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

Keywords: Longevity risk, stochastic mortality, longevity bonds, of risk: the risk of random fl uctuations of the observed mortality around the expected value, the risk of systematic de

The Securitization

of Longevity Risk

in Pension Schemes:

The Case of Italy

Susanna Levantesi, Massimiliano

Menzietti, and Tiziana Torri

CONTENTS

This 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 appro-priate hedging tools

Th e analysis refers to the Italian market adopting a Poisson Lee–Carter model to represent the evolution of mortality We describe the main fea-tures of longevity bonds a nd su rvivor s waps, a nd t he c ritical i ssue of the pricing models due t o t he i ncompleteness of t he mortality-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 Biffi s et al (2005) Finally, we calculate the risk-adjusted market price of a l ongevity bond with con-stant fi xed coupons and of a vanilla survivor swap

Keywords: Longevity risk, stochastic mortality, longevity bonds,

of risk: the risk of random fl uctuations of the observed mortality around the expected value, the risk of systematic deviations generated by an observed mortality trend diff erent f rom t he one forecast, a nd t he risk

of a sudden a nd short-term rise in the mortality frequency Th e risk of random fl uctuations, also called process risk, decreases in severity as the portfolio size increases Th e risk of systematic deviations can be decom-posed 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 lon-gevity risk, generated from possible divergences in the trend of mortality

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 reserving c alculations Unlike i n t he c ase of process r isk, risks of systematic deviations cannot be hedged by increasing the size of the portfolio; it rather increases with it Th e law of large numbers does not apply because the risk aff ects all the annuitants in the same direc-tion Understanding the risk, and determining the assets that the annu-ity providers have to deploy to cover their liabilities is a ser ious issue 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 Th e introduction of a specifi c law to regulate pen-sion funds in 1993 and the subsequent amendments in 2000 and 2005, contributed to the origin of the second and third pillars in the Italian pension s ystem At t he end of 2 008, t here were about 5 m illion i ndi-viduals contributing into pension schemes Out of this 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 capital Moreover, it was decided that only life insurance companies and pension plans with specifi ed characteristics are autho-rized to pay annuities Th e other operators have to transfer the accumu-lated capital to insurance companies at the moment of retirement.

A further development of the Italian annuities market is expected Th is induces pension funds and life insurance companies to be more responsi-ble in the management of the risks In this respect, some steps have already been t aken Th e Italian S upervisory A uthority o f t he I nsurance S ector (ISVAP) i ntroduced a n ew regulation (no 21/2008) a llowing i nsurance companies to revise t he demographic bases up to 3 y ears before retire-ment Consequently, the longevity risk is relegated only to the period of the annuity payment In addition, starting in 1998, the Italian Association

of Insurance Companies (ANIA) has developed projected mortality tables specifi c for the Italian annuities market (e.g., RG48 in 1998 and IPS55 in 2005) Th e 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 ance co mpanies a nd pens ion p lans sh ould m easure a nd ma nage i t To measure t he longevity r isk, a st ochastic mortality model t hat is able to

insur-fi t a nd forecast mortality is needed In t he last decades, ma ny tic m ortality m odels ha 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) Th e reader can also refer

stochas-to Cairns et al (2008b) for a de scription of selection criteria stochas-to choose a mortality model

However, t he pr oduction of pr ojected, a nd e ventually s tochastic, life tables is not suffi cient 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 dif-ferent countries, or through a suitable mix of insurance benefi ts within

a policy or a portfolio Th e more traditional way for transferring risks, through reinsurance, is not a viable solution Actually, reinsurance com-panies are reluctant to take on such a s ystematic and not diversifi able risk, consequently the reinsurance premiums are great An alternative way out is transferring part of the risk to annuitants selling annuities with payments linked to experienced mortality rates within the insured portfolio However, this solution is not always achievable

An a lternative a nd more attractive option may l ie i n t he t ransfer of the longevity risk into fi nancial markets via securitization Securitization

is a p rocess t hat consists i n isolating a ssets a nd repackaging t hem i nto securities that are traded on the capital markets Th e traded securities are dependent on an index of mortality, and are called mortality-linked secu-rities (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 Th ese 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-off s

Several mortality-linked securities have been proposed in the literature: longevity (or mor tality) bonds, survivor (or mor tality) swaps, mor tality futures, mortality forwards, mortality options, mortality swaptions and longevity (or survivor) caps and fl oors (see Blake et al (2006a) and Cairns

et al (2008a) for a detailed description)

Unlike reinsurance solutions, these fi nancial instruments, depending

on the selected index of mortality, involve returns to insurers and pension

funds n ot n ecessarily co rrelated w ith t heir l osses To ach ieve en ough liquidity, t he l ongevity ma rket w ill ha ve t o f ocus o n b road po pulation mortality indices while insurer and pension fund exposures might be con-centrated in specifi c regions or socioeconomic groups Th e fact that the cash fl ows from the fi nancial 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 as survivor swaps, would reduce the basis risk, but would increase the credit risk, the risk that one of the counterparties may not meet its obligations On t he other ha nd, t rading more 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 specifi c insured mortality

Several mortality-linked securities were suggested in the literature, but only f ew p roducts w ere i ssued i n t he ma rket N onetheless, a n i ncreas-ing attention toward mortality-linked securities is witnessed Signifi cant attempts to create products providing an eff ective transfer of the longevity risk have been observed among practitioners and investment banks (Biffi s

and Blake 2009) In March 2007, J.P Morgan launched Lifemetrics, a platform

for measuring and managing longevity and mortality risk (see Coughlan

et al 2007a, 2007b) It provides mortality rates and life expectancies for ferent countries (United States, England, a nd Wales) t hat can be u sed to determine the payoff of longevity derivatives and bonds In December 2007,

dif-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 securi-ties, exist also a l ively debate concerning the choice of the more appro-priate 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 for i nterest-rate der ivatives It is based on t he idea t hat both 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), Biffi s et al (2005), Biffi s 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 of mortality-linked sec urities is sc arcely d eveloped and hence incomplete, making it diffi cult to use arbitrage-free methods and impossible to estimate a unique risk-adjusted probability measure

An alternative approach is the distortion approach based on a d tion operator—the Wang transform (Wang 2002)—that distorts the dis-tribution 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)).

istor-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 aff ecting Italian annu-ity 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 al (2002) Due to the absence of a seco ndary annuities market in Italy necessary to calibrate prices, we extrapolate market data from the reference life table used by the Italian insurance companies for pricing and reserv-ing At the moment, the reference life table is the IPS55, a n ew projected life tables for annuitants developed in 2005 by ANIA (see ANIA 2005) Th e IPS55 is obtained by multiplying the national population mortality projec-tions, 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 ing on longevity bonds and survivor swaps Th e 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 Biffi s

focus-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

Th e 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, numer-ous works recently proposed in the literature were mainly concerned with the inclusion of stochasticity into the mortality models Stochastic mortality models can be f urther divided into continuous time models and discrete time models Th e former models include the one proposed by Milevsky and Promislow (2001), Dahl (2004), Biffi s (2005), Cairns et al (2006b), Biffi s and Denuit (2006), Dahl and Møller (2006), and Schrager (2006)

Th e 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) Th e model has been widely used in actuarial and demographic applications, and it can be co nsidered the 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 frame-work, Renshaw and Haberman’s work (2006) is one of the fi rst works that incorporates a cohort eff ect A more parsimonious model, including also a cohort component was later introduced by Cairns et al (2006b).

Th e 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 of mortality 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 et al (2002) Th ey proposed a diff erent procedure for the estimation of t he model, a lso substituting t he inappropriate assump-tion 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 of Koissi et a l (2006), we suggest a n original st rategy for dealing with u ncertainty u sing n onparametric boo tstrap tech niques o ver e ach component of the Lee–Carter model Th e identifi cation of such sources of uncertainty and their measurements are necessary for a correct manage-ment of the longevity risk

15.2.1 Model Framework and Fitting Method

Let t x q be t he probability t hat a n i ndividual of t he reference cohort, 0

aged x0 at time 0, will die before reaching the age x0 + t Given the

cor-responding survival probability t x p , the stochastic number of survivors 0

Var(l x t) l p x t x (1 t x p , where ) l is the initial number of individuals x0

in the reference cohort Th e expected number of survivors, ˆl x t0+, is obtained with the point wise projection of the death probability t x q ˆ 0

To o btain t he de ath p robabilities, t x q w e m odel a nd f orecast t he 0,

period central death rates at age x and time t, m x (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

Lee–Carter model Th erefore, they assumed that the number of deaths,

D x (t), is a random variable following a Poisson distribution:

N x (t) is the midyear population observed at age x and time t

m x (t) is the central death rate at age x and time t

Th e c entral de ath r ates m x (t) f ollow t he m odel su ggested b y L ee a nd

log-ous age profi le, as the parameter k t changes; and k t 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 infi nite number of possible sets of parameters, which would satisfy Equation 15.2

In order to overcome these problems of identifi ability, two constraints

on the parameters are introduced: ∑t k t =0 and ∑xβ =x 1

Th e parameters can be e stimated by ma ximizing the following likelihood function:

Jenkins 1976) With t he mere ex trapolation of t he t ime factor, k t, it is possible to forecast the entire matrix of future death rates.

15.2.3 Uncertainty

Th e diff erent sources of uncertainty need to be estimated and combined together: the Poisson variability enclosed in the data; the sample variabil-ity of t he parameters of t he Lee–Carter model a nd t he ARIMA model;

the uncertainty of the extrapolated va lues of the model’s time index k t

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 diff erent 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 the topic (Brouhns

et al (2005), Keilman and Pham (2006), and Koissi et al (2006) ) Th e strap is a co mputationally intensive approach used for the construction

boot-of prediction intervals, fi rst proposed by Efron (1979) A r andom vation is generated, wh ich s amples either f rom a n a ssumed pa rametric distribution (parametric b ootstrap), or f rom t he e mpirical di stribution

inno-of past fi tted errors (nonparametric or residual bootstrap) In this second approach, under the assumption of independence and identical distribu-tion 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 fi tted errors

Diff erently f rom previous st udies, we applied a n onparametric strap to compute parameters uncertainty of both Lee–Carter and asso-ciated A RIMA m odels Th e s imulation p rocedure w e f ollowed co nsists

boot-of two parts: fi rst, we evaluated the sampling variability boot-of the estimated coeffi cients of the model, sampling N times from the deviance residuals of the Lee–Carter model; second, for each of the N simulated k t 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

an a rray w ith N · M matrices of future period central death rates Aft er

selecting the diagonal of the matrices, corresponding to the cohort we are

interested in, we simulated from a b inomial distribution P paths of vivals, for each of the chosen diagonal Overall N · M · P simulations of

sur-survivors are performed

15.3 LONGEVITY RISK SECURITIZATION

As a lready s tated in t he introduction, s ecuritization i s a n inn ovative vehicle suggested to transfer the longevity risk into capital market through mortality-linked securities Among the diff erent mortality-linked securi-ties proposed in the literature, we focus our attention on longevity bonds and survivor swaps We will investigate their capability to hedge the lon-gevity risk faced by the Italian annuity providers, and make a comparison between the performances of the two products

Longevity bonds are mortality-linked securities, traded on organized exchanges, structured in a way that the payment of the coupons or prin-cipal is dependent on t he survivors of a g iven cohort in each year Th e literature about longevity bonds is quite extensive Th e fi rst longevity bond was su ggested by Bla ke a nd Burrows (2001), who proposed a l ongevity bond structure with annual payments attached to the survivorship of a reference population Lin and Cox (2005) proposed instead a bo nd with coupon payments equal to t he d iff erence be tween t he realized 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)

Th e 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 ful, academics as well as practitioners have paid considerable attention

unsuccess-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 not correlated enough w ith t he hedger’s mortal-ity experience To deal in part with this problem Cairns et al (2008a) suggested the use of longevity-linked securities built around a spec ial purpose vehicle (SPV) Th e SPV would arrange a swaps with the hedg-ers and then aggregates the swapped cash fl ows to pass them on to the market through a bond

Dowd et al (2006) suggested survivor swaps as a m ore advantageous derivative t han su rvivor b onds Th ey defi ned a su rvivor s wap a s “ an

agreement to exchange cash fl ows in the future based on the outcome of

at least one survivor index.” Th ey argued t hat survivor swaps are more tailor-made securities, which can be a rranged at lower transaction costs and are more easily cancelled than traditional bond contracts Survivor swaps require only the counterparties, usually life insurance companies,

to transfer their death exposure without need for the existence of a liquid market Survivor swaps were discussed in detail by Lin and Cox (2005), Dahl et al (2008), and Dawson et al (2008) Cox and Lin (2007) asserted that survivor swaps can be used by insurance companies to realize a natu-ral hedging through their annuity liabilities and life insurance

Th e fi rst mortality swap announced in the world was between Swiss

Re a nd Friends’ Provident (a U.K l ife assurer) i n April 2007 It was a pure longevity risk transfer in the form of an insurance contract, and was not tied to another fi nancial instrument or transaction Th e swap was based on the 78,000 Friends’ Provident pension annuity contracts written between July 2001 and December 2006 Swiss Re makes payments and a ssumes l ongevity r isk i n ex change f or a n u ndisclosed p remium (for further details see Cairns et al 2008a) Th e fi rst survivor swap traded

in the capital markets was in July 2008 between Canada Life and capital market investors with J.P Morgan as the intermediary (see Biffi s and Blake 2009)

A m ortality s wap c an be g enerally h edged by combining a ser ies of mortality forward contracts with diff erent ages (see Cairns et al 2008a) Th e mortality forwards involve the exchange of a realized mortality rate con-cerning a specifi ed population on the maturity date of the contract, in return for a fi xed mortality rate (the forward rate) agreed at the beginning of the contract Th ey were chosen by J.P Morgan as a solution for transferring longevity risk that on July 2007 announced the launch of the “q-forward” (see Coughlan et al 2007b) Many authors (see, e.g., Coughlan et al (2007b) and Biffi s and Blake (2009)) consider forward contracts as basic building blocks for a number of more complex derivatives

Sections 15.3.1 and 15.3.2 will describe longevity bonds and vanilla survivor swaps more in detail

15.3.1 Longevity Bonds

Th e longevity bond considered in this chapter is structured as the one posed by Lin and Cox (2005) It is a co upon-based longevity bond with coupons equal to t he d iff erence be tween t he realized a nd t he ex pected survivors of a cohort Considering the timing of longevity risk occurrence,

pro-we regard a coupon-at-risk longevity bond more appropriate for matching the potential losses ex perienced by t he i nsurer, t han a p rincipal-at-risk longevity bond.

A simple way to construct a longevity bond is through the tion of the cash fl ows of a st raight bond Given a st raight bond paying a

decomposi-fi xed a nnual c oupon C at each time t and the principal F at maturity T,

a special purpose company (SPC) would be responsible to split the claims into t wo su rvivor-dependent i nstruments, o ne acq uired b y t he i nsurer and the other by the investors (Blake et al 2006b) In formula we have for

t = 1,2,…,T :

t t

where

B t represents the benefi ts received by the annuity provider used to cover

the experienced loss up to the maximum level C

D t represents the payments received by the investors

Note that D t is specular to the payments B t Th erefore, the cash fl ows of the two survivor-dependent instruments depend on the realized mortality at

each future time t.

In the current work, we fi x the value of C at a constant value; however,

diff erent so lutions a re pos sible I n a p revious w ork, L evantesi a nd Torri (2008a) proposed a longevity bond paying annual fi xed coupons, set accord-ing to a predefi ned level of the insurance probability to experience positive losses in each year In this case, the cash fl ows from the bond could match the insurer losses—rather than the amount that the insurance company has

to pay to its annuitants—considerably reducing the cost of the product, still maintaining its effi ciency

Let us consider an annuity provider who has to pay immediate annuities, assumed to be co nstant, to a co hort of l annuitants all aged x x0 0 at initial time Let l x t0+ be the number of survivors to age x0 + t, for t = 1,2,…,ω − x0, where ω is the maximum attainable age in the cohort We set the annual

payment of the individual annuity at 1 monetary unit It follows that in t,

the annuity provider will pay the amount l x t0+, where l x t0+ is a random value

at evaluation time t = 0 L et ˆl x t0+ be the expected number of survivors to

age x0 + t that has been used for computing the premiums paid by the

annui-tants of the reference cohort Th e annuity provider is therefore exposed to the risk of systematic deviations between l x t0+ and ˆl x t0+ at each time t Th os e

deviations represent the losses experienced by the annuity provider at each

time t, used to defi ne the trigger levels in the longevity bond contract Th e

SPC pays the insurer the excess of the actual payments over the trigger, up

to a maximum amount

A coupon-based longevity bond with constant fi xed coupons C generates

the following cash fl ows for the annuity provider:

According to Equation 15.5, the benefi ts received by the investors D t are

equal to C − B t Th e insurer’s cash fl ows, that she has to pay to the

annui-tants at each time t, are off set by the benefi ts B t received from the SPC

Th erefore, the insurer’s net cash fl ows are equal to

Let W be the straight coupon bond price Let P be the premium that

the annuity provider pays to the SPC to hedge his/her longevity risk and

V be the price paid by the investors to purchase the longevity bond issued

from the SPC Th e structure of such transactions is shown in Figure 15.1

Straight bond

Th e SPC meets his/her obligations and has a profi t if P + V ≥ W In this chapter, we assume P + V = W.

15.3.2 Vanilla Survivor Swaps

A swap is an agreement between two counterparties to exchange one or more cash fl ows, where at least one is random Similarly, a survivor swap is

an agreement to exchange at least one random mortality-dependent cash

fl ow A ba sic survivor swap would involve t he exchange at some f uture time, of a single preset payment for a single random mortality-dependent payment (Dowd et al 2006)

A more complex structure, traded under the name of vanilla survivor swaps, is based on the agreement of the two counterparties to swap a series

of mortality-dependent cash fl ows, u ntil t he swap maturity Th ese tracts re semble t he a lready e xisting v anilla i nterest-rate s waps, c onsist-ing of one fi xed leg and one fl oating leg usually related to a ma rket rate

con-In the survivor swap case, the fi xed leg refers to the expected survivors according to a r eference life table, while the fl oating leg depends on the realized survivors at some future time (for other defi nitions, see Cairns

et al (2008a) and Dahl et al (2008))

As previously st ated, a m ortality s wap c an be c reated by combining together various mortality forwards as those launched in the market by J.P Morgan in July 2007, under the name of q-forwards (Coughlan et al 2007a,b) Biffi s and Blake (2009) provide a wide description of q-forwards

depending on the value of the LifeMetrics index.

Diff erent kinds of swaps are suggested in the literature A swap based

on a cohort of individuals from two diff erent countries or regions, ing that longevity risk is diversifi ed internationally A fl oating-for-fl oating survivor s waps, where a n a nnuity provider s waps w ith a l ife company, providing a natural hedging (Lin and Cox 2005)

assum-Survivor swaps have some advantages compared with longevity bonds

In fac t t hey i nvolve lower t ransaction cost s, t hey a re more fl exible and they can be tailor-made to meet diff erent needs, and they do not require the existence of a liquid market (Dowd et al 2006)

We describe now the survivor swap structure according to the notation previously used for longevity bonds Let ˆl x t0+ be t he fi xed payments of the

swap (equal to the expected number of survivors to age x0 + t for t = 1,2,…,

ω − x0 in the reference population) and let π be a fi xed proportional swap mium with positive, zero, or negative values Th e fl oating leg is equal to l x t0+

pre-corresponding to the realized number of survivors to age x0 + t Th e value of

the premium π is set in a way that the swap value is zero at inception, or ferently stated the market value of the fi xed and fl oating payments is equal.

dif-On e ach o f t he pa yment d ates t, t he fi xed pa yer pa ys t he a mount

+

+ π ˆ0

(1 )l to the fl oating payer and receives x t l x t0 + from the fl oating payer Overall the amount of money exchanged is equal to (1+ π)ˆl x t0+ −l if it x t0+

is positive, or l x t0+ − + π(1 )ˆl otherwise (see Dowd et al (2006) and Dahl x t0+

et al (2008)) Th e structure of such transactions is shown in Figure 15.2.15.4 PRICING MODEL

As previously observed, t he choice of a n appropriate pricing model for mortality-linked securities is a complicated issue To deal with the prob-lem of pricing, two alternatives approaches have been proposed in the literature: the distortion and the risk-neutral approach

Th e d istortion a pproach co nsists i n a pplying a d istortion o perator (the so -called Wang t ransform (Wang 2 002)) t o c reate a n eq uivalent risk-adjusted distribution, and obtain the fair value of the security under this risk-neutral measure Examples of this approach include Lin and Cox (2005), Dowd et a l (2006), a nd Denuit et a l (2007) Such a so lu-tion, even if appealing from a practical point of view, has diff erent draw-backs Specifi cally, it does not provide a universal framework for pricing

fi nancial and insurance risks and leads to diff erent market prices of risk when applied to d iff erent ages So it generates a rbitrage opportunities when tr ading m ortality-linked s ecurities o n diff erent c ohorts ( Bauer

et al 2008)

For these reasons, we price the survivor derivatives described above, by using the risk-neutral pricing model that adapt the arbitrage-free pricing framework of the interest-rate derivatives to the mortality-linked deriva-tives Th e price of such derivatives is given by the expected present value of

future cash fl ows, evaluated under a risk-neutral probability measure Q

Th e ch oice o f Q needs to be consistent with the market information,

so that the theoretical prices u nder Q match with the observed

mar-ket prices However, as already pointed out, the specifi cation of the neutral m easure i s p roblematic d ue t o t he l imited a mount o f ma rket

risk-Floating rate payer

information Th is approach is treated by Milevsky and Promislow (2001), Dahl (2004), Da hl a nd Møller (2006), Cairns et a l (2006a), a nd Biffi s

et al (2005), Biffi s and Denuit (2006)

Among the risk-neutral pricing models we will consider the time v ersion o f t he L ee–Carter m odel p roposed b y B iffi s e t a l (2005), extended within the Poisson framework In Biffi s e t al (2005), Biffi s and Denuit (2006), they describe a class of measure changes—i.e., equivalent martingale m easures—under wh ich st ochastic i ntensities o f m ortality remain of the generalized Lee–Carter type Consequently, they provide a risk-neutral version of the standard Lee–Carter model by a change in the intensity process as described in what follows

continuous-Let us defi ne a fi ltered probability space (Ω,,F,P), where all fi ltrations

are assumed to satisfy the conditions of right-continuity and completeness

We focus on a portfolio of insureds aged x i for i = 1,…,n at the time 0

At each time t, for a single insured, the death time is modeled as a F-stopping

time, τi, where τi is a nonnegative random variable and where the fi ltration

F = ( t)t∈[0,T*] carries i nformation about whether τi has occurred or not

by each time t in the interval [0,T*] We assume that F = G ∨ H, where

H =  ∈ is the minimal fi ltration making τi a

stop-ping time and G = ( t)t∈[0,T*] is a strict sub-fi ltration of F carrying tion about mortality dynamics and other relevant factors Under P the n

informa-stopping time, τi, is assumed to have stochastic intensities, µ ( )x i t , following

the generalized continuous-time Lee–Carter model:

esti-Following Biffi s et al (2005), it is possible to defi ne a probability measure

Q equivalent to the physical probability P on the space (Ω,  t ) Under Q, the