A Two-Factor Model for Stochastic Mortality with
Parameter Uncertainty: Theory and Calibration
Andrew J.G. Cairnsab, David Blakec, Kevin Dowdd
First version: February 2005
This version: June 28, 2006
In this paper we consider the evolution of the post-age-60 mortality curve in the
UK and its impact on the pricing of the risk associated with aggregate mortal-
ity improvements over time: so-called longevity risk. We introduce a two-factor
stochastic model for the development of this curve through time. The ﬁrst factor af-
fects mortality-rate dynamics at all ages in the same way, whereas the second factor
aﬀects mortality-rate dynamics at higher ages much more than at lower ages.
The paper then examines the pricing of longevity bonds with diﬀerent terms to
maturity referenced to diﬀerent cohorts. We ﬁnd that longevity risk over relatively-
short time horizons is very low, but at horizons in excess of 10 years it begins to
pick up very rapidly.
A key component of the paper is the proposal and development of a method for
calculating the market risk-adjusted price of a longevity bond. The proposed ad-
justment includes not just an allowance for the underlying stochastic mortality but
also makes an allowance for parameter risk. We utilise the pricing information con-
tained in the November 2004 European Investment Bank longevity bond to make
inferences about the likely market prices of the risks in the model. Based on these,
we investigate how future issues might be priced to ensure an absence of arbitrage
between bonds with diﬀerent characteristics.
Mortality risk; longevity risk; Perks model; longevity bond; survivor index; market
price of longevity risk; market price of parameter risk.
aActuarial Mathematics and Statistics, School of Mathematical and Computer Sciences, Heriot-
Watt University, Edinburgh, EH14 4AS, United Kingdom.
bCorresponding author: E-mail A.Cairns@ma.hw.ac.uk
cPensions Institute, Cass Business School, City University, 106 Bunhill Row, London, EC1Y
8TZ, United Kingdom.
dCentre for Risk & Insurance Studies, Nottingham University Business School, Jubilee Campus,
Nottingham, NG8 1BB, United Kingdom.
Recently it has become clear that mortality is a stochastic process: longevity has not
only been improving, but it has been improving, to some extent, in an unpredictable
way. These unanticipated improvements have proved to be of greatest signiﬁcance at
higher ages, and have caused life oﬃces (and pension plan sponsors in the case where
the plan provides the pension) to incur losses on their life annuity business. The
problem is that pensioners are living much longer than was anticipated, say, 20 years
ago. As a result life oﬃces are paying out for much longer than was anticipated,
and their proﬁt margins are being eroded in the process. The insurance industry
is therefore bearing the costs of unexpectedly greater longevity. Looking forward,
possible changes in lifestyle, medical advances and new discoveries in genetics are
likely to make future improvements to life expectancy highly unpredictable as well.
This, in turn, will lead to smaller books of life annuity business, smaller proﬁt
margins or both.
There are a number of possible types of systematic, mortality-related risks that
annuity providers and life insurers are exposed to. For the sake of clarity in this
paper we will use the following conventions:
• The term mortality risk should be taken to encompass all forms of uncertainty
in future mortality rates including increases and decreases in mortality rates.
• Longevity risk should be interpreted as uncertainty in the long-term trend in
mortality rates and its impact on the long-term probability of survival of an in-
dividual. Longevity risk is normally taken to mean the risk that survival rates
are higher than anticipated, although we strictly take it to mean uncertainty
in either direction.
• Short-term, catastrophic mortality risk should be interpreted as the risk that,
over short periods of time, mortality rates are much higher (or lower) than
would normally be experienced. Examples of such ‘catastrophes’ include the
inﬂuenza pandemic in 1918 or the tsunami in December 2004. Once the catas-
trophe has past we expect mortality rates to revert to their previous levels and
to continue along previous trends.1
The idea of using the capital markets to securitise and trade speciﬁc insurance risks
is relatively new, and picked up momentum in the 1990’s with a number of securi-
tisations of non-life insurance risks (see, for example, Lane, 2000). December 2003
saw the issue by Swiss Re of the ﬁrst bond to link payments to mortality risk: specif-
ically short-term, catastrophic mortality risk. A related capital market innovation,
the longevity bond, provides life oﬃces and pension plans with an instrument to
hedge the much-longer-term longevity risks that they face. The idea for longevity
1Note that long-term trends in mortality might, however, be aﬀected by certain types of catas-
trophe. For example, survivors of a severe outbreak of inﬂuenza might be weakened in some way
and more prone in the future to heart disease or cancer. In this sense, catastrophic mortality
events might be correlated with long-term trends.
bonds was ﬁrst published in the Journal of Risk and Insurance in 20012. Longevity
bonds are annuity bonds whose coupons are not ﬁxed over time, but fall in line
with a given survivor index3. For example, the survivor index might be based on
the population of 65-year-olds alive on the issue date of the bond. Each year the
coupon payments received by the life oﬃce or pension plan decrease by the per-
centage of the population who have died that year. If, after the ﬁrst year, 1.5% of
the population of what are now 66-year-olds have died, then the coupon payable at
the end of that ﬁrst year will fall to 98.5% of the nominal coupon rate. But this
is exactly what the life oﬃce or pension plan wants, since only 98.5% of their own
66-year old annuitants (assuming these are representative of reference population)
will be alive after one year, so they do not have to pay out so much.
In November 2004, BNP Paribas (in its role as structurer and manager) announced
that the European Investment Bank (EIB) would issue a longevity bond. The bond
had an initial market value of about £540m and a maturity of 25 years. Its coupon
payments are linked to a survivor index based on the realised mortality experience
of a cohort of males from England & Wales aged 65 in 2003 as published by the UK
Oﬃce for National Statistics (ONS). The intended main investors were UK pension
funds and life oﬃces.4 Although this issue was ultimately unsuccessful, there are
important issues to be learned about how to price such contracts (an issue which
we discuss at length in this paper) and about design issues (which are discussed
elsewhere: see, for example, Blake, Cairns and Dowd, 2006).
The basic cashﬂows under the EIB/BNP longevity bond, ignoring credit risk, are
described Appendix A. Our paper focuses on the mathematical modelling that
underpins the pricing of mortality-linked securities. For a full discussion of the
EIB/BNP bond as well as other types of mortality-linked security the reader is
referred to Cowley and Cummins (2005), Cairns, Blake, Dawson and Dowd (2005),
and Blake, Cairns and Dowd (2006).
A variety of approaches have been proposed for modelling the randomness in aggre-
gate mortality rates over time. A key earlier work is that of Lee and Carter (1992).
Their work focuses on the practical application of stochastic mortality and its sta-
tistical analysis. Aggregate mortality rates are, at best, measured annually and for
this reason Lee and Carter (1992) and later authors who adopted a similar approach
(see, for example, Brouhns, Denuit and Vermunt, 2002, Renshaw and Haberman,
2003, and Currie, Durban and Eilers, 2004) worked in discrete time. Models fol-
lowing the approach of Lee and Carter approach typically adapt discrete-time time
series models to capture the random element in the stochastic development of mor-
tality rates. Other authors have developed models in a continuous-time framework
(see, for example, Milevsky and Promislow, 2001, Dahl, 2004, Dahl and Møller,
2005, Miltersen and Persson, 2005, Biﬃs, 2005, and Schrager, 2006). For further
2Blake and Burrows (2001). See also Cox, Fairchild and Pedersen (2000).
3For this reason they are also known as survivor bonds (e.g. Blake and Burrows, 2001).
4The Swiss Re mortality bond and the EIB longevity bond were the ﬁrst exclusively to trade
mortality risk. However, there have been previous issues of securities that packaged together
several risks including mortality. The motivation for the issue of these securities goes beyond a
desire purely to hedge mortality risk. A full discussion of these securities can be found in Cowley
and Cummins (2005).
discussion and a review of previous work, the reader is referred to Cairns, Blake and
Continuous-time models have an important role to play in our understanding of
how prices of mortality-linked securities will develop over time. However, the rela-
tively intractability at the present time of such models is hindering their practical
implementation. In this paper, practical implementation of a model and statistical
analysis is very much at the forefront of what we wish to achieve. Consequently we
choose to develop a model in discrete time and adopt an approach that is similar in
vein to that of Lee and Carter (1992).
We propose a stochastic mortality model that we ﬁt to UK mortality data and show
how the calibrated model can be used to price mortality-linked ﬁnancial instruments
such as the EIB/BNP longevity bond. The model involves two stochastic factors.
The ﬁrst aﬀects mortality at all ages in an equal manner, whereas the second has an
eﬀect on mortality that is proportional to age. We present empirical evidence that
indicates that both these factors are needed to achieve a satisfactory empirical ﬁt
over the mortality term structure (that is, to model adequately historical mortality
trends at diﬀerent ages). The resulting model dynamics allow us to simulate cohort
survival rates, thereby enabling us to model longevity risk, and to model other
indices underlying alternative mortality-linked securities.
To price a mortality-linked security we adopt the risk-adjusted (or ‘risk-neutral’) ap-
proach to pricing adopted by, for example, Milevsky and Promislow (2001) and Dahl
(2004). Given the current dearth of market data, we propose a simple method for
making the adjustment between real and risk-adjusted probabilities, which involves
a constant market price for both longevity and parameter risk. The magnitude of
this adjustment is established by estimating the market prices of these two risks
implied by the proposed issue price of the EIB/BNP longevity bond.
Once a deep, liquid market in mortality-linked securities develops, however, we will
be able to determine more reliable estimates of these market prices of risk and,
indeed, to test the hypothesis that they are constant.
The layout of this paper is as follows. Section 2 outlines the model. Section 3 ﬁts
the model to English and Welsh mortality data, and discusses the plausibility of the
ﬁt. Section 4 then presents some simulation results for the survivor index based on
the calibrated model. Two alternative sets of simulation results are presented: ﬁrst,
results that do not take account of parameter uncertainty, and, second, results that
do take account of such uncertainty. Section 5 discusses the price of longevity risk
– that is to say, it discusses the premium that a life oﬃce or pension plan might
be prepared to pay to lay oﬀ such risk – and uses this to show how the EIB/BNP
bond might be priced in a risk-adjusted framework. Speciﬁcally we focus on the
market price of risk. It also presents some illustrative pricing results. Section 6
discusses the risk premium on new issues, and shows how the earlier results might
be used to price new longevity bonds with diﬀerent terms to maturity and following
diﬀerent cohorts. Section 7 comments brieﬂy on sensitivity of the results to changes
in interest rates. In Section 8 we discuss whether the market price of risk should
be positive or negative, bearing in mind the requirements of diﬀerent hedgers using
diﬀerent types on mortality-linked contract. In Section 9 we give a brief discussion
2 MODEL SPECIFICATION
of alternative models including some comments on the cohort eﬀect. Section 10
By analogy with interest-rate terminology, Cairns, Blake and Dowd (2006) used the
following notation for forward survival probabilities
p(t, T0, T1, x) = probability as measured at t that
an individual aged x at time 0 and still alive at T0
survives until time T1 > T0.
Let I(u) represent the indicator process that is equal to 1 at time u if the life aged
x at time 0 is still alive at time u, and 0 otherwise. Furthermore let Mu be the
ﬁltration generated by the development of the mortality curve up to time u.5 Then
p(t, T0, T1, x) = P r I(T1) = 1|I(T0) = 1, Mt).
Note that p(t, T0, T1, x) = p(T1, T0, T1, x) for all t ≥ T1, since the observation period
(T0, T1] is then past and not subject to any further uncertainty.
For simplicity in this exposition, we will deﬁne ˜
p(t, x) = p(t + 1, t, t + 1, x) to be the
realised survival probability for the cohort aged x at time 0. Additionally deﬁne the
realised mortality rate ˜
q(t, x) = 1 − ˜
In this paper, we adopt the following model6 for the mortality curve:
q(t, x) = 1 − p(t + 1, t, t + 1, x) =
1 + eA1(t+1)+A2(t+1)(x+t)
In this equation, A1(u) and A2(u) are stochastic processes that are assumed to be
measurable at time u. An example of a mortality curve is given in Figure 1. This
graph shows the ungraduated mortality rates above the age of 60 for England &
Wales males in 20027 along with the ﬁtted curve (ﬁtted using least squares applied
to (1)). The ﬁt is clearly very good. Simpler parametric curves can also be ﬁtted
(for example, qy = aA1+A2y) but the chosen curve gives a signiﬁcantly better ﬁt,
especially for higher ages.
5That is, Mu represents the history of the mortality curve up to time u.
6This is a special case of what are known as Perks models: see, for example, Perks (1932) or
Benjamin and Pollard (1993).
7Available from the Government Actuary’s Department website, www.gad.gov.uk .
2 MODEL SPECIFICATION
Year = 2002
q(t,y) (log scale)
Age of cohort at the start of 2002, y
Figure 1: Ungraduated mortality rates above the age of 60 for England & Wales
males for the year 2002 (dots) and ﬁtted curve eA1+A2y (1+eA1+A2y) for A1 = −10.95
and A2 = 0.1058.
1960 1970 1980 1990 2000
1960 1970 1980 1990 2000
Figure 2: Estimated values of A1(t) (left) and A2(t) (right) in equation (1) from
1961 to 2002 for England and Wales males.
3 STOCHASTIC MORTALITY
Estimated values for A1(t) and A2(t) for the years 1961 to 2002 are plotted in Figure
2.8 These results show a clear trend in both series. The downward trend in A1(t)
reﬂects general improvements in mortality over time at all ages. The increasing trend
in A2(t) means that the curve is getting slightly steeper over time: that is, mortality
improvements have been greater at lower ages. There were also changes in the trend
and in the volatility of both series. To make forecasts of the future distribution of
A(t) = (A1(t), A2(t)) , we will model A(t) as a two-dimensional random walk with
A(t + 1) = A(t) + µ + CZ(t + 1)
where µ is a constant 2 × 1 vector, C is a constant 2 × 2 upper triangular matrix9
and Z(t) is a 2-dimensional standard normal random variable. If we use data from
1961 to 2002 (41 observations of the diﬀerences) we ﬁnd that
−0.000 161 7
, and ˆ
V = ˆ
−0.000 161 7 0.000 002 590
If, on the other hand, we use data from 1982 to 2002 only (20 observations) then we
−0.000 093 9
, and ˆ
V = ˆ
−0.000 093 9 0.000 001 509
These results show a steepening of trends after 1982, with µ1 and µ2 both becoming
larger in magnitude. They also show that the volatilities in the later period were
notably smaller than in the earlier period.
An important criterion for a good mortality model (see, Cairns, Blake and Dowd,
2006, for a discussion) requires the model and its parameter values to be biologically
reasonable.10 The negative value for µ1 indicates generally improving mortality,
with this improvement strengthening after 1982. The positive value for µ2 means
that mortality rates at higher ages are improving at a slower rate. Indeed, above
the very high age of 113, the model predicts deteriorating mortality.11 This might
be perceived to be an undesirable feature of our model, but because this crossover
point is at such a high age it is not felt to be a serious problem here as the number
of lives involved is very low.
8For each t, A1 and A2 were estimated using least squares by transforming the ungraduated
mortality rates from qy to log qy/py = A1 + A2y + error.
9There are inﬁnitely many matrices C that satisfy V = CC , but the choice of C makes no
diﬀerence to our analysis. Provided the entries of C are all real valued, CC is always positive
semideﬁnite. The restriction of C to an upper-triangular triangular form means that C is straight-
forward to derive from V and that this (Cholesky) decomposition is unique.
10Experts in mortality will hold certain subjective views on how mortality rates might evolve
over time or how mortality rates at diﬀerent ages ought to relate to one another. Examples of such
criteria include: a requirement that the mortality curve in each calendar year is increasing with
age at higher ages; models that give rise to a strictly positive probability of immortality should be
ruled out. Many experts would agree with these criteria but others might not.
11In other words the mortality rate at ages > 113 is rising over time rather than lower.
3 STOCHASTIC MORTALITY
An additional criterion for biological reasonableness is that, in any given year in
the future, we should normally see mortality rates for older cohorts that are higher
than those for younger cohorts (that is, for ﬁxed t, ˜
q(t, x) should be an increasing
function of x). This criterion requires A2(t) to remain positive. In our model A2(t)
could, theoretically, become negative, but the positive value for µ2 and the initial
value for A2 in 2002 of 0.1058 means that A2(t) is very unlikely to do so. So the
possibility of a negative A2(t) is of little signiﬁcance and for all practical purposes
our model be regarded as satisfying this second criterion of biological reasonableness
In subsequent sections we will focus on the dynamics of a survivor index, S(t). This
is built up with reference to the mortality rates over time of one speciﬁc cohort,
and it makes sense, therefore, to look at cohort dynamics within the context of our
two-factor model. Investigating cohort dynamics also gives us the opportunity to
make a further check on biological reasonableness.
In some contexts following a cohort might mean analysing the force of mortality and
its dynamics over time. However, in the present paper we have chosen to work in
discrete time, so we will consider the dynamics of ˜
q(t, x) for a cohort aged x at time
0. It simpliﬁes matters if we consider
q(t + 1, x)/˜
p(t + 1, x) = A1(t + 1) + A2(t + 1)(x + t + 1)
= (1, x + t + 1) [A(t) + µ + CZ(t + 1)]
= log ˜
+ (µ1 + µ2 + A2(t)) + (1, x + t + 1) CZ(t + 1).
Now A2(t) is currently around 0.1058 and expected to increase slowly (µ2 > 0).
Furthermore, the standard deviation of A2(t) is very small over the time horizons
we are likely to consider (for example, the standard deviation of A2(25) is 0.006).
Thus µ1 + µ2 + A2(t) is initially positive and is expected to stay positive. As a
consequence, the cohort will experience generally increasing rates of mortality with
occasional falls in years when there is a large random mortality improvement across
the board (that is, when (1, x + t + 1) CZ(t + 1) < 0).
4 SIMULATION RESULTS FOR THE SURVIVOR INDEX S(T )
Simulation results for the survivor index S(t)
A longevity bond of the type proposed by the EIB/BNP indexes coupon payments
in line with a survivor index S(t) for a speciﬁed cohort of individuals.12
We now wish to determine the distribution for S(t) for the times t = 1, 2, . . . , 25 that
are relevant for the EIB/BNP bond. Even though the functional form for ˜
is relatively simple, its distribution for t > 2 is not analytically tractable, so we
resort to Monte Carlo simulation and obtain the simulated ˜
q(t, x) and S(t) from
simulations of the underlying process A(t).
Results with no allowance for parameter uncertainty
In our ﬁrst experiment, we simulated the A(t) according to equation (2) using esti-
mates for µ and V based on data from 1961-2002 and 1982-2002. These parameter
estimates were treated as if they were the true parameter values, implying that, to
begin with, we ignore parameter uncertainty. The results are plotted in Figure 3.
We can make the following observations:
• The solid curves plot the expected values of S(t). Measured at time 0, these
represent the ex ante probabilities of survival from time 0 to time t, p(0, 0, t, 65)
(which we refer to as spot survival probabilities). The mean trajectory based
on data from 1982-2002 (bottom plot) is slightly higher than that in the upper
plot (based on 1961-2002 data). This is because steepening trends in A1(t)
and A2(t) in the 1982-2002 data (Figure 2) signal greater improvements in the
• The dashed curves in each plot show the 5th and the 95th percentiles of the
distribution of S(t). We can observe that the resulting 90% conﬁdence interval
is initially quite narrow but becomes quite wide by the 25-year time horizon
(which is the maturity of the EIB/BNP longevity bond). We can also see that
the conﬁdence interval based on 1982-2002 data is a little narrower, reﬂecting
the smaller values on the diagonal of V .
• The conﬁdence interval for S(t) grows in quite a diﬀerent way from, say, that
associated with an investment in equities. This point is best illustrated by
looking at the variance of the logarithm of S(t), as illustrated in Figure 4.
We can see that this is very low in the early years indicating that we can
predict with reasonable certainty what mortality rates will be over the near
future. However, after time 10 the variance starts to grow very rapidly (almost
‘exponentially’). This contrasts with equities where we would expect to see
linear, rather than ‘exponential’, growth in the variance if the price process
follows geometric Brownian motion.
The explanation for this variance growth is that the longer-term survival prob-
abilities incorporate the compounding of year-by-year mortality shocks: the
12In the case of the EIB/BNP bond the reference cohort is the set of all England and Wales
males aged 65 in 2003. The method used to calculate S(t) for this cohort is given in Appendix A.
4 SIMULATION RESULTS FOR THE SURVIVOR INDEX S(T )
Data from 1961−2002
Data from 1982−2002
Figure 3: Mean and conﬁdence intervals for projected survival probabilities based on
data from 1961-2002 (top) or 1982-2002 (bottom). Each plot shows the mean (solid
curve) and the 5th and 95th percentiles (dashed curves) of the simulated distribution
of the reference index, S(t), with no allowance for parameter uncertainty.