behavioursbythe surrendersinacompetingrisksframework, … · 2015. 11. 25. · approach...

Surrenderbehaviours by thesubdistribution

approach

Xavier Milhaud

Introduction

Competing risksSurvival analysisBasicsApproach [FG99]

Whole LifeMain featuresSurrenders VSEconomyDescriptivestatistics

ApplicationNon-parametricresultsModellingValidation

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

AFIR - ERM - LIFE Lyon ColloquiaJune 25th, 2013

Xavier Milhaud1,2

Related to a joint work with D. Seror1 and D. Nkihouabonga1

1 ENSAE ParisTech, actuarial department2 CREST, financial & actuarial sciences lab

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Outline

1 Introduction

2 Competing risks and the subdistribution approach

3 Description of the product considered in the study

4 Application to our Whole Life contracts database

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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 asrisk 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 biasand that do not aim at predicting the timing of the surrender.

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2 Competing risks and the subdistribution approach

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Survival analysis: 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∑

i=1

Ni (t) where Ni (t) = 11{Ti≤t ; Ti≤Ci}.

To Ni (t) is associated the so-called “intensity process” Ai (t) s.t.

Ai (t) =

∫ t

0Yi (s)λ(s) ds, where

Yi (t): at-risk process (' exposure), λ(t): hazard rate such that

λ(t) =f (t)

S(t)= lim

∆→0

1∆

P(t < T ≤ t + ∆ |T > t).

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Classical estimators and most famous model class

Non-parametric unbiased estimators:

1 Kaplan-Meier estimator for the survival function,

S(t) =Y

T(i )<t

„1− 1

n − i + 1

«δiwhere δi = 11{Ti≤Ci }.

2 Nelson-Aalen estimator for the hazard rate,

λ(t) =nX

i=1

δi 11{Ti≤t}Pnj=1 11{Tj≥Ti }

.

Proportional hazards models (individual Cox-type modelling):

λi (t) = λ0(t) exp(XTi β)

where λ0(t) is the baseline hazard, Xi = (Xi1, ...,Xik) the krisk factors and β = (β1, ..., βk) the k regression coefficients.

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Competing risks: K mutually exclusive causes

T = min(T1, ...,TK ,C ) / Tj : lifetime before death from cause j .348 10. Competing Risks Model

Alive

0

Dead, cause KK

!!

!!

!!

!!!"

Dead, cause 11

##

##

##

###$

λ1(t)

λk(t)

!!!

FIGURE 10.1: Competing risks model. Each subject may die from k differentcauses

are the intensities associated with the K-dimensional counting process N =(N1, ..., NK)T and define its compensator

Λ(t) = (∫ t

0

λ1(s)ds, ...,

∫ t

0

λK(s)ds)T ,

such that M(t) = N(t) − Λ(t) becomes a K-dimensional (local squareintegrable) martingale. A competing risks model can thus be described byspecifying all the cause specific hazards. The model can be visualized asshown in Figure 10.1, where a subject can move from the “alive” state todeath of one of the K different causes.

Based on the cause specific hazards various consequences of the modelcan be computed. One such summary statistic is the cumulative incidencefunction, or cumulative incidence probability, for cause k = 1, .., K, definedas the probability of dying of cause k before time t

Pk(t) = P (T ≤ t, ε = k) =∫ t

0

αk(s)S(s−)ds, (10.1)

(Jt)t>0 is the competing risks process. It tells us in whichstate the ith policyholder is at time t (Jt ∈ {0, 1, ...,K}).τ is given by τ = inf{t > 0 | Jt 6= 0}.

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Main quantities of interest

1 The cause-specific hazard functions: ∀j ∈ {1, ..., p},

λj(t) = lim∆→0

P(t < T ≤ t + ∆ , J = j | T > t)

∆.

λ(t) =

p∑

j=1

λj(t), et S(t) = P(T > t) = e−R t0

Ppj=1 λj (s) ds .

2 The cumulative incidence functions (CIF):

Fj(t) = P(T ≤ t, J = j) =

∫ t

0fj(s) ds, where

fj (t) = lim∆→0

P(t<T≤t+∆ , J=j)∆

= λj (t) S(t)⇒ Fj (t) =R t0 λj (s) S(s−) ds.

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The subdistribution approach [FG99]

Context: Jt ∈ {0, 1, 2} (K = 2, event of interest is labeled “1”).

Idea: study a new process (ξt)t>0, derived from (Jt)t>0 andobtained by stopping adequately the latter:

ξt = 11{Jt=2} Jτ− + 11{Jt 6=2} Jt .

Interpretation: {Jt = 0} ' nothing happened until time t,whereas {ξt = 0} ' there was no event of interest until t.

Tool: consider ν = inf{t > 0 : ξt 6= 0}, the new random lifetimebefore the occurence of the event of interest (surrender).

ν =

{τ if Jτ = 1,∞ if Jτ = 2.

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Trick: ∀t ∈ [0,∞), P(ν ≤ t) = P(T ≤ t, J = 1) = F1(t).

Then, the subdistribution hazard of the event of interest follows

F1(t) = 1− S1(t) = 1− e−R t0 λ1(s) ds

and is finally given by

λ1(t) = lim∆→0

P(t < T ≤ t + ∆ , Jt = 1 | {T > t}∪ {T ≤ t, Jt 6= 1})∆

.

Novelty: ∀t, at-risk policyholders consist now in insureds still instate {0} at time t added to policyholders who have undergone acompeting risk before t.

Pros/Cons: not necessary to model every cause of failure / at-riskset is not really realistic, and not always known.

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3 Description of the product considered in the study

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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 equals0 after 2 years of contract duration,surrender option: can be exercised at any time.

The contract can be partially or totally surrendered: we focus hereon total surrenders (also other lapse causes: maturity, death, ...).

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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.

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History29 531 contracts, from 01/1995 to 05/2010.

Figure: Exposure (green), lapses (red), and surrender rate (black).

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Potential impact of financial markets (Dow Jones)

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First insights about the effect of risk factors

Figure: Statistics on contract lifetimes (in quarter) depending on thehealth diagnostic (covariate “risk state” hereafter).

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Summary of the descriptive analysis

→ Correlation between the variable of interest and some riskfactors: non-parametric and parametric tests.

Factor Age Health diagnostic Gender Living place UW year Prem. freq.H0 rejected rejected rejected not rejected rejected rejected

Table: χ2 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 KWH0 rejected rejected rejected rejected rejected rejected

Table: Independence tests (Kruskal-Wallis: KW) on contract lifetimes.

p-values suggest the following most discriminating features: healthdiagnostic (' premium), accidental death rider and premium freq.

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4 Application to our Whole Life contracts database

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General profile of hazard rates for competing risks

Figure: Adjusted non-parametric Nelson-Aalen estimator of thesubdistribution hazards depending on the cause of lapse. Baseline hazard

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Effects of risk factors on the lifetime distribution

Figure: Adjusted Nelson-Aalen estimator of surrender subdistributionhazard for policyholders with or without the accidental death rider.

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Cox model for the surrender subdistribution

We calibrate an extended Cox model for the subdistribution hazardassociated to the lifetime before surrender: for policyholder i ,

λi (t) = λ0(t) exp(XTi β + Z (t) η).

λ0(t): baseline hazard, non-parametric and unspecified.XT

i = (Xi1, ...,Xik) stands for the constant risk factors;

βT = (β1, ..., βk): corresponding regression coefficients;

Z (t): variation of the Dow Jones, and η its effect on λi (t).

√Correlation between covariates has initially been checked.√Assumption of PH was first validated (Schoenfeld residuals).

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Goodness-of-fit

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Other validation technique: survival curves

Accurate modelling in the first 8 years. Impact of risk factors: OK

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Issue and suggested improvement

Figure: Baseline hazard after the calibration of a Cox subdistributionhazard type for the surrender risk. To compare to Nelson-Aalen est.

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Comments and perspectives

→ This framework seems to be the most realistic for this problem,was not really investigated for life insurance lapses previously.

→ The subdistribution approach clearly allows us to reduce themodel risk, as it does not rely on modelling other causes of failure.

Nevertheless, it requiresmore work to do on the specification of the baseline hazard;

to perform further studies on the simulation of stochasticcounting processes in the subdistribution approach;

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.

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References

Domenico De Giovanni, Lapse rate modeling: A rational expectation approach, Finance Research

Group Working Papers F-2007-03, University of Aarhus, Aarhus School of Business, Departmentof Business Studies, 2007.

J.P. Fine and R.J. Gray, A proportional hazards model for the subdistribution of a competing

risk, Journal of the American Statistical Association 94 (1999), no. 446, 496–509.

Stephane Loisel and Xavier Milhaud, 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 InFinance, 2005.

Xavier Milhaud, 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, 76–81.

T. Martinussen and T.H. Scheike, Dynamic regression models for survival data, Springer, 2006.

Jean François Outreville, Whole-life insurance lapse rates and the emergency fund hypothesis,

Insurance: Mathematics and Economics 9 (1990), 249–255.

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behavioursbythe surrendersinacompetingrisksframework, … · 2015. 11. 25. · approach...

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