Pricing Survivor Derivatives With Cohort Mortality Dependence Under the Lee–Carter Framework

The Model

To capture the future mortality dynamics to price survivor derivatives, we analyze the mortality rates as a function of both age x and time t on a filtered probability space , where P is the physical probability measure, N is a positive integer, and is the information available at time t. Let m_x, t denote the central death rate at age x during calendar year t. To introduce the cohort mortality dependence in modeling longevity risk, we use the classical Lee–Carter model, which is

(1)

where the parameters b_x and k_t are subject to and , respectively, to ensure the model identification. This structure is designed to capture age and period effects, such that the a_x coefficients incorporate both the main age effects averaged over time and the actual force of mortality change according to an overall mortality index k_t, modulated by an age response b_x. The error term e_x, t reflects particular age-specific historical influences that are not captured by the model. In the classical Lee–Carter model, the error term is assumed to be a particular discrete-time Markov stochastic process, with mean change of 0 and a variance rate of . In other words, the change of , which is independent of , has a normal distribution with mean zero and variance . Different from the original Lee–Carter model, we additionally assume that the correlation coefficient between and is equal to , which captures age-specific mortality dependence between an x-aged person and y-aged person and thus contributes to cohort mortality dependence.

To forecast future mortality dynamics, Lee and Carter (1992) assume that a_x and b_x remain constant over time and forecast the future dynamics of the mortality index k_t using a standard ARIMA(p,1,q) time series model, as follows:

(2)

where p and q denote the autoregressive (AR) and moving average (MA), orders, respectively; , , and are the drift, AR, and MA parameters, respectively, with ; and , independent of e_t, is normally distributed with mean 0 and deterministic variance . In most applications, k_t is well modeled as a random walk with drift or referred to as the ARIMA(0,1,0) model: . In this case, the forecasted k_t changes linearly, and the corresponding forecasted mortality rate changes at a constant exponential rate. However, Lee (2000, p. 83) also suggests, “Sometimes a model of this general form, but with an added moving average term or autoregressive term, is superior.” We use the generalized ARIMA(p,1,q) model to describe the mortality forecast.

According to the generalized ARIMA(p,1,q) model, the easiest way to forecast the future mortality index is through the law of iterated projection (Hamilton, 1994; Huang and Wu, 2007). The increment of the time index at t₀ + n, (), can represent the sum of past mortality indexes prior to time t₀, such that

(3)

where

(4)

(5)

, , and for and . The proof of Equation (3) appears in Appendix A.

With the time index in Equation (3), we can express the increment of the logarithm mortality rate for the same age at t₀ + n as

(6)

Proposition 1: The future logarithm mortality rate for a person of age x at time t₀ + N, given the information at time t₀, can be expressed as

(7)

where denotes the expected cumulative rate of x-aged mortality improvement under the physical probability measure P and satisfies

(8)

as well as

(9)

where , or the variance of the logarithm of conditional on time t₀ information , satisfies

(10)

where is the mortality index volatility of ; is the age volatility of , n = 1, …, N; and takes the form:

(11)

Proof: Please see Appendix B.

By applying Proposition 1, we can project the future dynamics of mortality rates and the corresponding survival probability. That is, let denote the 1-year survival probability that an x₀-aged person in calendar year t₀ reaches age x₀ + 1. We assume that the age-specific mortality rates are constant within bands of age and time but may vary from one band to the next. Specifically, given any integer age x₀ and calendar year t₀, we suppose that

(12)

Thus, the 1-year survival probability can be calculated as under a constant force of mortality assumption. Let denote the n-year survival probability that an x₀-aged person in calendar year t₀ reaches age x₀ + n. We calculate the survival probability based on the cohort mortality rate,3 which is

(13)

where A_n, equal to , represents the arithmetic sum of the mortality rates.

To deal with the n-year survival probability for valuing survivor derivatives, we must obtain the first two moments of A_n. Therefore, we define the time n “pseudo-forward” of A_n as

(14)

where , and

(15)

If we divide A_n by F_n and refer to the result as , Proposition 2 provides the first two moments of .

Proposition 2: Let denote the normalized A_n, to calculate the survival probability in Equation (13). Under the physical probability measure P, the first two moments of take the form:

and

(16)

where denotes the cohort mortality correlation between ages x₀ + i and x₀ + j, based on the cohort to which a person aged x₀ at time t₀ belongs, which also corresponds to the cohort mortality dependence that we introduce.

The proof of Proposition 2 appears in Appendix C.

The cohort mortality dependence is defined as . Under the Lee–Carter model, conditional on time t₀ information , the cohort mortality dependence for people between the ages of x₀ + i and x₀ + j can be expressed as

(17)

Equation (17) shows that the cohort mortality correlation depends on the age-specific mortality correlation () between an x₀ + i-aged person and an x₀ + j-aged person. In addition, according to Equations (10) and (11), different ARIMA models contribute to different AR and MA parameters, as well as and the variance , which then lead to different cohort mortality dependence structures. If we calculate the survival probability in Equation (13), according to Proposition 2, we find that mortality improvements among different ages exhibit cohort mortality dependence.

Empirical Investigation of Cohort Mortality Dependence

We employ U.S. mortality data from the HMD () to illustrate the pattern of cohort mortality dependence. For mortality data of 1933 through 2007, we use a close approximation to singular value decomposition method4 to find the parameter estimates of the Lee–Carter model. The pattern of empirical mortality rates and the fitted values of a_x, b_x, and k_t appear in Figure 1. The mortality surface depicts the development of logarithm central death rates over calendar time; the downward slope of the time index k_t reflects the trend of mortality improvement.

The structure of cohort mortality dependence under the Lee–Carter model depends on how we project the future mortality rates. Time series analysis is widely employed to model k_t (Lee, 2000; Renshaw and Haberman, 2006; Denuit, Devolder, and Goderniaux, 2007). We also obtain the cohort mortality dependence by modeling the k_t as an ARIMA(p,1,q) process (see Equation (2)). Thus, we examine various ARIMA(p,1,q) models and evaluate their fit, according to the Akaike information criterion (AIC)5 and the log-likelihood. Table 1 lists the parameters for different ARIMA(p,1,q) models and the corresponding AIC and log-likelihood. Clearly, for the United States mortality experience, the ARIMA(2,1,2) model6 is the most appropriate for describing the mortality index; ARIMA(0,1,0) offers worse performance, according to the AIC and log-likelihood values.7

With the parameters in Table 1, we calculate the cohort mortality correlations between different ages based on Equation (17) using both the ARIMA(2,1,2) (solid line) and ARIMA(0,1,0) (dashed line) models. For illustrative purposes, Figure 2 only depicts the cohort mortality correlations for certain ages—66, 70, 75, 80, 85, and 89 with all other ages separately. It reveals two particularly important findings. First, the patterns of cohort mortality dependence for both ARIMA(0,1,0) and ARIMA(2,1,2) models are similar, but the cohort mortality correlations of the ARIMA(0,1,0) model are higher than those of the ARIMA(2,1,2) model, which means that the ARIMA model selection can affect cohort mortality dependence. Second, Figure 2 displays a remarkable phenomenon: the correlations between the mortality rates of younger ages and the mortality rates of older ages are lower, whereas those among older ages based on the same cohort are higher. For example, in the upper left-hand panel of Figure 2, the mortality correlations between persons aged 66 years and 80+ years are lower than 0.15. Therefore, a higher mortality rate at this age, such as might be due to accidents or catastrophic events (e.g., 2004 earthquake, tsunami), does not lead directly to higher mortality rates among older ages of the cohort. However, because most people die of natural causes, the mortality correlations at older ages should be higher than those in younger ages. In the lower right-hand panel of Figure 2, the cohort mortality correlation between ages 87 and 89 years equals 0.5296, and that between ages 88 and 89 years equals 0.4592.

We further take cohort mortality dependence into account to reveal the pattern of future survival probabilities. For comparison purposes, we illustrate different cohort mortality correlation assumptions of 0, 0.3, 0.6, and 0.9 for different ages and plot the corresponding 95 percent confidence interval of the simulated survival probabilities for a person aged 65 in Figure 3. A cohort mortality correlation of 0 implies mortality independence. The higher the cohort mortality correlation, the more uncertainty is involved in the future dynamics of mortality. Thus, ignoring cohort mortality dependence might underestimate longevity risk. We study the impact of cohort mortality dependence on pricing survivor derivatives.

Survivor Swaps Versus Survivor Floors

To investigate the effect of cohort mortality dependence on pricing survivor derivatives, we extend the literature by introducing a survivor floor for which the claim is contingent (i.e., an option-type contract). To clarify the mechanics of the survivor floor, we describe the survivor swap first and compare it with the proposed survivor floor. A survivor swap has been widely explored in the prior literature (Dawson, 2002; Blake, 2003; Dowd, 2003; Lin and Cox, 2007). Dowd et al. (2006) introduce the mechanism of a survivor swap for transferring longevity risk. A survivor swap is an agreement that involves the periodic exchange of a series of preset payments for a series of random mortality-dependent payments. On each payment date, the fixed-rate payer pays a preset amount, equal to the value of the notional principal multiplied by a fixed rate, and receives in return from the floating-rate payer a random mortality-dependent payment, equal to the value of the notional principal multiplied by the unexpected shock in survival probability (i.e., the difference between the actual survival probability and the reference survival probability). Thus, survivor swaps can be used to transfer the unexpected shock in mortality improvement. Down et al. point out that hedge is only as good as it is when the reference index is based is the insurer's own mortality experience. For more standardized reference indexes, if the expected reference indexes and insurers' own mortality experiences are highly correlated, the survivor swap can still hedge the insurer against a considerable amount of the aggregate mortality risk it faces.

Let be the total number of annuities of an insurer issued to a cohort that consists of persons aged x₀ at time t₀. To transfer longevity risk, the insurer could purchase an N-year survivor swap at time t₀ with a total nominal value , which is equal to the total number of annuity payments. The future actual payout to the annuitant at time t₀ + n is estimated at time t₀, denoted CF_n and given by

(18)

where represents the actual n-year survival probability for a person aged x₀ at time t₀ and we model using the proposed Lee–Carter model with cohort mortality dependence, as we described in the Lee–Carter model with cohort mortality dependence section.

The payout to the annuitant at time t₀ + n based on the reference mortality table at time t₀ is denoted as , which is given by

(19)

where is the n-year survival probability for a person aged x₀ at time t₀ based on the reference mortality table used by the insurer. We assume the insurer used the mortality experience in year 2007 (HMD, ) for the reference mortality table.

Let c_S denote the fixed swap rate and c_S n be the floating swap rate. The cash flows of a survivor swap for the fixed-rate payer and floating-rate payer are presented in Figure 4. At each payment date t₀ + n, , the insurer, which is also the longevity risk protection buyer, pays a prespecified fixed premium amount to a special purpose vehicle (SPV), which is the longevity risk protection seller, and sets up for this transaction and receives the floating payment amount, . In other words, the floating swap rate (c_S n) equals to . After using the survivor swap, the insurer can effectively manage the time t₀ + n aggregate cash outflow, equal to the scheduled payout plus the swap rate for .

In contrast, the survivor floor involves the exchange of a fixed series of payments for a series of basket survivor put options whose strike prices are equal to the reference survival probabilities on each payment date. The insurer can purchase survivor floors to hedge its longevity risk when annuitants live longer than predicted by the reference mortality rates. The cash flows of a survivor floor are shown in Figure 5. The insurer, a risk protection buyer, makes periodic fixed payments (premium leg) to an SPV, a risk protection seller, and receives floating payments at each payment date t₀ + n to offset the contingent loss. Assume the insurer purchases an N-year survivor floor at time t₀ with a total nominal value , equal to the total number of annuity payments. Then c_F denotes the floor rate and c_F n is the reference rate. The fixed payment is calculated as , and the floating payment is calculated as . The contingent loss for the insurer occurs when the annuitants live longer than that predicted by the reference mortality rates. The contingent loss at time t₀ + n can be expressed as

(20)

Equivalently,

(21)

The insurer can use the survivor floor to transfer the contingent loss from the issuer to the fixed-rate payer, if

(22)

After using the survivor floor, the insurer can effectively control the time t₀ + n aggregate cash outflow, as follows:

(23)

In brief, both survivor swaps and the survivor floors can effectively hedge the longevity risk of the insurer. The main difference is that the floor is an option and requires more premiums to hedge longevity risk. Under the no arbitrage condition, the hedge cost for the survivor swap therefore should be lower than the hedge cost for the survivor floor; accordingly, the swap rate (c_S) is lower than the floor rate (c_F).

Multivariate Wang Risk Measure

There is a pressing need for a universal framework to determine a fair value with financial and insurance risks, especially for pricing survivor derivatives or mortality-linked securities. Wang (2000) proposes a transformation for pricing contingent claims that can be traded or not. Because contracts contingent on mortality rates usually are not traded on financial markets, Wang's transformation helps value mortality-linked securities (Lin and Cox, 2007; Dowd et al., 2006; Liao, Yang, and Huang, 2007; Denuit, Devolder, and Goderniaux, 2007). Dowd et al. (2006) offer the first pricing model for a survivor swap based on the Wang transform (Wang 2000). To capture cohort mortality dependence between ages, we apply the multivariate extension of the Wang transformation (Cox, Lin, and Wang, 2006; Kijima, 2006) and obtain the mortality rate under the Wang risk measure Q, based on our proposed Lee–Carter model with cohort mortality dependence. For a multivariate setting, assume that follows a Gaussian copula with a correlation matrix . A multivariate extension of the Wang transformation is

(24)

where is the -variate standard normal distribution, and is the risk-adjustment parameter for the physical distribution of the jth variable, .

Consider a person aged x₀ + i, . Because and , , are normally distributed, the transformed distributions of and , , based on the multivariate Wang transformation, equal

(25)

such that represents the risk-adjustment parameter for the physical distribution of an x₀ + j-aged error term, and denotes the risk-adjustment parameter for the physical distribution of the mortality index. Therefore, the multivariate Wang transformation distorts the log-change in the x₀ + i-aged mortality rate during calendar year t₀ + n to another lognormal distribution.

Proposition 3: Under the Wang risk measure Q, the future logarithm mortality rate at time t₀ + N for a person aged x, given information at time t₀, as in Proposition 1, can be revised as follows:

(26)

where denotes the expected n-period cumulative rate of x-aged mortality improvement under Q, as given by

(27)

Proof: Substituting Equation (25) into Equation (7), we can obtain Equations (26) and (27).

Equivalently, given time t₀ information ,

(28)

There are two important implications of Equation (28). First, the expected rate of an x_i-aged mortality improvement, under the Wang risk measure Q, depends on the values and expected cumulative rate of x-aged mortality improvement for the physical probability measure P. Second, the x₀ + i-aged force of mortality during calendar year t₀ + n, conditional on , follows a new MA(n) process under the Wang risk measure Q, the MA order of which depends on the observed time span n. In addition, according to Proposition 3, we further express the risk-neutralized survival probability for the projected in the form of

(29)

where ; ; is the modified Bessel function of the second kind with order ; ; ; and . We use the reciprocal gamma distribution proposed by Milevsky and Posner (1998a, 1998b) as an approximation of the state-price density of A_n to derive Equation (29), and the Proof of Equation (29) is shown in Appendix D. Q.E.D.

Pricing Survivor Swaps and Survivor Floors With the Wang Risk Measure Q

The valuation formulas for both survivor swaps and survivor floors under the proposed Lee–Carter model with cohort mortality dependence are derived in this section. Let B(t,  T) denote the price of a zero-coupon bond issued at time t that pays one dollar at time T, ; r(t) is the risk-free rate. The price of a zero-coupon bond under the risk measure Q satisfies the following expression:

(30)

According to the no arbitrage theory, the fair price of the survivor swap at issue date t₀ is zero. Therefore, we have

(31)

To obtain the fair swap rate analytically, we let , where is the sum of force of mortality rates ranging from age x₀ to age x₀ + n − 1 based on the reference mortality table. Note that . Thus, we can express the floating swap rate as . Under the assumption that the mortality rate and financial risk are independent, we can derive the fair swap rate as follows:

(32)

Similarly, for the survivor floors, we have

(33)

Substituting and into Equation (22), we obtain

(34)

where .

Consequently, the fair floor rate c_F is of the form:

(35)

Equation (32) implies that the fair swap rate is determined by the moment-generating function of A_n. Similarly, Equation (35) implies that the fair floor rate is determined by a series of basket put options on mortality rates with the payoff , which is the survivor option. We use the reciprocal gamma distribution proposed by Milevsky and Posner (1998a, 1998b) as an approximation of the state-price density of A_n and thereby obtain an approximate closed-form expression for the fair valuation of survivor swaps and survivor floors.

Proposition 4: Under the Lee–Carter model with cohort mortality dependence, the approximate closed-form expression of the fair swap rate c_S is given by

(36)

Proof: Substituting Equation (29) into Equation (32), we can obtain Equation (36). Q.E.D.

Proposition 5: Under the Lee–Carter model with cohort mortality dependence, the approximate closed-form expression of the fair floor rate c_F is of the form

(37)

where BSP_n denotes the value of the embedded basket survivor put option and takes the form

(38)

where denotes the cumulative distribution of the gamma distribution evaluated at x.

Please see Appendix E for the proof.

Fair Swap Rates and Floor Rates

We examine the effects of cohort mortality dependence on the valuation of survivor swaps and survivor floors. The assumption of cohort mortality dependence directly follows the projection using the ARIMA(2,1,2) model. We consider a person aged 65 years in 2007, that is, x₀ = 65 and t₀ = 2007. The maturity of the survivor floor is 25 years. Without loss of generality, we assume that the initial term structure is flat and the risk-adjustment parameters are equal, so . We illustrate the result when is 0.05, 0.15, and 0.25 and when the initial term structure is flat at 1, 3, and 5 percent.

In Tables 2 and 3, we first examine the accuracy of the pricing formulas for both survivor swaps and survivor floors using simulations separately. For an interest rate of 3 percent and a risk premium of 0.15, the fair swap rate for a survivor swap is 108.3116 basis points (bps) according to the pricing formula and 108.3699 bps when we use simulations. In the same conditions, the fair floor rate for a survivor floor is 116.2810 bps with the pricing formula and 116.2175 bps using simulations. Thus, the results from the pricing formulas in Equations (37) and (38) are very close to those we obtain using simulations. Furthermore, the swap rate and floor rate both relate negatively to the interest rate and the risk premium, consistent with Dowd et al. (2006).

Next, we examine the effect of cohort mortality dependence on pricing survivor swaps and survivor floors using pricing formulas. Table 4 presents the fair swap rates for the survivor swap when we consider versus ignore cohort mortality dependence. With an interest rate of 3 percent and a risk premium of 0.15, the fair swap rate based on Equation (37) is 106.7155 bps when assuming the cohort mortality independence; it increases to 108.3116 bps when we account for cohort mortality dependence. It thus increases 1.5961 bps. No matter taking into account of cohort mortality dependence or not, the effects of the interest rate and risk premium on the fair swap rates change significantly. The effect also applies to the valuation of survivor floors. In Table 5, we calculate the floor rates for survivor floors based on Equation (38) when considering and ignoring cohort mortality dependence. The floor rate increases if we include cohort mortality dependence; for example, under an interest rate of 3 percent and a risk premium of 0.15, the fair floor rate is 107.9694 bps when assuming cohort mortality independence but 116.2810 bps when we take cohort mortality dependence into account, a difference of 8.3116 bps. Therefore, the fair swap rates and floor rates likely would be underestimated if we were to exclude cohort mortality dependence. Note that the effects of the cohort mortality dependence on the fair floor rates are more significant than that on the fair swap rates.

We also detail the impact of the duration of the survivor floor on the fair floor rate in Table 6, which shows that the fair floor rate is higher for a longer duration of the survivor floor. Longer maturity floors contain more options, each of which has a positive premium, so prices increase.

Sensitivity Analysis and Robustness Check

We conduct numerical sensitivity analyses and robustness checks but illustrate the effects only with survivor floors. We first investigate the difference related to the cohort mortality correlation assumption for pricing survivor floors. In Table 7, we report the impacts of the different cohort mortality correlations on the fair floor rates for survivor floors at different risk-adjustment parameters . The cohort mortality correlation relates positively to fair survivor floor rates, and the effect is significant, especially for a higher level of . For example, the fair floor rate increases 17.3046 bps (from 141.6884 bps to 158.9930 bps) with a risk premium of 0.05 and 31.9109 bps (from 74.9671 bps to 106.8780 bps) with a risk premium of 0.25 when the cohort mortality correlation increases from 0 to 1. Thus, it is important to incorporate the cohort mortality dependence structure into the pricing of survivor floors.

We also examine the impact of different ARIMA models on pricing results; we illustrate two different models, ARIMA(2,1,2) and ARIMA(0,1,0), in Table 8. As Figure 2 reveals, because the cohort mortality correlations of ARIMA(2,1,2) are lower than those of ARIMA(0,1,0), the floor rate calculated on the basis of ARIMA(2,1,2) is lower than that using the ARIMA(0,1,0) model. For a risk premium of 0.15 and interest rate of 3 percent, it decreases from 122.2333 bps to 116.2810 bps. Comparing the effect with cohort mortality correlation in Table 7, it is more significant than that using different ARIMA models, especially for a higher level of .

For a robustness check, we also use a different mortality experience. With the ARIMA(2,1,2) model, we estimate two different periods of mortality experiences, from 1933 to 2005 and from 1933 to 2007, then calculate the corresponding cohort mortality dependence. The impact of mortality experience on the fair survivor floor rates, in Table 9, reveal that the results are similar, which supports the robustness of our results for different periods of mortality experiences.