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Axa Sa — Technical Presentation (February 2008)
41 slides from Axa Sa's technical presentation, Feb 1, 2008
Slide 1: Life Annuities, Projected Life Tables, and Exchangeability: An Actuarial Analysis in the Lee-Carter Model
Workshop “Prospective Mortality Tables, Longevity and Mortality Linked Securities” LIFE ANNUITIES, PROJECTED LIFE TABLES, AND EXCHANGEABILITY: AN ACTUARIAL ANALYSIS IN THE LEE-CARTER MODEL MICHEL DENUIT Institut de Statistique & Institut des Sciences Actuarielles, UCL Louvain-la-Neuve, Belgique mich
Slide 2: Summary
Summary • Mortality projections • Type of dependence induced by random life tables • First-order mortality basis for life annuities • Systematic risk in large portfolios • Some references M. Denuit, Paris, February 1, 2008 – p. 2/41
Slide 3: Mortality Projections
MORTALITY PROJECTIONS M. Denuit, Paris, February 1, 2008 – p. 3/41
Slide 4: Life expectancies at birth and at age 65
Life expectancies at birth and at age 65 85 20 Males Females 80 18 Males Females 75 16 70 e65(t) e0(t) 65 14 60 12 55 50 10 1920 1940 1960 1980 2000 1920 1940 1960 1980 2000 t t Life expectancies at birth (on the left) and at age 65 (on the right), Belgian general population. Source: HMD. M. Denuit,
Slide 5: Standard deviation of the lifetime at birth and at age 65
Standard deviation of the lifetime at birth and at age 65 6.8 6.6 25 6.4 Stand. Dev. Stand. Dev. 20 6.2 6.0 15 5.8 5.6 10 1920 1940 1960 1980 2000 1920 1940 1960 1980 2000 Year Year Std deviations at birth (on the left) and at age 65 (on the right), for males (circles) and females (triangles), Belgi
Slide 6: Lee-Carter model for mortality projection
Lee-Carter model for mortality projection • Assume that we have at our disposal estimated death rates m b x (t) for x = xmin , . . . , xmax and t = tmin , . . . , tmax . • Lee & Carter (1992) suggested ln m b x (t) = ln mx (t) + ǫx (t), mx (t) = exp αx + βx κt , where ǫx (t) iid ∼ N or(0, σ 2 ). • T
Slide 7: Estimating the parameters involved
Estimating the parameters involved in ln m b x (t) = αx + βx κt + ǫx (t) • The parameters are obtained by OLS, i.e. α b, b, β and κ b minimize tX max xX max 2 b x (t) − αx − βx κt ln m . t=tmin x=xmin • Binomial maximum likelihood (Cossette et al., 2007). • Poisson maximum likelihood (Brouhns et al.
Slide 8: Particular features of the Lee-Carter model
Particular features of the Lee-Carter model • The OLS bκt ’s are then adjusted by refitting to the observed period life expectancies at age xmin , as suggested in Lee & Miller (2001). • The optimal fitting period is selected according to Booth, Maindonald & Smith (2002) or D & Goderniaux (2005). • I
Slide 9: A first look at HMD data: Belgian males
A first look at HMD data: Belgian males −2 −4 −6 −8 1920 1940 100 80 1960 60 t 1980 40 x 20 2000 M. Denuit, Paris, February 1, 2008 – p. 9/41
Slide 10: Lee-Carter parameters: Estimated alpha and beta
Lee-Carter parameters: Estimated αx ’s and βx ’s 0.04 −1 0.03 −2 0.02 Alpha Beta −3 0.01 0.00 −4 60 70 80 90 100 60 70 80 90 100 Age Age Ages 60 to 104, estimation based on rough data (dotted line) and on smoothed data bx ’s are on the left, βbx ’s are on the right. (circles), α M. Denuit, Paris, Fe
Slide 11: Lee-Carter parameters: Estimated kappa
Lee-Carter parameters: Estimated κt ’s 5 0 Kappa −5 −10 1975 1980 1985 1990 1995 2000 2005 Time Optimal starting year 1974 for males, estimation based on rough data (dotted line) and on smoothed data (circles) adjusted by refitting to the life expectancy at age 60. M. Denuit, Paris, February 1, 2008
Slide 12: Longevity fan chart: Cohort e65(t), Belgian males
Longevity fan chart (Blake, Cairns & Dowd, 2007): Cohort e65 (t), Belgian males 21 20 e65(t) 19 18 2010 2015 2020 2025 t M. Denuit, Paris, February 1, 2008 – p. 12/41
Slide 13: Comparison with other forecasts
Comparison with other forecasts • The present forecast for Belgium, general population, males (circle). • The official forecast performed by the Belgian Federal Planning Bureau (triangle). • The forecast by Andreev & Vaupel (2006) for SCOR using the Lee-Carter methodology (square). • The forecast by
Slide 14: Comparison with other forecasts: Cohort e65(t)
Comparison with other forecasts: Cohort e65 (t), Belgian males 22 21 20 e65(t) 19 18 17 2010 2015 2020 2025 t M. Denuit, Paris, February 1, 2008 – p. 14/41
Slide 15: Adverse selection: Life insurance market in 2005
Adverse selection: Life insurance market in 2005 −1 0 −2 −2 Smooth Log. Death Rate, Year 2005 Log. Death Rate, Year 2005 −3 −4 −4 −6 −5 −8 −6 −10 −7 30 40 50 60 70 80 90 100 40 50 60 70 80 90 100 Age Age General population (dotted line) and individual (circle) and group (triangle) life insurance mar
Slide 16: Relational models: Cohort life expectancies at 65
Relational models of Delwarde et al. (2004): Cohort life expectancies at 65, males, Belgian insurance market 28 26 24 e65(t) 22 20 18 2010 2015 2020 2025 t M. Denuit, Paris, February 1, 2008 – p. 16/41
Slide 17: Type of Dependence Induced by Random Life Tables
TYPE OF DEPENDENCE INDUCED BY RANDOM LIFE TABLES M. Denuit, Paris, February 1, 2008 – p. 17/41
Slide 18: Exchangeable life times in the Lee-Carter model
Exchangeable life times in the Lee-Carter model • Consider a group of n annuitants aged x0 in year t0 ; their respective remaining life times are T1 , . . . , Tn . • Given the κt0 +k ’s, the Ti ’s are independent with exp − ξmx0 (t0 |κ) if ξ ≤ 1, exp − (ξ − ⌊ξ⌋)m x0 +⌊ξ⌋ (t0 + ⌊ξ⌋|κ) ξ px0
Slide 19: Associated lifetimes
Associated lifetimes • Recall that X1 , . . . , Xn are associated if Cov [Ψ1 (X1 , · · · , Xn ), Ψ2 (X1 , · · · , Xn )] ≥ 0 for all non-decreasing functions Ψ1 and Ψ2 for which the covariances exist. • If βx ≥ 0 for all the ages x and κ is associated, D & Frostig (2007a) established that the lifetim
Slide 20: Associated annuity present values
Associated annuity present values • Assume that βx ≥ 0 for all x and that κ is associated. P⌊ξ⌋ • Let aξ| = k=1 v(0, k) denote an annuity certain, where ⌊ξ⌋ is the integer part of ξ . • Then, the present values of life annuity payments are associated, that is, the random variables aT1 | , . . . , aT
Slide 21: Convex order and association
Convex order and association • Recall that X CX Y ⇔ E[g(X)] ≤ E[g(Y )] for all the convex functions g such that the expectations exist. • Assume that βx ≥ 0 for all x and that κ is associated. • If T1⊥ , . . . , Tn⊥ are independent, each Ti⊥ being distributed as Ti , then n X n X aT ⊥ | CX aTi | . i
Slide 22: Risk measures
Risk measures • Here, a risk measure is meant as a functional ρ mapping a risk X (non-negative random financial loss) to the minimum extra cash which has to be added to X to make it “acceptable”. • Consider a non-decreasing distortion function g satisfying g(0) = 0 and g(1) = 1, and define the risk
Slide 23: Positively dependent lifetimes
Positively dependent lifetimes • Assume that βx ≥ 0 for all x and that κ is associated. • Then, " n # " n # X X ρg aT ⊥ | ≤ ρg aTi | i i=1 i=1 for any concave g . • In particular " n # " n # X X TVaR aT ⊥ | ; p ≤ TVaR aTi | ; p for any p. i i=1 i=1 M. Denuit, Paris, February 1, 2008 – p. 23/41
Slide 24: First-Order Mortality Basis for Life Annuities
FIRST-ORDER MORTALITY BASIS FOR LIFE ANNUITIES M. Denuit, Paris, February 1, 2008 – p. 24/41
Slide 25: Determination of first-order mortality basis
Determination of first-order mortality basis • Let us consider the cohort reaching age x0 (typically, retirement age) in year t0 . • For this cohort, D & Frostig (2007b) determine the conservative life table m[1] x0 +k , k = 1, 2, . . ., in order to satisfy [1] Pr[exp(αx0 +k +βx0 +k κt0 +k ) ≤ mx0 +
Slide 26: Determination of first-order mortality basis
Determination of first-order mortality basis [1] • Taking mx0 +k = πmref x0 +k with mref x0 +k = exp(αx0 +k + βx0 +k (κt0 + kd)), we thus require that Pr[βx0 +k (κt0 +k − (κt0 + kd)) ≥ ln π for all k = 1, 2, . . .] ≥ 1 − ǫmort . • The value of ln π can then be determined as a quantile of the random
Slide 27: First-order and second-order basis
First-order and second-order basis • Let P1 be the probability taken under the first-order mortality basis, and P2 be the probability taken under the second-order mortality basis. • Under P1 the lifetimes T1 , . . . , Tn are independent with death rates m[1] x0 +k , k = 1, 2, . . . • Under P2 the li
Slide 28: First-order and second-order basis (Ctd)
First-order and second-order basis (Ctd) • For any time horizon k and z ≥ 0, we have " n # " n # X X P2 aTi | ≤ z ≥ (1 − ǫmort )P1 aTi | ≤ z . i=1 i=1 • More generally, whatever the non-negative non-increasing function g : Rn → R+ , we have E2 [g(T1 , . . . , Tn )] ≥ E1 g(T1 , . . . , Tn ) (1 − ǫmor
Slide 29: Distribution of sum of annuity present values under P1
Pn Distribution of i=1 aTi | under P1 with n = 30, x0 = 65, t0 = 2005 ǫmort = 1% ⇔ π = 93.2%, obtained from Panjer algorithm 1.0 0.015 0.8 Prob. Mass Function 0.010 Cum. Dist. Function 0.6 0.4 0.005 0.2 0.000 0.0 0 100 200 300 400 500 0 100 200 300 400 500 600 Values Values M. Denuit, Paris, Februar
Slide 30: Systematic Risk in Large Portfolios
SYSTEMATIC RISK IN LARGE PORTFOLIOS M. Denuit, Paris, February 1, 2008 – p. 30/41
Slide 31: Large homogeneous portfolios
Large homogeneous portfolios • The diversification effect is apparent from n+1 1 X ax0 (t0 |κ) = E[aT1 | |κ] CX . . . CX aTi | n+1 i=1 n X 1 CX aTi | CX . . . CX aT1 | . n i=1 • For any concave g , we then have "n+1 # " n # " n # X X 1 X ρg aTi | − ρg aTi | ≤ ρg aTi | . n i=1 i=1 i=1 M. Denuit, Pari
Slide 32: The survival probability
The survival probability k px0 (t0 |κ) • For the cohort aged x0 in year t0 , we have k px0 (t0 |κ) = exp(−Sk ) with k−1 X Xk−1 Sk = exp αx0 +j + βx0 +j κt0 +j = δj exp(Xj ), j=0 j=0 where δj = exp(αx0 +j ) > 0 and Xj = βx0 +j κt0 +j ∼ N or(µj , σj2 ) with µj = βx0 +j E[κt0 +j ] and σj2 = (βx0 +j )2
Slide 33: Approximations for Sk
Approximations for Sk • Let Z ∼ N or(0, 1). • D & Dhaene (2007) proposed to approximate Sk with the help of Sku defined as k−1 X Sku = δj exp(µj + σj Z). j=0 • Sku appears as a sum of comonotonic variables and its quantiles are therefore easy to compute. M. Denuit, Paris, February 1, 2008 – p. 33/41
Slide 34: Approximations for Sk
Approximations for Sk • Define ri (k), i = 0, . . . , k − 1, as Pk−1 2 j=0 δ j exp(µ j )βx0 +i βx0 +j min{i, j}σ qP P . k−1 k−1 2 σi j=0 ℓ=0 δj δℓ exp(µj + µℓ )βx0 +j βx0 +ℓ min{j, ℓ}σ • D & Dhaene (2007) proposed to approximate Sk with the help of k−1 X 1 Skl = δj exp µj + rj (k)σj Z + (1 − (rj (k)
Slide 35: Approximation for annuity value
Approximation for ax0 (t0 |κ) • Following D & Dhaene (2007), D (2007a) proposed to use X ax0 (t0 |κ) ≈ exp(−Sku )v(0, k) k≥1 or X ax0 (t0 |κ) ≈ exp(−Skl )v(0, k). k≥1 • Quantile functions of these approximations are easily calculated. M. Denuit, Paris, February 1, 2008 – p. 35/41
Slide 36: Quantile function of a65(2005|kappa)
Quantile function of a65 (2005|κ) 15.2 12.6 12.4 15.0 12.2 14.8 12.0 14.6 11.8 11.6 14.4 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Individual life insurance market (left) and group life insurance market (right): the results obtained from 10,000 simulations correspond to the continuous line (—)
Slide 37: References
References - Andreev, K.F., & Vaupel, J.W. (2006). Forecasts of cohort mortality after age 50. MPIDR Working Paper 2006-12. - Blake, D., Cairns, A.J., & Dowd, K. (2007). Facing up to the uncertainty of life: the longevity fan charts. Pension Institute Discussion Paper 07-01. - Booth, H., Maindonald,
Slide 38: References
References - Brouhns, N., Denuit, M., & Vermunt, J.K. (2002b). Measuring the longevity risk in mortality projections. Bulletin of the Swiss Association of Actuaries, 105-130. - Cossette, H., Delwarde, A., Denuit, M., Guillot, F. & Marceau, E. (2007). Pension plan valuation and dynamic mortality tabl
Slide 39: References
References - Delwarde, A., Kachakhidze, D., Olie, L., & Denuit, M. (2004). Modèles linéaires et additifs généralisés, maximum de vraisemblance local et méthodes relationnelles en assurance sur la vie. Bulletin Français d’Actuariat 6, 77-102. - Denuit, M. (2007a). Comonotonic approximations to quanti
Slide 40: References
References - Denuit, M., & Frostig, E. (2007b). Prudential rules for life annuity pricing and reserving. WP 07-02, Institut des Sciences Actuarielles, UCL, Louvain-la-Neuve, Belgium. - Denuit, M., & Frostig, E. (2007c). Life insurance mathematics with random life tables. WP 07-08, Institut des Scien
Slide 41: References
References - Lee, R. & Miller, T. (2001). Evaluating the performance of the Lee-Carter approach to modelling and forecasting. Demography 38, 537-549. - Oeppen, J. & Vaupel, J.W. (2002). Broken limits to life expectancy. Science 296, 1029-1031. - Renshaw, A.E. & Haberman, S. (2003). Lee-Carter mortal