Consistently modeling unisex mortality rates. Dr. Peter Hieber, Longevity 14, University of Ulm, Germany

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1 Consistently modeling unisex mortality rates Dr. Peter Hieber, Longevity 14, University of Ulm, Germany

2 Seite 1 Peter Hieber Consistently modeling unisex mortality rates 2018 Motivation European Court of Justice, 2011 Different insurance premiums according to gender are prohibited (Gender directive 2004/113EC). But: Life insurance risk differs by gender (statistically significant).

3 Seite 2 Peter Hieber Consistently modeling unisex mortality rates 2018 Motivation, research question, introduction Consistent mortality models Numerical examples Consistency: Lee-Carter mortality model Reserves: (un)observed heterogeneity

4 Seite 3 Peter Hieber Consistently modeling unisex mortality rates 2018 Research question Given: Two groups with differing mortality risk and mortality model for each group. male/female smoker/non-smoker How to create unisex mortality models / unisex mortality tables that are consistent with a given male/female mortality model? male/female model for risk management. unisex model for premium calculation.

5 Seite 4 Peter Hieber Consistently modeling unisex mortality rates 2018 Portfolio at time t = 0 female/male portfolio (n = 20) unisex portfolio (n = 20) age y, survival probability T p y age z, survival probability T p z =? N y 0 = Nx 0 = 10 N z 0 = n = 20 age x, survival probability T p x

6 Seite 5 Peter Hieber Consistently modeling unisex mortality rates 2018 Portfolio at time t = T female/male portfolio (n = 20) unisex portfolio (n = 20) age y + T age z + T N y T = 8, Nx T = 9 N z T = 17 age x + T

7 Seite 6 Peter Hieber Consistently modeling unisex mortality rates 2018 Consistency: Example Consider an annuity portfolio of N y 0 females and Nx 0 males. Mortality risk is specified by two Lee-Carter models with parameters (A y t, By t, θy, cy) and (A x t, Bt x, θ x, c x). This implies a time-t -survival probability T p y := P( female survives T ) and T p x := P( male survives T ) For a unisex portfolio, this leads to the survival probability: T p z := Ny 0 P( female survives T ) + Nx N 0 x 0 P( male survives T ) +Ny N 0 0 x +Ny 0 What is the consistency error if we use a unisex Lee-Carter model with parameters (A z t, B z t, θ z, c z)? What happens if the group composition (N y 0, Nx 0 ) is not observable?

8 Seite 7 Peter Hieber Consistently modeling unisex mortality rates 2018 Consistency: Deterministic mortality tables In this talk, 2 consistent unisex mortality models are introduced. female/male portfolio unisex portfolio Consistency criterion 1 (unobservable) (C1) survival probability ˆξ 0 tp x + (1 ˆξ 0 ) tp y = tp z, for all t [0, T ]. (ˆξ 0 : initial guess of share of group x). Consistency criterion 2 (observable) (C1 ) portfolio members N x t + N y t = N z t, for all t [0, T ].

9 Seite 8 Peter Hieber Consistently modeling unisex mortality rates 2018 Demography in Germany M1

10 Seite 9 Peter Hieber Consistently modeling unisex mortality rates 2018 Motivation, research question, introduction Consistent mortality models Numerical examples Consistency: Lee-Carter mortality model Reserves: (un)observed heterogeneity

11 Seite 10 Peter Hieber Consistently modeling unisex mortality rates 2018 Survival curve { t p y } t [0,T ], { t p x } t [0,T ] 100% 80% female survival probability 60 year old life expectancy female male survival probability 60 year old life expectancy male survival probability 60% 40% 20% 0% age DAV 2004R, annuity table (includes risk margins).

12 Seite 10 Peter Hieber Consistently modeling unisex mortality rates 2018 Survival curve survival probability 100% 80% 60% 40% female survival probability 60 year old life expectancy female male survival probability 60 year old life expectancy male unisex survival probability 60 year old life expectancy unisex 20% 0% age DAV 2004R, annuity table (includes risk margins).

13 Seite 10 Peter Hieber Consistently modeling unisex mortality rates 2018 Unisex survival curve { t p z } t [0,T ] 100% 80% unisex survival probability 60 year old life expectancy unisex survival probability 60% 40% 20% 0% age For an initial share of males ξ 0, choose: tp z = ξ 0 tp x + (1 ξ 0 ) tp y.

14 Seite 11 Peter Hieber Consistently modeling unisex mortality rates 2018 Stochastic mortality rates a 13 x male mortality intensity female mortality intensity mortality intensity time t Plots {λ x t } t [0,20] (male) and {λ y t } t [0,20] (female). Survival curves t p x := e t 0 λx s ds, t p y := e t 0 λy s ds.

15 Seite 12 Peter Hieber Consistently modeling unisex mortality rates 2018 Assumption (Mortality model) For i {x, y, z}, we assume that Given the survival curve { tp i } t [0,T ], individual deaths are independent. Choose tp i := e t 0 λ i s ds. Number of survivors at ti e t > 0 is binomially distributed: N i t Bin ( N i 0, tp i ). Randomness in the survival curve { tp i } t [0,T ] (systematic mortality risk) is conditionally independent of the binomial distribution (unsystematic mortality risk). The intensity {λ i t} t 0 is continuous.

16 Seite 13 Peter Hieber Consistently modeling unisex mortality rates 2018 Definition (Unisex mortality model (unobservable)) For initial share of males ˆξ 0, define λ z t := ˆξ 0 e t 0 λx s ds ˆξ 0 e t 0 λx s ds + (1 ˆξ 0 ) e t 0 λy s ds }{{} time-t share of males + λ x t (1 ˆξ 0 ) e t 0 λy s ds ˆξ 0 e t 0 λx s ds + (1 ˆξ 0 ) e t λ y 0 λy s ds t. }{{} time-t share of females We obtain: N z T Bin ( n, ˆξ 0 tp x + (1 ˆξ 0 ) tp y ). How to obtain λ z : Solve t p z = ˆξ 0 tp x + (1 ˆξ 0 ) tp y for λ z t.

17 Seite 14 Peter Hieber Consistently modeling unisex mortality rates 2018 Definition (Unisex mortality model (observable)) For initial share of males ˆξ 0, define N x t µ z t := Nt x + N y λ x t + t Nt x + N y λ y t. (1) t }{{}}{{} time-t share of males time-t share of females N y t We obtain: N z T = Nx t + N y t, where Nx t Bin ( ξ 0 n, t p x ) and N z t Bin ( (1 ξ 0 )n, t p y ). (µ z t is still the instantaneous death probability, but does not define a mortality model).

18 Seite 15 Peter Hieber Consistently modeling unisex mortality rates 2018 For the observable case, it is necessary, to observe deaths immediately (no reporting delays etc.) and to observe the group membership. For the unobservable case, we do not observe the group membership or deaths immediately.

19 Seite 16 Peter Hieber Consistently modeling unisex mortality rates 2018 Implications for risk management M2 Unisex portfolio: N z T Bin ( N z 0, T p z ), where tp z = ξ 0 tp x + (1 ξ 0 ) tp y. Female/male portfolio: N y T Bin ( N y 0, T p y ), N x T Bin ( N x 0, T p x ). Lemma (Prudence of the unisex mortality model (C1)) E[N z T ] = E[N x T + N y T ], (2) Var(N z T ) Var(N x T + N y T ). (3) Proof: special cases: e.g. Feller [1950].,

20 Seite 17 Peter Hieber Consistently modeling unisex mortality rates 2018 Motivation, research question, introduction Consistent mortality models Numerical examples Consistency: Lee-Carter mortality model Reserves: (un)observed heterogeneity

21 Seite 18 Peter Hieber Consistently modeling unisex mortality rates 2018 Consistency: Lee-Carter mortality model T survival probability IABE unisex T survival probability IABE male T survival probability 0.1 IABE female T survival probability consistent unisex T survival probability time to maturity T consistency error ν 0 2.0% 1.6% 1.2% 0.8% 0.4% 0.0% 0.4% time to maturity T Parameters: Belgian Actuarial Society, IA BE (available online).

22 Seite 19 Peter Hieber Consistently modeling unisex mortality rates 2018 Reserves: (un)observed heterogeneity Consider a portfolio of n pure endowment insurance contracts with survival benefit S =e 1 at maturity T = 10. Risk-free rate r = 0%. 10% of the portfolio is disabled with life expectancy: T p disabled z = 60% T p z. We choose the standard deviation principle and define the per-contract actuarial reserve (in % of the contract s nominal e 1) as R j := 1 n α Var ( NT z 2 ). (4)

23 Seite 20 Peter Hieber Consistently modeling unisex mortality rates 2018 Reserves: (un)observed heterogeneity 10 year survival probability non disabled group disabled group initial age z actuarial reserves (in % of notional) 1.9% 1.7% 1.5% 1.3% 1.1% 0.9% 0.7% 0.5% 0.3% 0.1% n = 10, unobserved group membership n = 10, observed group membership n = 100, unobserved group membership n = 100, observed group membership initial age z Reserves annuity portfolio with 10% disabled persons. M2

24 Seite 21 Peter Hieber Consistently modeling unisex mortality rates 2018 Conclusion How to create unisex mortality models / unisex mortality tables that are consistent with a given male/female mortality model? M1 M2 Change/stochasticity in male/female mortality rates affects also male/female share in the annuity portfolio (also stochastic!). Observed heterogeneity reduces mortality risks (e.g. the portfolio s variance), compare two consistency criteria. Further interesting aspects: adverse selection, effect of portfolio size n.

25 Seite 22 Peter Hieber Consistently modeling unisex mortality rates 2018 Literature T. R. Bielecki and M. Rutkowski. Credit risk: modeling, valuation and hedging. Springer, E. Biffis. Affine processes for dynamic mortality and actuarial valuations. Insurance: Mathematics & Economics, Vol. 37:pp , E. Biffis, M. Denuit, and P. Devolder. Stochastic mortality under measure changes. Scandinavian Actuarial Journal, Vol. 4:pp , A. Chen and E. Vigna. A unisex stochastic mortality model to comply with EU gender directive. Insurance: Mathematics & Economics, Vol. 73:pp , M. Dahl. Stochastic mortality in life insurance: market reserves and mortality-linked insurance contracts. Insurance: Mathematics & Economics, Vol. 35, No. 1:pp , M. Dahl and T. Møller. Valuation and heding of life insurance liabilities wih systematic mortality risk. Insurance: Mathematics & Economics, Vol. 39:pp , P. Hieber. Modeling unisex mortality rates: A discussion of consistency, (un)observed heterogeneity and adverse selection. Working Paper, 2018.