insurer o insurer. Consequenly, since he aim of he paper is o find he maximum price of longeviy risk for a generic insurer, naural hedging beween longeviy and moraliy as well as oher risk diversificaion sraegies) canno be included in he model. If he capial associaed wih longeviy, risk was evaluaed in ligh of he capial allocaion of he oal diversified SCR; he view on he maximum price would change. Alhough i is generally correc o ake accoun of diversificaion effecs, he paper focuses on he siuaion of an insurer ha wishes o hedge a ne longeviy posiion considering he SCR for he sand-alone longeviy risk, ne of diversificaion effecs accordingly. The longeviy risk can be miigaed and he SCR for longeviy risk reduced, if he insurer eners ino a longeviy-linked securiy. If an insurer is compleely hedged from longeviy risk afer enering a longeviy-linked securiy, SCR for longeviy risk reduces o zero, and so does he corresponding RM. Therefore, he risk margin measures he benefi o he insurer from longeviy risk hedging obained hrough securiizaion. In oher words, he insurer is ineresed in securiizaion if is cos is less han he saving of RM ha he/she could obain from is use; oherwise, he/she gives up o cover he longeviy risk and holds he regulaory capial necessary o address i. Therefore, he RM for longeviy risk can be considered an esimaion of he maximum price ha he insurer would be willing o pay o ransfer longeviy risk. Noe ha we provide esimaes of he maximum accepable price of longeviy risk, wihou aking ino accoun he minimum price. From he insurer s poin of view, on which he paper focuses, here is no minimum price of longeviy risk below which he/she would no ransfer he risk o he
Risks 2017, 5, 29 3 of 21 capial markes. The minimum price can only be deermined from he poin of view of he longeviy insurance seller, bu if his eniy is no an insurance/reinsurance underaking and herefore no regulaed by Solvency II, a mehodological framework based on he RM is no usable o deermine he minimum price. When available, upper and lower bounds for he risk premium specify a price range for negoiaion beween sellers and buyers Cui 2008) and any price wihin his range can be an accepable ransacion price Barrieu and Loubergé 2013). The approach we propose could also be applied o insurers subjec o oher solvency regimes, e.g., o hose subjec o he Swiss Solvency Tes ha provides a marke value margin calculaed wih he cos of capial approach similarly o Solvency II) ha would be reduced o zero in he case of perfec hedging. We will no deal wih his aspec here. For furher informaion on he Swiss Solvency Tes, he reader can refer o he Swiss Federal Office of Privae Insurance SWISS 2006) and Eling e al. 2008). However, longeviy-linked securiies are affeced by he counerpary defaul risk ha can increase heir coss and implies a specific capial charge for he insurer enering in he ransacion see Biffis e al. 2016) for an analysis of counerpary defaul risk in longeviy swaps). The calculaion of he risk margin for counerpary defaul risk in longeviy-linked securiies is beyond he scope of his paper. If he counerpary defaul risk were inroduced, i would reduce he SCR savings achieved by he longeviy risk hedging and would increase he cos of hedging. If such risk is aken ino accoun, he maximum price should be less. Exising lieraure see, e.g., Li and Hardy 2011)) shows ha he longeviy hedge is affeced by basis risk, he risk caused by he difference beween he reference populaion of he hedging insrumen and he insured populaion. The presence of basis risk decreases he hedge effeciveness of he hedging insrumen, and a larger solvency capial requiremen is necessary o cover fuure liabiliies. However, in his paper, we do no ake ino accoun basis risk, as we assume ha he insurer s reference populaion is he same as he longeviy-linked insrumens. To deermine he maximum price of longeviy risk, we consider a survivor forward S-forward), an agreemen beween wo counerparies o exchange a mauriy a fixed survival-dependen paymen for a paymen depending on he realized survival of a given cohor of individuals. An insurer wishing o hedge longeviy risk may ener ino a porfolio of S-forwards aking a long posiion, i.e., assuming an obligaion o pay fixed paymens in reurn for variable paymens. The S-forward premium is measured as he difference beween he fixed and he expeced value of fuure cash flows and can be represened by a spread. We propose a mehod o deermine he maximum spread ha he longeviy risk hedger is willing o pay o cover unexpeced losses. I is calculaed such ha, a incepion, he RM of a given insurance conrac hedged by he S-forward is equal o he RM of he conrac wihou longeviy hedging, so ha he wo soluions are indifferen. We conduc he analysis by considering hree differen calculaion mehods of he SCR allowed under he Solvency II projec: he sandard formula, he value a risk VaR) approach and he sressed-rend approach, so ha he assessmen is consisen wih he measuremen of he SCR and RM chosen by he insurance company. Moraliy dynamics are modeled via a sochasic projecion model o ake ino accoun he uncerainy in he fuure evoluion of moraliy. To find he model ha bes fis Ialian daa, we compare he performances of seven differen models. The Cairns Blake Dowd model Cairns e al. 2006a) wih a cohor effec is he model ha bes fis he daa. Our resuls show ha he maximum spread increases wih he saring age of he S-forward underlying cohor, as well as wih he mauriy, due o he increasing uncerainy of survival probabiliies for older ages and longer ime. For example, under he sandard formula approach, he maximum risk premium acceped by insurers for an S-forward wrien on a cohor of Ialian males aged 65 a he issue of he annuiy is 0.3% for five-year mauriy and 15.4% for a 20-year mauriy. We esimae he maximum price of longeviy risk according o a risk-neural pricing framework ha includes a risk adjusmen ino he real-world probabiliy disribuion. In he risk-neural valuaion, he moraliy dynamics is specified under a risk-adjused measure ha is equivalen o he real-world measure, and he expeced cash flows can be discouned o he risk-free rae. Once having defined
Risks 2017, 5, 29 4 of 21 he moraliy model under he real-world probabiliy measure, we use he RM approach o calculae he maximum value of he risk-neural survival probabiliies and find he maximum marke price of longeviy risk minimizing he squared errors beween his value and he real-world survival probabiliies. In he same pricing framework, he upper limi of he longeviy risk marke price can be used o price oher and more complex longeviy-linked securiies, such as survivor swaps and q-forwards. To deermine he longeviy risk premium, Milevsky e al. 2005) proposed he Sharpe raio approach based on boh expeced value and he volailiy of paymens. Under his approach, he expeced reurn on he longeviy-linked securiies is equal o he risk-free rae plus he Sharpe raio muliplied hree imes is sandard deviaion. The work in Milevsky e al. 2006) saed ha for he longeviy marke, he Sharpe raio can be se equal o ha from he S&P500 index, i.e., 0.25. In his conex, we firsly deermine he maximum price of a se of q-forwards hrough our model based on he RM approach and hen esimae he implied maximum Sharpe raio ha can be considered accepable by an annuiy provider for hedging longeviy risk. Alhough he use of RM o price he longeviy-linked securiies has been suggesed in he lieraure, he mehod we propose is original, as i provides an exac measuremen of he maximum price of hese hedging insrumens, allowing insurers o assess heir acual convenience o longeviy risk securiizaion and hus promoing he longeviy marke developmen. The paper is organized as follows. Secion 2 describes he feaures of S-forwards. In Secion 3, we presen he mehod o deermine he maximum price of an S-forward conrac from he RM in he Solvency II framework, as well as he procedure o esimae he corresponding marke price of longeviy risk. Secion 4 describes he moraliy model adoped in he paper and provides he resuls of he model fiing applied o he Ialian male populaion. The risk-neural version of he model necessary for pricing S-forwards is also shown. Secion 5 gives numerical resuls for he maximum price of S-forwards wrien on he Ialian populaion for differen cohors and mauriies and he underlying marke price of longeviy risk. Conclusions are given in Secion 6. 2. S-Forwards: Definiion and Noaion Consider he siuaion of an insurer wih a porfolio of pure endowmens paying a lump sum a mauriy T, if he insured is sill alive. We assume he same sum insured for all of he policyholders. In his regard, where he sums insured are differen, heir heerogeneiy may affec he riskiness of he insurance porfolio. Acually, a sronger dispersion in he insured amouns leads o a higher capial requiremen due o random flucuaions in moraliy. Oherwise, as poined ou by Piacco e al. 2009) see page 317 for furher deails), longeviy risk is independen of he heerogeneiy of he porfolio. Therefore, as our paper focuses on longeviy risk, we do no consider he effec of he heerogeneiy of he insured amouns on he porfolio s riskiness. Suppose ha he/she decides o ransfer he longeviy risk arising from his porfolio o he capial marke hrough a se of S-forwards, as a saic hedge. An S-forward is an agreemen beween wo counerparies o exchange a a fixed mauriy T an amoun equal o he realized survival probabiliy of a given populaion cohor floaing rae paymen), in reurn for a fixed survival probabiliy agreed a he incepion of he conrac fixed rae paymen). Among he longeviy-linked securiies we have chosen he S-forwards for several reasons: hey are cerainly very promising insrumens hanks o heir lower ransacions coss, high flexibiliy and easiness o be canceled, and hey require only he will of he counerparies o ransfer heir deah exposure wihou he need of a liquid marke. Furher, hey involve lower basis risk compared wih oher derivaives. As saed in he Inroducion, we assume ha he insurer s reference populaion is he same as he S-forward s one.
Risks 2017, 5, 29 5 of 21 Consider a cohor of idenical policyholders, l x,0, all aged x a iniial Time 0. For easiness of represenaion, lump sums are fixed o one moneary uni m.u.). A mauriy T, he number of survivors o age x + T boh realized and expeced) is: l x+t,t = l x,0 T p x,0 realized) 1a) ˆl x+t,t = E P l x+t,t ) = l x,0 T ˆp x,0 expeced) 1b) where T p x,0 is he survival rae of an individual aged x a Time 0 o be alive a age x + T, E P is he expecaion operaor associaed wih real world probabiliy measure P and T ˆp x,0 = E P T p x,0 ). If he difference beween he realized and he expeced survivor a ime T is posiive, i.e., l x+t,t ˆl x+t,t > 0, he insurer experiences a porfolio loss a ime T. Le 1 + π)ˆl x+t,t be he fixed leg of he S-forward a ime T, where π is he fixed proporional risk premium required by invesors o ake on longeviy risk and l x+t,t he floaing leg. A paymen dae T, he cash-flow is equal o he difference beween he fixed leg and he floaing leg Figure 1). Figure 1. S-forward cash flows. As observed in he Inroducion, he pricing of longeviy-linked securiies is far from being rivial. The exising lieraure on his opic is quie exensive, and here is a lively debae concerning he performance of differen approaches for pricing such securiies. Acually, here are a leas hree pricing approaches for longeviy-linked securiies: he Wang ransform, he insananeous Sharpe raio and he risk-neural approach. The firs one is he disorion approach based on he Wang ransform Wang 2002) providing a disorion operaor ha ransforms he deah probabiliy disribuion o generae risk-adjused deah probabiliies discouned by risk-free ineres raes e.g., see Lin and Cox 2005)). The Wang ransform has been criicized by many auhors see, e.g., Siu-Hang Li and Cheuk-Yin Ng 2011); Bauer e al. 2010)) because i is no a linear funcional and, herefore, yields arbirage opporuniies ha do no lead o a price consisen wih he arbirage-free price. Moreover, as poined ou by Cairns e al. 2006b), i is no clear how differen ransforms for differen cohors and erms o mauriy relae o one anoher and form a coheren whole. All of he aforemenioned auhors conclude ha he Wang disorion approach does no provide a universal framework for pricing longeviy-linked securiies and is no a suiable choice in a framework wih muliple cohors. As briefly described in he Inroducion, he second approach applies he Sharpe raio rule assuming ha he risk premium required by invesors o ake on longeviy risk is equal o he Sharpe raio for oher undiversifiable financial insrumens Milevsky e al. 2005). This approach corresponds o a change of he probabiliy measure from he real-world measure o a risk-adjused measure provided by a consan marke price of risk Bauer e al. 2010). As menioned in he Inroducion, Milevsky e al. 2006) propose for he longeviy marke a 0.25 level of he Sharpe raio according o sock marke daa. However, Bauer e al. 2010) saed ha empirical sudies show ha he risk premium for socks is considerably higher han for oher securiies. The Sharpe raio can be alernaively calibraed o a suiable annuiy quoe, bu again, he quesion relies on he adequacy of he annuiy marke o price longeviy-linked securiies. In his regard, i should be kep in mind ha he annuiy provider, in addiion o longeviy risk, is also subjec o oher sources of risk, and he resuling risk premium should be greaer han he risk premium for a longeviy-linked securiy. The hird approach consiss of adaping he risk-neural pricing framework developed for ineres-rae derivaives o longeviy-linked securiies for example, see Dahl 2004); Cairns e al. 2006a);
Risks 2017, 5, 29 6 of 21 Biffis e al. 2010)). As poined ou by Barrieau e al. Barrieu and Loubergé 2013) he main underlying assumpion of his approach is ha i is possible o replicae cash flows of a given ransacion dynamically using basic raded securiies in a highly liquid marke. Therefore, he applicaion of a risk-neural approach requires a highly liquid underlying marke ha is necessary o build a replicaing sraegy. Neverheless, he longeviy marke a presen is far from being liquid, and he specificaion of he risk-neural measure is difficul due o he limied daa on he marke price of longeviy risk. Our model allows esimaing he maximum marke price of longeviy risk from he informaion enclosed in he RM calculaed wih he Solvency II sandard formula. Therefore, i allows applying he risk-neural approach ha can be generally considered he mos adequae mehodology because i is suied for muliple cohors and ages; unlike he Sharpe raio approach, i is able o capure he dependence beween differen cohors Meyricke and Sherris 2014) and, unlike, he Wang ransform, provides a uniquely-defined price and hen a universal framework for pricing longeviy-linked securiies. We define a risk-neural probabiliy measure Q ha is equivalen o he curren real-world measure P, and we assume ha he ineres rae is independen of he moraliy rae. We se π in such a way ha he S-forward value is zero a he incepion dae: we are assuming ha he marke values of he fixed and floaing legs are equal. Therefore, he following condiion holds: S-forward value = [ ] E Q l x+t,t ) 1 + π) E P l x+t,t ) d0, T) = 0 2) where d0, T) is he price a Time 0 of a zero-coupon bond paying a fixed amoun of one a ime T. From Equaions 1) and 2), we can find he fixed proporional S-forward premium, π, as follows: π = E Q T p x,0 ) T ˆp x,0 1. 3) Noe ha 1 + π) can be wrien as e δt, where δ is he spread required by he floaing leg payer expressed as a coninuously compounding rae. Hence, δ is given by: 3. Pricing S-Forwards via he Risk Margin [ ] EQ T p x,0 ) 1 δ = ln T. 4) T ˆp x,0 Under he Solvency II projec, he marke value of he liabiliies relaed o unhedgeable risks is se equal o he sum of is bes esimae BE) and a risk margin RM), represening a risk adjusmen of he BE. The RM is hen deermined by a cos-of-capial CoC) approach, i.e., by he cos of providing an amoun of capial equal o he SCR necessary o fulfill he insurance obligaions over ime. In he definiion proposed by Solvency II, he SCR a Time 0 is he capial required o cover, wih 99.5% probabiliy, he unexpeced losses on a one-year ime horizon. The CoC rae represens he average spread over he risk-free rae, which he marke requires as earning on insurance companies equiy. As observed by Zhou and Li 2013), he profi margin, which an insurance company aking on risk mus include in is insurance premiums, should ypically be no less han he risk margin. However, an exac compuaion of he RM would require he deerminaion of SCR, condiional on he moraliy evoluion up o ime see Chrisiansen and Niemeyer 2014) for he SCR definiions for fuure poins in ime). For pracical purposes, he compuaion of SCR fuure values is oo complex, so ha simplificaions are needed Börger 2010). In he EIOPA Guidelines on he valuaion of echnical provisions EIOPA 2014b), some approximaions are lised see Guidelines 62). In he following, we will consider he approximaion of he SCR by assuming a moraliy evoluion up o ime according o is bes esimae. In he formula, he RM a ime is defined as:
Risks 2017, 5, 29 7 of 21 T 1 RM = 6% i= SCR i d, i + 1), 5) where he CoC rae is se o 6% as fixed in he curren Solvency II sandard formula, he SCR i is he SCR for year i and d, i + 1) is he discoun facor calculaed wih he risk-free ineres rae erm srucure. Solvency II provides several mehods o calculae he SCR: full inernal model, sandard formula and parial inernal model, sandard formula wih underaking-specific parameers, sandard formula simplificaion. In he following, we will consider he sandard formula, as well as wo parial inernal models. The esimaes derived from he parial inernal models are compaible wih he overall calibraion objecives for he sandard formula i.e., a value a risk wih a 99.5% confidence level over a one-year ime horizon) as required by Solvency II. a) SCR under he sandard formula: The Solvency II projec suggess a sandard formula o compue he above-defined SCR. If we jus consider he SCR relaed o longeviy risk and assume no oher risk exiss, he SCR required a for an insurance company wih a porfolio of pure endowmens is equal o: SCR shock = BOF longeviy shock = V shock ˆV, 6) b) where BOF longeviy shock = BOF BOF shock denoes he change in he value of is basic own funds a ime due o a longeviy shock; V shock is he value of he echnical provisions afer experiencing a longeviy shock; and ˆV is he BE of he echnical provisions. The longeviy shock in he sandard formula is a permanen reducion by 20% of he BE of he moraliy raes a each age EIOPA 2014a). SCR under he VaR measure: Insead of he sandard formula ha ses a deerminisic longeviy shock, a more appropriae model o capure he insurer risk can be based on he calculaion of BOF in a sochasic seing as a funcion of value a risk VaR) of liabiliies given a fixed confidence level. In general form, he VaR measures he poenial loss in value of a risky asse or liabiliy over a defined ime horizon for a given confidence level, say 99.5% i.e., wih a 0.5% ruin probabiliy). The SCR VaR can be defined as: SCR VaR = BOF longeviy VaR = V VaR ˆV, 7) c) where V VaR = VaR 99.5% V ) is he quaniles a he 99.5% confidence level of echnical provisions. SCR under he sressed-rend approach: As longeviy risk arises from he uncerainy in changes of he long-erm moraliy rend, Richards e al. 2013) argue ha a sressed-rend approach is he mos appropriae way o invesigae longeviy rend risk. This rend is he resul of an accumulaion of small changes over many years; herefore, echnical provisions should be calculaed via a long-erm sress projecion from a sochasic moraliy model, under a sressed-rend approach. There has been pleny of feedback from QIS4 and QIS5 paricipans on he adequacy of he one-off shock approach for longeviy risk recommended by he sandard formula. As he curren shock was only a shock on he level, i failed o adequaely ake ino accoun rend risk: underakings fel a sress on he fuure improvemen raes would be more appropriae EIOPA 2011). We define he SCR under he sressed-rend approach, SCR sress, as: SCR sress = BOF longeviy sressed-rend = V sress ˆV, 8)
Risks 2017, 5, 29 8 of 21 where V sress is he echnical provision calculaed by he sressed-rend moraliy projecion obained by deducing he Φ 1 99.5%) projecion sandard error from he cenral projecion, where Φ is he cumulaive disribuion funcion of a normal 0,1) see Richards e al. 2013) for furher deails). The projecion sandard error here encloses parameer risk and no process risk, i.e., he risk arising from he variabiliy of he ime processes used o represen he biomeric rends. I is usually denoed as he risk of random flucuaions and is a diversifiable risk, because i decreases, in relaive erms, by increasing he porfolio size. Observe ha, while he sandard formula is based on a deerminisic approach, he compuaion of he SCR for longeviy risk via he VaR approach and he sressed-rend approach requires a sochasic moraliy model. Under a given approach a, b or c) and assuming a moraliy evoluion up o ime according o is bes esimae, he SCR for a porfolio of pure endowmens is given by: SCR = V ˆV = l x,0 ˆp x,0 T p x, T ˆp x,0 ) d, T). 9) where T ˆp x,0 is he BE of survival probabiliies and T p x, are he survival probabiliies a ime calculaed according o a defined approach a, b or c): T p x, = T p shock x, T p VaR x, T p sress x, where T p VaR x, are he quaniles a he 99.5% confidence level of he sochasic survival probabiliies ha an individual alive a a age x + will be alive a T a age x + T and T p sress x, are he sressed-rend survival probabiliies. Assuming ha an insurer is compleely hedged agains longeviy risk, by assumpion he only risk source, here is no need for he company o se any solvency capial aside. Boh he SCR and he RM reduce o zero. Reasonably, we can assume ha an insurer migh be ineresed in securiizing his/her longeviy risk if he ransacion price is lower or equal o he presen value of he fuure CoC required upon longeviy risk. Following his poin of view, he RM can be considered as he maximum price ha an insurance company would pay for longeviy risk securiizaion see Börger 2010)). As argued in he Inroducion, if we ake ino accoun he counerpary defaul risk, we should include he RM for such a risk. In his case, he maximum price should be less. Consider an insurance company enering an S-forward wih mauriy T. The SCR a ime afer he hedging is equal o: a) b) c) SCR S = l x,0 l S x,0 ) ˆp x,0 T p x, T ˆp x, ) d, T), 11) where l x,0 and lx,0 S are he survivors in he porfolio of pure endowmens and hose underlying he S-forward, respecively. Noe ha if hese wo quaniies coincide, l x,0 = lx,0 S, hen SCRS = 0. Subsiuing Equaions 9) and 11) in Equaion 5), we derive he following values of he RM in = 0 before and afer he hedging, respecively: 10) T 1 RM 0 = 6% i=0 SCR i d0, i + 1) = 6% l x,0 d0, T) T 1 i ˆp x,0 T i p x,i T i ˆp x,i )di, i + 1), i=0 12)
Risks 2017, 5, 29 9 of 21 RM0 S T 1 = 6% SCRi S d0, i + 1) = 6% l x,0 lx,0 S ) d0, T) i=0 T 1 i ˆp x,0 T i p x,i T i ˆp x,i )di, i + 1). i=0 The difference beween he RM before and afer he longeviy hedging RM 0 = RM 0 RM S 0 ) provides a measure of he amoun of RM saved by he insurer hedging he longeviy risk hrough an S-forward: RM 0 = 6% l S x,0 d0, T) T 1 i=0 13) i ˆp x,0 T i p x,i T i ˆp x,i )di, i + 1). 14) Clearly, when l x,0 = lx,0 S, he RM saved by he insurer exacly maches he RM for longeviy risk see Equaion 12)). We assume ha such savings corresponds o he maximum premium ha an insurer would pay for hedging longeviy risk. This leads o: RM 0 π ˆl x+t,t S d0, T). 15) where ˆl S x+t,t = ls x,0 T ˆp x,0. From Equaions 14) and 15), we can obain he maximum price π max for he S-forward fixed payer or he corresponding spread required by he floaing leg payer, δ max ): π max = e δmaxt 1 = δ max = ln = 6% T 1 i=0 i ˆp x,0 T i p x,i di, i + 1) T ˆp x,0 [ 6% T 1 i=0 i ˆp x,0 T i p x,i di, i + 1) + T ˆp x,0 1 6% T 1 i=0 T ˆp x,0 ) T 1 di, i + 1). i=0 ] di, i + 1)) 16) 1 T. 17) This resul is valid for basic insrumens wih a single mauriy, as an S-forward. A presen, he marke of S-forwards is far from being developed. Therefore, π max or he spread δ max ), calculaed hrough he RM, can be used for pricing hese derivaives. Thus, he hedger should no be ineresed in enering ino a ransacion if he price is higher han ha calculaed wih his approach. Besides, he π max calculaed on differen mauriies can be used o find he maximum price of vanilla survivor swaps. 3.1. The Marke Price of Longeviy Risk Given a sochasic moraliy model wih n facors, he dynamics of he model under he risk-adjused pricing measure Q depends on a vecor of he marke price of longeviy risk λ = λ 1, λ 2,..., λ n ), associaed wih he n facors. Assuming ha π = π max and reminding abou he relaion beween π and he raio beween risk-neural and real-world survival probabiliies see Equaion 3)), we obain a maximum value for E Q T p x,0 ): EQ max T p x,0 ) = 1 + π max T 1 ) T ˆp x,0 = 6% i ˆp x,0 T i p x,i di, i + 1)+ i=0 ] T 1 + T ˆp x,0 [1 6% i=0 di, i + 1). 18) Now, we can find he marke price of longeviy risk λ minimizing he squared errors beween he maximum value of he risk-neural survival probabiliies calculaed via he RM approach, EQ max T p x,0 ), and he risk-neural survival probabiliies depending on he moraliy, model, E Qλ) T p x,0 ), so ha:
Risks 2017, 5, 29 10 of 21 λ = arg min λ m j=1 [ E max Q T j p xj,0 ) E Qλ) T j p xj,0)] 2. 19) where m is he number of S-forwards he model is calibraed o and T j and x j are respecively he mauriy and he age a issue of he reference populaion of he j-h S-forward. Noe ha he values of λ could be used o find he maximum price of more complex longeviy-linked securiies. In he nex secion, we describe he moraliy model we use in he paper. 4. The Moraliy Model We define D x, as he deahs occurring during year a age x, assumed o follow a Poisson disribuion, E x, he exposed-o-risk aged x during year so ha he cenral deah rae a age x and ime are esimaed as m x, = D x, E x,. Le q x, be he probabiliy o die in year for an individual aged x and p x, = 1 q x, he corresponding survival probabiliy. We make he usual assumpion ha he force of moraliy remains consan over each year of ineger age and over each calendar year. This implies ha he relaionship beween q x, and m x, is given by q x, = 1 e m x,. Various sochasic moraliy models have been se forh in recen years. We selec he model among seven sochasic models ha bes fis our daa according o he Bayes informaion crierion BIC), an objecive model selecion crierion based on he saisical qualiy of fi 1.The seven models we compared are aken from he paper of Cairns e al. 2009) and include he Lee Carer model LC), is exension proposed by Renshaw and Haberman RH), he age-period-cohor model inroduced by Currie, he Cairns Blake Dowd model CBD) and hree exensions ha add a cohor effec, a quadraic erm o he age effec and a differen form of he cohor effec, respecively. We fi he seven models o he moraliy daa of he Ialian populaion downloaded from he Human Moraliy Daabase HMD 2010), relaed o he age range 60 90 and he period 1974 2007 for males. Daa sar from year 1974 as he firs year of available ime series on moraliy ables, annually processed by he Ialian Naional Insiue of Saisics ISTAT) according o a uniform mehodology. The fiing resuls are shown in Table 1 where he exension of he Cairns Blake Dowd model including a cohor componen, here indicaed as he CBD-1, is characerized by he highes value of he BIC saisic. In order o ensure compleeness, we included he corresponding maximum log-likelihood MLL) esimaes for all of he analyzed models. The CBD-1 model is described by he following equaion: ) qx, logiq x, ) = ln = k 1) 1 q + k 2) x x) + γ c 3). 20) x, where x is he average of he ages used in he daase, k 1) and k 2) reflec period-relaed effecs and γ c 3) represens he cohor-relaed effec, wih c = x. To avoid any idenifiabiliy problems, wo consrains were inroduced: c C γ c 3) = 0 and c C c γ c 3) = 0, where C is he se of cohor years of birh ha have been included in he analysis 1884 1947). 1 The BIC is defined as: BIC = l ˆρ) 0.5KlnN), where l ˆρ) is he log likelihood funcion, ρ is he se of parameers o be esimaed wih he likelihood funcion, ˆρ is he maximum likelihood esimae of he parameers vecor, N is he number of observaions and K he effecive number of esimaed parameers. The log likelihood funcion of he model is defined as: lρ; D, E) = x, {D x, ln [E x, m x, ρ)] E x, m x, lnd x,!)}.
Risks 2017, 5, 29 11 of 21 Table 1. Maximum log-likelihood esimaes, BIC saisic and rank for each model calculaed on moraliy raes of Ialian males, ages 60 90, years 1974 2007. Model MML BIC Rank MLL) Rank BIC) LC 9823 10149 7 7 RH 6105 6754 1 4 Currie APC 7049 7486 5 5 CBD 9691 9927 6 6 CBD-1 6230 6681 3 1 CBD-2 6122 6688 2 2 CBD-3 6250 6709 4 3 In Figure 2 we show he maximum-likelihood esimaes of k 1), k 2) and γ c 3). We can see a disincive cohor effec in he parameer γ c 3), showing ha cohor moraliy is rising a a higher rae for males born afer 1920. In he esimaion of he cohor effec, we excluded cohors for which here were less han five observaions: cohors birh in 1884 1887 and 1944 1947. a) b) c) Figure 2. Esimaed parameers of he CBD-1 model, Ialian male populaion, ages 60 90, years 1974 2007. a) Parameer k 1) ; b) Parameer k 2) ; c) Parameer γ c 3). 4.1. Moraliy Projecions In order o make forecass of he CBD-1 model, we selec a mulivariae ARIMA model for he hree processes of he esimaed parameers k 1), k 2) and γ c 3). In more general noaion, he mulivariae ARIMA model can be described as: K s+1 = K s + φ K s K s 1 ) + µ + C Z s+1 21) where K s is he vecor of parameers k 1), k 2) and γ 3) a sep s, where s indicaes he ime or he cohor c according o he case; φ is he vecor of parameers of he auoregressive par of he model; µ is a 3 1 vecor of he drifs of he model; C is a 3 3 consan upper riangular marix so ha CC is he covariance marix; and Z is a 3 1 vecor of sandard normal random variables. Balancing beween he bes seleced ARIMA models fiing he parameers 2 and he desire o deal wih a simple model, we chose an ARIMA0,1,0) o describe he firs wo parameers, k 1) and k 2), and an ARIMA1,1,0) for he cohor parameer γ c 3). We assume he exisence of correlaion beween 2 The goodness of fi is assessed hrough boh he maximum log-likelihood funcion and Akaike s informaion crierion AIC = 2l + 2K, where l is he log likelihood funcion and K he effecive number of esimaed parameers.
Risks 2017, 5, 29 12 of 21 parameer k 1) and k 2), while we adop he usual assumpion ha γ c 3) is independen of he ohers. Resuls of he fiing procedure on Ialian daa are repored in Table 2. Table 2. Fied parameers of he ARIMA models. Parameer ARIMA σ 2 µ φ k 1) 0,1,0) 0.000783 0.020394 0 k 2) 0,1,0) 0.000001 0.001015 0 γ 3) 1,1,0) 0.000871 0.002610 0.644393 The fied marix C is as follows: 0.027975 0 0 C = 0.000476 0.001094 0. 0 0 0.029513 Noe ha under he sressed-rend approach, he moraliy model is given by logi sress q x, ) = logi q x, ) σ x, Φ 1 99.5%), where q x, is he moraliy rae from he cenral projecion and σ x, is he volailiy arising from he uncerainy originaing from he ARIMA parameers esimae. 4.2. The Risk-Neural Moraliy Model In he following, we specify he risk-neural approach o he moraliy model described in he previous subsecion, which so far was defined under he real-world probabiliy measure P. For he mulivariae ARIMA model specified by Equaion 21) under he risk-neural measure Q, we propose: K s+1 = K s + φk s K s 1 ) + µ + CZ Q s+1 λ) 22) where he vecor λ = λ 1, λ 2, λ 3 ) is he marke price of longeviy risk associaed wih processes Z 1), Z 2) and Z 3) under he measure Q, respecively. Following Cairns e al. 2006a), in he absence of complee marke price daa, we assume ha λ is consan over ime. Vecor λ should be calibraed on he marke price of longeviy-linked securiies when available. The marke of longeviy-linked securiies being incomplee, i is no always possible o find real marke daa. Following he approach proposed in Secion 3, we find he marke price of longeviy risk, λ = λ 1, λ 2, λ 3 ) according o Equaion 19). To solve he minimizaion problem in such an equaion, we have used he opim) funcion in he saisical compuing sofware R R Core Team 2015)). The marke price of longeviy risk affecs he logi of deah probabiliies under he risk-neural measure in he following way see Figure 3): λ 1 : posiive values produce a downward shif and a lower slope of he curve he laer effec is due o he presence of a posiive correlaion beween parameers k 1 and k 2 ); λ 2 : posiive values produce a lower slope of he curve and a roaion around he poin x, logiq x )); λ 3 : if we consider cohor daa, posiive values produce a downward shif of he curve only for fuure cohors, while in he case of cross-secional daa, posiive values increase he slope of he curve for fuure cohors, while he slope does no change for hisorical cohors.
Risks 2017, 5, 29 13 of 21 a) b) c) Figure 3. Logi of projeced deah probabiliies year 2023) under he real-world probabiliy measure and risk-neural probabiliy measure wih differen λ values. a) λ = 1, 0, 0); b) λ = 0, 1, 0); c) λ = 0, 0, 1). 5. Numerical Resuls In his secion, we show he resuls of he numerical applicaion of our model for he Ialian populaion. Tables 3 5 show he values of δ max evaluaed in he year 2008 for he cohors of individuals aged 65, 70, 75, 80, 85 and for differen mauriies T, according o he hree specificaions of he SCR. Table 3. Values of δ max for differen cohors and mauriies. Sandard formula approach. Age: 65 70 75 80 85 T δ max : sandard formula approach a) 5 0.000533 0.000903 0.001652 0.003176 0.00568 10 0.001306 0.002337 0.004465 0.008911-15 0.003161 0.005533 0.010492 - - 20 0.006731 0.011762 - - - 25 0.013637 - - - - Table 4. Values of δ max for differen cohors and mauriies. VaR approach. Age: 65 70 75 80 85 T δ max : VaR approach b) 5 0.000261 0.000479 0.000984 0.002044 0.003711 10 0.000969 0.001857 0.003860 0.007883-15 0.003080 0.005889 0.011727 - - 20 0.008129 0.014827 - - - 25 0.018580 - - - -
Risks 2017, 5, 29 14 of 21 Table 5. Values of δ max for differen cohors and mauriies. Sressed-rend approach. Age: 65 70 75 80 85 T δ max : Sressed-rend approach c) 5 0.000456 0.000787 0.001475 0.002891 0.005216 10 0.001593 0.002885 0.005607 0.011253-15 0.004670 0.008326 0.015965 - - 20 0.011347 0.020044 - - - 25 0.024735 - - - - Values of δ max do no increase linearly wih he S-forward mauriy see Figure 4 for he VaR approach; plos for oher approaches are similar) as a naural consequence of he increasing uncerainy of survival probabiliies for longer ime horizons. Moreover, δ max increases wih he saring age of he underlying cohor due o he increasing uncerainy of survival probabiliies for older ages. This means ha hedgers are disposed o accep a higher spread when he cohor is older or he mauriy is longer. Figure 4. Values of δ max for he cohors aged 65 85 in 2008 and all of he mauriies. Red lines: cohors aged 65, 70, 75, 80 and 85. To beer undersand he differences in he maximum price δ max among he considered approaches, in Figure 5, we plo he survival probabiliies, T p x,, implied in he calculaion for five cohors aged 65, 70, 75, 80, 85 in year 2008). For example, if we consider he cohor born in 1933, we can observe ha he five-year survival probabiliy under he shock approach is higher han he corresponding survival probabiliy calculaed using VaR and sress-rend approaches. This causes a higher δ max in he shock approach for S-forwards wih a five-year mauriy compared o he oher approaches. The opposie scenario occurs if we look a he 15-year survival probabiliy.
Risks 2017, 5, 29 15 of 21 a) Cohor 1943 b) Cohor 1938 c) Cohor 1933 d) Cohor 1928 e) Cohor 1923 Figure 5. Values of T ˆp x,0, T p shock x,0 and T p VaR x,0 for differen cohors. As described in Secion 3.1, we obain he vecor of he marke price of longeviy risk λ minimizing he squared errors beween he risk-neural survival probabiliies and he corresponding value calculaed via he RM approach see Equaion 19)). The λ vecor is calibraed on he maximum S-forward prices evaluaed in 2008 considering he se of ages 65 85 and he mauriies from 5 25 years. Resuls are given in Table 6.
Risks 2017, 5, 29 16 of 21 Table 6. Marke price of longeviy risk λ. Parameers λ a) Sandard formula b) VaR c) Sressed-rend k 1) 0.421670 0.410675 0.652299 k 2) 0.009506 0.352785 0.294197 γ 3) 0.198082 0.207165 0.501635 The hea map of he relaive percenage error RPE), Emax Q T j p xj,0 EQ max ) E Qλ) T j p xj,0 T j p xj,0 ) ), is ploed in Figure 6. I helps o idenify he difference beween he risk-neural probabiliies by age verical lines) and mauriy horizonal lines) wih ligh grey areas indicaing posiive values, while dark grey areas show negaive values. In all of he graphs of Figure 6, he mos negaive values are found for older ages and smaller mauriies, while he mos posiive values are found for he younger ages and larger mauriy. This implies ha he model underesimaes he risk-neural probabiliies for conracs wih smaller mauriies and older cohors, and conversely, i overesimaes he risk-neural probabiliies for conracs wih larger mauriies and younger cohors. a) Sandard formula approach b) VaR approach Figure 6. Values of Emax Q c) Sressed-rend approach T j p xj,0 EQ max ) E Qλ) T j p xj,0 T j p xj,0 ) ) according o differen approaches. We also analyzed he behavior of π max and λ when he CoC rae varies in he range of 4 8% for each of he approaches here considered. Obviously, π max values show an increase/decrease
Risks 2017, 5, 29 17 of 21 proporional o he increase/decrease of he CoC rae see Equaion 16)). From he π max values, we hen esimaed he λ vecor saisfying Equaion 19) resuls are shown in Figure 7). a) λ 1 b) λ 2 c) λ 3 Figure 7. Values of λ = λ 1, λ 2, λ 3 ) according o differen approaches and differen cos-of-capial CoC) 4%, 6%, 8%). In Figure 8, we sudy he impac of differen CoC raes on he logi ransform of deah probabiliies evaluaed in 2008) and on he logi ransform of projeced deah probabiliies evaluaed in year 2023) under boh he real-world and risk-neural probabiliy measure. Resuls show ha an increase in he CoC rae produces, wihin he sandard formula approach, a downward shif of he logiq x, ) under he risk-neural measure. This shif is mainly obained by an increase of λ 1 ; however, a change in λ 1 also produces a roaion around he poin x, logiq x )) ha is compensaed by a reducion of λ 2. Under he sandard formula approach, λ 3 has minor changes. A similar behavior, bu a a greaer level, is observed under he sressed-rend approach. Conversely, under he VaR approach, an increase of he CoC rae involves a downward shif of he logiq x, ) under he risk-neural measure accompanied by a decrease in he slope for cohors prior o 1943 age less han 80 years in 2023). In hese cases, λ 3 has negaive values, decreasing along wih he CoC raes. a) b) c) Figure 8. Logi ransform of projeced deah probabiliies year 2023) under he real-world probabiliy measure and risk-neural probabiliy measure wih differen CoC values. a) Sandard formula approach; b) VaR approach; c) Sressed-rend approach.
Risks 2017, 5, 29 18 of 21 5.1. q-forward Pricing As menioned above, he marke price of longeviy risk provided by our model could be used o price oher longeviy-linked securiies. Up o now, a wide range of longeviy-linked securiies has been proposed on he marke, e.g., longeviy bonds, longeviy or survivor swaps and q-forwards. Among hese, q-forwards have several advanages: hey are based on sandardized indices reflecing he experience of a large populaion, and hey are less difficul o price, less expensive and more liquid. For hese reasons, we decided o focus our analysis on q-forwards. A q-forward is a zero-coupon swap ha involves he exchange a mauriy dae of a fixed amoun for a random amoun ha is proporional o a moraliy index for he reference populaion a mauriy. The fixed paymen is proporional o he forward moraliy rae for he reference populaion and is se so ha he q-forward value is zero a incepion see Coughlan e al. 2007) for furher deails on he mechanism of q-forwards). The payoff of q-forwards usually depends on he average moraliy rae for age-buckes of five or 10 years. In our model, we obain he values of q-forwards as he risk-neural expecaion, E Q q x,t ) while heir prices are usually calculaed according o he Sharpe raio approach see, e.g., he LifeMerics echnical documen Coughlan e al. 2007)), as follows: q f x,t = 1 SR T σ x)q e x,t 23) qx,t where T is he ime o mauriy, SR is he required annualized Sharpe raio and σ x = sd q x,t ) is he hisorical sandard deviaion of changes in he moraliy rae. We consider a se of hree q-forwards for age-buckes of five years, from age 65 up o age 79 wih differen mauriies. Resuls obained wih our model are shown in Figure 9 compared wih he expeced moraliy rae q e x,t. q-forwards values calculaed under boh he sandard formula and VaR approaches are quie similar, while hey are lower under he sressed-rend approach, consisen wih he resuls in Figure 5. a) Age 65-69 b) Age 70-74 c) Age 75-79 Figure 9. Q-forward prices by differen age groups. Now, we can esimae he implied maximum Sharpe raio, SR = 1 Tσ x 1 q f x,t q e x,t ), ha an annuiy provider should be willing o pay o hedge longeviy risk from he q-forward prices repored in Figure 9. The Sharpe raio values for he se of hree q-forwards wih a 10-year mauriy are repored in Table 7. Resuls differ from one approach o he oher, bu he sandard formula and VaR approaches produce values ha are quie similar o 0.25, usually suggesed in he lieraure see, e.g., Milevsky e al. 2006); Loeys e al. 2007)).
Risks 2017, 5, 29 19 of 21 Table 7. Sharpe raio. Age Group Sandard Formula VaR Sress-Trend 65 69 0.21 0.21 0.31 70 74 0.23 0.29 0.37 75 79 0.23 0.32 0.39 6. Conclusions In his paper, we have invesigaed he possibiliy o price S-forwards using he informaion enclosed in he risk margin and he sandard formula under he Solvency II projec. Our approach provides esimaes of he maximum price ha he fixed payer is disposed o pay for hedging longeviy risk. If he required S-forward price were higher, he annuiy provider would have he convenience of keeping longeviy risk. The maximum price is calculaed under he Solvency II sandard formula and wo parial inernal models one based on he VaR and he oher one on he sressed-rend approach) for differen ages and mauriies, as well as for differen levels of CoC rae. Once a moraliy projecion model is defined under he real-world probabiliy measure, we calibrae he upper limi of marke price of longeviy risk, λ, associaed wih he risk-neural moraliy model. In he same pricing framework, λ values can be used o find he maximum price of oher longeviy-linked securiies. Specifically, we price q-forwards, and we calibrae he implied maximum Sharpe raio ha an annuiy provider should be willing o pay o hedge longeviy risk. The prices provided by our model seem consisen wih hose obained by oher commonly-used approaches e.g., he Sharpe raio) and offer a poin of reference for insurance companies or pension funds in defining longeviy risk hedging policies hrough longeviy-linked securiies. In fac, hese prices represen he hreshold above which i is no convenien o use hese hedging ools, and risk reenion requiring an appropriae level of solvency capial becomes preferable. In he numerical example here provided, only he Cairns Blake Dowd wih cohor effec) moraliy projecion model is used. I would be ineresing o analyze he effecs of differen moraliy models on he marke price of longeviy risk, aking ino consideraion he model risk in he pricing. This aspec will be objec of fuure research. One shall noice ha he approach suggesed in his paper depends on wo fundamenal assumpions. Firs of all, we adop he CoC mehod o calculae RM, while one migh consider oher mehods and assess wheher he resuls obained were similar o he CoC mehod boh numerically and in erms of he considered variables. Secondly, we assume ha all insurance companies accep he same marke price of longeviy risk implici in he RM calculaion under Solvency II. However, diversificaion effecs, sraegic reasons and aiude oward risk could induce he insurer o accep a differen marke price of longeviy risk see Börger 2010)). Moreover, we have no considered he counerpary defaul risk ha would increase he coss of longeviy-linked securiies and require a specific capial charge for he insurer. The effec of his risk on he risk margin and he maximum price of longeviy risk should be he objec of furher research. Finally, i should be kep in mind ha he proposed mehodology highly depends on he individual siuaion of he insurance company and ha he paper aims o esablish a maximum price for he marke of longeviy-linked securiies, ailored o he insurer s specific porfolio. Alhough our approach has some limiaions, i has he advanage of working in he known and possibly sandardized framework of Solvency II, and he RM can be considered a possible ool o esimae a maximum value for he marke price of longeviy risk. Acknowledgmens: The auhors hank he anonymous referees for helpful commens. Auhor Conribuions: The wo auhors have equally conribued o he paper. Conflics of Ineres: The auhors declare no conflic of ineres.
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