On the Calibration of Mortality Forward Curves

Transcription

1 On the Calibration of Mortality Forward Curves Wai-Sum Chan, Johnny Siu-Hang Li and Andrew Cheuk-Yin Ng Abstract In 2007, a major investment bank launched a product called q-forward, which may be regarded as a forward contract on a mortality rate. The pricing of mortality forwards is similar to the pricing of other forward-rate contracts, such as interest-rate forwards or foreign exchange forwards. In particular, since investors require compensation to take on longevity risk, the forward mortality rate at which q-forward contracts will trade will be smaller than the expected mortality rate. The relationship between the forward rate and the time to maturity is called a mortality forward curve. In this paper, we contribute a method for calibrating mortality forward curves. This method consists of two parts, one of which is the generation of a distribution of future mortality rates, and the other of which is the transformation of the distribution into its risk-neutral counterpart, using the idea of canonical valuation developed by Stutzer (1996, The Journal of Finance, 51, ). To illustrate the method, mortality forward curves for English and Welsh males are calibrated. Keywords: Canonical valuation; Maximum entropy measure; Mortality derivatives; The Cairns-Blake-Dowd model Presenting author, chanws@cuhk.edu.hk. Wai-Sum Chan is Professor of Finance in the Department of Finance at The Chinese University of Hong Kong, Hong Kong, PR China. Johnny Siu-Hang Li is Fairfax Chair in Risk Management in the Department of Statistics at the University of Waterloo, Waterloo, Ontario, Canada. Andrew Cheuk-Yin Ng is Associate Professor in the Department of Finance at The Chinese University of Hong Kong, Hong Kong, PR China. 1

2 1 Introduction The trading of longevity risk, that is, the risk that people live longer than expected, has recently become a hot topic in finance. 1 Through trading in the capital markets, pension plan providers and annuity writers can lay off their longevity risk exposures that introduce uncertainty about the period of time over which benefits must be paid. They view capital market solutions as a plausible alternative to reinsurance, where capacity is often limited, and a good opportunity to enhance their credit profiles. On the other hand, some investors including hedge funds are interested in acquiring an exposure to longevity risk, provided that expected returns are reasonable. This is because longevity securities have a very low correlation to virtually every other asset class in the market, be it inflation, foreign exchange, commodities, equities or credit. 2 The Life and Longevity Markets Association (LLMA) estimates that the potential size of the longevity securities market exceeds 2 trillion in the UK alone, and exceeds $25 trillion worldwide. 3 The first generation of longevity securities is largely bond-based (see, e.g., Blake and Burrows (2001); Blake et al. (2006b)). In November 2004, a longevity bond to be issued by the European Investment Bank (EIB) was announced by BNP Paribas. The bond, with a maturity of 25 years, was intended for UK life insurance companies and pension funds with exposure to longevity risk. Coupon payments from the bond were linked to a survivor index, which was based on the realized mortality rates of a certain cohort of English and Welsh males. The issue, however, was withdrawn in late A possible reason for its failure is that the amount of capital required is high relative to the reduction in risk exposure, making the bond capital-expensive as a risk management tool. The second generation is derivatives-based, usually in the form of a swap or forward (see, e.g., Dowd (2003); Dowd et al. (2006)). Derivative structures are often considered more favorable than bonds because the requirement to invest a large amount of capital up front in purchasing the bond is eliminated. To date we have 1 Discussions on this topic can be found in the following articles: Longevity and Mortality Risk Market Takes Shape, Financial Times, August ; Newcomers Pile into Longevity Risk Market, Financial Times, May ; Lifeline for Longevity Risk Trading, Financial Times, December The low correlation between longevity risk and the risks associated with other asset classes is further discussed by Blake et al. (2006a) and analyzed statistically by Ribeiro and di Pietro (2009). 3 See Coughlan (2010) and LLMA s website ( 2

3 Notional 100 fixed mortality rate FIXED RATE PAYER (Investor) FIXED RATE RECIEVER (Hedger) Notional 100 realized mortality rate Figure 1: Settlement of a q-forward contract at maturity. seen a number of mortality swap transactions. A recent example is the longevity swap agreed between Babcock International and Credit Suisse in Under the terms of the contract, Babcock s pension plan will swap pre-agreed monthly payments to Credit Suisse in return for monthly payments dependent on the longevity of the plan s 4,500 retired members. We call this contract a bespoke longevity swap, by which we mean the arrangement is based on the actual number of survivors in the hedger s portfolio. While hedgers may prefer a bespoke security that is based on the concept of survival rates, investors and intermediaries favor more standardized instruments which are easier to analyze and more conductive to the development of liquidity. Moreover, derivatives that are linked to survival rates (or the number of survivors in the reference population) may be considered path dependent, because a survival rate is essentially a function of several mortality rates at different ages and different times. Investors may find it difficult to evaluate the underlying risk. The perceived complexity would certainly limit investors appetite, posing barriers to market growth. As a precursor to a liquid longevity market, in July 2007, JPMorgan launched a product called q-forward, a forward contract on a mortality rate at a certain age. 4 The derivative involves an exchange of the realized mortality rate at some future date, in return for a pre-agreed fixed mortality rate (see Fig. 1). Provided that it is fairly priced, there is no payment changes hands at the inception of the trade. A pension plan wishing to hedge longevity risk could execute q-forwards in which it receives fixed mortality rates and pays realized mortality rates. At maturity, the forward contracts will pay out to the pension plan a net amount that increases as mortality rates fall to offset the correspondingly higher value of pension liabilities. 4 It is so named because the letter q is the notation used by actuaries to denote mortality rates. 3

4 Mortality forwards are simpler than longevity bonds and swaps, because they are based on a single mortality rate rather than a survival rate that depends on a series of mortality rates. Another advantage of q-forward is that it is more transparent to investors. In particular, its floating leg is linked to a broad-based index (the LifeMetrics index), which is derived from national population statistics. Relevant data are available in the public domain for investors reference. In the early stages of the market s development, transactions are restricted to a limited number of standardized contracts, with a range of maturities and reference ages, in which liquidity can be concentrated. Pension plans and annuity writers can use the available q-forwards to build a longevity hedge with a reasonable effectiveness (see, e.g., Cairns et al. (2008); Coughlan (2009); Li and Hardy (2009)). In a fair market, the fixed mortality rate agreed at inception defines the forward mortality rate for the population in question. Because the investors (the fixed rate payers) require compensation to take on longevity risk, the forward mortality rate will lie below the corresponding expected mortality rate. In this way, on average (i.e., if mortality is realized as expected), a net payment will be made from the hedgers to the investors. Therefore, the difference between the expected and forward mortality rates can be regarded as the expected risk premium to the investors who acquire a longevity risk exposure. The relationship between the forward mortality rate and the maturity date is called a mortality forward curve (see Fig. 2). This curve, which contains all information about the prices of q-forwards with different times to maturity, is the main focus of this study. It is not straightforward to identify fair mortality forward curves in an incomplete market. In this paper, we contribute a method that allows us to calibrate mortality forward curves. Specifically, the method is derived from the idea of canonical valuation, developed by Stutzer (1996). This approach does not strictly require the use of security prices to predict other security prices, making it particularly useful in today s market where we have only a handful of mortality-linked securities available. Although we may calibrate the forward curves by actuarial approaches that are based on a distortion operator, such as the Wang transform (Wang, 1996), there is usually no strong case for the choice of the distortion operator. On top of that, additional subjective decisions are needed if a more sophisticated model is assumed for the evolution of mortality rates (see, e.g., Chen and Cox, 2009). Such subjectivity can be avoided when canonical valuation is used. 4

5 Expected Forward Mortality rate Risk premium Time to maturity (in years) Figure 2: Illustrative expected and forward mortality curves. Empirical findings indicate that canonical valuation performs well in pricing options on equity indexes. Stutzer (1996) reports that, in a simulated market governed by the Black-Scholes assumptions, canonical valuation produces prices close to Black- Scholes prices, even without using any of the simulated market prices in the valuation process. Gray and Newman (2005) show that, in a stochastic volatility environment, canonical valuation clearly outperforms the historic-volatility-based Black-Scholes estimator for most combinations of moneyness and maturity. Canonical valuation has also been applied to different derivative securities including soybean futures options (Foster and Whiteman, 1999) and bond futures options (Stutzer and Chowdhury, 1999). Results suggest that canonical valuation has merits in both applications. Further extensions of canonical valuation has been considered by Alcock and Carmichael (2008), Alcock and Auerswald (2010), Haley and Walker (2010) and Liu (2010). The rest of this article is organized as follows: Section 2 presents a general set-up of canonical valuation, with an explanation to the intuitions behind; Section 3 presents a stochastic mortality model that is fitted to historic data for the population in question, and describes how an empirical distribution of mortality scenarios can be generated from the model; Section 4 explains how we may transform the empirical distribution to its risk-neutral counterpart with canonical valuation; Section 5 calibrates mortality 5

6 forward curves on the basis of the estimated risk-neutral density; finally, Section 6 concludes the paper. 2 Canonical Valuation: A General Set-up Let us consider a market in which there are m distinct primary securities, whose values evolve according to the state of nature ω. We assume that the ith security, where i = 1, 2,..., m, has a time-zero price of F i and, at the risk-free interest rate, a random discounted payoff of f i (ω). Let P be the real-world probability measure and Q be the set of all measures equivalent to P and satisfying E Q [f i (ω)] = F i, i = 1, 2,..., m, (1) for any Q in Q. That is, Q is the set of all equivalent martingale measures. Assume further that there are a finite number N of states of nature. If m = N, then we say the market is complete. In a complete market, the equivalent martingale measure is unique. However, if m < N, which happens when there are only a few securities trading in the market, then we say the market is incomplete. Market incompleteness implies there are infinitely many equivalent martingale measures. To price a derivative in an incomplete market, we need to choose an equivalent martingale measure that is justifiable. canonical valuation. This important step may be accomplished by using the principle of The principle of canonical valuation is heavily based on the Kullback-Leibler information criterion (Kullback and Leibler, 1951). Denote by [ ] dq D(Q, P ) = E P dq ln dp dp the Kullback-Leibler information criterion of measure Q from measure P. Under the principle of canonical valuation, we should choose the equivalent martingale measure Q 0 that minimizes the Kullback-Leibler information criterion, that is, Q 0 = arg min Q Q D(Q, P ), subject to the constraints specified in equation (1). We call Q 0 the canonical measure. When measure P is discrete uniform, the set-up above is equivalent to the maximization of the Shannon entropy in physics. Therefore, this principle is sometimes 6

7 referred to as the principle of maximum entropy. Interested readers are referred to Jaynes (1957) and Kapur (1989) for applications of this principle in physical science. In statistics, the Kullback-Leibler information criterion D(Q, P ) represents the information gained by moving from measure P to measure Q. From a Bayesian viewpoint, we may regard the real-world probability measure P as the prior distribution. In the absence of any information about market prices, the real-world probability measure P is the only measure we can use. Given the prices of the m primary securities, we can update the prior assessment by incorporating the information contained in equation (1). However, no information other than equation (1) should be incorporated in the update. As a result, we choose the measure that minimizes D(Q, P ), the resulting gain in information, subject to the price constraints in equation (1). Generally speaking, there are two widely used approaches to price contingent claims: Replication: When the market is complete, then every contingent claim can be replicated perfectly, and the price of the contingent claim is just the cost of the hedging strategy. Expected utility maximization: Alternatively, risk-neutral valuation can also be derived from general market equilibrium. By assuming no arbitrage and some mild conditions on the underlying utility function, one can use an expected utility maximization argument to derive a risk-neutral measure based on marginal utility. Risk-neutral valuation can thus be understood as a method to utilityweight the probabilities rather than utility-weight the random cashflows. See, for example, Appendix 11B in McDonald (2006) for a one-period binomial tree setting and Corollary 3.10 in Föllmer and Schied (2004) for a general setting in discrete-time. In our pricing problem, the market is incomplete. Mortality forwards (and any mortality derivative in general) cannot be replicated perfectly. However, one can consider expected utility maximization. The method of canonical valuation can be justified from this viewpoint. Specifically, as stated in proposition 3.1 in Frittelli (2000), when one assumes exponential utility, minimizing the Kullback-Leibler information criterion is equivalent to maximizing utility. As a result, the price obtained from the canonical measure fits into the framework of utility maximization. 7

8 To implement canonical valuation, we are required to generate a number of scenarios with equal probability. This step can be accomplished either non-parametrically or semi-parametrically. In the non-parametric approach, which is used in the original work of Stutzer (1996), we generate realizations of the random variable in question by drawing with replacement directly from the associated data sample. In contrast, in the semi-parametric approach, we first fit a parametric model to the associated data sample, and then we generate realizations of the random variable in question from the fitted model. In this study we use the semi-parametric approach to generate scenarios of future mortality. The scenarios generated may be regarded as a collection of all states of nature. As a result, if N scenarios are generated, then the probability mass function for the state of nature ω under the real-world probability measure P is given by Pr(ω = ω j ) = π j = 1, j = 1, 2,..., N. N The above is often called the empirical probability distribution or the ungrouped histogram of ω. Let π j, j = 1, 2,..., N, be the probability distribution of ω under an equivalent martingale measure Q. We can rewrite the constraints in equation (1) as N f i (ω j )πj = F i, i = 1, 2,..., m, (2) j=1 and the Kullback-Leibler information criterion as N j=1 π j ln π j π j. As such, to find the canonical measure Q 0, we solve the following constrained minimization problem: Q 0 = arg min π j N j=1 π j ln π j π j such that N j=1 π j = 1 and (2) holds. Given Q 0, it is straightforward to place a value on a derivative security. Let us consider a security that has a payoff, discounted to time-zero at the risk-free interest rate, of g(ω j ) in scenario j. The price of this security is simply N j=1 g(ω j) π j, where π j, j = 1, 2,..., N, is the probability distribution of ω under Q 0. 3 Modeling Mortality Dynamics When we calibrate mortality forward curves, the random variable in question is the mortality rate q x,t. This is the probability that an individual aged exactly x at exact 8

9 time t will die between t and t + 1. Due to longevity improvement, for any given age x, the value of q x,t is different at different times. Our goal is to model the evolution of the vector q t = (q x1,t, q x2,t,..., q xn,t) over time. In this study we set x 1 = 60 and x n = 90, as hedgers (pension plans and annuity writers) are most interested in this age range. In subsequent sections, the longevity securities presented are linked to the future mortality rates for English and Welsh males. Here we estimate a stochastic mortality process from the associated data sample, which is obtained from the Human Mortality Database (2010). 5 Specifically, we consider a generalized version of the celebrated Cairns-Blake-Dowd model (Cairns et al., 2006). Blake-Dowd model can be expressed as ( ) qx,t ln = κ (1) t + κ (2) t (x x), 1 q x,t In its original form, the Cairns- where x is the average age over the age range we consider, and κ (1) t and κ (2) t are period indexes. The model assumes that, at a fixed time t, ln(q x,t /(1 q x,t )) is a linear function of the mean corrected age x x, with κ (1) t κ (2) t being the slope. Intuitively, we may interpret κ (1) t mortality level at time t and κ (2) t curve (in logit 6 scale) at time t. being the y-intercept and as an indicator of the overall as an indicator of the steepness of the mortality The original Cairns-Blake-Dowd model has two limitations. First, it fails to capture any potential curvature in the mortality curve (in logit scale). Second, it does not incorporate any cohort effect, which refers to the observed phenomenon that people born in certain years have experienced more rapid improvement than people born in other years. The Continuous Mortality Investigation (CMI) Bureau (2002) noted that cohort effects are highly significant in the mortality experience of UK male pensioners and insured lives. Their findings suggest that a model that permits cohort effects should be considered in this study. For the reasons above, we base our calibration work on the following generalization of the Cairns-Blake-Dowd model: ( ) qx,t ln = κ (1) t + κ (2) t (x x) + κ (3) t ((x x) 2 ˆσ 1 q x) 2 + γ t x, (4) (3) x,t 5 Alternatively, the data may be obtained from the United Kingdom s Office for National Statistics or LifeMetrics ( 6 We call the transformation ln(y/(1 y)) a logit transformation. 9

10 where κ (1) t, κ (2) t, and κ (3) t are period risk factors, γ (4) t x is a cohort risk factor, and ˆσ 2 x is the mean of (x x) 2 over the age range we consider. This model is called Model M7 in Cairns et al. (2009). The generalized version contains a cohort risk factor γ (4) t x that is explicitly linked to the year of birth, t x. Parameter γ (4) t x captures the cohort (year-of-birth) effect by taking different values for different years of birth. The generalized version also includes a quadratic term κ (3) t ((x x) 2 ˆσ x) 2 to capture the potential curvature in the relationship between ln(q x,t /(1 q x,t )) and x. As with many other stochastic mortality models, this model has an identifiability problem. In more detail, if κ (1) t, κ (2) t, κ (3) t, and γ (4) t x are model parameters, then it can be shown that κ (1) t = κ (1) t + ψ 1 + ψ 2 t + ψ 3 t 2 + ψ 3ˆσ 2 x, and κ (2) t = κ (2) t ψ 2 2ψ 3 t, κ (3) t = κ (3) t + ψ 3, γ (4) t x = γ (4) t x ψ 1 ψ 2 (t x x) ψ 3 (t x x) 2 are also parameters of the model. following constraints : x,t γ (4) t x = 0, (t x)γ (4) t x = 0, x,t (t x) 2 γ (4) t x = 0. x,t To stipulate parameter uniqueness, we use the The use of these constraints is equivalent to setting ψ 1, ψ 2, and ψ 3 to 0, thereby ensuring the fitted γ (4) t x will fluctuate around zero and will have no discernible linear trend or quadratic curvature. Note that we place no restriction on the signs of the model parameters. We can estimate the model parameters by the method of maximum likelihood. The details regarding parameter estimation are provided in the Appendix. 10

11 Having fitted equation (3) to historic data, the period indexes are modeled by a trivariate random walk with drift: κ t+1 = κ t + µ + CW t+1, (4) where κ t = (κ (1) t, κ (2) t, κ (3) t ), µ is a constant 3 1 vector, C is a constant 3 3 upper triangular matrix, and W t is a 3-dimensional standard normal random vector. The cohort index γ t x, (4) which has no long-term trend, is modeled by an AR(1) process: γ (4) c = φ 0 + φ 1 γ (4) c 1 + a t, where c = t x, φ 0 and φ 1 are constants, and {a t } is a sequence of iid normal random variables with zero mean and constant variance. Although early 20th century data are available, it may not be desirable to use all available data to estimate the model. This is because in doing so, the period indexes may significantly depart from linearity, violating the assumption behind equation (4). The assumption of linearity can be better met by appropriately restricting the fitting period, as Booth et al. (2002) suggest. According to the decision rule suggested by Booth et al. (2002), we find that it is most appropriate to begin the fitting period in year The estimated parameters are show diagrammatically in Figure 3. As expected, the estimated values of γ (4) t x fluctuate around zero and have no discernible linear trend. The spike of γ (4) t x at t x = 1918 has captured the effect of the Spanish flu epidemic, which might have affected people who were born in 1918 (see, e.g., Holmes, 2004). We use the Bayes Information Criterion (BIC; Schwarz, 1978) to evaluate the estimated model. In general terms, the BIC for a fitted model is defined by BIC = ˆl 0.5j ln(n), where ˆl is the maximized log-likelihood, j is the number of effective parameters, and n is the number of observations. The BIC merits goodness-of-fit (measured by the log-likelihood) but penalizes for the use of additional parameters. A model with a higher BIC value is preferred. In Table 1 we show the BIC values for the estimated Cairns-Blake-Dowd model, both in its original and generalized form. The values indicate the need for using the generalized version rather than the simpler original version. 11

12 κ (1) t t κ (2) t t κ (3) t 0 x t γ (4) t x t x Figure 3: Parameter estimates for the generalized Cairns-Blake-Dowd model. The need for a cohort index and a quadratic term in the model can also be seen from the standardized residuals, which can be expressed as Z x,t = D x,t ˆD x,t, ˆDx,t where D x,t is the actual death count at age x from time t to t + 1, and ˆD x,t is the corresponding death count estimated from the model. If the model is appropriate, Z x,t will be approximately iid standard normal random variables, and the pattern of Z x,t should be random and have little clustering. In Figure 4 we show the contour plots of Z x,t for both versions of the Cairns-Blake-Dowd model. We observe large clusters in the plot for the original version, indicating that it fails to give an adequate fit. In sharp contrast, the pattern for the generalized version looks a lot more random and contains much less clustering. The variance of Z x,t (see Table 1) also points to the same conclusion. The generalized version gives a variance of Z x,t that is significantly smaller and closer to 1. 12

13 Model BIC Var(Z x,t ) Original Cairns-Blake-Dowd Generalized Cairns-Blake-Dowd Table 1: Values of BIC and Var(Z x,t ). 90 The Original Cairns Blake Dowd 90 The Generalized Cairns Blake Dowd x 75 x t t Figure 4: Contour plots of the standardized residuals, Z x,t, from the original and generalized Cairns-Blake-Dowd models. It is noteworthy that in the contour plots, diagonals from bottom left to top right correspond to different years of birth. For the simpler model, the contour plot contains obvious diagonal bands, indicating that there are some year-of-birth specific effects that cannot be captured by the model. On the other hand, for the model with a cohort effect term (γ (4) t x), the contour plot contains no apparent diagonal bands, indicating that γ (4) t x has successfully captured the year-of-birth specific effects. We acknowledge that the generalized Cairns-Blake-Dowd model is not the only model that can model cohort effects. For instance, Richards et al. (2006) model cohort effects with a P-splines regression. However, this approach is not applicable here, because we cannot generate from a P-splines regression sample paths of future mortality, which are necessary for implementing canonical valuation. Another 13

14 alternative model is the age-period-cohort log-bilinear model proposed by Renshaw and Haberman (2006). Although this model can generate sample paths, it is not considered in this study because the algorithm for estimating its parameters has a slow rate of convergence. When fitted to the data we consider, this model would contain 36 more parameters than the model we use in our calculations. Further, on the basis of several selection criteria, Cairns et al. (2009) conclude that the generalized Cairns-Blake-Dowd model is more preferred when considering English and Welsh males data. 4 Deriving the Canonical Measure To carry out canonical valuation, we need to generate a large number of mortality scenarios. Given the generalized Cairns-Blake-Dowd model, we generate mortality scenarios by simulating future values of the period indexes, κ (1) t, κ (2) t, and κ (3) t, and the cohort index, γ t x. (4) Given a realization of κ (1) t, κ (2) t, κ (3) t, and γ t x, (4) we can obtain a scenario of future mortality rates by substituting the values into equation (3). Note that the generated mortality scenarios are equally probable. Suppose that N scenarios are generated. Under the real-world probability measure, the probability of having a mortality rate of q x,t (ω j ) (i.e., being in the state of nature ω j ), for j = 1, 2,..., N, is simply 1/N. Other than the simulated mortality scenarios, we need at least one price constraint in order to implement canonical valuation. BNP/EIB longevity bond announced in 2004 to illustrate. In this paper we use the price of the The BNP/EIB longevity bond is a 25-year amortising bond (i.e., a bond without principal repayment) with coupon payments that are linked to a survivor index, which is based on the realized mortality rates of English and Welsh males aged 65 in The index I(t) on which the coupon payments are based is defined as follows: I(t) = I(t 1)(1 m 64+t,2002+t ), t = 1, 2,..., 25, where I(0) = 1, and m x,t = ln(1 q x,t ) is the central death rate at age x and in year t. In each year t, t = 1, 2,..., 25, from inception, the bond pays a coupon of 50 I(t) million. According to EIB, the issue price was determined by discounting the expected coupon payments (in the real-world probability measure) at LIBOR minus 35 basis 14

15 points. Cairns et al. (2006) observed that the EIB curve typically stands about 15 basis points below the LIBOR curve. The difference (20 basis points) can be regarded as the longevity risk premium. Cairns et al. (2006) further assume that the EIB curve is flat at the level of 4% per annum. Discounting the expected coupon payments at 4% less the risk premium of 20 basis points, they obtain an estimated time-0 price of ( ) million. For each simulated mortality scenario, we have an array of future mortality rates from which we can calculate I(t) at t = 1, 2,..., 25. Let I(t, ω j ) be the value of the longevity index at time t in the jth scenario. Then in the jth scenario, the payoffs, discounted to time zero at the risk-free interest rate, from the longevity bond is given by 25 v(ω j ) = 50 B(0, t)i(t, ω j ), t=1 where B(0, t) is the price at time 0 of a risk-free zero-coupon bond maturing for 1 at time t. 7 Under the real-world probability measure P, the probability of having a discounted payoff of v(ω j ) from the longevity bond is π j = 1/N, for j = 1, 2,..., N. We now transform this probability distribution into its risk-neutral counterpart. We let πj be the probability associated with v(ω j ) (i.e., the jth scenario) under an equivalent martingale measure Q. Under Q, the expectation of v(ω) must be the same as the market price of the longevity bond at time zero. In other words, the following constraint must be satisfied: N v(ω j )πj = (5) j=1 The canonical measure is then found by minimizing the Kullback-Leibler information criterion, subject to N j=1 π j = 1 and equation (5). We solve this problem with the method of Lagrange multipliers, which says the constrained minimization is 7 We define here a risk-free bond by a bond that is free of longevity risk; that is, its payoff is the same regardless of what mortality scenario it turns out to be. On this basis of our definition, a risk-free bond may be subject to other types of risk, for example, counterparty default risk. Such a bond could be one that is issued by EIB (or an institution with a similar credit rating) and is not mortality-linked. In the rest of this article, the risk-free rate refers to the interest rate on such a bond. 15

16 equivalent to minimizing ( N N ) L = πj ln πj λ 0 πj 1 λ 1 j=1 j=1 N j=1 ( v(ωj )π j ). Let π j, j = 1, 2,..., N, be the solution, that is, the canonical measure Q 0. We require it to satisfy the first-order conditions: ln π j + 1 λ 0 λ 1 v(ω j ) = 0, j = 1, 2,..., N, or equivalently, π j = exp(λ 0 + λ 1 v(ω j ) 1), j = 1, 2,..., N, which means π j is proportional to exp(λ 1 v(ω j )). It follows from N j=1 π j = 1 that π j = exp(λ 1 v(ω j )) N j=1 exp(λ, j = 1, 2,..., N. (6) 1v(ω j )) What remains is the Lagrange multiplier λ 1, which can be determined by substituting equation (6) into equation (5) or by the following expression: λ 1 = arg min γ N exp(γ(v(ω j ) 572.1)). j=1 Figure 5 shows the resulting canonical measure, which is obtained from N = 5000 mortality scenarios. Given π j, for j = 1,..., N, we can price another mortality-linked security easily. Suppose that, in the jth scenario, a mortality-linked security has a payoff, discounted to time-0 at the risk-free rate, of h(ω j ). The time-0 price of such a security is simply given by N j=1 h(ω j) π j. We acknowledge that basing the price constraint on the BNP/EIB longevity bond is subject to the following limitations: 1. Information contained in this price, which was announced in 2004, might have become obsolete. 2. The bond did not actually trade, and therefore its price may not truly reflect market participants aversion to longevity risk. 16

17 π * j j Figure 5: The canonical measure. However, as of when the article is written, the BNP/EIB longevity bond seems to be the only suitable security with pricing information available in the public domain. Other mortality-linked securities, such as the 3-year bond sold by Swiss Re in 2003, are short-term securities for hedging the risk of catastrophic events. They are therefore not suitable for deriving the canonical measure which will then be used to price mortality forward contracts for hedging the uncertainty in long-term longevity improvement. 5 Mortality Forward Curves Let us recall how a mortality forward contract is structured. Suppose that the contract is linked to the mortality rate at age x and at a future time T. Then at maturity, which is usually a few months after time T due to the lag in the availability of official data, the fixed rate payer (the investor) pays to the fixed rate receiver (the hedger) an amount proportional to the forward mortality rate, q f x,t, which is agreed at inception. In return, the hedger pays to the investor an amount proportional to the realized value of the reference mortality rate, q x,t. The settlement that takes place at maturity is 17

18 based on the net amount payable and is proportional to the difference between the transacted forward mortality rate and the realized reference mortality rate. Since there is no price paid to enter into a forward contract, we have B(0, T + δ)e Q [q x,t q f x,t F 0] = 0, where T + δ is the maturity date, δ is the lag in the availability of official data, and F t is the filtration generated by the development of the mortality curve up to time t. As a result, we can calculate the forward mortality rate at which the contract will transact by the following equation: q f x,t = EQ [q x,t F 0 ]. All that then remains is to obtain E Q [q x,t F 0 ], the risk-neutral expectation of q x,t. When we use canonical valuation, the expectation can be calculated as follows: N q x,t (ω j )πj, j=1 where q x,t (ω j ) is the value of q x,t in the jth mortality scenario, and πj is the probability associated with the jth scenario under the canonical measure Q 0, which we identified in Section 4. In Figure 6 we show the calibrated mortality forward curves, that is, the relationships between q f x,t and T, for some selected ages. As expected, the expected and forward mortality curve diverge as T increases, since the uncertainty associated with the underlying mortality rate increases as we project the rate further into the future. We also observe that the spread is significantly larger at higher ages, for which mortality rates are more volatile and more difficult to predict. The relationship between the expected risk premium (i.e., the difference between the expected and forward mortality rates) and age can be seen more easily from Figure 7, in which we show the expected and forward mortality rates at a given time to maturity T = 20. The increasing divergence with age indicates that investors demand a higher risk premium when taking on longevity risk associated with more advanced ages. For the reader s reference, we summarize in Table 2 the expected risk premia for different combinations of reference age and time to maturity. On top of the limitations we mentioned in Section 4, the time-0 price ( million) of the BNP/EIB longevity bond is just an estimate, which is calculated 18

19 Age 65 Age 70 Mortality rate Expected Forward Mortality rate Expected Forward Mortality rate Maturity (years) Age 75 Expected Forward Mortality rate Maturity (years) Age 80 Expected Forward Mortality rate Maturity (years) Age 85 Expected Forward Mortality rate Maturity (years) Age 90 Expected Forward Maturity (years) Maturity (years) Figure 6: Calibrated mortality forward curves for English and Welsh males at representative ages. 19

20 Expected Forward 0.12 Mortality rate Age Figure 7: Expected and forward mortality rates at T = 20, English and Welsh males. under the assumption of a flat EIB curve and a risk premium of 20 basis points. It is worthwhile to perform a sensitivity analysis to understand how the forward mortality rates may change when the estimated price is varied. The sensitivity test is performed by considering two other risk premia: 10 basis points and 30 basis points. For each of these risk premia, the time-0 price of the BNP/EIB longevity bond is estimated by discounting the expected coupon payments at 4% less the assumed risk premium. The estimated time-0 prices are million and million, respectively. We then use each of these two prices as a price constraint to recalibrate the mortality forward curves. The resulting risk premia on mortality forwards with different reference ages and times to maturity are summarized in Tables 3 and 4. 6 Concluding Remarks This study presents a method for calibrating mortality forward curves. The method presented is derived from a semi-parametric implementation of canonical valuation. 20

21 Age / Maturity (years) Table 2: Expected risk premia on mortality forwards with different reference ages and times to maturity. The price constraint is formulated with an assumed price of million. Age / Maturity (years) Table 3: Expected risk premia on mortality forwards with different reference ages and times to maturity. The price constraint is formulated with an assumed price of million. In particular, we utilize a generalized Cairns-Blake-Dowd model to form an empirical distribution of future mortality rates, and then we transform the empirical distribution to its risk-neutral counterpart by maximizing the associated Shannon entropy. As an illustration, we calibrate mortality forward curves for English and Welsh males. From these curves hedgers and investors can access the expected risk premium associated with a specific mortality forward contract. Over the past few years, the advancement of the longevity market has been supported by the development of various tools that are designed to improve the degree of transparency, education and standardization. These tools include mortality forecasting software provided by the CMI Bureau and the LifeMetrics, historic mortality data supplied by organizations such as the Max Planck Institute for Demographic 21

22 Age / Maturity (years) Table 4: Expected risk premia on mortality forwards with different reference ages and times to maturity. The price constraint is formulated with an assumed price of million. Research and the CMI Bureau, and standardized longevity indexes offered by, for example, Credit Suisse, Deutsche Börse, Goldman Sachs and JPMorgan. However, there is still a lack of valuation methods that are applicable to the current market situation. This study partially fills in this gap, and we believe that significant further research on the topic of pricing is necessary. Standardized longevity securities such as mortality forwards have both pros and cons. While investors and intermediaries often view standardized securities more favorably as a means of trading longevity risk, they cannot take away all of the longevity risk involved in the hedger s portfolio. In particular, a hedge formed by standardized securities will always leave a residual exposure to population basis risk, which refers to the mismatch between the longevity experience of individuals associated with the hedger s portfolio and that associated with the index to which the security is linked. We may minimize population basis risk by constructing the hedge carefully. By sensitivity matching, Coughlan (2009) demonstrates that a hedge effectiveness of 86% can be achieved. With more advanced hedging strategies, we can even obtain an over 90% hedge effectiveness (see Li and Hardy, 2009). Given a hedging strategy and the calibrated mortality forward curves, a hedger such as a pension plan can cost out the longevity hedge formed by mortality forward contracts. The pension plan can compare this cost against the costs associated with other options, which include bespoke longevity swaps and buy-ins, that is, the seeking a financial institution to insure (lock-in) its pension liabilities. A financially optimal decision can then be made. 22

23 In the original work of Stutzer (1996), canonical valuation is implemented nonparametrically. A non-parametric approach is not used here, in large part because it is unclear about how to incorporate cohort effects if mortality scenarios are bootstrapped directly from the data. We acknowledge that, by using a semi-parametric approach, the calibration work is subject to parameter risk, that is, the uncertainty involved in estimating the model parameters. Such uncertainty may be taken into account by using the parametric bootstrap, which is based on a sequence of generated pseudo samples. In using the parametric bootstrap, we estimate one set of model parameters from each of the pseudo samples, yielding a distribution, rather than just a single point estimate, of model parameters. Each mortality scenario is derived from a specific set of parameter estimates, and therefore, parameter uncertainty is incorporated in the resulting empirical distribution on which canonical valuation is based. On the basis of this method, Li (2010) found that longevity bond prices that are calculated with the incorporation of parameter uncertainty can be significantly higher than those without. It is expected that the forward mortality rates will be lower if parameter risk is included in our calculations. This issue deserves further investigation in future research. References Alcock, J. and Auerswald, D. (2010). Empirical Tests of Canonical Nonparametric American Option-Pricing Methods. Journal of Futures Markets, 30, Alcock, J. and Carmichael, T. (2008). Nonparametric American Option Pricing. Journal of Futures Markets, Blake, D. and Burrows, W. (2001). Survivor Bonds: Helping to Hedge Mortality Risk. Journal of Risk and Insurance, Blake, D., Cairns, A.J.G., and Dowd, K. (2006a). Living with Mortality: Longevity Bonds and Other Mortality-Linked Securities. British Actuarial Journal, 12, Blake, D., Cairns, A.J.C., Dowd, K. and MacMinn, R. (2006b). Longevity Bonds: Financial Engineering, Valuation and Hedging. Journal of Risk and Insurance, 73,

24 Booth, H., Maindonald, J. and Smith, L. (2002). Applying Lee-Carter under Conditions of Variable Mortality Decline. Population Studies, 56, Cairns, A.J.G., Blake, D., and Dowd, K. (2006). A Two-Factor Model for Stochastic Mortality with Parameter Uncertainty: Theory and Calibration. Journal of Risk and Insurance, 73, Cairns, A.J.G., Blake, D. and Dowd, K. (2008). Modelling and Management of Mortality Risk: A Review. Scandinavian Actuarial Journal, 108, Cairns, A.J.G., Blake, D., Dowd, K., Coughlan, G.D., Epstein, D., Ong, A., and Balevich, I. (2009). A Quantitative Comparison of Stochastic Mortality Models Using Data from England and Wales and the United States. North American Actuarial Journal, 13, Chen, H., and Cox. S.H. (2009). Modeling Mortality With Jumps: Applications to Mortality Securitization. Journal of Risk and Insurance, 76, Continuous Mortality Investigation Bureau (2002). An Interim Basis for Adjusting the 92 Series Mortality Projections for Cohort Effects. CMI Working Paper 1. London: Institute of Actuaries and Faculty of Actuaries. Coughlan, G. (2010). Life and Longevity Markets Association: The Development of a Longevity and Mortality Trading Market. Presentation in the Sixth International Longevity Risk and Capital Markets Solutions Conference, Sydney, Australia, September Coughlan, G. (2009). Longevity Risk Transfer: Indices and Capital Market Solutions. In Barrieu, P. and Albertini, L. (eds.), The Handbook of Insurance-Linked Securities. West Sussex: John Wiley & Sons. Dowd. K. (2003). Survivor Bonds: A Comment on Blake and Burrows. Journal of Risk and Insurance, 70, Dowd, K., Blake, D., Cairns, A.J.G. and Dawson, P. (2006). Survivor swaps. Journal of Risk and Insurance, 73, Föllmer, H., and A. Schied. (2004). Stochastic Finance An Introduction in Discrete Time, 2nd ed. Number 27 in de Gruyter Studies in Mathematics. Walter de Gruyter. Foster, F.D. and C.H. Whiteman. (1999). An Application of Bayesian Option Pricing to the Soybean Market. American Journal of Agricultural Economics, 81,

25 Frittelli, M. (2000). The Minimal Entropy Martingale Measure and the Valuation Problem in Incomplete Market. Mathematical Finance, 10, Gray, P. and Newman, S. (2005). Canonical Valuation of Options in the Presence of Stochastic Volatility. Journal of Futures Markets, 25, Haley, M.R. and Walker, T.B. (2010). Alternative Tilts for Nonparametric Option Pricing. Journal of Futures Markets, in press. Holmes, E.C. (2004). Enhanced: 1918 and All That. Science, 303, Human Mortality Database. University of California, Berkeley (USA), and Max Planck Institute of Demographic Research (Germany). Available at or (data downloaded on 1 Jan 2010). Jaynes, E.T. (1957). Information Theory and Statistical Mechanics. Physical Review, 106, Kapur, K. N. (1989). Maximum-Entropy Models in Science and Engineering. New York: Wiley Eastern, Ltd. Kullback, S. and Leibler, R. A. (1951). On Information and Sufficiency. Annals of Mathematical Statistics, 22, Li, J.S.-H. (2010). Pricing Longevity Risk with the Parametric Bootstrap: A Maximum Entropy Approach. Insurance: Mathematics and Economics, 47, Li, J.S.-H. and Hardy, M.R. (2009). Measuring Basis Risk Involved in Longevity Hedges. Paper presented in the Fifth International Longevity Risk and Capital Markets Solutions Conference, New York, USA, September Liu, Q. (2010). Pricing American Options by Canonical Least-Squares Monte Carlo. Journal of Futures Markets, 30, McDonald, R.L. (2006). Derivatives Markets, 2nd ed. Boston: Addison-Wesley. Renshaw, A.E. and Haberman, S. (2006). A Cohort-based Extension to the Lee-Carter Model for Mortality Reduction Factors. Insurance: Mathematics and Economics, 38, Richards, S.J., Kirkby, J.G., and Currie, I.D. (2006). The Importance of Year of Birth in Two-Dimensional Mortality Data. British Actuarial Journal, Ribeiro, R. and di Pietro, V. (2009). Longevity Risk and Portfolio Allocation. Available at 25

26 Stutzer, M. (1996). A Simple Nonparametric Approach to Derivative Security Valuation. Journal of Finance, 51, Stutzer, M. and Chowdhury, M. (1999). A Simple Nonparametric Approach to Bond Futures Option Pricing. Journal of Fixed Income, 8, Schwarz, G. (1978). Estimating the Dimension of a Model. Annals of Statistics, 6, Wang, S. (1996). Premium Calculation by Transforming the Layer Premium Density. ASTIN Bulletin, 26(1), Appendix Maximum Likelihood Estimation for the Generalized Cairns-Blake-Dowd Model Let us define D x,t by the number of deaths at age x from time t to t + 1, and E x,t by the corresponding exposures to the risk of death. To construct the likelihood function, we treat D x,t as independent Poisson responses, that is, D x,t Poisson(E x,t m x,t ), where m x,t is the central death rate. The product E x,t m x,t is the expected number of deaths at age x and in year t. This gives the following log-likelihood: l = (D x,t ln(e x,t m x,t ) E x,t m x,t ln(d x,t!)), (7) x,t where D x,t! stands for D x,t factorial. The generalized Cairns-Blake-Dowd model is based on death probability q x,t rather than the central death rate m x,t. To relate these two quantities, we use the following relation: m x,t = ln(1 q x,t ), (8) which holds if we assume that the force of mortality is constant over each year of integer age and over each calendar year. So the required likelihood function can be obtained by substituting equation (3) into equation (8) and then into equation (7). 26

27 Parameter estimates can be obtained by maximizing the corresponding likelihood function. The maximization can be accomplished by an iterative Newton-Raphson method, in which parameters are updated one at a time. The updating of a typical parameter θ proceeds according to u(θ) = θ l/ θ 2 l/ θ 2, where u(θ) is the updated value of θ in the iteration. The parameter constraints are applied at the end of each iteration. 27