1999 Proceedings of Amer. Stat. Assoc., pp STATISTICAL ASPECTS OF JOINT LIFE INSURANCE PRICING

Transcription

1 1999 Proceedings of Amer. Stat. Assoc., pp STATISTICAL ASPECTS O JOINT LIE INSURANCE PRICING Heeyung Youn, Arady Shemyain, University of St.Thomas Arady Shemyain, Dept. of athematics, U. of St.Thomas, 2115 Summit Ave., OSS201, St.Paul, N, Key Words: life insurance, joint survival functions, copula functions, age difference 1. Introduction Joint life insurance policies can be issued to married couples so that the insurance payoff is due at the time of the first death of a spouse (joint-life policy) or the second death of a spouse (last survivor policy). In recent years, a certain attention in the actuarial industry has been attracted to the latter. These policies are generally used by older couples for estate tax purposes and carry large amounts of insurance. To simplify actuarial calculations in the pricing of joint life policies it has traditionally been assumed that individual mortalities of spouses are independent. However, research over the past 40 years suggests otherwise. While the independence assumption tends to overestimate the mortality cost for joint first death insurance and joint annuity products, it underestimates the cost for joint last survivor policies. Considering the fact that insurance amounts in these policies are typically large, this unrealistic assumption could have a large financial impact on the insurance industry as company portfolios change with the aging population s insurance choices and growing wealth. The problem of fair pricing for such policies deals with the construction of a two-dimensional survival function for a married couple. Special attention should be paid to adequately reflect the association between two important variables: the husband s age at death and the wife s age at death. The models currently used for construction of two-dimensional survival functions are based on spouses chronological age only. Both nonparametric models (Dabrowsa (1988), Pruitt (1993)) and parametric copula models (Hougaard (1986), Anderson et al. (1992), Hougaard et al. (1992), rees et al. (1996), rees and Valdez (1998)) have been used. We believe that they tend to underestimate the role of factors associating the two spousal deaths in real (calendar) time. The deaths of two spouses resulting from a common disaster and the death of a second spouse due to broen-heart syndrome are recognized as examples of such association. In the present paper we introduce an additional variable age difference between the spouses. It is observable at the policy issue date. Then we build the twodimensional survival function of the spouses conditional on age difference. This allows us to reflect the role of calendar-time related factors in the model. An alternative approach to incorporation of real (calendar) time-scale is also discussed. The copula model presented is based on marginal male and female survival functions with the association parameter depending on the age difference between the spouses. or an illustration we use data from a large Canadian insurance company, the same set of data used by rees et al. (1996). In Section 2 we address the problem of estimation of joint and joint conditional survival functions used in actuarial computations. We concentrate on copula models and problems of joint last survivor insurance. In Section 3 we discuss the sources of association in bivariate mortality and justify the introduction of an additional age difference variable. Then in Section 4 we proceed with the correlation analysis of the data set used in our illustration. Section 5 contains the description of the suggested parametric copula model. inally, our conclusions are given and supported with numerical results. 2. Conditional Survival Probabilities Let us denote by S (, ) PX ( xy, y) (1) the joint survival function of the spouses, where X and Y respectively are wife s and husband s ages at death, (X,Y) being a dependent random vector. The premium computations for the joint husband-wife policies (see e.g., Bowers et al. (1997)) require the estimation of survival probabilities p p P X x+, Y y+ X x, Y y, p 1 c for the joint-life status (active until the first death in the family) and h

2 p Pc{ X x+ } U{ Y y+ } X x, Y yh, (2) p p 1 for the last survivor status (active until both spouses die). Using (1), we obtain the following formula for calculating p : p S( x+, y) + S( x, y+ ) S( x+, y+ ) S( x, y) (3) In some recent studies (see Hougaard (1986), Hougaard et al. (1992), rees et al. (1996)), the method of copula functions was suggested for the construction of joint survival functions. According to this method, the joint survival function of X and Y is represented as S( x, y) C( S( x), S( y)), where S ( x) P( X x) and S ( y) P( Y y) are female and male marginal survival functions, and C(u,v) is a copula a function with special properties, mixing up the marginals with a certain association parameter. rees et al. (1996) suggest using twoparameter Gompertz or Weibull (defined below) marginal female and male survival functions and ran s copula α u 1 ( e 1)( e Cuv (, ) u+ v 1 ln 1+ α α e 1 HG ( 1 ) α ( 1 v ) 1) with the association parameter α < 0. Then the maximum lielihood estimator is constructed for the 5- dimensional vector parameter, where the first 4 components correspond to parameters of Gompertz or Weibull marginal survival functions, and the last one is the parameter of association. The estimate $ α corresponding to the value 0.49 of Spearman s correlation coefficient indicates a strong statistical dependence between the husband s and wife s mortality. However, we believe that some additional dependence could be captured if another variable D - the age difference between a husband and a wife the husband s age minus wife s age, is introduced into the picture. To illustrate this point, let us consider the value S( x+ 1, x) + S( x, x+ 1) S( x+ 1, x+ 1) S( x, x) If we assume, for example, that x y, i.e., D 0, this is the conditional probability that both spouses (last survivor status) survive one calendar year, from age x to I KJ age x +1. This probability is liely to be influenced by such factors as common disaster (accidents etc.) and broen-heart syndrome, when the death of a surviving spouse follows the first death closely. However, if we assume x y +10, i.e., D 10, this probability addresses the event of two spouses death at the same age, which in real time has to do with two events 10 calendar years apart from each other. In this case, it is not liely to be influenced by common disaster and broen heart. However, the joint survival function defined in (1) is estimated unconditionally on D, so we do not discriminate between these two cases. The following discussion of possible sources of dependence and the correlation analysis of the available data set gives additional justification for the use of conditioning on D in the related survival function analysis. 3. Sources of Association Since most of the customers of last survivor insurance policies represent married couples, we can list several possible sources of association between husbands and wives mortality: - common lifestyle, - common disaster, - broen-heart factor. The first of these three factors is related to the physical age of partners and directly affects the correlation between X and Y their ages at death. However, the other two factors correspond to events happening simultaneously (common disaster) or close in calendar time (broen-heart factor). Since the calendar time does not coincide with the time scale for X and Y, their effect on the correlation between X and Y depends on this time scale difference. We approach this issue by introducing the new variable D (see above). This variable is always observable at the policy issue date. If we estimate the conditional bivariate survival functions S ( x, y ) d S ( x, yd d ), these estimates may give us more specific information than does the joint survival function S (, ) PX ( xy, y) alone. An alternative apporoach to dealing with this issue would be to shift the time scale introducing a new variable Z wife s age at the time of husband s death husband s lifelength since wife s date of

3 birth. It is easy to see that Z Y D. The effect of calendar-time sources of association translates naturally into correlation between X and Z. However, obtaining a meaningful marginal distribution for Z might be difficult. 4. Correlation Analysis of Data The data set we use for illustration comes from 14,947 joint and last-survivor annuity contracts of a large Canadian insurer. The contracts were in payoff status over the observation period December 29, 1988 through December 31, or each contract, we have information on the date of birth, date of death (if applicable), date of contract initiation (entry age) and sex of each annuitant. Contracts with same sex annuitants (a total of 58 contracts) were eliminated from the study. or couples with multiple contracts all but one of those (3,432 contracts) were also eliminated from the study. This way each couple was included in the data set only once. In the following, we study the mortality dependence relation within a male and female pair, based on information from 11,457 pairs (married couples). In order to estimate the conditional survival function from our data set, we will have to round the variables X and Y to the nearest integer number of years. Then we can consider subsamples corresponding to integer values of D. Aggregate data for larger (more than 1000 data points) subsamples corresponding to the values D 0,1,2,3,4, are presented in Table 1 below. Table 1. Number of Policies by ortality Status and Age Difference D. All D<0 D0 D1 Data Observed (Total) emale Deaths (D) ale Deaths (D) Both Died (BD) Ratio ( BD Total)/( D D) Table 1 shows the number of female and male annuitants and the number of deaths that occurred during the observation period. Ratios between the observed number of double deaths (BD) and the expected number of double deaths ( D D / Total ) under the assumption of independence between female and male mortality rates are given for the entire data set and for subsamples grouped by age differences. The table shows that the actual number of deaths is about 4 times the number expected, which casts doubt on the assumption of independence. Using the data points for which both partners die during the observation period (no censoring), we calculate full and partial nonparametric correlations (Kendall s tau and Spearman s rho) between random variables X, Y, and Z Y D. Calculations are carried out (a) for the entire sample and (b) for subsamples corresponding to D 0,1,2,3,4, which are the only values of D providing for a statistically sufficient number of observations. Standard deviation for the Spearman s sample correlation is estimated as ( n 1) 1/2, where n is the corresponding subsample size. We expect sample correlations between the spouses ages at death to be increased by a certain positive correlation in the entry ages. If we calculate the partial correlation of X and Y controlling for the entry ages, we get the value of ~ ρ.41. However, in order to detect the effect of the variable D, we return to full bivariate nonparametric correlations. The main results are summarized in the following table. Table 2. Nonparametric Correlations Variables X,Y X,Z D0 D1 D2 D3 D4 Kendall's $τ Ratio $τ / (X,Y) Spearman's $ρ Ratio $ρ /(X,Y) Sample Size St. Deviation for $ρ D2 D3 D4 D>4 Observed (Total) emale Deaths (D) ale Deaths (D) Both Died (BD) Ratio ( BD Total)/( D D) Though small subsample sizes bring about high standard deviations and do not allow us to mae any conclusive remars, we may observe that the sample correlations between X and Y within the subsamples of fixed D are uniformly higher than over the entire data set. Also, correlation between X and Z is higher than between X and Y, and the ration between corresponding correlations is almost stable. An explanation of this phenomenon may be provided by the following

4 relationships for Pearson s correlation coefficient under the assumption of X and D being independent: ρ( XYD, d) ρ( XZ, + d) ρ( XZ, ); Cov( X, Z + D) ρ( XY, ) ρ( XZ, + D) σ σ X Z+ D (4) The marginals do not change with d, so the only parameter depending on age difference is association. It reflects the belief that for large values of d, as the distance between the physical age time scale of Y and the calendar time scale of Z Y D increases, the association should go down. In order to capture this downward trend, we wor out the following sequence of actions: odel A: Cov( X, Z) ρ( X, Z) σ σ + σ 1+ σ / σ X Z D D Z This gives evidence that some additional association in our data is captured when we shift the time scale or group the data set according to the value of the variable D. Accordingly, we will concentrate on the estimation of the conditional survival function S ( x, y ) d S ( x, yd d ) rather than on S( x, y ). 5. Description of the odel We will use a Weibull-Hougaard copula model with one important addition: the association parameter α is allowed to depend on d the observed value of D. The choice of Hougaard s copula versus ran s copula is explained by the simpler overall expression when used with Weibull marginals, which was essential in our search for a convenient functional form of dependence for α α( d ). The joint conditional survival function of X and Y is represented as S ( x, y) C ( S ( x), S ( y)), d d (i) (ii) (iii) run complete LE for (5), when we do not assume parametric dependence of α on d, and estimate the vector parameter ( β, γ, β, γ,{ α( d)} d R ), where R is the set of all possible values of D in the data; using the least-square fit for a reasonable subset of R (in our case we use all subsamples Dd with more than 100 observations, total count 94.3% of the data set), choose an appropriate parametric model for α α( d ) reflecting desirable properties of this function: attaining maximum at d 0, decreasing for d > 0, even, converging to 1 at ±. Our choice based on the study data was θ function α( d; λ, θ) 1 + with hyperparameters ( λ, θ ); 1 + λd 2 run a maximum lielihood model and obtain estimates for the vector parameter ( β, γ, β, γ, λ, θ), which results in a family of conditional survival functions S ( x, y) C ( S ( x), S ( y)). d d where γ S ( x) exp( ( x/ β ) ), γ S ( y) exp( ( y/ β ) ), are Weibull survival functions, and Cd ( u, v) is a Hougaard s copula with association parameter α ( d ) > 1. The associated conditional survival function is then: S ( x, x+ d) C ( S ( x), S ( x+ d)) R L S T N d d HG α( d ) γ α( d ) γ x x d exp H G I + K J + β β I KJ O QP 1/α ( d ) U V W (5) 6. Numerical Results and Conclusions The results of implementation of the model developed in the previous section to the data set described in Section 4 are presented in Table 3 below. As a benchmar, we include the results from odel B: maximim lielihood estimation according to Weibull- Hougaard model where α is fixed (not allowed to depend on d) and odel C: no association, marginal survival functions estimated separately under an assumption of independence using maximum lielihood estimation for a Weibull parametric model.

5 Table 3. LE of the Parameters for Three odels of the Conditional Survival unction Sd ( x, y) Parameters odel A odel B odel C emale β γ ale β γ Association α ( 0 ) α(5) α( 10) λ θ AIC The AIC (Aaie Information Criterion) values indicate that the trade-off between increased number of parameters (6 instead of 5 or 4) and lielihood values is still in our favor. Let us see how the difference in parameter estimates is reflected in the insurance premium values. In the premium computations for the joint last survivor insurance, we use the following formula for the last survivor insurance premium A derived through the calculation of annuity values && a : A i i a a 1 1 && ; where && p 1+, (6) 1+ i 0 where x and y respectively are female and male ages at the policy issue date, i is the interest rate, and formula (3) is used for evaluation of conditional survival probabilities with S( x, y) replaced by S ( x, y ) d S ( x, yd d ) defined by (5) with parameter values from Table 3. Then the ratios of premium values are analyzed for most typical policy issue ages with results presented in Tables 4A and 4B. Table 4A. Ratio of Premium Values (odel A to odel C) D X H I K Table 4B. Ratio of Premium Values (odel B to odel C) D X These results demonstrate that odel A captures some additional association between survival functions of the spouses which is not reflected in more conventional copula models (e.g., odel B). This additional association may have a large impact on last survivor insurance pricing. Acnowledgements The research project was supported by the Society of Actuaries (CKER). We express sincere gratitude to the project oversight group for its support and valuable comments. We also wish to than Thomas Louis for the most helpful discussions and John Kemper for his careful reading the manuscript. References ANDERSON, J.E., LOUIS, T.A., HOL, N.V. and HARVALD B. (1992) Time Dependent Association easures for Bivariate Survival unctions, Journal of the American Statistical Association, Vol. 87, 419. BOWERS, N., GERBER, H., HICKAN, J., JONES, D. and NESBITT, C. (1997) Actuarial athematics, Schaumburg, Ill.: Society of Actuaries. DABROWSKA, D. (1988) Kaplan-eier Estimate on the Plane, Annals of Statistics, Vol. 16, 4, REES, E., CARRIERE, J. and VALDEZ, E. (1996) Annuity Valuation with Dependent ortality, Journal of Ris and Insurance, Vol. 63, 229. REES, E. and VALDEZ, E. (1998) Understanding Relationships Using Copulas, Actuarial Research Clearing House, Proceedings, 32 nd Actuarial Research Conference, August 6-8, 1997, 5. HOUGAARD, P. (1986) A Class of ultivariate ailure Time Distributions, Biometria, Vol. 73, 671 HOUGAARD, P., HARVALD, B., and HOL, N.V. (1992) easuring the Similarities Between the Lifetimes of Adult Twins Born , Journal of the American Statistical Association, Vol. 87, 17. PRUITT, R.C. (1993) Small Sample Comparison of Six Bivariate Survival Curve Estimators, J. Statist. Comput. Simul., Vol. 45, 147.

6