A Two-Factor Model for Stochastic Mortality with Parameter Uncertainty: Theory and Calibration

Introduction

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, twenty 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 article we will use the following conventions.

The idea of using the capital markets to securitize and trade specific insurance risks is relatively new, and picked up momentum in the 1990s with a number of securitizations 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: specifically 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 bonds was first published in the Journal of Risk and Insurance in 2001.² Longevity bonds are annuity bonds whose coupons are not fixed over time, but fall in line with a given survivor index.³ 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 percentage 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 twenty-five years. Its coupon payments were to be linked to a survivor index based on the realized 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 article) 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 in Appendix A. Our article 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, et. al. (2005), and Blake, Cairns, and Dowd (2006).

A variety of approaches have been proposed for modelling the randomness in aggregate 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 statistical 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; Currie, Durban, and Eilers, 2004) worked in discrete time. Models following the approach of Lee and Carter typically adapt discrete-time time series models to capture the random element in the stochastic development of mortality 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; Schrager, 2006). For further discussion and a review of previous work, the reader is referred to Cairns, Blake, and Dowd (2006).

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 relative intractability at the present time of such models is hindering their practical implementation. In this article, practical implementation of a model and statistical analysis are 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”) approach 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 that the hypothesis are constant.

The layout of this article is as follows. The “Model Specification” section outlines the model. The “Stochastic Mortality” section fits the model to English and Welsh mortality data, and discusses the plausibility of the fit. The next section 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. “The Price of Longevity Risk” 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. “The Risk Premium on New Issues” shows how the earlier results might be used to price new longevity bonds with different terms to maturity and following different cohorts. “Sensitivity to the EIB Interest Rate” comments briefly on sensitivity of the results to changes in interest rates. In the following section, 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 the “Alternative Models” section, we give a brief discussion of alternative models including some comments on the cohort effect. The final section concludes.

Model Specification

By analogy with interest-rate terminology, Cairns, Blake, and Dowd (2006) used the following notation for forward survival probabilities

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 be the filtration generated by the development of the mortality curve up to time u.⁵ Then

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 to be the realized survival probability for the cohort aged x at time 0. Additionally, define the realized mortality rate .

In this article, we adopt the following model⁶ for the mortality curve:

(1)

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, q_y=a) but the chosen curve gives a significantly better fit, especially for higher ages.

Stochastic Mortality

Estimated values for A₁(t) and A₂(t) for the years 1961–2002 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 drift. Specifically,

(2)

where μ is a constant 2 × 1 vector, C is a constant 2 × 2 upper triangular matrix⁹ and Z(t) is a two-dimensional standard normal random variable. If we use data from 1961 to 2002 (41 observations of the differences) we find that

(3)

If, on the other hand, we use data from 1982 to 2002 only (20 observations) then we find that

(4)

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.

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 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 as well.

Cohort Dynamics

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 analyzing the force of mortality and its dynamics over time. However, in the present article, we have chosen to work in discrete time, so we will consider the dynamics of for a cohort aged x at time 0. It simplifies matters if we consider

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).

Simulation Results for The Survivor IndexS(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 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 estimates for μ and V based on data from 1961–2002 to 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 to 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 future.

• •

  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 probabilities incorporate the compounding of year-by-year mortality shocks: the survival probability for year t depends on shocks applied to mortality rates in each of the years 1 to t, and each individual shock affects survival probabilities in all subsequent years.¹³

Results with Parameter Uncertainty

We consider next the impact of parameter uncertainty. It is clear that we have a limited amount of data and so the parameter estimates above must inevitably be subject to some degree of uncertainty. We will analyze this using standard Bayesian methods.¹⁴

Recall that we have assumed that the process A(t) is subject to i.i.d. multivariate normal shocks with mean μ and covariance matrix V. In the absence of any clear prior beliefs about the values of μ and V we will use a non-informative prior distribution. A common prior for the multivariate normal distribution in which both μ and V are unknown is the Jeffreys prior

where |V| is the determinant of the matrix V.¹⁵ With this prior and with n observations D={D(1), … , D(n)} (where D(t) =A(t) −A(t− 1)), it is known that (see, for example, Gelman et al., 1995) the posterior distribution¹⁶ for μ, V |D is:

(5)

(6)

In what follows, we will restrict ourselves to an analysis based on data from 1982 to 2002.

For each simulated sample path of A(t), we simulate first μ and V from the Normal-Inverse-Wishart distribution and use these values for the whole of that sample path. The results of these simulations can be seen in Figures 5 and 6. In Figure 5, we can see the impact of parameter uncertainty on the confidence interval: specifically that parameter uncertainty becomes much more significant as a source of uncertainty in S(t) as t increases. We can see that 25 years ahead parameter uncertainty accounts for about half of the uncertainty in S(t).¹⁷ In Figure 6, we plot the variance of log S(t), and the use here of a log scale allows us to see clearly that for smaller values of t parameter uncertainty is much less important (that is, the difference between the two curves is quite small).

The Price of Longevity Risk

Now consider the price that a life office or pension fund might be prepared to pay to lay off its exposure to longevity risk. From 3-6 we can infer that if premiums are to be paid in respect of each future year, the premium will be much larger for the 25-year payment than, say, the 10-year payment. Furthermore, a reasonable proportion of this premium might be in respect of the desire to eliminate exposure to parameter uncertainty.

Pricing Using Risk-Adjusted Probability Measures

We propose to specify the dynamics under a risk-adjusted pricing measureQ that is equivalent to, in the probabilistic sense, the current real-world measure (which we shall refer to as P).¹⁸ The measure Q is also commonly referred to as the risk-neutral measure¹⁹ or as an equivalent-martingale measure.

Recall that we worked earlier with the following dynamics under P:

where Z(t+ 1) is a standard 2-dimensional normal random variable under P.

Under the risk-adjusted measure Q(λ) we propose²⁰ that:

In this equation is a standard two-dimensional normal random variable²¹ under Q. The vector λ= (λ₁, λ₂) represents the market prices of longevity risk associated with the processes Z₁(t) and Z₂(t), respectively. Under the chosen decomposition for the matrix C (upper triangular), λ₁ is associated with level shifts in mortality (specifically log q/p), while λ₂ is associated with a tilt in log q/p. We assume (as part of our model) that λ is constant rather than time dependent: indeed it is difficult to propose anything more sophisticated for λ in the absence of any market price data.

We can make the following points about Q(λ):

• •

  Complete market models such as the Black–Scholes option-pricing model force upon us a unique choice of measure Q. In contrast, here we have an incomplete market, and a range of possibilities for Q(λ).

• •

  If there exists some form of market in mortality-linked securities then the choice of Q(λ) needs to be consistent with this (limited) market information (so that theoretical prices under Q(λ) match observed market prices).

• •

  Beyond these restrictions, the choice of Q(λ) becomes a modelling assumption. Thus, here we have postulated that the market price of risk, λ, might be constant over time (in the same way that the market price of risk is normally assumed to be constant in the Black–Scholes model).²²

• •

  The assumptions embedded in Q(λ) form a testable hypothesis. However, the assumptions can only be tested once a market develops in a range of mortality-linked securities and with sufficient liquidity over time that historical price data can be used to test if the assumption that the market price of risk is constant or time varying.

Example: The EIB/BNP Longevity Bond

As an example, consider the 25-year EIB/BNP longevity bond announced in November 2004, with an issue price based on a yield of 35 basis points below LIBOR. The appropriate starting point is the EIB curve for conventional fixed-interest bonds issued typically at 15 basis points below LIBOR. This means that the new longevity bond was priced at 20 basis points below standard EIB rates. This spread below standard EIB rates will be denoted by δ in the equations that follow. We will now make the following assumptions:

• •

  The projected survival rates used in the pricing of the bond (in the case of the EIB/BNP bond, this is the projection made by the UK Government Actuary's Department) are unbiased estimates at time 0 under the real-world measure P of the survival rates.

• •

  The spread of 20 basis points below the standard EIB curve is accounted for entirely by the market price of longevity risk.

• •

  The development of mortality rates over time is independent of the dynamics of the interest-rate term structure over time.²³

We will refer to as the survivor index based upon the latest GAD projections available at time 0.²⁴ Assumption 1 implies that .

Next, let us refer to P(0, T) as the price at time 0 of a fixed-principle zero-coupon bond issued by the EIB that pays 1 at time T. The basis declared by the EIB and BNP for the initial price of the bond was , where δ is the spread (expressed as a continuously compounding rate) below the EIB curve used in pricing the bond.²⁵ Given assumption 1 this is equivalent to²⁶

(7)

Values for based upon parameters in (4) and without parameter uncertainty are given in Table 1, column 1. For example, if we assume that implied EIB zero-coupon prices are given by P(0, T) = 1.04^−T and if we set δ= 0.0020 and λ= (0, 0)′ then we find that the price at issue of the bond on this contractual basis (Equation 7) is V(0) = 11.442.

Table 1.
Longevity Bond Expected Cashflows Under the Risk-Neutral Measure Q(λ): and Market Prices Under Various Assumptions for the Market Prices of Longevity and Parameter Risk

	Column Reference
	1	2	3	4	5	6	7
Parameter uncertainty:	N	Y	N	N	N	Y	Y
λ1	0	0	0.375	0	0.175	0	0
λ2	0	0	0	0.316	0.175	0	0
λ3	–	0	–	–	–	1.684	0
λ4	–	0	–	–	–	0	1.419
t
1	0.9836	0.9836	0.9837	0.9836	0.9836	0.9837	0.9836
2	0.9661	0.9661	0.9664	0.9662	0.9663	0.9664	0.9662
3	0.9475	0.9475	0.9482	0.9477	0.9479	0.9482	0.9477
4	0.9278	0.9278	0.9289	0.9281	0.9285	0.9289	0.9281
5	0.9068	0.9068	0.9086	0.9074	0.908	0.9086	0.9074
6	0.8845	0.8845	0.8872	0.8856	0.8863	0.8872	0.8856
7	0.861	0.8609	0.8646	0.8626	0.8635	0.8646	0.8626
8	0.836	0.8359	0.8408	0.8384	0.8395	0.8407	0.8383
9	0.8095	0.8095	0.8157	0.8129	0.8142	0.8156	0.8129
10	0.7816	0.7815	0.7893	0.7862	0.7877	0.7892	0.7861
11	0.7522	0.752	0.7616	0.7583	0.7599	0.7615	0.7582
12	0.7213	0.721	0.7326	0.7292	0.7308	0.7325	0.729
13	0.6888	0.6885	0.7023	0.6989	0.7004	0.7021	0.6987
14	0.6548	0.6545	0.6707	0.6675	0.6689	0.6704	0.6672
15	0.6195	0.6191	0.6378	0.635	0.6362	0.6374	0.6346
16	0.5828	0.5823	0.6036	0.6015	0.6024	0.6032	0.6011
17	0.5448	0.5443	0.5684	0.5672	0.5676	0.5679	0.5667
18	0.5059	0.5052	0.5321	0.5321	0.532	0.5315	0.5316
19	0.4661	0.4654	0.495	0.4965	0.4957	0.4944	0.4959
20	0.4258	0.4251	0.4573	0.4606	0.459	0.4566	0.4599
21	0.3853	0.3847	0.4191	0.4245	0.422	0.4185	0.4238
22	0.345	0.3445	0.3809	0.3885	0.3851	0.3803	0.3879
23	0.3054	0.305	0.3428	0.353	0.3486	0.3424	0.3524
24	0.2667	0.2668	0.3054	0.318	0.3128	0.3052	0.3177
25	0.2297	0.2302	0.2689	0.2841	0.278	0.269	0.284
Price
δ= 0	11.240	11.237	11.442	11.442	11.442	11.439	11.439
δ= 0.0020	11.442	11.439	–	–	–	–

The risk-adjusted approach to pricing assumes that (exploiting assumption 3 above)

(8)

A comparison of Equations (7) and (8) shows that δ can be interpreted as an average risk premium per annum. We shall see later that this risk premium will depend upon the term of the bond and on the initial age of the cohort being tracked.

We can now ask the question: What values for the market prices of risk λ₁ and λ₂ satisfy V_λ(0) =V(0)? Put another way, under what circumstances does the risk-adjusted price (Equation 8) match the issue price quoted in the contract (Equation 7)?

With no parameter uncertainty, and δ= 0 we found that we could obtain V_λ(0) = 11.442 with (λ₁, λ₂) = (0.375, 0) and (0, 0.316). For these two values for λ the values for are given in Table 1 columns 3 and 4. In column 5, we have also given an intermediate value for λ between these two extremes.²⁷ Here we can achieve V_λ(0) =V(0) with λ₁=λ₂= 0.175.^28,29

We next introduce parameter uncertainty into the analysis. We first simulate under P with full parameter uncertainty and the values for are given in Table 1, in column 2 with λ= (0, 0, 0, 0)′. We have seen in Figure 6 that parameter uncertainty presents a significant risk to annuity providers. It follows that they will be prepared to pay a premium to reduce this risk in the same way that they are prepared to pay to reduce the impact of longevity risk.

In this analysis, we concentrate on introducing a market price of risk for the mean μ. In the case of no parameter uncertainty we assume that . With parameter uncertainty, the basic model (see Appendix B) simulates first V, then calculates the upper-triangular matrix C that satisfies V=CC′, and then simulates

where Z_μ is a standard bivariate normal random variable. Once again we have to identify possible equivalent measures. We propose here a similar restriction that, under the equivalent measure, Z_μ is still bivariate normal with unit variances but a shifted mean. Thus we now simulate μ with reference to two additional market prices of risk λ₃ and λ₄:

where λ_μ= (λ₃, λ₄)′ and .

We now have four market prices of risk to play with to match the single price derived by discounting expected cashflows under P at EIB minus 20 basis points. With parameter uncertainty included, the expected cashflows under P change very slightly (see Table 1, column 2), as does the price of V(0) = 11.439. The values for λ₁ and λ₂ required to match this price are essentially unchanged from the values that were determined before (Table 1, columns 3 and 4) and are consequently not repeated in the table. The required values for λ₃ and λ₄ were, respectively, 1.684 and 1.419, with the corresponding values for quoted in Table 1, columns 6 and 7.

For the various cases presented in Table 1 we have plotted in Figure 7 the expected value under P or Q(λ) of S(t) for t= 1, … , 25. This plot helps us to analyze the impact of using the different measures and, in particular, to see where most of the additional value in the longevity bond resides. The expected values in the upper plot show us two things. First, the inclusion of parameter uncertainty has almost no effect on the expected values under P. Second, the expected values under the different Q(λ) measures look similar, and all show up the largest differences compared with the P measure near t= 25.

The lower plot in Figure 7 allows us to differentiate more easily between the different Q(λ) measures. Here we have plotted the expected risk premium per annum on a zero-coupon longevity bond that makes a single payment of S(t) at time t that is held from time 0 through to maturity at t. This is calculated by converting the ratio of two expected values into an additional rate of return per annum as follows:

From this lower plot we can see that the level of the risk premium depends to some extent on the choice of Q(λ). However, we can note from both Figure 7 and Table 1 that cases 3 and 6 produce very similar results and that cases 4 and 7 also produce very similar results.³⁰ The reason for these similarities is not, in fact, difficult to explain. Recall that we have

Thus, relative to simulation under P with parameter uncertainty, when we simulate under Q(λ), λ₁ will have the same effect as n^−1/2λ₃ and λ₂ will have the same effect as n^−1/2λ₄. We can check this by comparing the values of λ₁ to λ₄ in Table 1. Projections have been made on the basis of n= 20 observations of A(t+ 1) −A(t), and we can see that the ratios of λ₃ to λ₁ and λ₄ to λ₂ are both close to (20)^1/2 as predicted.

Now return to the results presented in Table 1. Why are the required values of λ₁ to λ₄ positive? And why does the average risk premium per annum plotted in Figure 7 differ in the way that it does for λ₁ and λ₂ (curves A and B)?

To answer these questions, we need to analyze the impact on the mortality curve of changes in A₁(t) and A₂(t). Recall that we have

Now, if we replace A₁(t) +A₂(t)(x+t) by where and , then for a suitable choice of x₀ (specifically, ) the processes and become independent random walks with drift. has Z₁(t) as its driver with market price of risk λ₁ and has Z₂(t) as its driver with market price of risk λ₂. Under this transformation we have

We can now see that, since both diagonal elements of are positive, a positive shock Z₁(t) will produce a level shift in over all ages x: that is, an unanticipated deterioration in longevity. A positive value of λ₁, in contrast, causes to be pushed downwards over time thereby enhancing improvements in longevity. So λ₁ > 0 is required to produce a positive risk premium (that is, higher expected values of S(t) under Q(λ)).

We have to be slightly more careful when we analyze the impact of positive shocks Z₂(t). Specifically, for x+t above age x₀= 62.2, a positive value for Z₂(t) will increase (particularly so at high ages). However, the same positive value for Z₂(t) will cause to fall for values of x+t less than 62.2. Now in our analysis we are considering a cohort who are all aged 65 at time 0, so that S(T) is constructed from the experienced mortality rates . Since the minimum age is 65, a positive shock in Z₂(t) will cause an increase in each of , everything else being equal. Thus, we infer that λ₂ must also be positive to produce a positive risk premium.

This discussion also helps us to explain the difference between the curves corresponding to (λ₁, λ₂) = (0.375, 0) and (λ₁, λ₂) = (0, 0.316) in the lower half of Figure 7. Specifically risk adjustments to the dynamics of A₂(t) through the use of λ₂ have proportionately a much greater effect on higher-age mortality than adjustments to A₁(t) through λ₁. This means that the probability of survival to higher ages is much more sensitive to λ₂ than to λ₁. Thus we see that curve A in Figure 7 corresponding to λ₂ is flatter than curve B initially but then picks up at a much faster rate, ending up at a higher level.

The Risk Premium on New Issues

The announcement in 2004 of the 25-year EIB longevity bond will, we hope, be followed by other issues with different maturity dates and which will follow different cohorts.

Recall that the 25-year bond following the age-65 cohort (we will refer to this as the (T= 25, x= 65) bond), had a 20 basis-point risk premium per annum. The question now is: What risk premiums are appropriate for bonds with different terms to maturity or that follow older or younger cohorts? It is important to address this question to ensure that possible future bonds are priced in a consistent fashion.

This question can be answered relatively easily. The key is that the market prices of risk λ₁ and λ₂ used in pricing the (T, x) bond must be the same as those used in pricing the (25, 65) bond. Thus for each (T, x) we calculate the price of the bond by determining expectations under Q(λ) and then discounting at EIB rates as before. We then calculate the price of the bond using expectations under P, but then discounting at EIB rates minus the risk premium δ as in Equation (7). We then need to find the value of δ that equates the two prices under P and Q(λ).³¹

Recall that the only longevity bond so far proposed does not allow us to determine λ uniquely. Instead, for any other proposed bond (T, x), the risk premium δ(T, x, λ) will depend on λ.

Risk premia on (T, x) bonds are given in Tables 2, 3, and 4.³² We can make the following observations:

Table 5 shows the impact in the truncated expected future lifetime e(x, T) when we move from the real-world measure, P, to the risk-neutral measure Q(λ) when λ= (0.175, 0.175).³⁴ The trends in this table match those in Table 4 with the exception of the trend along the diagonal from (x, T) = (60, 30) to (70, 20) where the trend is reversed. As we move upwards along the diagonal we have the same two factors working in opposite directions as before: decreasing term and increasing age. In Tables 2–4, the impact of discounting was sufficient to allow the increasing-age effect to dominate. In Table 5 the absence of discounting means that the decreasing-term effect is dominant.

Table 5.
Effect of the Change of Measure from P to Q(λ) for λ= (0.175, 0.175) on Expected Future, Truncated Lifetimes: . Upper Part of Table Shows e(x, T) Under the Real-World Measure P and the Lower Part of the Table Shows the Increase in e(x, T) When We Change to the Risk-Neutral Measure Q(λ).

		Initial Age of Cohort, x
		60	65	70
		eP(x, T)
Maximum	20	16.95	15.15	12.74
Years	25	19.59	16.78	13.45
T	30	21.30	17.53	13.64
	∞	22.43	17.79	13.66
		e Q(λ)(x, T) −eP(x, T)
Maximum	20	0.12	0.20	0.28
Years	25	0.28	0.40	0.47
T	30	0.54	0.65	0.60
	∞	1.22	1.02	0.66

Sensitivity to the EIB Interest Rate

We can also investigate the impact of a change in interest rates. Specifically, let us take λ as given but change the EIB interest rate from 4% to 5% per annum. In this case we find that the impact on the risk premium is relatively small. Specifically, if λ= (0.375, 0) then δ(25, 65) = 19.1 basis points and if λ= (0, 0.315) then δ(25, 65) = 18.9 basis points. This reduction in the risk premium reflects the relative lowering, in present-value terms, of the later, more-uncertain cashflows under the bond.

The Sign of the Market Price of Risk

In previous sections, we focused on the EIB/BNP longevity bond and used the information contained in the offer price to make inferences about the market price of risk. We concluded that this particular bond had a negative risk premium: that is, holders of the bond were being asked to pay a premium in order to reduce their exposure to longevity risk.³⁵

Now consider a bond that allows life insurers to hedge their exposure to short-term catastrophic mortality risk in their term-insurance portfolios (for example, the Swiss Re mortality bond issued in 2003³⁶). One might argue that life insurers will be prepared to pay a premium to reduce their exposure to the risk of high mortality rates. Indeed this is the case with the Swiss Re mortality bond (see, for example, Beelders and Colarossi, 2004). The problem is that this suggests that the market prices of risk in our model should take the opposite sign to those estimated in the section titled “Example: the EIB/BNP Longevity Bond.”

We can offer some partial answers to this apparent paradox.

Alternative Models

Conclusions

In this article, we have used a simple two-factor model for the development of the mortality curve over time that seems, nevertheless, to fit the data well. The model allows us to simulate the distribution of a survivor index over various time horizons under both the real-world probability measure and under a variety of possible risk-adjusted measures. By taking expectations under the latter measure, this model enables us to price the longevity risk inherent in longevity bonds, given the known longevity risk premium (of 20 basis points) contained in the world's first longevity bond, namely the November 2004 EIB 25-year bond designed by BNP Paribas with a reference cohort of 65-year-old English and Welsh males. The chosen model is well suited to pricing longevity bonds. For other types of contracts, that involve, for example, derivative characteristics on future mortality rates (such as guaranteed annuity options) models formulated within the forward-mortality model (such as the Olivier–Smith model described in Olivier and Jeffrey, 2004) or the mortality-market model (Cairns, Blake, and Dowd, 2006) frameworks are likely to prove more efficient to implement.

We find that the premium increases with both term and the initial age of the reference cohort. In the latter case, this is caused by the greater volatility that is associated with the higher mortality rates of older people compared with younger people. For example, in the worst-case scenario considered (where the entire longevity risk premium is associated with the second (i.e., volatility) factor), the premium for a 30-year bond with a reference cohort aged 70 is 42.3 basis points.

Another key finding of the article is that the reference cohort's initial age is more important for determining the premium than the bond's maturity. To illustrate, again in the context of the worst-case scenario, the premium for a 20-year bond with a reference cohort aged 70 is 26.1 basis points, whereas the premium for a 30-year bond with a reference cohort aged 60 is 15.0 basis points. This shows that the greater uncertainty in death rates at higher ages dominates the greater discounting of the more distant cash flows of longer maturing bonds.

These findings suggest that open-ended survivor bonds that continue to pay out as long as members of the reference cohort are still alive would not have an excessively high longevity risk premium. However, they might be unattractive in other respects, such as the administrative inconvenience associated with paying very small coupons fifty years or so after the bond was issued. So fixed-term longevity bonds might well dominate for practical considerations. Our results also suggest that fixed-term longevity bonds might also be favored by investors wishing to avoid ultra-long longevity risk being dominated by parameter risk.

We propose in future research to investigate alternatives to the random walk model with drift used here. Possibilities include models drawn from the ARIMA class of time series models.³⁹ By taking this approach, we will be investigating the important issue of model risk in addition to the parameter risk considered in this article.

In this article, we have assumed that the fitted values of A(t) are known with certainty. A further line of research is to relax this assumption and to use instead filtering approaches or Markov Chain Monte Carlo (MCMC) methods to estimate jointly the posterior distribution of the parameters and of the current values of A(t).