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

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339

339
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341
344
345
349
350
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331

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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 ha 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

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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 2 008, t here w ere abo ut 5 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. 2 1/2008) a llowing i nsurance
companies t o r evise t he dem ographic ba ses u p t o 3 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

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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 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). 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

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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 ha 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 ha 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.

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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).

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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 0 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 0 +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 0 is the initial number of individuals

in the reference cohort. The expected number of survivors, lˆx 0 +t, is obtained
with the point wise projection of the death probability t qˆ x 0.
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

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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 = 0 and ∑ x β x = 1.
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 ) .

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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

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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

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The Securitization of Longevity Risk in Pension Schemes ◾ 341

agreement to exchange cash flows in the future based on the outcome of
at l east o ne su rvivor i ndex.” They a rgued t hat su rvivor s waps a re m ore
tailor-made securities, which can be a rranged at lower transaction costs
and a re more e asily c ancelled t han t raditional 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 natural hedging through their annuity liabilities and life insurance.
The first mortality swap announced in the world was between Swiss
Re a nd F riends’ Pr ovident (a U.K. l ife a ssurer) i n A pril 2 007. It wa s a
pure longevity r isk t ransfer i n t he form of a n i nsurance contract, a nd
was not t ied to a nother fi nancial i nstrument or t ransaction. The swap
was ba sed on t he 78,000 Friends’ Provident pension a nnuity 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). The first 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 Biffis and
Blake 2009).
A m ortality s wap c an be g enerally h edged b y co mbining a ser ies o f
mortality forward contracts with different ages (see Cairns et al. 2008a). The
mortality forwards involve the exchange of a realized mortality rate concerning a specified population on the maturity date of the contract, in return
for a fixed mortality rate (the forward rate) agreed at the beginning of the
contract. They 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
Biffis 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 w ill de scribe longevity bonds a nd va nilla
survivor swaps more in detail.
15.3.1 Longevity Bonds
The longevity bond considered in this chapter is structured as the one proposed by L in a nd C ox (2005). It is a co upon-based longevity bond w ith
coupons eq ual t o t he d ifference be tween t he r ealized a nd t he ex pected
survivors of a cohort. Considering the timing of longevity risk occurrence,

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342 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling

we regard a coupon-at-risk longevity bond more appropriate for matching
the po tential l osses ex perienced b y t he i nsurer, t han a p rincipal-at-risk
longevity bond.
A simple way to construct a longevity bond is through the decomposition of the cash flows of a st raight bond. Given a st raight bond paying a
fi xeda nnualc 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 :
C = Bt + Dt (1

5.5)

where
Bt represents the benefits received by the annuity provider used to cover
the experienced loss up to the maximum level C
Dt represents the payments received by the investors
Note that Dt is specular to the payments Bt. Therefore, the cash flows of the
two survivor-dependent instruments depend on the realized mortality at
each future time t.
In the current work, we fix the value of C at a co nstant value; however,
different 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 fixed coupons, set according to a predefined level of the insurance probability to experience positive
losses in each year. In this case, the cash flows 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 efficiency.
Let us consider an annuity provider who has to pay immediate annuities,
assumed to be co nstant, to a co hort of l x 0 annuitants all aged x0 at i nitial
time. Let l x 0 +t be the number of survivors to age x0 + t, for t = 1,2,…,ω − x0,
where ω is the ma ximum attainable age in the cohort. We set the annual
payment of the individual annuity at 1 m onetary unit. It follows that in t,
the annuity provider will pay the amount l x 0 +t, where l x 0 +t is a random value
at evaluation time t = 0. L et ˆl x 0 +t be the expected number of survivors to
age x0 + t that has been used for computing the premiums paid by the annuitants of the reference cohort. The annuity provider is therefore exposed to
the risk of systematic deviations between l x 0 +t and lˆx 0 +t at each time t. Thos e

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The Securitization of Longevity Risk in Pension Schemes ◾ 343

deviations represent the losses experienced by the annuity provider at each
time t, used to define the trigger levels in the longevity bond contract. The
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 fixed coupons C generates
the following cash flows for the annuity provider:
l x 0 +t − lˆx 0 +t > C
− lˆ ≤ C
0
⎧C
⎪⎪
Bt = ⎨ l x 0 +t − lˆx 0 +t

⎪
⎪⎩0

x 0 +t

(15.6)

x 0 +t

l x 0 +t − lˆx 0 +t ≤ 0

According to Equation 15.5, the benefits received by t he investors Dt are
equal to C − B t. The insurer’s cash flows, that she has to pay to the annuitants at each time t, are offset by the benefits Bt received from the SPC.
Therefore, the insurer’s net cash flows are equal to

lˆx 0 +t

⎧l x +t − C
⎪⎪ 0
− Bt = ⎨lˆx 0 +t
⎪
⎪⎩l x 0 +t

l x 0 +t − lˆx 0 +t > C
− lˆ ≤ C
0x 0 +t

l x 0 +t

(15.7)

x 0 +t

− lˆx 0 +t ≤ 0

when no basis risk is involved in the transaction, and the longevity bond
covers the same risk of the insurer.
Let W be the straight coupon bond price. Let P be t he 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. The structure of such transactions is shown in Figure 15.1.

Straight
bond
Price
W
Annuity
provider

Coupon
C
Longevity bond
price V

Premium P
SPC
Payments Bt

FIGURE 15.1

Longevity bond cash flows.

© 2010 by Taylor and Francis Group, LLC

Investors
Coupon Dt


344 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling

The SPC meets his/her obligations and has a p rofit 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 flows, where at least one is random. Similarly, a survivor swap is
an agreement to exchange at least one random mortality-dependent cash
flow. A ba sic su rvivor s wap would i nvolve t he ex change 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 m ortality-dependent c ash flows, u ntil t he s wap ma turity. These contracts re semble t he a lready e xisting v anilla i nterest-rate s waps, c onsisting of one fi xed leg and one floating leg usually related to a ma rket rate.
In t he su rvivor s wap c ase, t he fi xed leg refers to t he ex pected su rvivors
according to a r eference l ife table, while t he floating leg depends on t he
realized survivors at s ome f uture t ime (for ot her definitions, see Cairns
et al. (2008a) and Dahl et al. (2008)).
As p reviously st ated, a m ortality s wap c an be c reated b y co mbining
together va rious mortality forwards as t hose launched i n t he ma rket by
J.P. Morgan in July 2007, under the name of q-forwards (Coughlan et al.

2007a,b). Biffis and Blake (2009) provide a wide description of q-forwards
depending on the value of the LifeMetrics index.
Different kinds of swaps are suggested in the literature. A s wap based
on a cohort of individuals from two different countries or regions, assuming that longevity risk is diversified internationally. A floating-for-floating
survivor s waps, wh ere a n a nnuity p rovider s waps w ith a l ife co mpany,
providing a natural hedging (Lin and Cox 2005).
Survivor swaps have some advantages compared with longevity bonds.
In fac t t hey i nvolve l ower t ransaction cost s, t hey a re m ore flexible and
they can be t ailor-made to meet different 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. L et lˆx 0 +t be t he fixed payments of t he
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 fixed proportional swap premium with positive, zero, or negative values. The floating leg is equal to l x 0 +t
corresponding to the realized number of survivors to age x0 + t. The value of

© 2010 by Taylor and Francis Group, LLC


The Securitization of Longevity Risk in Pension Schemes ◾ 345
(1 + π)lˆx0+t
Fixed rate payer

Floating rate payer
lx0+t

FIGURE 15.2

Survivor swap cash flows.

the premium π is set in a way that the swap value is zero at inception, or differently stated the market value of the fixed and floating payments is equal.
On e ach o f t he pa yment d ates t, t he fi xed pa yer pa ys t he a mount
(1 + π)lˆx0 +t to t he floating payer a nd receives l x 0 +t from t he floating payer.
Overall the amount of money exchanged is equal to (1 + π)ˆlx0 + t − lx0 + t if it
is positive, or lx0 + t − (1 + π)lˆx0 + t otherwise (see Dowd et al. (2006) and Dahl
et al. (2008)). The structure of such transactions is shown in Figure 15.2.

15.4 PRICING MODEL
As p reviously o bserved, t he ch oice o f a n a ppropriate p ricing m odel f or
mortality-linked securities is a complicated issue. To deal with the problem of pricing, two alternatives approaches have been proposed in the
literature: the distortion and the risk-neutral approach.
The d istortion a pproach co nsists i n a pplying a d istortion o perator
(the so -called W ang t ransform (W ang 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 e t a l. (2006), a nd Denuit e t a l. (2007). S uch a so lution, even if appealing from a practical point of view, has different drawbacks. Specifically, it does not provide a universal framework for pricing
fi nancial and insurance risks and leads to different market prices of risk
when a pplied t o d ifferent a ges. S o i t g enerates a rbitrage o pportunities
when tr ading m ortality-linked s ecurities o n different 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 derivatives. The price of such derivatives is given by the expected present value of
future cash flows, evaluated under a risk-neutral probability measure Q.
The ch oice o f Q needs to be consistent with the market information,
so th at th e th eoretical p rices u nder Q m atch wi th th e o bserved m arket prices. However, as already pointed out, the specification of the riskneutral m easure i s p roblematic d ue t o t he l imited a mount o f ma rket

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346 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling

information. Th is approach is treated by Milevsky and Promislow (2001),
Dahl (2004), Da hl a nd M øller (2006), C airns e t a l. (2006a), a nd B iffis
et al. (2005), Biffis and Denuit (2006).
Among the risk-neutral pricing models we will consider the continuoustime v ersion o f t he L ee–Carter m odel p roposed b y B iffis e t a l. ( 2005),
extended within the Poisson framework. In Biffis e t al. (2005), Biffis and
Denuit (2006), they describe a cla ss 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.
Let us define a filtered probability space (Ω,,F,P), where all filtrations
are assumed to satisfy the conditions of right-continuity and completeness.
We focus on a portfolio of insureds aged xi for i = 1,…,n at the time 0.
At each time t, for a single insured, the death time is modeled as aF-stopping
time, τi, where τi is a nonnegative random variable and where the filtration
F = (t)t∈[0,T*] c arries i nformation abo ut wh ether τi ha s occ urred o r n ot
by e ach t ime t in t he interval [0,T*]. We a ssume t hat F = G ∨ H, where
n
H = ∪ i =1H i and H i = (ti )t ∈[0,T *] is the minimal filtration making τi a stopping time and G = (t)t∈[0,T*] is a strict sub-filtration of F carrying information about mortality dynamics and other relevant factors. Under P the n
stopping time, τi, is assumed to have stochastic intensities, µ xi (t ), following
the generalized continuous-time Lee–Carter model:
µ xi (t ) = exp(α xi + t + β xi + t kt )

(15.8)

for some continuous functions α(.) a nd β(.) a nd Rd-valued G-predictable
process k. We assume that k has the dynamics described by the following
stochastic differential equation:

dk t = δ(t , k t )dt + σ(t , k t )dWt (1

5.9)

where W is a d-dimensional standard Brownian motion generating the
filtration G.
In this framework αx and βx of Equation 15.2 are t he point w ise estimates of the functions α(.) and β(.).
Following Biffis et al. (2005), it is possible to define a probability measure
Q equivalent to the physical probability P on the space (Ω, t). Under Q, the

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The Securitization of Longevity Risk in Pension Schemes ◾ 347

n stopping times, τi, have stochastic intensity of the generalized Lee–Carter
type as follows:
µ xi (t ) = µ xi (t )(1 + φi ) (1

5.10)

where φi is a strictly positive process given by
φi = exp(axi +t + bxi +t kt ) − 1 (1

5.11)

Assuming that α = α + a and β = β + b , we can rewrite Equation 15.10 in
the following manner:
µ xi (t ) = exp(α xi + t + β xi + t kt )

(15.12)

Under Q, the dynamics of the time-trend k are described, instead of (15.9)
that holds under F, by the following differential equation:
dkt = [δ(t , kt ) − ηt σ(t , kt )]dt + σ(t , kt )dWt (1

5.13)

The cha nge of measure a ffects t he d rift of t he t ime i ndex k through the
process η as well as the intensity process itself through the process φi.
The dy namics o f µ under Q c an be r epresented, i nstead o f E quation
15.12, by
µ xi (t ) = exp(α xi +t + β xi +t kt )

(15.14)

where k˜t under Q follows the dynamics:
dkt = δ(t , kt )dt + σ(t , kt )dWt

(15.15)

The eq uality be tween E quations 1 5.12 a nd 1 5.14 i s v erified f or
[β x +t + bx +t ]kt = β x +t kt an d bx +t = β x +t [σ(t , kt ) σ(t , kt ) − 1]. Ther efore, to
provide e stimates o f t he f unctions ax+t an d bx+t, a nd t he pa rameter ηt
changing the drift of k under Q, we fi x the parameters βx+t estimated from
mortality data, and estimate the α x +t’s and the time index k˜t from market.
Given k˜t ’s parameters, we are able to calculate ηt:
ηt =

δ(t , kt ) δ(t , kt )

−
σ(t , kt ) σ(t , kt )

© 2010 by Taylor and Francis Group, LLC

(15.16)


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⎟ .
⎝ 0
⎠
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 ⎟ ⎥ .
⎝ 0
⎠⎦
⎣
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

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(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 )
0

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.

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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, 2 004), 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 2 005). 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.

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352 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling

1
0.06
0
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

0

−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

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The Securitization of Longevity Risk in Pension Schemes ◾ 353
0.030
0

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

0

−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 6 5 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 0 = 10,000 Italian males, aged 65 in the
year 2 005. 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 4 0 t imes t he f uture n umber o f su rvivors f rom a b inomial
distribution. O verall w e per formed 6 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 2 005, 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

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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