A Two-Population Mortality Model to Assess Longevity Basis Risk
AA Two-Population Mortality Model to Assess Longevity BasisRisk
Selin ¨Ozen , S¸ule S¸ahin Karabuk University, Department of Actuarial Sciences, Karabuk, TURKEY Department of Mathematical Sciences, Institute for Financial and ActuarialMathematics, University of Liverpool, Liverpool, UK
Abstract
Index-based hedging solutions are used to transfer the longevity risk to the capi-tal markets. However, mismatches between the liability of the hedger and the hedg-ing instrument cause longevity basis risk. Therefore, an appropriate two-populationmodel to measure and assess the longevity basis risk is required. In this paper, weaim to construct a two-population mortality model to provide an effective hedgeagainst the longevity basis risk. The reference population is modelled by using theLee-Carter model with renewal process and exponential jumps proposed by ¨Ozenand S¸ahin [26] and the dynamics of the book population are specified. The analysisbased on the UK mortality data indicate that the proposed model for the referencepopulation and the common age effect model for the book population provide abetter fit compared to the other models considered in the paper. Different two-population models are used to investigate the impact of the sampling risk on theindex-based hedge as well as to analyse the risk reduction regarding hedge effective-ness. The results show that the proposed model provides a significant risk reductionwhen mortality jumps and the sampling risk are taken into account.
Keywords:
Longevity basis risk; mortality jumps; two-population mortality model.
Longevity risk can be defined as the risk that members of some reference population mightlive on average longer than anticipated. It is a crucial financial concern for both pensionplans and life insurers since the institutions might have to make higher payments thanexpected due to the longevity risk. Life expectancy continues to rise in association withimprovements in nutrition, hygiene, medical knowledge, lifestyle, and health care. Uncer-tainty about future mortality improvements might have significant economic implicationsfor annuity providers, pension providers, and social insurance programs. Although theindividuals have different lifespan, longevity risk might affect all pension plans and lifeinsurers, and hence it is not possible to diversify it with an increase in portfolio size.Therefore hedging of the longevity risk is of critical importance for both pension planproviders and life insurance companiesVarious solutions have been presented to manage and mitigate the longevity risk.Index-based hedging solutions, which include longevity-linked securities and derivatives,provide more advantages over other hedging solutions, such as faster execution, greatertransparency, liquidity potential, and lower costs [23]. Due to offering significant capitalsavings and providing effective risk management, index-based longevity instruments at-tract increased interests from within and outside of the worlds of insurance and pensions. [email protected] [email protected] a r X i v : . [ q -f i n . R M ] J a n he first step of the assessment of longevity risk and thus the valuation of index-basedfinancial products is the mortality modelling. The choice of the appropriate model iscrucial to quantify the risk and provide a foundation for pricing and reserving. Due tothe inadequacy of the quality and the size of the portfolio, a reference population indexis commonly used by hedgers in index-based hedging solutions. The payments of thefinancial products are associated with this reference population index, but not the (book)population that underlies the portfolio that is being hedged. Therefore, longevity risktrading usually entails two different populations: the first is affiliated with the portfolioof the hedger, while the other is linked to the hedging instrument [30]. There wouldthen be a potential mismatch between the hedging instrument and the portfolio, due tocertain demographic differences (e.g. age profile, sex, socioeconomic status). This mightgive rise to longevity basis risk, the assessment of which is under research in the latestactuarial literature [23]. Hence, a multi-population mortality model is required to providean accurate mortality model for measuring the basis risk.Several multi-population mortality models have recently been presented while only [30]consider the transitory mortality jump effects in the modelling process. It is importantto incorporate the mortality jumps to estimate the uncertainty surrounding a centralmortality projection. Incorporating the jumps into the modelling process allows us toestimate the probability of catastrophic mortality deterioration when pricing securitiesfor hedging extreme mortality risk [30]. In this paper, a different approach proposed by¨Ozen and S¸ahin [26], has been used for modelling jump effects. This approach includes thehistory of catastrophic events in the jump frequency modelling process by using renewalprocess as well as a specification of the Lee–Carter (LC) model for mortality.The aim of this paper is to build an appropriate two-population mortality modelincorporating mortality jumps to assess the longevity basis risk for pricing longevity-linked financial products. Such a model provides a basis to effective risk managementstrategies. To illustrate the impact of our proposed mortality model in hedge effectiveness,we consider a hedge for a hypothetical pension plan. Moreover, we take sampling risk intoaccount since the available historical data is usually small for a pension plan. Therefore,the size of a pension plan is examined in regard to hedge effectiveness. We also comparethe hedge effectiveness of our model with the other two commonly used mortality models.The results show that our proposed model provides a better risk reduction.The remainder of the paper is structured as follows. Section 2 introduces some helpfulnotations. In Section 3, an overview of the existing multi-population mortality modelsis provided. The steps for building a two-population mortality model are described inSection 4. Section 5 applies the proposed model to a hypothetical pension plan andexamines the effectiveness of the hedge. Finally, Section 6 concludes the paper. We begin with introducing some helpful notations adopted from Villegas et al. [29]. Letus denote the reference population by R that is backing the hedging instrument, and B is used for the book population whose longevity risk is going to be hedged. Time will bemeasured in units of years, and year t will refer to time interval [ t, t + 1]. For the referencepopulation, D Rx,t and E Rx,t show the death counts and exposure to risk at age x at lastbirthday in year t . Central mortality rates for any individual of the reference populationof age x in year t will be denoted by m Rxt and computed as m Rx,t = D Rx,t /E Rx,t . Likewise,the same values for the book population are given here as D Bx,t , E Bx,t and m Bx,t = D Bx,t /E Bx,t .2 further assumption being made here is that the data for the reference and bookpopulations can be different regarding specified sets of ages and specified amounts of years.For instance, we have D Rx,t and E Rx,t for consecutive ages x = x , ..., x n R and consecutivecalender years t = t , ..., t n R in the reference population, while D Bx,t , E Bx,t are available forages x , ..., x n B and calender years t = u , ..., t n B in the book population.The reference population’s data might be provided for a longer time frame than thatof the book population, which is n R ≥ n B . Moreover, the calendar years of data in abook may be provided as a subset of the comparable calendar years for the referencepopulation, t n B (cid:54) = t n R . Also, the ages provided by the book might constitute a smallerportion of those that are provided for the reference population. We need to specify an appropriate two-population model for m Rx,t and m Bx,t which has theability to capture the trends present within both the book and reference populations. It iscrucial to incorporate these trends since the mortality trends of the reference populationsupport the hedging instrument while the trends in the book population are significant forthe longevity basis risk to be hedged. Future mortality will be forecasted by the specifiedmodel in a consistent way.Several models have been developed to display the mortality evolution of two relatedpopulations. These models usually derived by expanding the previous single-populationmodels by incorporating the correlations and interactions existing between populations.Although the majority of research on modelling multi-population has been conductedrelatively recently, the seeds are traced back to the influential paper published by Carterand Lee [8]. The paper introduced feasible approaches for the extension of the authors’single-population model for differences in US mortality between men and women. Themodel suggests applying independent Lee–Carter models to individual populations as thefirst approach for multi-population modelling. Afterwards, the joint- κ model, based onthe assumption that populations’ mortality dynamics are driven by one commonly sharedtime-varying factor, was developed. The third approach was based on an extension ofthe Lee–Carter model, applying co-integration techniques and estimating the populationsjointly. Brief descriptions of the new models established on the basis of the Lee–Cartermodel are given below: Independent Modelling:
In this approach, mortality is modelled with the utilisation oftwo independent Lee–Carter models. Let m ix,t be the central death rate for population i in year t at age x . The model can then be expressed as follows:ln( m ix,t ) = a ix + b ix k it + e ix,t , i = R, B. (1)All of those parameters hold the same meanings that they possess in the originalLee–Carter model. It is possible to estimate the model parameters with the applicationof singular value decomposition, the Markov Chain Monte Carlo approach, or maximumlikelihood estimation. A mortality index can be modelled using two independent ARIMAprocesses for forecasting purposes. Although the model is easily applicable it ignoresthe dependency between the mortality rates of the populations. Hence, it might lead an3verestimation of the basis risks.
The Joint-k Model:
This model is based on the assumption that the mortality rates ofboth populations being driven by one single mortality index. This model may be expressedin the following way: ln( m ix,t ) = a ix + b ix k t + e ix,t , i = R, B. (2)In the joint-k model, the mortality index is the driving force of the changes in mortalityrates for both populations. Model parameters are estimated as in the previous approachwhile the mortality index k t is modelled based on an appropriate ARIMA process. How-ever, the model assumes that the mortality improvements of the populations are perfectlycorrelated and the existence of the common factor suggests identical advancements inmortality for both populations for all periods. Hence, the assumption is not realistic. [22]introduced a population-specific factor for this model, which is referred to as the “aug-mented common factor model”. Augmented Common Factor:
For the first approach, that of the two independent Lee–Carter models, life expectancy divergence increases in the long run. The joint-k modelcannot completely resolve this issue, since discrepancy between two populations in termsof parameter b ix could generate divergences in the mortality predictions.Li and Lee [22] present criteria for the divergence problem, as given below:- b Rx = b Bx for all x.- k Rt and k Bt have identical drift terms of the ARIMA process.Given these conditions, Li and Lee [22] introduced a specific factor for the Lee–Cartermodel: ln( m ix,t ) = a ix + b x k t + b ix k it + e ix,t , i = R, B. (3) b ix k it term serves to capture variations in the changing rate of mortality of population i from the long-term mortality change tendencies suggested by the common factor, b x k t .The k it factors are modelled using the AR(1) process to ensure the avoidance of anydivergence from the mortality projections [21].Another modelling approach for two-population mortality is the extension of theCairns–Blake–Dowd (CBD) mortality model for a single population [7]. A version ofthe CBD model for two populations and its variants were introduced by Li et al. [24]. Forexample the two-population variant of the CBD model with the incorporation of quadraticeffects, known as the M7 model, can be described as follows:logit q ix,t = κ i, t + ( x − ¯ x ) κ i, t + (cid:0) ( x − ¯ x ) − σ x (cid:1) κ i, t + γ it − x , i = R, B, (4)where ¯ x denotes average age and σ x is the average value of ( x − ¯ x ) . κ ,it and κ ,it are twostochastic processes which represent the two time indices of the model. Time index κ ,it reflects the level of mortality measured at time t , while κ ,it shows the slope and affectsevery age differently. γ it − x parameter represents the cohort effect. Li et al. [24] considered4hree different approaches, which were presented in the work of [31] to forecast futuremortality rates.The use of an age-period-cohort (APC) model with two populations was presented byCairns et al. [5] and Dowd et al. [14]. The model is expressed in the following way:log m ix,t = a ix + k it + γ it − x , i = R, B. (5) a ix , k it and γ it − x are the age, period and cohort effects of the populations.Spreads that exist between the state variables can be modelled as a mean-revertingprocess for each population so that the short-term trends in the mortality rates can vary,whereas there are parallel long-term improvements. In Cairns et al. [5], a Bayesianframework which allows to estimate non-observable state variables and the underlyingparameters of the stochastic process in one stage, is used. Moreover, Dowd et al. [14]developed a gravity approach in which the mortality rates of two populations experienceattraction to one another determined by a dynamic gravitational force. The force dependson the comparative sizes of the populations in question [29].Jarner and Kryger [17] and Cairns et al. [5] recognised the comparative value of thereference population supporting the index and the population whose longevity risk is beinghedged. Their approach centres on the reference population at the beginning, after whichthe dynamics of book mortality must be given for the incorporation of characteristics fromthe reference population. This relative method has important aspects such as it permitsthe mismatching of data between the book and reference population and it is applicablein the typical case in which a book population is significantly smaller than a referencepopulation [15]. The mortality models that are used in the relative method are presentedin Table 1 [29]. Table 1: Mortality Models for The Relative Method Original Model Name Reference Book-ReferenceModel Population Difference FormulaCommon Factor CF+Cohorts LC+Cohorts a Bx Common Age Effect CAE+Cohorts LC+Cohorts a Bx + β Rx k Bt Relative LC with Cohorts RelLC+Cohorts LC+Cohorts a Bx + β Bx k Bt Gravity Gravity (APC) APC a Bx + k Bt + γ Bt − x Two-population M5 M7-M5 M7 κ (1 ,B ) t + ( x − ¯ x ) κ (2 ,B ) t Two-population M6 M7-M6 M7 κ (1 ,B ) t + ( x − ¯ x ) κ (2 ,B ) t + γ Bt − x Two-population M7 M7-M7 M7 κ (1 ,B ) t + ( x − ¯ x ) κ (2 ,B ) t + (( x − ¯ x ) − ˆ σ x ) κ (3 ,B ) t + γ Bt − x Saint Model M7-Saint M7 κ (1 ,B ) t + ( x − ¯ x ) κ (2 ,B ) t + (( x − ¯ x ) − ˆ σ x ) κ (3 ,B ) t Plat Relative Model M7-Plat M7 − x − ¯ x κ (1 ,B ) t There are other multi-population applications of well-known single-population models.For instance, Biatat and Currie [2] expanded the P-spline approach to encompass scenarioswith two populations; previously, it had been utilised with success for cases of singlepopulations. Hatzopoulos and Haberman [16] and Ahmadi and Li[1] applied a multivariategeneralised linear model (GLM) for obtaining coherent forecasting of mortality in casesof multiple populations [29].However, to our knowledge, only Zhou et al. [30] incorporates jumps that are due tointerruptive events such as the Spanish flu epidemic in 1918 to two-population mortalitymodel. Their model can be regarded as a two-population generalisation of the model inChen and Cox [9]. They assumed that the mortality of a population is either jump-free5r subject to one transitory mortality jump. The severity of a mortality jump is normallydistributed.Although many multi-population mortality models exist, only a few investigates howto measure the longevity basis risks. Some of the earlier research designed for quantifyingbasis risk, such as Cairns et al. [6], Ngai and Sherris [25], and Li and Hardy [21], haveapplied the original framework constructed by Coughlan et al. [10].
The first step in pricing the longevity-linked products is to establish a two-populationmortality model in order to measure the longevity basis risk. A relative approach isapplied in this paper, as in Haberman et al. [15], since it has many advantages overjoint modelling. However, the modelling framework is slightly different from the originalformulations used for the reference model.
All of the examples provided in the paper utilise historical UK mortality data, which werecollected from the Continuous Mortality Investigation (CMI) and the Human MortalityDatabase (HMD). The first data represent the mortality experience of CMI assured malelives that are being hedged. The subsequent dataset is for the reference population,which provides the mortality experience of male lives in England and Wales (EW). Forthe reference population, a sample period from 1961 to 2016 is considered, while for thebook population, the sample period comprises the years of 1961-2005. The sample agerange being considered is 65 to 89.
The model considered in the paper is a Lee–Carter model with exponential transitoryjumps and renewal process. By using renewal process, we attempted to include thehistory of catastrophic events into the mortality modelling process. In ¨Ozen and S¸ahin[26], the proposed model was compared to other mortality models with jump effects. Theanalysis has shown that the arrivals between two catastrophic events is important andthe proposed model provides a better fit to the historical data (see ¨Ozen and S¸ahin [26]for more details). Moreover, as indicated before, mortality jumps have important impactson mortality dynamics and it is essential that they are incorporated into the modellingprocess. Hence we use the Lee–Carter model with exponential transitory jumps andrenewal process as our reference population mortality model.Here, we assume that transitory jumps are only valid for the reference populationbecause of the quality and size of the available data for the national population. Theproposed model is given by the following: log ( m Rx,t ) = a Rx + b Rx k Rt , (6) k Rt = k R + ( µ − σ ) t + σW ( t ) + N ( t ) (cid:88) i =1 Y i . (7)6ere, m Rx,t denotes the central death rate in year t for age x , a Rx represents the agepattern of the death rates, k Rt reflects variations that exist across time in the log mortalityrates, b Rx represents the mortality rates’ sensitivity to changes in time-varying mortalityindex k Rt , W ( t ) signifies standard Brownian motion, N ( t ) denotes the renewal process,and, finally, Y ( i ) denotes a sequence of iid exponential random variables representing thesize of the jumps.There are two identifiability constraints, which means that unique solutions exist forall of the model’s parameters. These identifiability constraints are given as follows: (cid:88) x b Rx = 1 , (cid:88) t k Rt = 0 . We will estimate the model’s parameters using the MLE method. First, reference pop-ulation parameters a Rx , b Rx , and k Rt are estimated. Afterwards, Equation (7) is used tocalibrate the time-varying mortality index. We need to find the density function of theindependent one-period increments, ∆ k Ri = r i = k Ri − k Ri − , to estimate the parameters ofthe calibrated model.Let D = { k , k , ..., k T } represent the mortality time series at times of t = 1 , , ..., T ,which have equal spacing. The one-period increments are independent and identicallydistributed (iid). Unconditional density for the one-period increment f ( r ) may be givenas follows: f ( r i ) = P (0) f ( r i |
0) + N ( t ) (cid:88) n =1 P ( n ) f ( r i | n ) , (8)where P (0) = 1 − F ( t ), P ( n ) = (cid:82) t P n − ( t − s ) f ( s ) ds are the probability of no jump and n jumps occur in the renewal process, where F ( t ) and f ( t ) are the distribution and densityfunctions of inter-arrival times of between two jumps. f ( r i | , f ( r i | n ) are conditionaldensities for a one-period increment; more specifically, they are conditional on the givennumbers of jumps and expressed as: f ( r i |
0) = 1 √ πσ e − ( r − µ +0 . σ σ f ( r i | n ) = (cid:90) ∞ η n ( n − X n − e − ηX √ πσ e − ( r − X − µ +0 . σ σ dx = η n ( n − √ πσ (cid:90) ∞ X n − e − ηX − σ ( r − X − µ +0 . σ ) dx Then, we can write the log-likelihood of the model as follows: L ( D ; µ, σ, η, α, β ) = T (cid:88) i =1 ln( f ( r i )) . The estimated a Rx , b Rx , µ, σ, η, α, β parameter values are shown in Table 2, while time-varying index k Rt is illustrated in Figure 1.Given the estimated parameters, the closed-form expression for the expected futurecentral death rates can be derived as follows: E [ ˆ m Rxt ] = exp ( a Rx + b Rx ( k R + ( µ − σ ) t + σW ( t ) + N ( t ) (cid:88) i =1 Y i )) . (9)7able 2: Estimated Parameters for the UK Age a x b x Age a x b x
60 -4.2486 0.0388 75 -2.7879 0.035661 -4.1505 0.0391 76 -2.6909 0.034962 -4.0451 0.0399 77 -2.6061 0.033563 -3.9482 0.0402 78 -2.5122 0.032564 -3.8408 0.0408 79 -2.4167 0.031465 -3.7472 0.0409 80 -2.3246 0.029866 -3.6598 0.0401 81 -2.2401 0.027867 -3.5517 0.0410 82 -2.1366 0.027268 -3.4593 0.0404 83 -2.0461 0.025769 -3.3607 0.0401 84 -1.9495 0.025070 -3.2684 0.0392 85 -1.8587 0.023371 -3.1758 0.0378 86 -1.7637 0.022772 -3.0687 0.0381 87 -1.6793 0.021373 -2.9749 0.0379 88 -1.5959 0.019574 -2.8755 0.0369 89 -1. 5088 0.0179Jump Diffusion Parameters µ =-0.2640 σ = 0.2764 η =1.4792 α =0.0015 β =0.6173 With the reference population in hand, it is now time to investigate the book population’smortality dynamics. Estimating the reference population first allows us to make knowl-edgeable decisions regarding the model’s book part, and we can also incorporate featuresfrom the reference population [29].The dynamics of the book population’s mortality are specified through the log mor-tality differences of the book population and the reference population. In this paper,we compare the most commonly used models which are the Lee–Carter model, the age-period-cohort (APC) model, the Cairns–Blake–Dowd (CBD) model, and common ageeffect models to model the book population.Note that for all the models being compared we assume that D Bx,t ∼ P oisson ( E Bxt , q
Bxt ). The dynamics of the book population are given as follows: log ( m Bx,t ) − log ( m Rx,t ) = a Bx + b Bx k Bt . (10)The term a Bx denotes the difference in the book population’s level of mortality comparedto that of the reference population for age x . Thus, we can conclude that the mortalitylevel in the book equals a Rx + a Bx . Time index k Bt contributes to establishing the differencethat exists in the mortality trends. The b Bx term shows us how differences in mortalityfor age x will respond if any change occurs in time index k Bt [15].8igure 1: Estimated Values of k Rt . This model may be seen as an extension of the Lee–Carter model that possesses a commonage effect. It can be given by the following equation: log ( m Bx,t ) − log ( m Rx,t ) = a Bx + b Rx k Bt . (11)The a Bx and k Bt parameters here are the same as in the LC model for the book population.Different from the LC model, there is a common age effect parameter, b Rx , which is thesame as for the reference model. The APC model was introduced by Currie [11] and it is widely used in the literature. Itcan be regarded as a generalization of the LC model and a two-population version of thismodel may be written in the following way: log ( m Bx,t ) − log ( m Rx,t ) = a Bx + k Bt + γ Bt − x . (12) a Bx , k Bt , and γ Bt − x respectively represent the age, the period, and the cohort effects of thebook population [11]. The γ Bt − x term is utilized here in order to account for differencesthat exist in the cohort effect in the two populations of interest. These parameters reflectthe mortality differences between the two populations. Cairns et al. [7] introduced the following model with the aim of fitting the mortality rates: logit ( q Bx,t ) − logit ( q Rx,t ) = κ ,Bt + ( x − ¯ x ) κ ,Bt . (13) κ ,Bt and κ ,Bt are two stochastic processes and represent the time indices of book pop-ulation. These parameters reflect the mortality differences between the two populationsas in the APC model. 9he analysis of the models considered in this section becomes something of a challengedue to the CBD model directly modelling one-year death rate q x,t while the others thatare being considered in the paper model central death rates m x,t . In order to comparethe models in a consistent way, we must introduce an additional step for the modelling of q x,t . We transform the one-year death probabilities in the central death rates using theidentity m x,t = − log(1 − q x,t ). For all mentioned models, the parameters are estimated bytwo main steps. As indicated before, the parameters of the book population are estimatedconditional on the parameters of the reference population. Under Poisson assumption,the log-likelihood function of the book population is as follows: l B = (cid:88) x,t (cid:0) D Bx,t ln E Bx,t + D Bx,t ln m Bx,t − E Bx,t m Bx,t − ln( D Bx,t !) (cid:1) . We estimate the parameters by applying the maximum likelihood method. The parame-ters obtained for the book population are given in Figure 2 for different mortality models.According to Figure 2, the a Bx parameter shows that the younger ages reveal lowerrates of mortality while the older ages reveal higher mortality. The positive values of b Bx demonstrate that mortality decreases for all ages. These results are valid for all a Bx and b Bx parameters for all mortality models of the book population. The mortality index, k Bt ,reflects the changes in mortality rates over the years for the LC, Common age and APCmodels. The γ Bt − x parameter represents the cohort related effects in the book population.The negative values of κ ,Bt parameter in the CBD model indicate the lower mortality rateswhile the positive values reflect the higher mortality rates. The κ ,Bt parameter controlsthese lower and higher mortality rates in the CBD model for the book population.The BIC values obtained from the fitted models for book population mortality aregiven in Table 3. The common age effect model has the lowest BIC value according toTable 3. Therefore, we model the book population’s mortality using the common ageeffect model. Table 3: BIC Values for the Book Population Models LC Model Common Age Effect Model APC Model CBD Model12684.89
Finally, we complete the modelling framework by specifying the period’s dynamicsand the cohort terms, which will be used to forecast and simulate the future rates ofmortality. A detailed analysis regarding the selection of the time series to be used in thedynamics can be found in the work of Li et al. [24]. This part of the study confines itselfto focusing on the models that are commonly applied in the literature. We assume thatthe two populations will experience similar improvements in the long run and thus weassume that the spread in both time indices and cohort effects should be modelled as astationary process.In this paper, the time-varying mortality indices of the book population k Bt aremodelled as an autoregressive process of order one; we are thus able to write k Bt = ψ + ψ k Bt − + ξ t for the LC, the common age effect, and the APC models. In the longterm, the mean of k Bt equals ψ / (1 − ψ ) if | ψ | <
1. The autocorrelation depends on thesize of ψ . More technical aspects of time-series modelling can be found in the work ofTsay [28]. 10igure 2: Estimated Parameters of the Book Population Models In evaluating the uncertainty of future outcomes and finding the optimal model to assesslongevity basis risk, it is necessary to address all of the parameter errors , process errors ,and model errors from a modelling or a regulatory perspective such as that of SolvencyII [23]. Parameter error refers to the uncertainty in estimating model parameters, while process error arises from variations that exist within the time series. Finally model error reflects the uncertainty that is present in the model selection.In the literature, a number of studies have been proposed to allow for both processerror and parameter error in index-based hedging. For instance, Brouhns et al. [4] useda parametric Monte Carlo simulation method for the generation of examples of modelparameters following a multivariate normal distribution. Later, in a subsequent work,Brouhns et al. [3] also explored a semi-parametric bootstrapping procedure designedfor the simulation of death rates from the Poisson distribution with the obtained meanequaling the observed numbers of deaths. On the other hand, Renshaw and Haberman1127] utilized fitted numbers of deaths by using the Poisson process. In another study,Koissi et al. [18] used a bootstrap method for the residuals of a fitted Lee–Carter model.Different from the existing methods, Czado et al. [12] and Kogure et al. [19] suggestedthe application of Bayesian adaptations of the LC model. Li [20] quantitatively comparedpossible methods for simulations; according to the conclusions of that study, samplingresults will all be relatively close to each other regardless of whether the method appliedis parametric, semi-parametric, Bayesian, or residual bootstrapping. All of these varioussimulation methods possess individual advantages and disadvantages. In this study, thebootstrapping method of Brouhns et al. [3] has been selected due to its ability to helpfullyinclude both parameter errors and process errors in simulating future mortality rates. Thebootstrapping procedure is detailed as follows: • Estimation of the parameters of the LC model is performed by using original data.Once they are obtained, those estimated parameters are then applied for estimatingthe numbers of deaths for both the reference and the book population by ˆ m Rx,t E Rx,t ,ˆ m Bx,t E Bx,t . • The new data on numbers of deaths are simulated from a binomial distribution forthe book population to include the sampling risk and Poisson distribution is used forthe reference population. The newly simulated data will then be used for estimationof the reference and book populations’ mortality parameters. Incorporating this stepmeans that the model can allow for parameter error. • Next, we must fit time-series processes to the new data sample’s temporal modelparameters, k Rt and k Bt , since we want to be able to simulate their future values.Furthermore, the inclusion of this step means that the model can allow for processerror. k Rt is modelled by using the proposed model and k Bt is modelled by usingAR(1). • We generate future mortality rate samples for all x and future t with the incorpora-tion of the parameters obtained in step (2) and the simulated values that we gainedin step (3) into log ( m Rx,t ) and log ( m Bx,t ). As a result, our set of future mortality rateswill form one random future scenario.5. We repeat steps (1) to (4) until we have produced a total of 10,000 random futurescenarios.Different from Haberman et al. [15], in this paper, the parameter errors of the referencepopulation has been considered by applying bootstrapping to both reference and bookpopulation models estimations.A sample from the simulated mortality paths are shown in Figure 3. The mortalitypaths enable us to obtain projected mortality rates, hence future liabilities of pensionplan and hedging instrument.
The finite sizes of the book and reference populations and the randomness of the outcomesof the individual lives cause the sampling risk. If the size of populations are infinite, thefuture outcomes will converge the true expected values according to the law of largenumbers. Nevertheless, the size of the populations is limited in reality. Although the12igure 3: Sample Paths of m x,t .bigger countries have very large population sizes, the annuity or pension portfolio’s size isusually small. Hence the book and reference populations’ outcomes will deviate randomlyfrom their true expected values and also from each other. To reflect the effect of theportfolio size, the number of lives is simulated as: l Bx +1 ,t +1 ∼ Binomial( l Bx,t , − q Bx,t ) (14) l Bx,t is the future number of lives aged x at time t of the book population. q Bx,t is thefuture mortality rate at age x at time t and it is simulated from the semi-parametricbootstrapping method. Simulating the number of lives of the book population by usingthe binomial distribution provides protection from the sampling risk [23]. After constructing a two population mortality model, we need to compare the proposedmodel with other mortality models and show its effectiveness. Therefore, we consider twoadditional two-population mortality models that are commonly used in the literature.First model is the
LC model with jumps and the second one is the
LC with common ageeffect model called as
LC+Cohorts . LC with jumps model is very similar to Zhou etal. [30], however we use the relative approach to estimate the parameters of the models.Thus our mortality data could be based on different sizes of periods for reference andbook population. We use the same notation with Zhou et al. [30] for the
LC model withjumps and the model is as follows:ln( m ix,t ) = a ix + b ix k it + e ix,t , i = 1 , . (15)where a ix and b ix are the same as in the original LC model. The index k it is decomposedinto the sum of two components, k it + N it Y it . The first component, k it , is the time- t value13f an unobserved period effect index that is free of jumps, while the second term, N it Y it indicates the jump effect at time t . The model allows the two populations to have differentjump times, jump frequencies and jump severities. They allow a maximum of one jumpin a given year and the jump severity Y it follows a normal distribution with mean µ Y andvariance V Y (see for more model details Zhou et al. [30]).The second mortality model that we consider here is the LC+Cohorts model thatis given in Section 3. The parameters of the models are estimated by using maximumlikelihood method.The estimated parameters for these two models are given in Appendix A.
In this section, we consider a hedging strategy to assess longevity basis risk and to measureeffectiveness of the hedge while taking mortality jumps and sampling risk into account.The effectiveness of a hedge can be described as how much longevity risk is transferredaway. Following formula can be used to define the level of longevity risk reduction for thehedge as in Coughlan et al. [10]:longevity risk reduction = (cid:18) − risk (hedged)risk (unhedged) (cid:19) × risk (unhedged) and risk (hedged) represent the appropriate dispersion-based risk measures for the aggregate longevity of the portfolio before and after the hedg-ing. A perfect hedge would have a longevity risk reduction equal to 1 and a good hedgewill have a risk reduction degree close to 1; a risk reduction degree close to 0 indicatesan ineffective hedge [13]. In this paper, the variance risk measure is used to minimise thevariations in the expected future cash flows of the hedging instrument.A simple hypothetical case study based on a pension plan is considered to illustratethe effect of the proposed mortality models and different volumes of book populationdata on hedge effectiveness. The pension plan members are assumed to have underlyingmortality rates that are same as the CMI male assured lives dataset. Suppose all thepensioners in the plan are aged 65 and pays £ L ( t ), is given as below: L ( t ) = (cid:88) t =1 l B t,t (1 + r ) − t (17)As a floating-leg receiver, the present value of the longevity swap’s future cash inflows,S(t), can be written as S ( t ) = (cid:88) t =1 (cid:0) t p R − t p R ; forward (cid:1) (1 + r ) − t (18)14or this equation, we calculate random future survivor index t p R and forward sur-vivor index t p R ; forward by applying the survival probability formula, as follows: t p R =(1 − q R , )(1 − q R , ) ... (1 − q R t − ,t − ). Furthermore, the present value of the aggregatepension plan position after longevity hedging (hedged position) may be expressed withthe following statement: (cid:88) t =1 l B t,t (1 + r ) − t − w (cid:88) t =1 (cid:0) t p R − t p R ; forward (cid:1) (1 + r ) − t (19)where weight w denotes the notional amount of longevity swap necessary for successfulhedging to be performed [23].Moreover, in order to take the sampling risk into account, we use the binomial deathprocess for the book population as given in Equation (14). To emphasise the role of thesize of the population on hedge effectiveness, we produce three simulated distributions as l (65) = 5 , l (65) = 10 ,
000 and l (65) = 100 , l (65) LC with Renewal P. & LC with Jumps LC+CohortsExponential Jumps5000 0.4507 0.2317 0.133510000 0.7602 0.5713 0.2605100000 0.8569 0.7392 0.6328 Index-based hedging solutions have many advantages. In such capital market solutions,it is possible to transfer the longevity risk to capital markets at lower costs. However,the potential differences between hedging instruments and pension or annuity portfoliocause longevity basis risk. In this paper, we construct a two-population mortality modelto measure and manage the longevity basis risk.15n appropriate two-population model was built for EW male lives and CMI assuredmale lives to measure longevity basis risk, and the relative approach to model the popu-lations has been adopted. The modelling process of the reference population was followedby the modelling of the dynamics of the book population’s mortality. The reference pop-ulation is modelled by using the LC model with renewal process and exponential jumpsproposed by ¨Ozen and S¸ahin [26] and the common age effect model outperformed amongthe others to model the book population.The bootstrap approach of Brouhns et al. [3] was applied in order to include bothparameter error and process error in the simulation of future mortality rates. The Poissondistribution is used for the simulation of the reference population’s lives and the binomialdistribution is used for the simulation of the book population’s lives to consider thesampling risk.Furthermore, the impact of the proposed mortality model and sampling risk to hedgeeffectiveness is examined. For this purpose, a hypothetical pension plan and hedgingstrategy which consists of 10-year longevity swap is considered based on the three differenttwo-population mortality models namely the proposed LC model with renewal process andexponential jumps, LC with jumps model and LC with common age effect model. Thenthe hedge effectiveness is calculated by using these three mortality models to compare therisk reduction caused by the models. The analysis suggests that the proposed mortalitymodel provides a more effective risk reduction for mortality jump risk and sampling riskthan the other two models.A possible future study can be to construct an optimal hedging framework with collat-eralisation to obtain reasonable risk reduction rates by using the proposed two-populationmodel.
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Appendix A
In this section, the parameters of the LC model with jumps and LC+Cohorts are pre-sented.The a Rx and b Rx parameters are the same for all models.In [30], the period effect indices are modeled by following set of equations:ˆ k t +1 = ˆ k t + µ k + Z k ( t + 1) ,k t +1 = ˆ k t +1 + N t +1 Y t +1 , k ( t ) = ˆ k t − ˆ k t , ˆ∆ k ( t + 1) = µ ∆ k + φ ∆ k ˆ∆ k ( t ) + Z ∆ k ( t + 1) ,k t +1 = ˆ k t +1 + N t +1 Y t +1 The estimated a Bx and b Bx parameters are given in Table 5.Table 5: Estimated Parameters for the LC with Jumps ModelAge a x b x Age a x b x
60 -0.8348 0.0234 75 -0.6154 0.018861 -0.8006 0.0230 76 -0.6021 0.016662 -0.7823 0.0217 77 -0.5548 0.016763 -0.7775 0.0222 78 -0.5528 0.015964 -0.7879 0.0225 79 -0.5282 0.014665 -0.8082 0.0234 80 -0.4969 0.015366 -0.7920 0.0205 81 -0.4566 0.013667 -0.8199 0.0199 82 -0.4905 0.012668 -0.7798 0.0199 83 -0.4426 0.013169 -0.7650 0.0194 84 -0.4414 0.012470 -0.7193 0.0189 85 -0.4493 0.012471 -0.6876 0.0177 86 -0.4449 0.014072 -0.6941 0.0191 87 -0.4244 0.013173 -0.6655 0.0175 88 -0.3931 0.010674 -0.6572 0.0168 89 -0.4136 0.0113The parameters of the jump component of the model are presented in Table 6.Table 6: Estimated Parameters for the LC with jumps model µ k = − . µ Y = 4 . µ Y = 4 . µ ∆ k = − . φ ∆ k = 0 . V Y = 0 . V Y = 0 . V Z = 0 . P r ( N t = 0 , N t = 0) = 0 . P r ( N t =0 , N t = 1) = 0 . P r ( N t = 1 , N t = 1) = 0 . a x Age ˆ a x
60 -0.5431 75 -0.393061 -0.5123 76 -0.388662 -0.4981 77 -0.354563 -0.4897 78 -0.356964 -0.4995 79 -0.341965 -0.5207 80 -0.317166 -0.5223 81 -0.289367 -0.5495 82 -0.320168 -0.5135 83 -0.282869 -0.5032 84 -0.280170 -0.4664 85 -0.298871 -0.4513 86 -0.290472 -0.4500 87 -0.284673 -0.4293 88 -0.263974 -0.4287 89 -0.2944Figure 4: Estimated k tt