Surrender tables for lapse risk management ininsurance with competing risks

European Actuarial Journal ConferenceLeuven, 09/12/2018

Xavier Milhaud1

Joint work with C. Dutang2

1 LSAF, ISFA, Université Claude Bernard Lyon 1, Lyon, France2 Université Dauphine, Paris, France

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TWO WORDS ON THE SURRENDER RISK

What is the surrender risk in life insurance ? [De 07],[Out90]

Some key points :

1 risk factors are “market-specific” [MGL10] :need to integrate product and country characteristics as riskfactors into the surrender behaviours modelling [MMDL11].

2 timing is a key-point to recover administration costs...

⇒ Regressions (avoid GLM, whose use introduce a selectionbias) that aim at predicting the timing of the surrender.

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DESCRIPTION OF WHOLE LIFE PRODUCTS

Lump sum at death (with minimum guaranteed return),

Cyclical level premiums, depending on gender, age, andhealth (tobacco consumption, medical examination, ...) ;

Commission : 50% 1st year, then 4% each y., and 0 after 10y,

Surrender option can be exercised at any time, with surrendervalue depending on market performance...

N.B. : contracts can be partially or totally surrendered : totalsurrenders here (also other lapse causes : maturity, death, ...).

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EMPIRICAL LAPSE TRAJECTORY (almost 30 000 contracts)

Why to focus on surrenders ? SURRENDERS DRIVE LAPSES !

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INFORMATION IN THE DATABASE

Variable Statistics CommentsIssue Date between 01/01/1995 for policies not terminated in Dec. 2008,

and end of 2008 we have no information : fixed right censoredTime duration min : 0.01 ; max : 62.09 ; unknown if policy not terminated(in quarters) mean : 30.26 ; std : 18.78Gender male : 50.05% ; female : 49.95% no missing valuePayment frequency infra annual : 61.37% ; annual : 23.44% ; Infra-annual : monthly, quarterly, semi-annual.

other (supra annual) : 15.19%Risk state non smoker : 63.01% ; smoker : 36.99% no missing valueUnderwriting age young : 47.46% between 0 and 34 years old

middle : 34.04% between 35 and 54 years oldold : 18.50% between 55 and 84 years old

Living place east coast : 20.62% ; west coast : 4.60% : no missing valueother : 74.78%

Annual premium min : -1.07 ; median : -0.30 ; max : 12.13 this variable has been standardized.mean : $560.88 ; std : $526.5870 (in original dollar scale)

DowJones Index variation min : -4.53 ; median : -0.38 ; max : 2.43 this variable has been standardized.mean : 0.001781 ; std : 0.049413 (in original scale)

Accidental death rider yes : 16.42% ; no : 83.58%Termination cause 0 : 49.06% in force

1 : 38.22% surrender2 : 12.72% cancellation (other causes : death, term,. . .)

Death indicator 0 : 95.62% alive1 : 4.38% dead

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BIVARIATE DESCRIPTIVE STATISTICS / STATISTICAL TESTS

→ Example of statistics concerning the contract durationdepending on premium frequency / policyholder’s age / ...

Figure 3: Surrender rates depending on premium frequency and policyholder’s age.

and the mean of the contract lifetime (in quarters). Graphically, the e↵ect is clear: both the paymentfrequency and the age have a significant impact. For instance, people that pay premiums more thanonce a year are more likely to surrender, and middle-aged policyholders exhibits the highest surrenderrates. This last statement is not surprising since people from 35 to 54 years old usually have morefrequent liquidity needs for personal projects. However, despite that they tend to surrender more thanthe others, they also have higher mean contract lifetime.

3 Survival analysis: cause-specific or subdistribution approach?

Recall that our goal is to predict individual contract lifetimes. In order to achieve this, there existtwo standard approaches in survival analysis: the cause-specific one, and the subdistribution one. Thegoal of this section is twofold: to present the probabilistic details of both models, and to make a choicebetween both, based on the specificity of our data.

Let us denote by T the random variable of the contract lifetime. In a survival analysis, thedistribution of T is generally specified by its hazard function, or its survival function. Indeed, thesurvival function defined as ST (t) = P (T > t) characterizes the distribution, as well as the hazardrate defined as �T (t) = �S0

T (t)/ST (t). Typically, for an exponential distribution of rate �, �T (t) = �;and for a Weibull distribution of scale � and shape k, �T (t) = (k/�) (t/�)k�1. The survival functioncan be derived from the hazard function by ST (t) = exp(�

R t0 �T (s)ds) since �T (t) = �d ln ST (t)/dt.

In practice lifetimes are may be censored, that is we observe the random variable Y = min(T, C)where T is the variable of interest and C is the censoring variable. C may be random or deterministicdepending on the type of studies: hereafter, C is considered to be a random variable. Assumingthat C is independent from T , it is easy to show that �T (t) = �Y (t) + �C(t). So a straightforwardconclusion is that random censoring without the information whether T is censored or not leads tofalse estimation of �T (t) when estimating �Y (t). However, when we know if T is censored or not, thecouple (T, 11TC) has the targeted hazard rate

�T (t) = limdt!0

P (t T < t + dt | T � t)

dt.

Hence, there is no particular bias when dealing with censoring as long as we know which data are

7

p-values of χ2 and Kruskal-Wallis tests suggest the following mostdiscriminating features : health diagnostic (' premium), accidentaldeath rider and premium frequency.

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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 T̄ = min(T ,C).

For right censored data, the corresponding counting process isN(t) =

∑ni=1 Ni(t) where Ni(t) = 11{T̄i≤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

P(t ≤ T < t + ∆ |T ≥ t)∆

.

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MUTUALLY EXCLUSIVE CAUSES : COMPETING RISKS348 10. Competing Risks Model

Alive

0

Dead, cause KK

!!

!!

!!

!!!"

Dead, cause 11

##

##

##

###$

!1(t)

!k(t)

!!!

FIGURE 10.1: Competing risks model. Each subject may die from k di!erentcauses

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

"(t) = (

! t

0

!1(s)ds, ...,

! t

0

!K(s)ds)T ,

such that M(t) = N(t) ! "(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 di!erent 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, " = k) =

! t

0

#k(s)S(s!)ds, (10.1)

T̄ = min(T1, ...,TK ,C), Tj : lifetime before death from cause j ;

(Jt )t>0 is the competing risks process : tells us in which statethe ith policyholder is at time t (Jt ∈ {0, 1, ...,K }).

τ is given by τ = inf{t > 0 | Jt , 0}.

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QUANTITIES OF INTEREST

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

λj(t) = lim∆→0

P(t ≤ T < t + ∆ , JT = j | T ≥ t)∆

, λ(t) =

p∑j=1

λj(t)

et S(t) = P(T > t) = e−∫ t0∑p

j=1 λj(s) ds.

2 The cumulative incidence functions (CIF) :

Fj(t) = P(T ≤ t , JT = j) =

∫ t

0fj(s) ds =

∫ t

0S(s−) λj(s) ds,

=

∫ t

0e−∫ s0∑p

j=1 λj(u) duλj(s) ds “ = 1 − e−

∫ t0 αj(s) ds ”

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FIRST COMMENTS

→ Ideally : assess with high precision every cause of failure to getthe overall hazard rate⇒ recover the survival function.

→ Some individual hazard rates can be hard to estimate in smallportfolios (e.g. death, ...) ;

→ Interpretability of the impact of covariates on each CIF is verytricky since all causes of failure are mixed...

→ Take care of compensation effects.

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CAUSE-SPECIFIC APPROACH - ESTIMATION ISSUE

→ Death depending on (feature considered) : smoker status.⇒ Compensation effects !

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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,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 , 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 , JT = 1) = F1(t).

Then, the subdistribution hazard of the event of interest follows

F1(t) = 1 − S1(t) = 1 − e−∫ t0 λ1(s) ds

and is finally given by

λ1(t) = lim∆→0

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

.

Novelty : ∀t , at-risk policyholders (PH) are those in state 0 at time t+ PH who have undergone a competing risk before t .Pro : not necessary to model every cause of failure !

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SOME COMMENTS

→ Intuition is simple : get back to the case of a 1-cause of failure...

→ Gain of interpretability : cause of interest has been put appart...Immediate link between the hazard rate and the CIF of interest(understanding about the impact of covariates is made simple) ;

→ F&G model is not about making the model become morecomplex, but rather about changing exposure...⇒ hereafter, classical Cox model for modelling surrender h.r. (butalso estimate more complex models with time-varying effects...).

→ Statistics are more robust since exposure is higher than in amulti-cause context.

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RESULTS AND IMPACTS

After rigorous optimization (backward model selection), we get :

Covariate Coefficient Standard Error p-valueAccidental Death Rider - Yes -0.191 0.0371 2.5×10−7

Gender - Female -0.0784 0.0256 2.2×10−3

Premium Frequency - Annual -0.222 0.0316 2.13×10−12

Premium Frequency - Other -0.395 0.0398 0Risk State - Smoker -0.136 0.0269 4.7×10−7

Underwriting Age - Middle 0.121 0.0293 3.83×10−5

Underwriting Age - Old -0.2 0.0389 2.73×10−7

Annual Premium 0.142 0.0116 0Dow Jones variation 0.889 0.0186 0

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PREDICTIVE POWER by BACKTESTS (HISTORY VS MODEL)

→ Independent test dataset to do backtesting (9772 PH, 1/3).→ Censoring = 49.77% (49.06% of lapsed contracts for learning).

→ (I) “Overall” quality of predictions : for each quarter and each PHi, compute the proba. that the PH makes the decision to terminateher contract in this period, i.e.

P(d1 < Ti ≤ d2, JTi = 1 |Xi , Ti > d1),

where d1 and d2 are adequate durations.

→ (II) “Local” quality of predictions : surrender rates by risk profile.Surrender rate for a given profile is estimated as in (I), but Rt ishere the set of PH having a given set of characteristics.

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(I) RE-BUILDING THE SURRENDER TRAJECTORY

Concretely, we have

P(d1 < Ti ≤ d2, JTi = 1 |Xi , Ti > d1) =P(d1 < Ti ≤ d2, JTi = 1, Ti > d1 |Xi)

P(Ti > d1, JTi = 1 |Xi)

=FT ,1(d2;Xi) − FT ,1(d1;Xi)

1 − FT ,1(d1;Xi).

Prediction of surrender rate r̂t : sum estimated probabilities withineach period (divided by the exposure on the same period) :

r̂t =1nt

∑i∈Rt

P̂(t < Ti ≤ t + 1, JTi = 1 |Xi , Ti > t),

where Rt = pop. at-risk at the beginning of quarter t , and nt its size.

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(I) RE-BUILDING THE SURRENDER TRAJECTORY (cont’d)

1995 2000 2005

0.00

0.02

0.04

0.06

Surrender rate predictions

Date

Qua

rterly

sur

rend

er ra

te

1995 2000 2005

0.00

0.02

0.04

0.06

1995 2000 2005

0.00

0.02

0.04

0.06

1995 2000 2005

0.00

0.02

0.04

0.06

EmpiricalNonparametricIntermediateFine and Gray

Figure: Comparison between the observed and predicted surrender ratesobtained by , models on the test set, from 01-01-1995 to 12-31-2008.

→ Statistical issues appears in nonparametric/semiparametricmodels...(DJ impact depends on elapsed time, and is clearlyoverestimated at 60 quarters⇒ wrong local effects)

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CREATING SURRENDER TABLES

→ Aim : propose an experimental table based on historical data.This tool could help operational teams to manage the surrenderrisk in a day-to-day task and make some easier ALM predictions.

→ In our context, the table provides the surrender rate by durationof the contract (in month), for the 14 first years (our experience).

→ It corresponds to surrender rates for the reference profile.

→ Some adjustments have then to be made depending on theprofile of the policyholder. In other words, it represents theaverage surrender risk for the current portfolio composition.

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# Months Rate (%)1-12 0.8500 0.7665 0.4990 0.5199 0.5052 0.5057 0.5155 0.5505 0.5640 0.5904 0.5738 0.490713-24 0.4824 0.4858 0.4741 0.4927 0.5071 0.5097 0.4925 0.5104 0.5097 0.5034 0.5070 0.434925-36 0.4368 0.4398 0.4429 0.4311 0.3982 0.3928 0.3839 0.3373 0.3514 0.4092 0.4476 0.434237-48 0.3966 0.3970 0.3878 0.3470 0.3615 0.3665 0.3752 0.3766 0.3103 0.3063 0.3085 0.329349-60 0.3267 0.3390 0.3125 0.3085 0.2993 0.3154 0.3177 0.3187 0.2878 0.2924 0.2946 0.300661-72 0.2912 0.3024 0.3163 0.3200 0.3223 0.3168 0.3323 0.3347 0.3212 0.2930 0.3045 0.316273-84 0.4045 0.3940 0.4078 0.3591 0.3536 0.3439 0.3409 0.3338 0.3336 0.3263 0.3413 0.341185-96 0.3479 0.3970 0.3844 0.4001 0.4273 0.4621 0.4470 0.4317 0.3857 0.3872 0.3770 0.404897-108 0.4153 0.4155 0.4024 0.4041 0.4042 0.4058 0.3728 0.3681 0.3695 0.3708 0.3875 0.3860109-120 0.4014 0.4014 0.3470 0.3248 0.3368 0.3395 0.3691 0.3800 0.3941 0.3973 0.3780 0.3232121-132 0.3355 0.3367 0.3394 0.3650 0.3566 0.3693 0.3707 0.3555 0.3236 0.3363 0.3374 0.3403133-144 0.3229 0.3257 0.3267 0.3295 0.3170 0.3231 0.3190 0.3200 0.3107 0.3220 0.3092 0.3014145-156 0.3041 0.2945 0.3042 0.3069 0.2229 0.2092 0.2167 0.2172 0.2177 0.2110 0.2312 0.2425157-168 0.2449 0.2364 0.2442 0.2467 0.2327 0.2078 0.2173 0.2160 0.2164 0.2169 0.2118 0.2104

Table: Surrender rates for each month of lifetime for the reference profile.

Covariate Coefficient to be applied Type of effect How much ?Accidental Death Rider - Yes 0.8261 decrease risk ' 17%

Gender - Female 0.9245 decrease risk ' 8%Premium Frequency - Annual 0.8009 decrease risk ' 20%Premium Frequency - Other 0.6736 decrease risk ' 33%

Risk State - Smoker 0.8728 decrease risk ' 13%Underwriting Age - Middle 1.1286 increase risk ' 13%

Underwriting Age - Old 0.8187 decrease risk ' 18%Annual Premium (+1 unit) 1.1525 increase risk ' 15%

Dow Jones variation (+1 unit) 2.4326 increase risk ' 143%

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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 enables to reduce themodel risk, as it does not rely on modelling other causes of failure.→ Side effects (commission here) can lead more complex modelsto give odd predictions : standard Cox model remains powerful.

Nevertheless, it requires

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.

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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, AarhusSchool of Business, Department of Business Studies, 2007.

J P Fine and R J Gray, A Proportional Hazards Model for the Subdistribution of aCompeting Risk, J. Am. Stat. Assoc. 94 (1999), no. 446, 496–509.

A J McNeil, R Frey, and P Embrechts, Quantitative Risk Management, PrincetonSeries In Finance, 2005.

Xavier Milhaud, M-P. Gonon, and Stephane Loisel, Les comportements de rachat enAssurance Vie en régime de croisière et en période de crise, Risques 83 (2010),76–81.

X Milhaud, V Maume-Deschamps, and S Loisel, Surrender triggers in Life Insurance :what main features affect the surrender behavior in a classical economic context ?,Bull. Français d’Actuariat 22 (2011), 5–48.

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 fundhypothesis, Insur. Math. Econ. 9 (1990), no. 4, 249–255.

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