Surrenders in a competing risks framework, application with the [FG99] model

Transcription

1 s in a competing risks framework, application with the [FG99] model AFIR - ERM - LIFE Lyon Colloquia June 25 th, ,2 Related to a joint work with D. Seror 1 and D. Nkihouabonga 1 1 ENSAE ParisTech, actuarial department 2 CREST, financial & actuarial sciences lab 1 / 26

2 Outline 1 2 and the 3 Description of the product considered in the study 4 to our contracts database 2 / 26

3 Two words on the surrender risk First, what is the surrender risk in life insurance? [DG07], [Out90] Some key points: 1 major or minor topic? depending on the business line... 2 risk factors are market-specific [MGL10]: clear need to integrate product and country characteristics as risk factors into the surrender behaviours modelling [LM11]. 3 timing is a key-point to recover administration costs... Regressions (avoid GLM, whose use introduce a selection bias and that do not aim at predicting the timing of the surrender. 3 / 26

4 2 and the 4 / 26

5 : theoretical background [MS06] T : unobservable lifetime, with density f (survival function S). C: contract duration until censorship (administrative here). The actual observation is given by T = min(t, C). For right censored data, the corresponding counting process follows N(t) = n N i (t) where N i (t) = 1 { T i t ; T i C i }. i=1 To N i (t) is associated the so-called intensity process A i (t) s.t. A i (t) = t 0 Y i (s) λ(s) ds, where Y i (t): at-risk process ( exposure), λ(t): hazard rate such that λ(t) = f (t) S(t) = lim 1 P(t < T t + T > t). 0 5 / 26

6 Classical estimators and most famous model class unbiased estimators: 1 Kaplan-Meier estimator for the survival function, Ŝ(t) = Y T (i ) <t «δi 1 1 where δ i = 1 {Ti Ci }. n i Nelson-Aalen estimator for the hazard rate, ˆλ(t) = nx i=1 δ i 1 {Ti t} P n j=1 1. {Tj T i } Proportional hazards models (individual Cox-type modelling): λ i (t) = λ 0 (t) exp(x T i β) where λ 0 (t) is the baseline hazard, X i = (X i1,..., X ik ) the k risk factors and β = (β 1,..., β k ) the k regression coefficients. 6 / 26

7 : K mutually exclusive causes Competing Risks Model T = min(t 1,..., T K, C) / T j : lifetime before death from cause j. 0 Alive λ 1 (t) λ k (t) 1 Dead, cause 1 K Dead, cause K (J t ) t>0 is the competing risks process. It tells us in which state the i th policyholder is at time t (J t {0, 1,..., K}). FIGURE 10.1: model. Each subject may die from k different causes τ is given by τ = inf{t > 0 J t 0}. are the intensities associated with the K-dimensional counting process N = (N 1,...,N K) T and define its compensator t t 7 / 26

8 Main quantities of interest 1 The cause-specific hazard functions: j {1,..., p}, λ(t) = P(t < T t +, J = j T > t) λ j (t) = lim. 0 p j=1 λ j (t), et S(t) = P(T > t) = e R t 0 P p j=1 λ j (s) ds. 2 The cumulative incidence functions (CIF): F j (t) = P(T t, J = j) = t 0 f j (s) ds, where P(t<T t+, J=j) f j (t) = lim = λ 0 j (t) S(t) F j (t) = R t λ 0 j(s) S(s ) ds. 8 / 26

9 The [FG99] Context: J t {0, 1, 2} (K = 2, event of interest is labeled 1 ). Idea: study a new process (ξ t ) t>0, derived from (J t ) t>0 and obtained by stopping adequately the latter: ξ t = 1 {Jt=2} J τ + 1 {Jt 2} J t. Interpretation: {J t = 0} nothing happened until time t, whereas {ξ t = 0} there was no event of interest until t. Tool: consider ν = inf{t > 0 : ξ t 0}, the new random lifetime before the occurence of the event of interest (surrender). ν = { τ if Jτ = 1, if J τ = 2. 9 / 26

10 Trick: t [0, ), P(ν t) = P(T t, J = 1) = F 1 (t). Then, the hazard of the event of interest follows and is finally given by F 1 (t) = 1 S 1 (t) = 1 e R t λ1(s) ds 0 P(t < T t +, J t = 1 {T > t} {T t, J t 1}) λ 1(t) = lim. 0 Novelty: t, at-risk policyholders consist now in insureds still in state {0} at time t added to policyholders who have undergone a competing risk before t. Pros/Cons: not necessary to model every cause of failure / at-risk set is not really realistic, and not always known. 10 / 26

11 3 Description of the product considered in the study 11 / 26

12 General product description We consider WL contracts with the following characteristics: lump sum at death of the insured, guaranted return during the contract lifetime, fiscality constraints: TAMRA law, cyclical level premiums, whose amount depends on insured s gender and age, the policyholder s health (potential medical examination), the tobacco consumption. commission depends on the distribution channel, but equals 0 after 2 years of contract duration, surrender option: can be exercised at any time. The contract can be partially or totally surrendered: we focus here on total surrenders (also other lapse causes: maturity, death,...). 12 / 26

13 The surrender value combines 3 components 1 lump sum at death, embedding a guaranted return: 2 final capped dividends depending on the sum insured; 3 stochastic dividends during the contract lifetime (based on the profitability of the company).! Financial markets are likely to impact the surrender behaviours. 13 / 26

14 History contracts, from 01/1995 to 05/2010. Figure: Exposure (green), lapses (red), and surrender rate (black). 14 / 26

15 Potential impact of financial markets (Dow Jones) )$%%%%& ($#%%%& ($%%%%& "$#%%%& "$%%%%& '$#%%%& '$%%%%& %$#%%%&!"#$%&#'()*%+(,-(.*/0*)(-)("*.1*&#'(2345(.-)*.,(6(7#18( %$%%%%&!"#$%& '"*''*'++)& %,*%,*'++-& %)*%#*"%%%& "+*%'*"%%(& "#*'%*"%%#& "'*%-*"%%,& '-*%)*"%''& '#$%& '%$%& #$%& %$%&!#$%&!'%$%&!'#$%&!"%$%&./ &0451& 6789&(:&;451& 15 / 26

16 First insights about the effect of risk factors Figure: Statistics on contract lifetimes (in quarter) depending on the health diagnostic (covariate risk state hereafter). 16 / 26

17 Summary of the descriptive analysis Correlation between the variable of interest and some risk factors: non-parametric and parametric tests. Factor Age Health diagnostic Gender Living place UW year Prem. freq. H 0 rejected rejected rejected not rejected rejected rejected Table: χ 2 tests (binary surrender decision VS categorical risk factors). Factor Age class Health diagnostic Gender Living place Acc. rider Prem.freq. Test KW KW Wilcoxon KW Wilcoxon KW H 0 rejected rejected rejected rejected rejected rejected Table: Independence tests (Kruskal-Wallis: KW) on contract lifetimes. p-values suggest the following most discriminating features: health diagnostic ( premium), accidental death rider and premium freq. 17 / 26

18 4 to our contracts database 18 / 26

19 General profile of hazard rates for competing risks Figure: Adjusted non-parametric Nelson-Aalen estimator of the hazards depending on the cause of lapse. Baseline hazard 19 / 26

20 Effects of risk factors on the lifetime distribution Figure: Adjusted Nelson-Aalen estimator of surrender hazard for policyholders with or without the accidental death rider. 20 / 26

21 Cox model for the surrender We calibrate an extended Cox model for the hazard associated to the lifetime before surrender: for policyholder i, λ i (t) = λ 0 (t) exp(x T i β + Z(t) η). λ 0 (t): baseline hazard, non-parametric and unspecified. X T i = (X i1,..., X ik ) stands for the constant risk factors; β T = (β 1,..., β k ): corresponding regression coefficients; Z(t): variation of the Dow Jones, and η its effect on λ i (t). Correlation between covariates has initially been checked. Assumption of PH was first validated (Schoenfeld residuals). 21 / 26

22 Goodness-of-fit 22 / 26

23 Other validation technique: survival curves Accurate modelling in the first 8 years. Impact of risk factors: OK 23 / 26

24 Issue and suggested improvement Figure: Baseline hazard after the calibration of a Cox hazard type for the surrender risk. To compare to Nelson-Aalen est. 24 / 26

25 Comments and perspectives This framework seems to be the most realistic for this problem, was not really investigated for life insurance lapses previously. The clearly allows us to reduce the model risk, as it does not rely on modelling other causes of failure. Nevertheless, it requires more work to do on the specification of the baseline hazard; to perform further studies on the simulation of stochastic counting processes in the ; to better integrate correlation between behaviours, [MFE05]: common shocks model, adding a frailty variable into the hazard definition, use survival mixtures. Final goal: should improve the day-to-day ALM of the company. 25 / 26

26 References Domenico De Giovanni, Lapse rate modeling: A rational expectation, Finance Research Group Working Papers F , University of Aarhus, Aarhus School of Business, Department of Business Studies, J.P. Fine and R.J. Gray, A proportional hazards model for the of a competing risk, Journal of the American Statistical Association 94 (1999), no. 446, Stephane Loisel and, From deterministic to stochastic surrender risk models: Impact of correlation crises on economic capital, European Journal of Operational Research 214 (2011), no. 2. A.J. McNeil, R. Frey, and P. Embrechts, Quantitative risk management, Princeton Series In Finance, 2005., M-P. Gonon, and Stephane Loisel, Les comportements de rachat en assurance vie en régime de croisière et en période de crise, Risques (2010), no. 83, T. Martinussen and T.H. Scheike, Dynamic regression models for survival data, Springer, Jean François Outreville, Whole-life insurance lapse rates and the emergency fund hypothesis, Insurance: Mathematics and Economics 9 (1990), / 26