Chapter 15 the securitization of longevity risk in pension schemes; the case of italy

CHAPTER 15 The Securitization of Longevity Risk in Pension Schemes: The Case of Italy Susanna Levantesi, Massimiliano Menzietti, and Tiziana Torri CONTENTS 15.1 I ntroduction 15.2 Stochastic Mortality Model 15.2.1 Model Framework and Fitting Method 15.2.2 F orecasting 15.2.3 U ncertainty 15.3 Longevity Risk Securitization 15.3.1 L ongevity Bonds 15.3.2 Vanilla Survivor Swaps 15.4 P ricing Model 15.5 N umerical Application 15.5.1 D ata 15.5.2 Real-World and Risk-Adjusted Death Probabilities 15.5.3 Longevity Bond and Vanilla Survivor Swap Price 15.6 C onclusions References 36 332 336 337 339 339 340 341 344 345 349 350 351 354 359 331 © 2010 by Taylor and Francis Group, LLC 332 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling T his ch a pter f ocu ses o n t he sec uritization o f l ongevity r isk i n pension sch emes t hrough m ortality-linked sec urities A mong t he alternative mortality-linked sec urities p roposed i n t he l iterature, w e consider a longevity bond and a vanilla survivor swap as the most appropriate hedging tools The analysis refers to the Italian market adopting a Poisson Lee–Carter model to represent the evolution of mortality We describe the main features o f l ongevity bo nds a nd su rvivor s waps, a nd t he c ritical i ssue o f the p ricing m odels d ue t o t he i ncompleteness o f t he m ortality-linked securities market and to the lack of a secondary annuity market in Italy, necessary to calibrate the pricing models For pricing purposes, we refer to the risk-neutral approach proposed by Biffis et al (2005) Finally, we calculate t he r isk-adjusted ma rket price of a l ongevity bond w ith constant fi xed coupons and of a vanilla survivor swap Keywords: Longevity risk, stochastic mortality, longevity bonds, survivor swaps 15.1 INTRODUCTION During t he t wentieth c entury, mortality s be en cha racterized by a n unprecedented decline at all ages, and, conversely, by a very steep increase in life expectancy Knowledge on future levels of mortality is of primary importance for life insurance companies and pension funds, whose calculations are based on those values However, even though mortality has been forecast, the risk that the random values of future mortality will be different than expected remains Th is is called mortality risk Mortality risk itself belongs to the wider group of underwriting risk that, together with c redit, o perational, a nd ma rket r isks, co nstitute t he f our ma jor risks a ffecting i nsurers Mortality r isk i ncludes t hree d ifferent sources of risk: the risk of random fluctuations of the observed mortality around the expected value, the risk of systematic deviations generated by an observed m ortality t rend d ifferent f rom t he o ne f orecast, a nd t he r isk of a sudden a nd short-term rise in the mortality frequency The risk of random fluctuations, also called process risk, decreases in severity as the portfolio size increases The risk of systematic deviations can be decomposed into model risk and parameter risk, which combined are referred to as uncertainty risk, alluding to the uncertainty in the representation of a phenomenon Under the heading of uncertainty risk is the so-called longevity risk, generated from possible divergences in the trend of mortality © 2010 by Taylor and Francis Group, LLC The Securitization of Longevity Risk in Pension Schemes ◾ 333 at adult and old ages In practice, it refers to the risk that, on average, the annuitants might live longer than the expected life duration involved in pricing a nd r eserving c alculations Unlike i n t he c ase o f p rocess r isk, risks of systematic deviations cannot be hedged by increasing the size of the portfolio; it rather increases with it The law of large numbers does not apply because t he risk affects a ll t he annuitants in t he same direction Understanding the risk, and determining the assets that the annuity providers have to deploy to cover t heir l iabilities i s a ser ious i ssue Increasing attention has been devoted to the longevity risk in the recent years Th is is also the case in Italy where it has been fi nally observed in the development of the annuity market Indeed, before the 1990s, when major pension reforms were implemented, the Italian annuities market was hardly developed The introduction of a specific law to regulate pension funds in 1993 and the subsequent amendments in 2000 and 2005, contributed to t he origin of t he second a nd t hird pillars i n t he Italian pension s ystem A t t he en d o f 008, t here w ere abo ut m illion i ndividuals contributing into pension schemes Out of t his number, nearly 3.5 million contributed into pension funds and the rest into individual pension schemes Within these regulations, the Italian legislator decided that participants of pension schemes must annuitize at least half of the accumulated c apital Moreover, it wa s dec ided t hat only l ife i nsurance companies a nd pension plans w ith spec ified characteristics are authorized to pay annuities The other operators have to transfer the accumulated capital to insurance companies at the moment of retirement A further development of the Italian annuities market is expected This induces pension funds and life insurance companies to be more responsible in the management of the risks In this respect, some steps have already been t aken The I talian S upervisory A uthority o f t he I nsurance S ector (ISVAP) i ntroduced a n ew r egulation (no 1/2008) a llowing i nsurance companies t o r evise t he dem ographic ba ses u p t o y ears bef ore r etirement Consequently, t he longevity risk is relegated only to t he period of the annuity payment In addition, starting in 1998, the Italian Association of Insurance Companies (ANIA) has developed projected mortality tables specific for the Italian annuities market (e.g., RG48 in 1998 and IPS55 in 2005) The more recent IPS55 is the reference life table currently used by insurance companies for pricing and reserving A responsible management of the longevity risk implies that life insurance co mpanies a nd pens ion p lans sh ould m easure a nd ma nage i t To measure t he l ongevity r isk, a st ochastic m ortality m odel t hat i s ab le t o © 2010 by Taylor and Francis Group, LLC 334 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling fit a nd f orecast m ortality i s n eeded I n t he la st dec ades, ma ny st ochastic m ortality m odels ve be en de veloped: e.g , L ee a nd C arter (1992), Brouhns e t a l ( 2002), M ilevsky a nd Pr omislow ( 2001), Rensha w a nd Haberman (2003, 2006), Cairns et al (2006b) The reader c an a lso refer to Cairns et al (2008b) for a de scription of selection criteria to choose a mortality model However, t he pr oduction of pr ojected, a nd e ventually s tochastic, life tables is not sufficient for the management of the longevity risk In fact, although annuity providers can partially retain the longevity risk, “a legitimate business risk which they understand well and are prepared to assume” (Blake et al 2006a), they should transfer the remaining risk to a void ex cessive ex posure W ith t his r espect, a lternative so lutions exist Natural hedging is obtained by diversifying the risk across different countries, or through a suitable mix of insurance benefits within a policy or a po rtfolio The more traditional way for transferring risks, through reinsurance, is not a viable solution Actually, reinsurance companies a re reluctant t o t ake on such a s ystematic a nd not d iversifiable risk, consequently t he reinsurance premiums a re g reat A n a lternative way out i s t ransferring pa rt of t he r isk to a nnuitants sel ling a nnuities with payments linked to experienced mortality rates within the insured portfolio However, this solution is not always achievable An a lternative a nd m ore a ttractive o ption ma y l ie i n t he t ransfer o f the longevity risk into financial markets via securitization Securitization is a p rocess t hat co nsists i n i solating a ssets a nd r epackaging t hem i nto securities that are traded on the capital markets The traded securities are dependent on an index of mortality, and are called mortality-linked securities (for an overview on securitization of mortality risk see Cowley and Cummins 2005) By investing in mortality-linked securities, an annuity provider has the possibility to hedge the systematic mortality risk inherent in their annuities These contracts are also interesting from the investor’s point of view, since they allow an investor to diversify the asset portfolio and improve their risk-expected return trade-offs Several mortality-linked securities have been proposed in the literature: longevity (or mor tality) b onds, s urvivor (or mor tality) s waps, mor tality futures, mortality forwards, mortality options, mortality s waptions a nd longevity (or survivor) caps and floors (see Blake et al (2006a) and Cairns et al (2008a) for a detailed description) Unlike reinsurance solutions, t hese financial i nstruments, depending on the selected index of mortality, involve returns to insurers and pension © 2010 by Taylor and Francis Group, LLC The Securitization of Longevity Risk in Pension Schemes ◾ 335 funds n ot n ecessarily co rrelated w ith t heir l osses T o ach ieve en ough liquidity, t he l ongevity ma rket w ill ve t o f ocus o n b road po pulation mortality indices while insurer and pension fund exposures might be concentrated i n speci fic regions or socioeconomic g roups The fact that the cash flows from the financial derivatives are a function of the mortality of a population that may not be identical to the one of the annuity provider creates basis risk, the risk associated with imperfect hedging Trading c ustom-tailored der ivatives f rom o ver-the-counter ma rkets, such a s su rvivor swaps, would reduce t he ba sis r isk, but would i ncrease the c redit r isk, t he r isk t hat one of t he counterparties may not meet its obligations On t he o ther nd, t rading m ore st andardized der ivatives, like longevity bonds, decreases the credit risk but increases the basis risk Generally, they are focused on broad population mortality indices instead of being tailored on a specific insured mortality Several mortality-linked securities were suggested in t he literature, but only f ew p roducts w ere i ssued i n t he ma rket N onetheless, a n i ncreasing attention toward mortality-linked securities is witnessed Significant attempts to create products providing an effective transfer of the longevity risk have been observed among practitioners and investment banks (Biffis and Blake 2009) In March 2007, J.P Morgan launched Lifemetrics, a platform for measuring a nd ma naging longevity a nd mortality r isk (see C oughlan et al 2007a, 2007b) It provides mortality rates and life expectancies for different co untries ( United S tates, E ngland, a nd Wales) t hat c an be u sed t o determine the payoff of longevity derivatives and bonds In December 2007, Goldman Sachs launched a monthly index called QxX.LS (www.qxx-index com) in combination with standardized 5- and 10-year mortality swaps Parallel to the choice of the more appropriate mortality-linked securities, ex ist a lso a l ively debate concerning t he choice of t he more appropriate p ricing a pproach f or m ortality-linked sec urities On e o f t hese approaches i s t he ad aptations o f t he r isk-neutral p ricing f ramework developed f or i nterest-rate der ivatives It i s ba sed o n t he i dea t hat bo th the force of mortality and the interest rates behave in a similar way: they are pos itive st ochastic p rocesses, bo th e ndowed w ith a t erm st ructure (see Milevsky and Promislow (2001), Dahl (2004), Biffis et al (2005), Biffis and Millossovich (2006), and Cairns et al (2006a)) Nevertheless, such an approach is not universally accepted Unlike the interest-rate derivatives market, t he ma rket o f m ortality-linked sec urities is sc arcely d eveloped and hence i ncomplete, ma king it d ifficult to u se a rbitrage-free methods and impossible to estimate a unique risk-adjusted probability measure © 2010 by Taylor and Francis Group, LLC 336 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling An a lternative approach is t he distortion approach based on a d istortion operator—the Wang transform (Wang 2002)—that distorts the distribution of the projected death probability to generate risk-adjusted death probability that can be discounted at the risk-free interest rate (see Lin and Cox (2005), Cox et al (2006), and Denuit et al (2007)) In t his cha pter, w e f ocus o n l ongevity bo nds a nd su rvivor s waps a s instruments that are able to hedge the longevity risk affecting Italian annuity providers To represent the evolution of mortality we rely on a d iscrete time stochastic model: the Lee–Carter Poisson model proposed by Brouhns et a l (2002) Due to t he absence of a seco ndary a nnuities ma rket i n Italy necessary to calibrate prices, we extrapolate market data from the reference life t able u sed by t he Italian i nsurance companies for pricing a nd reserving At t he moment, t he reference l ife t able i s t he I PS55, a n ew projected life tables for annuitants developed in 2005 by ANIA (see ANIA 2005) The IPS55 is obtained by multiplying the national population mortality projections, performed by the Italian Statistical Institute (see ISTAT 2002) and the self-selection fac tors o btained f rom t he E nglish ex perience, d ue t o t he paucity of Italian annuity market data The chapter is organized as follows: In Section 15.2, we present the stochastic mortality model, used to estimate future mortality rates In Section 15.3, we describe the longevity risk securitization mainly focusing on longevity bonds and survivor swaps The structure and features of these securities are also presented in this section Section 15.4 is devoted to evaluate longevity bonds and survivor swaps in an incomplete market To price the securities we refer to the risk-neutral measure proposed by Biffis et al (2005) that is described in this section In Section 15.5, we present a numerical application on Italian data Final remarks are presented in Section 15.6 15.2 STOCHASTIC MORTALITY MODEL The need of stochastic models, and not only deterministic models, is widely recognized, if we want to measure the systematic part of the mortality risk present in the forecast (Olivieri and Pitacco 2006) As a consequence, numerous works recently proposed in the literature were mainly concerned with the inclusion of stochasticity into the mortality models Stochastic mortality models c an be f urther d ivided i nto continuous t ime models a nd d iscrete time models The former models include the one proposed by Milevsky and Promislow (2001), Dahl (2004), Biffis (2005), Cairns et al (2006b), Biffi s and Denuit (2006), Dahl and Møller (2006), and Schrager (2006) © 2010 by Taylor and Francis Group, LLC The Securitization of Longevity Risk in Pension Schemes ◾ 337 The la tter m odels a re a st raightforward co nsequence o f t he k ind o f data available, generally annual data subdivided into integer ages In this framework, the earliest and still the most popular model is the Lee–Carter model (Lee and Carter 1992) The model has been widely used in actuarial and demographic applications, and it can be co nsidered t he standard in modeling and forecasting mortality A review of the variants proposed to the model is presented in Booth et al (2006) In the same discrete framework, Renshaw and Haberman’s work (2006) is one of the first works that incorporates a cohort effect A more parsimonious model, including also a cohort component was later introduced by Cairns et al (2006b) The L ee–Carter m odel r educed t he co mplexity o f m ortality o ver both a ge a nd t ime, su mmarizing t he l inear decl ine o f m ortality i nto a single time index, further extrapolated to forecast mortality We will consider a generalization of the original Lee–Carter model introduced by Brouhns e t a l (2002) They proposed a d ifferent procedure for t he estimation of t he m odel, a lso s ubstituting t he in appropriate a ssumption of homoscedasticity of the errors Forecasting m ortality o bviously l eads t o u ncertain o utcomes, a nd sources of uncertainty need to be e stimated and assessed Following the work o f K oissi e t a l (2006), w e su ggest a n o riginal st rategy f or de aling with u ncertainty u sing n onparametric boo tstrap tech niques o ver e ach component of the Lee–Carter model The identification of such sources of uncertainty and their measurements are necessary for a correct management of the longevity risk 15.2.1 Model Framework and Fitting Method Let t q x be t he p robability t hat a n i ndividual o f t he r eference co hort, aged x0 at time 0, will die before reaching the age x0 + t Given t he corresponding survival prob ability t p x 0, t he stochastic number of su rvivors l x +t follows a binomial distribution with parameters E(lx0 +t ) = lx0 t px0 and Var(lx0 +t ) = lx0 t px0 (1 − t px0 ), where l x is the initial number of individuals in the reference cohort The expected number of survivors, lˆx +t, is obtained with the point wise projection of the death probability t qˆ x To o btain t he de ath p robabilities, t qx0 , w e m odel a nd f orecast t he period c entral de ath r ates a t a ge x and time t, mx(t), with the Poisson log-bilinear model suggested by Brouhns et al (2002) Considering the higher va riability o f t he o bserved de ath r ates, a t a ges w ith a s maller number of deaths, they assumed a Poisson distribution for the random component, a gainst t he a ssumed nor mal d istribution i n t he or iginal © 2010 by Taylor and Francis Group, LLC 338 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling Lee–Carter model Therefore, they assumed that the number of deaths, Dx(t), is a random variable following a Poisson distribution: Dx (t ) ~ Poisson(N x (t ) ⋅ mx (t )) = [N x(t ) ⋅ mx(t )]Dx (t ) e −[N x (t )⋅mx (t )] Dx (t )! (15.1) where Nx(t) is the midyear population observed at age x and time t mx(t) is the central death rate at age x and time t The c entral de ath r ates mx(t) f ollow t he m odel su ggested b y L ee a nd Carter (1992): ln[mx (t )] = α x + β x kt + ε x ,t (15.2) where the parameter αx refers to the average shape across ages of the logmortality schedule; βx describes the pattern of deviations from the previous age profile, as the parameter kt changes; and kt can be seen as an index of the general level of mortality over time Without f urther co nstraints, t he m odel i s u ndetermined I n o ther words, there are an infinite number of possible sets of parameters, which would satisfy Equation 15.2 In order to overcome these problems of identifiability, two constraints on the parameters are introduced: ∑ t kt = and ∑ x β x = The pa rameters c an be e stimated by ma ximizing t he following loglikelihood function: log L(α, β, k) = ∑ Dx (t )(α x + β x kt ) − N x (t )e α x +βx kt + C (15.3) x ,t Because of the presence of the bilinear term, βxkt, an iterative method is used to solve it To evaluate the fitting of the Poisson model, the deviance residuals are calculated: ⎡ ⎤ ⎛ D (t ) ⎞ rD = sign[Dx (t ) − Dˆ x (t )]⋅ ⎢ Dx (t )ln ⎜ x ⎟ − (Dx (t ) − Dˆ x (t ))⎥ ⎝ Dˆ x (t ) ⎠ ⎣ ⎦ ˆ where Dˆ x (t ) = N x (t )e mx (t ) © 2010 by Taylor and Francis Group, LLC 0.5 (15.4) The Securitization of Longevity Risk in Pension Schemes ◾ 339 15.2.2 Forecasting To o btain t he f uture va lues o f t he c entral de ath r ates, L ee a nd C arter (1992) assume that the parameters αx and βx remain constant over time and forecast future values of the time factor, kt, intrinsically viewed as a stochastic process, using a standard univariate time series model Box and Jenkins identification procedures are used here to estimate and forecast the autoregressive integrated moving average (ARIMA) model (Box and Jenkins 1976) W ith t he m ere ex trapolation o f t he t ime fac tor, kt, i t i s possible to forecast the entire matrix of future death rates 15.2.3 Uncertainty The different sources of uncertainty need to be e stimated and combined together: the Poisson variability enclosed in the data; the sample variability o f t he pa rameters o f t he L ee–Carter m odel a nd t he A RIMA m odel; the u ncertainty of t he ex trapolated va lues of t he model’s t ime i ndex kt An analytical solution for the prediction intervals, that would account for all the three sources of uncertainty simultaneously, is impossible to derive due to the very different sources of uncertainty to combine An empirical solution to the problem is found through the application of parametric and nonparametric bootstrap, following recent works on t he topic (Brouhns et al (2005), Keilman and Pham (2006), and Koissi et al (2006) ) The bootstrap i s a co mputationally i ntensive approach u sed for t he construction of prediction intervals, first proposed by E fron (1979) A r andom i nnovation i s g enerated, wh ich s amples ei ther f rom a n a ssumed pa rametric distribution ( parametric b ootstrap), o r f rom t he e mpirical di stribution of past fitted errors (nonparametric or residual bootstrap) In this second approach, under the assumption of independence and identical distribution of the residuals, it is assumed that the theoretical distribution of the innovations is approximated by the empirical distribution of the observed deviance residuals Random innovations are generated by sampling from the empirical distribution of past fitted errors Differently f rom p revious st udies, w e a pplied a n onparametric boo tstrap t o compute pa rameters u ncertainty of both L ee–Carter a nd a ssociated A RIMA m odels The s imulation p rocedure w e f ollowed co nsists of two parts: first, we evaluated the sampling variability of the estimated coefficients of the model, sampling N times from the deviance residuals of the Lee–Carter model; second, for each of the N simulated kt parameters, we evaluated the variability of the projected model’s time index, sampling M times from the residuals of the ARIMA model Overall, we simulated © 2010 by Taylor and Francis Group, LLC 340 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling an a rray w ith N ·M matrices of future period central death rates After selecting the diagonal of the matrices, corresponding to the cohort we are interested in, we simulated from a b inomial distribution P paths of survivals, for each of t he chosen diagonal Overall N · M · P simulations of survivors are performed 15.3 LONGEVITY RISK SECURITIZATION As a lready s tated in t he in troduction, s ecuritization i s a n inn ovative vehicle suggested to transfer the longevity risk into capital market through mortality-linked securities Among the different mortality-linked securities proposed in the literature, we focus our attention on longevity bonds and survivor swaps We will investigate their capability to hedge the longevity risk faced by the Italian annuity providers, and make a comparison between the performances of the two products Longevity bonds a re mortality-linked securities, t raded on organized exchanges, structured in a way that the payment of the coupons or principal i s depen dent o n t he su rvivors o f a g iven co hort i n e ach y ear The literature about longevity bonds is quite extensive The first longevity bond was su ggested b y Bla ke a nd B urrows (2001), wh o p roposed a l ongevity bond structure with annual payments attached to the survivorship of a reference population Lin a nd Cox (2005) proposed instead a bo nd w ith coupon pa yments eq ual t o t he d ifference be tween t he r ealized a nd t he expected survivors of a given cohort Longevity bonds are also discussed by Blake et al (2006a,b) and Denuit et al (2007) The fi rst and the only longevity bond launched on the market was the so-called EIB/BNP longevity bond (see Azzopardi 2005 for more details), it was launched in 2004 and withdrawn in 2005 Although unsuccessful, academics as well as practitioners have paid considerable attention to t his p roduct a nd defi ned i ts p roblems ( see Bla ke e t a l ( 2006a,b), Cairns et al (2006b), and Bauer et al (2008)) One of the problems with the EI B/BNP l ongevity bo nd wa s t he p resence o f t he ba sis r isk: t he reference i ndex wa s n ot co rrelated en ough w ith t he h edger’s m ortality experience To deal in part with this problem Cairns et al (2008a) suggested t he use of longevity-linked securities built a round a spec ial purpose vehicle (SPV) The SPV would arrange a swaps with the hedgers and then aggregates the swapped cash flows to pass them on to the market through a bond Dowd et a l (2006) suggested su rvivor swaps a s a m ore adva ntageous derivative t han su rvivor b onds They defined a su rvivor s wap a s “ an © 2010 by Taylor and Francis Group, LLC 348 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling Now, let us define r(t) the interest-rate process adapted to G and consider an in surance m arket in volving m ortality-linked s ecurities w ith p rices adapted to F We have the mortality dynamics of the reference population ⎛ t ⎞ that is described by P (τ > t t ) = exp ⎜ − µ(s)ds⎟ ⎝ ⎠ The price of the straight bond used to construct a longevity bond and paying a fi xed annual coupon C at each time t and the principal F at maturity T, is given by ∫ T W = Fd(0, T ) + C ∑ d(0, t ) t =1 (15.17) where t he r isk-free d iscount fac tor a t t ime z ero, d(0,t), i s g iven b y ⎡ ⎛ t ⎞⎤ EQ ⎢exp ⎜ − r (s )ds ⎟ ⎥ ⎝ ⎠⎦ ⎣ Under this framework, assuming the independence between interest rate and mortality, the market prices P of the premium that the annuity provider pays to hedge his/her longevity risk and V of the longevity bond, are given by ∫ T P = R ∑ EQ (Bt t )d(0, t ) t =1 T V = R Fd(0, T ) + R∑ EQ (Dt t )d(0, t ) t =1 (15.18) (15.19) where EQ(Bt|t) and EQ(Dt|t) are the expected values of payments Bt and Dt received by the annuity provider and investors, respectively, under the risk-neutral measure Q conditional on sub-filtration t With reference to the vanilla survivor swap, the swap value at time zero to the fi xed-rate payer is Swap value = V[lx0 + t ] − V[(1 + π)lˆx0 + t ] (15.20) where V[(1 + π)lˆx0 +t ] and V[lx0 +t ] are the market prices at time zero of the fi xed leg and of the floating leg, respectively Under this framework, assuming the independence between interest rate and mortality, V[(1 + π)lˆx0 +t ] is the expected present value of the fi xed leg, under the real-world probability measure P: T V[(1 + π)ˆlx0 +t ] = (1 + π)∑ ˆlx0 + t d(0, t ) t =1 © 2010 by Taylor and Francis Group, LLC (15.21) The Securitization of Longevity Risk in Pension Schemes ◾ 349 and V[lx0 +t ] is the expected present value of the floating leg under the riskadjusted probabilities measure Q conditional on sub-filtration t and the risk-free discount factor, d(0,t) T V[lx0 +t ] = ∑ EQ[lx0 +t t ]d(0, t ) t =1 (15.22) Correspondingly, the value of the premium π, set so that the swap value at inception is zero, is equal to T π= ∑ EQ[lx +t t ]d(0, t ) t =1 T ∑ lˆx0 +t d(0, t ) −1 (15.23) t =1 15.5 NUMERICAL APPLICATION The analysis is carried out on Italian data, with the focus on the annuity market In particular, we look at the life table used on the Italian market to price a nnuities (IPS55) a nd to a m ore realistic l ife t able t hat we computed on t he basis of self-selection factors recently published by t he Working group on annuitants’ life expectancy in Italy (hereafter called “Working group,” see Nucleo di osservazione della durata di vita dei percettori di rendite (2008)) Working w ith t he t wo l ife t ables, w e a re ab le t o o btain a n ex plicit measure of the price implicitly charged by the insurance companies for carrying t he l ongevity r isk The s ame m easure w ill be cha rged on t he price of t he products t hat t he i nsurance company buys to t ransfer t he risk We consider a longevity bond and a vanilla survivor swaps build on Italian population data The choice to use population data is motivated by the lack of official life tables of annuitants in Italy Moreover, national statistics are more reliable and easily accessible from investors, eliminating any possible moral hazard However, we cannot deny the presence of the basis risk introduced using the general population data and not the one specific of the insured population Section 15.5.1 describes the data more in detail The following sections describe t he results of t he stochastic mortality model previously provided, under the real-world and the risk-neutral setting Prices of the two products are also evaluated on a co hort of Italian males aged 65 in the year 2005 © 2010 by Taylor and Francis Group, LLC 350 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling 15.5.1 Data The data we used refer to the mortality of the Italian population, derived from t he H uman M ortality Da tabase ( HMD, 004), a n o pen so urce database, wh ose d ata co me d irectly f rom t he o fficial v ital s tatistics a nd the c ensus counts published by ISTAT (the Italian National I nstitute of Statistics) Male population on the 1st of January and male death counts are considered over the period 1950–2005, by single year of age, in the age range 50–110 The self-selection factors provided by the Working group, and applied on the death probabilities of the general population intend to reproduce the mortality of annuitants in Italy These factors are the result of the analysis over the period 1980–2004, on the mortality of pensioners in Italy It is a common practice for the Italian annuity providers to price their annuities with the IPS55 life table, the one recommended by the National Association of I nsurance C ompanies (ANIA 005) The I PS55 l ife t able is ba sed o n m ortality p rojections per formed b y t he I talian N ational Statistical I nstitute (ISTAT), referring t o t he cohort of i ndividuals born in 1955 These projections are obtained by applying the Lee–Carter model to the death rates of the Italian general population ANIA applied to the projected death probabilities self-selection factors taken from the English experience, a nd not exactly representative of t he Italian experience Life tables for cohorts other than 1955 are obtained through a cohort-specific age-correction t hat increases or decreases t he insured’s age w ith respect to the real age However, the current approximation, providing mortality rates constant at intervals, could be t he reason for unsatisfactory results In fact, as already obtained in a previous work by Levantesi et al (2008b), using the IPS55 life table to calibrate the risk-neutral intensities, returns inconsistent r esults F or a spec ific co hort o f i ndividuals, t he a uthors obtained r isk-neutral de ath p robabilities h igher t han t he r eal-world ones, when it is well known that risk-neutral death probabilities should be lower because investors require a risk premium for the longevity risk To overcome such a problem, we suggest an alternative use to the projected de ath p robabilities o f t he g eneral po pulation, f urther co rrected with t he sel f-selection fac tors p rovided b y A NIA We c an st ill s ay t hat these tables represent the mortality of annuity providers In what follows, we call these tables “IPS55 adjusted.” Figure 15.3 plots the self-selection factors assumed both by ANIA and the Working group We can observe a smaller self-selection assumed from the Working group with respect to the one assumed by ANIA © 2010 by Taylor and Francis Group, LLC The Securitization of Longevity Risk in Pension Schemes ◾ 351 100 90 % 80 70 60 Working group IPS55 50 40 50 60 70 80 Age 90 100 110 Self-selection factors used in the IPS55 life table based on English experience, and self-selection factors derived by the Working group from Italian pensioner data FIGURE 15.3 15.5.2 Real-World and Risk-Adjusted Death Probabilities The c urrent sec tion p resents t he r esults o btained a pplying t he P oisson Lee–Carter model described in Section 15.2 to the data mentioned earlier Figure 15.4 plots t he pa rameters αx, βx, a nd kt of t he L ee–Carter model (see Equation 15.2) estimated on Italian male death rates, over the period 1950–2004 and the age range 50–110 The corresponding 95% confidence intervals, obtained from the first phase of the bootstrap, are also included in the plots As already anticipated, we calibrate the risk-neutral intensities using the “IPS55 adjusted” mortality table This table are very close to the one used for modeling future mortality, with the exception of the self-selection factors The parameters of the Poisson Lee–Carter model of Equation 15.14 are plotted in Figure 15.5, while the adjustment functions used to calibrate the risk-neutral intensities are plotted in Figure 15.6 Future va lues o f m ortality a re o btained t hrough t he f orecast o f t he parameters kt ke eping co nstant t he o ther t wo pa rameters o f t he m odel Future values of kt and k˜t are obtained applying the ARIMA(0,1,0) model to the series of parameters Parameters σ and δ under both the physical and risk-neutral measures, introduced in Equations 15.9 and 15.15, are given in Table 15.1 © 2010 by Taylor and Francis Group, LLC 352 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling 0.06 0.04 βx αx –1 –2 0.02 –3 0.00 –4 –5 −0.02 50 60 (a) 70 80 Age 90 100 50 110 60 (b) 70 80 90 100 110 Age 10 kt −10 −20 −30 1950 1960 (c) 1970 1980 Year 1990 2000 Parameters αx, βx, and kt of t he Poisson Lee–Carter model, w ith 95% confidence intervals Italian males FIGURE 15.4 We o bserved th at th e process of k˜t i s v ery cl ose t o t he p rocess o f kt under the real-world measure as a consequence of the similarity in underlying mortality t ables Applying E quation 15.16, we obtain t he va lue of the pa rameter ηt, that is changing the d rift of k under Q, a nd i s eq ual to −0.0168 Even though ηt is very small, we should not forget how the change of measures affecting the dynamics of µ under Q occurs through the process φi (see Equations 15.10 and 15.11) We can indeed see in Figure 15.5 how in this case the change of measure is mainly due to the change in parameter α Results combined together provide the death probabilities © 2010 by Taylor and Francis Group, LLC The Securitization of Longevity Risk in Pension Schemes ◾ 353 0.030 0.025 0.020 βx αx −2 0.015 0.010 −4 0.005 0.000 −6 50 60 70 80 Age (a) 90 100 50 110 60 70 (b) 80 Age 90 100 110 10 kt −10 −20 1950 1960 1970 1980 1990 2000 (c) Year Parameters o f t he P oisson L ee–Carter mo del e stimated o n t he mortality o f t he g eneral p opulation m ultiplied b y t he s elf-selection f actors provided by the Working group (black line), and on the mortality of the “IPS55adjusted” life table (grey line) Italian males FIGURE 15.5 0.0 0.0015 −0.1 0.0010 b a −0.2 −0.3 0.0005 −0.4 −0.5 0.0000 −0.6 50 (a) 60 70 80 Age 90 50 100 110 60 70 (b) 80 90 Age 100 110 FIGURE 15.6 Adjustment functions ax+t and bx+t used to calibrate the risk-neutral intensities obtained from the Lee–Carter model Italian males © 2010 by Taylor and Francis Group, LLC 354 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling for t he co hort o f i ndividuals a ged i n 2005, for both the real-world and risk-neutral approaches, as shown in Figure 15.7 As can be observed using the “IPS55 adjusted” the risk-neutral de ath p robabilities a re a lways lower than the real ones, ensuring a positive risk premium TABLE 15.1 Estimated Parameters σ and δ of the Model under Both Physical and Risk-Neutral Measures kt k˜t 1.9246 −0.5940 2.0290 −0.5921 Parameter σ δ 15.5.3 Longevity Bond and Vanilla Survivor Swap Price We consider a l ongevity bond a nd a va nilla survivor swap both structured on a n i nitial cohort of l x = 10,000 Italian males, aged 65 in the year 005 The pa th o f f uture su rvivors i s g enerated f rom a b inomial distribution, u sing t he f uture de ath p robabilities o btained i n Section 15.2 For each of t he 15,000 (N · M = 100 · 150) death probabilities, we generate t imes t he f uture n umber o f su rvivors f rom a b inomial distribution O verall w e per formed 00,000 pa ths o f f uture su rvivors (N · M · P= 100 · 150 · 40) 1.0 0.8 t qx 0.6 0.4 0.2 Real-world approach Risk-neutral approach 0.0 70 80 90 Age 100 110 Death probabilities for t he generation a ged 65 i n 005, together with 95% prediction intervals Results are relative to the real-world (black lines) and risk-neutral (grey lines) approaches Italian males FIGURE 15.7 © 2010 by Taylor and Francis Group, LLC The Securitization of Longevity Risk in Pension Schemes ◾ 355 To obtain t he price of t he t wo mortality-linked securities, we have to make some further specifications We set the maturity T equal to 25 years for both securities Formulae t o c alculate t he p rice o f a l ongevity bo nd w ith co upons a t risk a re presented i n Equations 15.17 t hrough 15.19 of Section 15.4 The longevity bond is built in a way that the value of the constant fixed coupon C is redistributed between the two counterparties, depending on the mortality experienced by the reference cohort However, prices are evaluated considering the risk-adjusted expected value of the cash flows, generated with the risk-adjusted death probabilities The constant fixed coupon C of the longevity bond is calculated as the product of the coupon rate c, equal to the par yield (the bond is issued at par), and the face value F of the straight bond The value of the par yield is the result of the term structure taken from the Committee of European Insurance and Occupational Pensions Supervisors (see CEIOPS 2007) and is equal to 3.77% The same term structure is used to discount the future cash flows of the two products, when computing the prices According t o t he n umber o f a nnuitants, w e se t t he fac e va lue o f t he longevity bond refunded at maturity equal to F = 10,000 Euros Ther efore, the price W of the straight bond is equal to 10,000 Euros This is divided between t he p remium P pa id b y t he a nnuity p roviders a nd i s eq ual t o 5026.07 a nd l ongevity bo nd p rice V pa id b y t he i nvestors a nd i s eq ual to 4973.93 The present value of the assured principal refunded to investors accounts for about 80% of the longevity bond price V The remaining 20% of t he price accounts for t he pa rt of t he coupons t hat t he investors will receive on average The expected value of the cash flows received both by the annuity provider and the investors, under both the real-world and risk-neutral probability measures, is reported in Figure 15.8 The grey and increasing line in Figure 15.8 represents the expected value of the cash flows received by the annuity provider It is growing consistently with the growing exposure to longevity r isk C orrespondingly, t he ex pected va lue of t he c ash flows received b y i nvestors dec reases w ith t ime The s ame va lues i n t he r iskneutral f ramework are plotted in Figure 15.8b The existing relationship between t he t wo l ines is completely reversed This is due to t he i mplicit inclusion of a price for the risk that has been transferred from the annuity provider to the investors The premium, π, of t he su rvivor swap is presented i n E quation 15.23 of Section 15.4 and is equal to 0.0738 As described in Section 15.3.2, the © 2010 by Taylor and Francis Group, LLC 356 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling 300 300 E(D(t)) E(B(t)) Ct 200 100 100 0 65 (a) E(D(t)) E(B(t)) Ct 200 70 75 80 85 Age 65 (b) 70 75 80 Age 85 FIGURE 15.8 E[Bt] a nd E[Dt] u nder b oth t he r eal-world ( a) a nd r isk-neutral (b) probability measures fi xed payer pays the amount (1 + π)ˆlx0 +t, and receives l x +t from the floating payer The difference between the expected value of the floating legs under the risk-neutral measure, EQ[lx0 +t Gt ], and the expected value of the fixed legs of the swap, (1 + π)ˆlx0 + t, is reported in Figure 15.9 This difference is positive between years 11 a nd 25 ( i.e., between ages 75 and 89), the period of maximum exposure of the annuity provider to longevity r isk Such a r esult is consistent w ith t he t rend of EQ[Bt] i n t he longevity bond (see Figure 15.8) We p resented i n t he w ork t wo a lternative i nstruments a pt t o t ransfer t he l ongevity r isk c arried b y a n a nnuity p rovider I n o ne c ase, t he annuity provider partially transfers the longevity risk to investors via a longevity bond; in the other case, the annuity provider transfers all the longevity risk to a floating ratepayer v ia a su rvivor swap The longevity bond a ssumes t he payment of a p remium at t ime z ero, to cover f uture losses up to t he ma ximum level C, while no premiums a re pa id w ithin the survivor swap Under t he r isk-adjusted m easure, t he ex pected p resent va lues o f future payments charged to the annuity provider in all the situations are obviously t he s ame Under t he r eal-world m easure, t he v olatility o f t he expected present value of future payments and the corresponding losses experienced are different Ignoring for the moment the risk of default implied by one or the other instrument as well as the basis risk, we can make some considerations on © 2010 by Taylor and Francis Group, LLC The Securitization of Longevity Risk in Pension Schemes ◾ 357 600 400 200 −200 −400 −600 66 68 70 72 74 76 78 80 Age 82 84 86 88 90 FIGURE 15.9 Expected value of the difference between floating leg and fi xed leg under the risk-neutral measure the preference of one or t he other product, looking at t he losses ex perienced from the annuity provider’s in any of the cases Therefore, we concentrate on losses distribution with attention to three specific cases: a The annuity provider retains the longevity risk b The annuity provider partially transfers the longevity risk to investors via a longevity bond c The a nnuity p rovider t ransfers a ll t he l ongevity r isk t o a floating ratepayer via a survivor swap © 2010 by Taylor and Francis Group, LLC 358 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling The corresponding losses experienced by the annuity provider in each of the three cases are a l x +t − lˆx +t b l x +t − lˆx +t − Bt c π lˆx +t Specifically, in case (c) the random payment of the annuity provider to the annuitants is perfectly offset by the floating leg of the survivor swap The payments he/she is responsible for a re fi xed at issuing t ime a nd not random a ny more Hence, losses π lˆx +t a re proportional to t he ex pected value of survivors at each time t In case (b) the annuity provider’s risk is reduced up to a maximum level C, still existing the possibility that his/her future payments will exceed the maximum level On t he other nd, he/she could a lso ex perience gains due to unexpected increases in mortality Therefore, the longevity bond is not as effective as the survivor swap in reducing the risk of negative fluctuations, but allows for possible gains Certainly the choice of one of the two products is preferable to the case (a) where no product is acquired In fact, in situations (b) and (c) the payments received by the annuity provider from the SPC and the floating leg payer, respectively, reduce the randomness of the losses he/she may experience A p lot o f t he m ean va lues a nd t he 9th per centile o f t he l osses distribution is provided in Figure 15.10 The 9th p ercentile of losses i n cases (a) a nd ( b) p resents, a s w e w ere i ntuitively ex pecting, h igher va lues in case (a) where no instrument is used to transfer the longevity risk A complete transfer of the risk is instead assumed in case (b) for the first 11 years approximately We should not forget here t hat t he reduction i n future losses experienced in case (b) comes at the cost of the premium P previously calculated In c ase (c) wh ere l osses a re c ertain, w e o bserve h igher va lues o f t he losses i n t he first lf o f t he per iod, w ith r espect t o t he 9% per centiles o f t he l osses i n t he o ther t wo c ases This be havior is s ubsequently inverted in the second half of the period, where substantial lower values of losses are observed Recalling the behavior of the longevity risk previously described, we can fairly say that the swap generates losses that not follow the occurrence of the risk, but rather go in an opposite direction © 2010 by Taylor and Francis Group, LLC The Securitization of Longevity Risk in Pension Schemes ◾ 359 1,200 Loss 99%—case (a) Loss 99%—case (b) Loss—case (c) 1,000 800 600 400 200 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 t The 99th percentile of the losses distribution in cases (a) and (b) and the mean values of the losses distribution in case (c) FIGURE 15.10 In fac t, t he a nnuity p rovider ex periences dec reasing l osses o ver t ime against an increasing trend of the longevity risk 15.6 CONCLUSIONS Longevity bo nds a nd su rvivor s waps, a lthough sha ring s imilarities, a re essentially d ifferent The l ongevity bo nd i s f ramed i nto t he m ore r igid structure of a straight bond, characterized by a low risk of default and limited coverage In fact, the cash flows received by the two counterparties are proportional to the coupons, and hence limited to the value of the coupons On t he o ther nd, t he va nilla su rvivor s waps a re m ore flexible instruments, easily cancelled, but characterized by a higher risk of default In this case, the cash flows not have any upper or lower bound, which in the case of the longevity bond was between zero and the value of the coupon Longevity bonds and survivor swaps represent an interesting solution to manage the longevity risk of annuity providers However, they are difficult to price due to the incompleteness of the mortality-linked securities market As far as the Italian population is considered, additional problems arise from the lack of a secondary annuity market not allowing an estimation of the market price of risk © 2010 by Taylor and Francis Group, LLC 360 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling REFERENCES ANIA 2005 IPS55 Basi demografiche per le assicurazioni di rendita Consultation document Azzopardi, M 2005 The lo ngevity b ond Presentation a t t he F irst International Conference on Longevity Risk and Capital Market Solutions, London, U.K Bauer, D., Boerger, M., and Russ, J 2008 On the pricing of longevity-linked securities Working Paper, University of Ulm, Ulm, Germany, available at: www mortalityrisk.org Biffis, E 2005 Affine processes for d ynamic mortality and ac tuarial valuations Insurance: Mathematics and Economics 37: 443–468 Biffis, E and Blake, D 2009 Mortality-linked securities and derivatives Discussion Paper PI-0901, Pensions Institute, London, U.K Biffis, E a nd D enuit, M 2006 L ee–Carter g oes r isk-neutral: An a pplication t o the Italian annuity market Giornale dell’Istituto Italiano degli Attuari LXIX: 33–53 Biffis, E and Millossovich, P 2006 The fair value of guaranteed annuity options Scandinavian Actuarial Journal 1: 23–41 Biffis, E., D enuit, M., a nd D evolder, P 2005 S tochastic mo rtality under me asure changes Discussion Paper PI-0512, Pensions Institute, London, U.K Blake, D and Burrows, W 2001 Survivor bonds: Helping to hedge mortality risk Journal of Risk and Insurance 68: 339–348 Blake, D., Cairns, A J G., and Dowd, K 2006a L iving with mortality: Longevity bonds a nd o ther mo rtality-linked s ecurities British A ctuarial J ournal 12: 153–228 Blake, D., Cairns, A J G., D owd, K., a nd MacMinn, R 2006b Longevity bonds: Financial en gineering, val uation a nd hedgin g The J ournal o f R isk a nd Insurance 73(4): 647–672 Booth, H., H yndman, R J., Tickle, L., a nd de J ong, P 2006 L ee–Carter mortality f orecasting: A m ulti-country co mparison o f va riants a nd ext ensions Demographic Research 15: 289–310 Box, G E and Jenkins, G M 1976 Time Series Analysis: Forecasting and Control, Holden-Day, San Francisco, CA Brouhns, N., Denuit, M., and Vermunt, J K 2002 A Poisson log-bilinear approach to t he co nstruction o f p rojected lif e t ables Insurance: M athematics a nd Economics 31: 373–393 Brouhns, N., D enuit, M., a nd Van Keilegom, I 2005 B ootstrapping the Poisson log-bilinear model for mortality forecasting Scandinavian Actuarial Journal 3: 212–224 Cairns, A J G., Blake, D., and Dowd, K 2006a Pricing death: Frameworks for the valuation and securitization of mortality risk Astin Bulletin 36: 79–120 Cairns, A J G., Blake, D., and Dowd, K 2006b A two-factor model for stochastic mortality with parameter uncertainty: Theory and calibration The Journal of Risk and Insurance 73(4): 687–718 Cairns, A J G., B lake, D., and Dowd, K 2008a M odelling and management of mortality risk: A review Scandinavian Actuarial Journal 2–3: 79–113 © 2010 by Taylor and Francis Group, LLC The Securitization of Longevity Risk in Pension Schemes ◾ 361 Cairns, A J G., B lake, D., D owd, K et al 2008b Mortality den sity f orecasts: An analysis of six st ochastic mortality models Discussion Paper PI-0801, Pensions Institute, London, U.K CEIOPS (C ommittee o f E uropean I nsurance a nd O ccupational P ensions Supervisors) 2007 Third Quantitative Impact Study (QIS3), Term structures, available on the web site: http://www.ceiops.org/ Coughlan, G., Epstein, D., Ong, A et al 2007a LifeMetrics: A toolkit for measuring a nd ma naging lo ngevity a nd mo rtality r isks Technical cument, J.P Morgan Pension Advisory Group, available at http://www.jpmorgan.com/ Coughlan, G., Epstein, D., Sinha, A., and Honig, P 2007b q-forwards: Derivatives for transferring longevity and mortality risks J.P Morgan Pension Advisory Group, London, U.K Cowley, A a nd Cummins, J D 2005 S ecuritization of life insurance assets and liabilities Journal of Risk and Insurance 72: 193–226 Cox, S.H., Lin, Y., and Wang, S 2006 Multivariate exponential tilting and pricing implications for mortality securitization Journal of Risk and Insurance 73(4): 719–736 Cox, S H a nd L in, Y 2007 N atural hedgin g o f lif e a nd a nnuity r isks North American Actuarial Journal 11: 1–15 Dahl, M 2004 S tochastic mo rtality in lif e in surance: M arket r eserves a nd mortality-linked insurance contracts Insurance: Mathematics and Economics 35: 113–136 Dahl, M a nd M øller, T 2006 V aluation a nd hedgin g o f lif e in surance r isks with systematic mortality risk Insurance: Mathematics and Economics 39: 193–217 Dahl, M., M elchior, M., a nd Møller, T 2008 On syst ematic mortality risk and risk-minimization wi th sur vivor swa ps Scandinavian A ctuarial J ournal 108(2–3): 114–146 Dawson, P., Dowd, K., C airns, A J G., a nd Blake, D 2008 C ompleting the survivor der ivatives ma rket: A g eneral p ricing f ramework, Discussion P aper PI-0712, Pensions Institute, London, U.K Denuit, M., D evolder, P., and Goderniaux, A C 2007 S ecuritization of longevity risk: Pricing sur vivor bonds with Wang transform in t he Lee–Carter framework Journal of Risk and Insurance 74(1): 87–113 Dowd, K., Blake, D., Cairns, A J G., and Dawson, P 2006 Survivor swaps Journal of Risk and Insurance 73: 1–17 Efron, B 1979 Bootstrap methods: Another look for the jack-knife The Annals of Statistics 7: 1–26 HMD 2004 Human Mortality Database, University of California, Berkeley (USA) and Max Planck Institute for Demographic Research, Rostock (Germany), available at: http://www.mortality.org or http://humanmortality.de ISTAT 2002 P revisioni della p opolazione residente p er s esso, et e r egione dal 1.1.2001 al 1.1.2051 Settore Popolazione ISTAT, Roma Keilman, N a nd Pham, D Q 2006 P rediction intervals for L ee–Carter-based mortality f orecasts Proceedings of the Eur opean P opulation C onference 2006, Liverpool, U.K © 2010 by Taylor and Francis Group, LLC 362 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling Koissi, M C., S hapiro, A F., and Högnäs, G 2006 E valuating and extending the Lee–Carter model for mortality forecasting: Bootstrap confidence intervals Insurance: Mathematics and Economics 38(1): 1–20 Lee, R D and Carter, L R 1992 Modelling and forecasting U.S mortality Journal of the American Statistical Association 87: 659–675 Levantesi, S and Torri, T 2008a Setting the hedge of longevity risk through securitization Proceedings of the 10th Italian-Spanish Congress of Financial and Actuarial Mathematics Cagliari, Italy, June 23–25, 2008 Levantesi, S., Menzietti, M., and Torri, T 2008b Longevity bonds: An application to the Italian annuity market and pension schemes Proceedings of the 18th International AFIR Co lloquium Rome, Italy, September 30th–October 3rd, 2008, available at: http://actuaries.org/AFIR/Colloquia/Rome2/papers.cfm Lin, Y and Cox, S H 2005 Securitization of mortality risks in life annuities Journal of Risk and Insurance 72(2): 227–252 Milevsky, M A and Promislow, S D 2001 Mortality derivatives and the option to annuitise Insurance: Mathematics and Economics 29: 299–318 Nucleo di oss ervazione della d urata di vi ta dei p ercettori di r endite 2008 L e tendenze demogra fiche dei p ercettori di r endite in I talia 1980–2004 e proiezione 2005–2035 Consultation document, available at: http://www ordineattuari.it Olivieri, A and Pitacco, E 2006 L ife annuities and longevity dynamics Working Paper 36, Centro di Ricerche Assicurative e Previdenziali, Università Bocconi, Milano, Italy Renshaw, A E and Haberman, S 2003 Lee–Carter mortality forecasting with age-specific enha ncement Insurance: M athematics a nd Ec onomics 33: 255–272 Renshaw, A E and Haberman, S 2006 A cohort-based extension to the Lee–Carter model for mortality reduction factors Insurance: Mathematics and Economics 38: 556–570 Schrager, D F 2006 Affine stochastic mortality Insurance: M athematics a nd Economics 38: 81–97 Wang, S 2002 A uni versal framework for pricing financial and insurance risks ASTIN Bulletin 32(2): 213–234 © 2010 by Taylor and Francis Group, LLC ... decomposed into model risk and parameter risk, which combined are referred to as uncertainty risk, alluding to the uncertainty in the representation of a phenomenon Under the heading of uncertainty risk. .. to the longevity risk in the recent years Th is is also the case in Italy where it has been fi nally observed in the development of the annuity market Indeed, before the 1990s, when major pension. .. provider has the possibility to hedge the systematic mortality risk inherent in their annuities These contracts are also interesting from the investor’s point of view, since they allow an investor