# Modelling spousal mortality dependence: evidence of heterogeneities and implications

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1 1/23 Modelling spousal mortality dependence: evidence of heterogeneities and implications Yang Lu Scor and Aix-Marseille School of Economics Lyon, September 2015

2 2/23 INTRODUCTION

3 3/23 Motivation It has been documented that the two spouses have positive mortality dependence. Many pension funds/annuity providers offer reversionary benefit. The aim of this paper is to model such a dependence, and discuss its impact on pricing.

4 4/23 Cash flow of a joint annuity: Case A 65 T 2 T 1 age Figure: In Case A, the principal beneficiary dies after the spouse (T 1 > T 2 ). No additional benefit for the spouse.

5 5/23 Case B 65 T 1 T 2 age Figure: In Case B, the principal beneficiary dies before the spouse (T 1 < T 2 ). Spouse continuous to be receive a reduced annuity.

6 /23 Different types of dependencies The mortality jump can be due to two reasons: The broken-heart syndrome, that is a causal effect. Spurious risk correlation due to similar risk factors, i.e. observed and unobserved heterogeneities. Current literature usually neglects the heterogeneities. It is important to disentangle these different types of dependencies [see Gourieroux and Lu (2015 IME)].

7 7/23 Outline We estimate a model which distinguishes them. We find that heterogeneities explain a significant proportion of the dependency. These heterogeneities can be either observed or unobserved. We find important over/under estimations of the standard model depending on the characteristics of the couple.

8 8/23 THE MODEL

9 9/23 The status of the couple both alive wife died first husband died first both died

10 0/23 Before the first death: spurious risk correlation The mortality intensities are: θ 1 (t U, V, T 1 > t, T 2 > t, X, c 1, c 2 ) = λ 1 (t + c 1 ) exp(β 1 X)U, θ 2 (t U, V, T 1 > t, T 2 > t, X, c 1, c 2 ) = λ 2 (t + c 2 ) exp(β 2 X)V, where: T 1, T 2 are duration, c 1, c 2 are ages at inception; X is the covariate(s), including ages at inception and dates of birth; Thus there can be duration effects, age effects, and even cohort effects; U, V are the unobserved individual heterogeneities; they introduce spurious dependence between T 1, T 2 ; λ 1, λ 2 are baseline intensities; This is called mixed proportional hazard competing risks model.

11 1/23 After the first death: the broken-heart syndrome After the death of male (resp. female), the mortality of the spouse is multiplied by a constant γ 1 (resp. γ 2 ). These parameters capture the real duration dependence, the contagion". Usually γ 1 γ 2.

12 12/23 (Conditional) transition intensities both alive wife died first λ 1 (t)φ 1 (X)U λ 2 (t)φ 2 (X)V γ 1 λ 1 (t)φ 1 (X)U γ 2 λ 2 (t)φ 2 (X)V husband died first both died

13 13/23 The parameters of interest are: Baseline intensities λ 1, λ 2, which we specify as exponential functions. The distribution of (U, V ), which we specify in a semi-parametric way. Regression coefficients β 1, β 2. Jump parameters γ 1, γ 2.

14 14/23 The data A French joint annuity portfolio kindly provided by Scor. Population size: couples. Observation period: Available information: ages at inception, amount of the annuity, dates of birth, potential dates of death (with truncations and censoring). The model is estimated by maximum likelihood. See the paper for details.

15 15/23 EMPIRICAL RESULTS

16 6/23 What we find Asymmetric jumps: γ 1 = 1.53 > γ 2 = The unobserved heterogeneities of the two spouses are highly positively correlated. They account for a significant portion of the dependency that cannot be explained by the broken-heart syndrome. Moreover the parameters γ 1, γ 2 are over-estimated if unobserved heterogeneities are omitted.

17 17/23 Comparison of three models Model unobserved heterogeneity mortality jump 1 Yes Yes 2 No Yes 3 No No Table: Summary of the three models We compare their forecasts of the following quantities: Residual life expectancies. The price of Contract 1 (resp. 2), for a male (female) principal beneficiary, with a 1/3 discount for the spouse if she/he survives longer.

18 18/23 Case 1: Male is aged 60, female 65 male residual female residual Price Price Model life life Contract Contract expectancy expectancy No. 1 No Case 2: Male is aged 70, female 65 male residual female residual Price Price Model life life Contract Contract expectancy expectancy No. 1 No

19 18/23 Case 1: Male is aged 60, female 65 male residual female residual Price Price Model life life Contract Contract expectancy expectancy No. 1 No Case 2: Male is aged 70, female 65 male residual female residual Price Price Model life life Contract Contract expectancy expectancy No. 1 No

20 19/23 Case 1: Male is aged 60, female 65 male residual female residual Price Price Model life life Contract Contract expectancy expectancy No. 1 No Case 2: Male is aged 70, female 65 male residual female residual Price Price Model life life Contract Contract expectancy expectancy No. 1 No

21 20/23 Discussions The standard model that omits dependencies lead to huge error. This error can be either positive, or negative, depending on the age of both spouses. Omitting unobserved heterogeneities leads to under-estimation of the life expectancies.

22 21/23 CONCLUSION

23 2/23 We estimated a model which captures the joint dependency of the two lifetimes. We found evidence of heterogeneities on the mortality rates, besides the broken-heart syndrome. We assessed their impacts. Finally, the findings are also useful for the pricing of single life insurance.

24 23/23 THANKS FOR YOUR ATTENTION. QUESTIONS/SUGGESTIONS?

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