# Statistical Analysis of Life Insurance Policy Termination and Survivorship

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1 Statistical Analysis of Life Insurance Policy Termination and Survivorship Emiliano A. Valdez, PhD, FSA Michigan State University joint work with J. Vadiveloo and U. Dias Session ES82 (Statistics in Actuarial Science) ERCIM 2013, London, UK, Dec 14-16, 2013 EA Valdez (MSU) Life Insurance Policy Termination and Survivorship ERCIM / 31

2 Preliminaries Preliminaries An individual has a future lifetime random variable T and is exposed to two possible reasons to fail: withdrawal (policy termination) or mortality (death). Denote the cause of failure by J with: J = w indicates failure due to withdrawal, and J = d indicates failure due to death. Convenient to introduce theoretical net lifetime random variables: T w and T d. Assume their respective distribution, survival and density functions exist: F j, S j and f j, for j = w, d. Competing Risk Models: T w and T d are never observed simultaneously, but only (T, J) where T = min(t w, T d ). Model identifiability is a common issue here: one approach is to specify the joint distribution or copula function associated with (T w, T d ). See Tsiatis (1975). EA Valdez (MSU) Life Insurance Policy Termination and Survivorship ERCIM / 31

3 Preliminaries Competing risk models Competing risk models can be applied in several disciplines: actuarial science: life insurance contracts economics: duration till employment, cause of leaving employment medical statistics: clinical trials epidemiology: occurrence/recovery of diseases engineering: time/cause of failure of a mechanical system In actuarial science, some of the literature: Carriere (1994, 1998), Valdez (2000), Tsai, Kuo and Chen (2002) Actuarial students study what is called Multiple Decrement Models. Plenty of literature here. EA Valdez (MSU) Life Insurance Policy Termination and Survivorship ERCIM / 31

4 Preliminaries Outline Motivation Model calibration Data characteristics Distribution of face amount Parametric models Time-until-withdrawal Age-at-death Calibration results Time-until-withdrawal Age-at-death Implications of results Mortality selection Financial cost Concluding remarks Acknowledgement EA Valdez (MSU) Life Insurance Policy Termination and Survivorship ERCIM / 31

5 Motivation Motivation for model constructions Data-driven. Our observables are best illustrated by the following figure: This diagram provides an illustration of the observed times until withdrawal and times until death. EA Valdez (MSU) Life Insurance Policy Termination and Survivorship ERCIM / 31

6 Motivation Model calibration Data source used in the calibration A sub-sample from a portfolio of life insurance contracts from a major insurer. detailed information on the type of policies (e.g. PAR, TERM, UL, CONV) and additional characteristics sub-sample consists of 65,435 terminated single-life insurance contracts with mortality dates tracked from the US Social Security System administration office our data file recorded a 1918 as the year with the earliest policy issue date and the end of the observation period is 14 February 2008 Our policy record indicates 61,901 of the total observations are censored, representing about 94.6% of the observation. For each contract observed, we have policy effective (issue) date, the termination date and the date of death, if applicable. EA Valdez (MSU) Life Insurance Policy Termination and Survivorship ERCIM / 31

7 Motivation Data characteristics Policy characteristics and other observable information Categorical variables Description Proportions PlanType Type of insurance plan: PlanTypeP 42.4% PlanTypeT 28.0% PlanTypeO 29.6% RiskClass Insured s assigned risk class: RiskClass = N 72.0% RiskClass = Y 28.0% Sex Insured s sex: Male = % Female = % Smoker Smoker class: Non-smoker = N 66.6% Smoker = S 12.4% Combined = C 21.0% Censor Censoring indicator for death: Censor = % Censor = 0 5.4% Continuous variables Minimum Mean Maximum IssAge The policyholder s issue age Face Amount The policy s insured amount 1 213,000 60,000,000 Temp FEAmt Temporary flat extra amount (per 1000) Perm FEAmt Permanent flat extra amount (per 1000) MEFact Extra mortality factor Dates IssDate Policy effective or issue date BDate Insured s date of birth WDate Policy withdrawal or lapse date DDate Insured s date of death, if applicable EA Valdez (MSU) Life Insurance Policy Termination and Survivorship ERCIM / 31

8 Motivation Distribution of face amount Count and face amount Number of policies and average face amount by plan type, sex and issue age Issue Age Males Females Total Plan Type > > 70 PlanTypeP Count 6,461 8,476 2, ,401 4,545 1, ,776 Face Amount 46, , , ,028 35, , , , ,605 PlanTypeT Count 1,130 9,557 1, , ,333 Face Amount 323, , ,320 1,461, , , , , ,264 PlanTypeO Count 2,076 7,314 3, ,516 3,789 1, ,326 Face Amount 124, , , ,704 79, , , , ,690 EA Valdez (MSU) Life Insurance Policy Termination and Survivorship ERCIM / 31

9 Parametric models Time-until-withdrawal A class of duration models for time-until-withdrawal Suppose we can write T w as T w = exp(µ)t σ 0 for some non-negative rv T 0. With log-transformation, log(t w ) = µ + σ log(t 0 ) = µ + σλ, where Λ = log(t 0 ), µ and σ are location and scale parameter provided σ 0 to avoid a degenerate distribution. Because we can write the survival distribution function of T w as ( ) log(t) µ S Λ, σ > 0 σ S w (t) = ( ) log(t) µ 1 S Λ, σ < 0 σ where S Λ denotes the survival function of Λ, the distribution of T w belongs to a log-location-scale family of distributions. EA Valdez (MSU) Life Insurance Policy Termination and Survivorship ERCIM / 31

10 Parametric models Time-until-withdrawal Covariates Introduce covariates through the location parameter µ. With x as a vector of covariates, such as policyholder characteristics, and β, the vector of linear coefficients. Then replace µ = x β. We have T w = exp(x β)t σ 0 and log(t w ) = x β + σ log(t 0 ) = x β + σλ, which generalizes the ordinary regression model. This specification is a special case of the Accelerated Failure Time (AFT) model commonly studied in survival analysis. EA Valdez (MSU) Life Insurance Policy Termination and Survivorship ERCIM / 31

11 Parametric models Time-until-withdrawal Distribution of the time-until-withdrawal Straightforward to find explicit form of the distribution of T w in terms of the distribution of T 0. The survival function of T w can be expressed as Its density can be expressed as S w (t) = S 0 ((e µ t) 1/σ). f w (t) = 1 σ t (e µ t) 1/σ f 0 ( (e µ t) 1/σ), where S 0 and f 0 are respectively the survival and density functions of T 0. Within this class of models, oftentimes more straightforward to specify the distribution of T 0 rather than of its logarithm. EA Valdez (MSU) Life Insurance Policy Termination and Survivorship ERCIM / 31

12 Parametric models Time-until-withdrawal Class of distribution models considered Log-Normal Distribution: T 0 has a log-normal distribution with parameters 0 and 1. [ 1 f w (t) = exp 1 ( ) ] log(t) µ 2. 2πσt 2 σ Generalized Gamma Distribution: T 0 is a standard Gamma with scale of 1, shape parameter m. f w (t) = 1 1 [ σ t Γ(m) (e µ t) m/σ exp (e µ t) 1/σ]. GB2 Distribution: T 0 has a Beta of the second kind (B2) density with parameters γ 1 and γ 2. f w (t) = 1 1 (e µ t) γ 1/σ [ σ t B(γ 1, γ 2 ) 1 + (e µ t) 1/σ] γ 1 +γ 2. EA Valdez (MSU) Life Insurance Policy Termination and Survivorship ERCIM / 31

13 Parametric models Age-at-death Survival models Let the (fixed) issue age be z and X d the age-at-death r.v. so that provided T wd > 0. X d z = z + T w + (T d T w ) = z + T w + T wd, If T w is known, then (X d z, T w = t w ) = z + t w + T wd. Thus, we have P (T wd > t wd z, T w = t w ) = P (T d > T w + t wd z, T w = t w ) = P (X d > z + t w + t wd ) P (X d > z + t w ) = S d(z + t w + t wd ), S d (z + t w ) where S d is the survival function of X d. EA Valdez (MSU) Life Insurance Policy Termination and Survivorship ERCIM / 31

14 Parametric models Age-at-death Survival models considered Gompertz Distribution: Survival function has the form [ ( S d (x) = exp e m /σ 1 e x/σ )], where m > 0 is mode and σ > 0 is dispersion about this mode. See Carriere (1992). With B = 1 σ exp( m /σ ) and c = exp(1/σ ), it leads us to the hazard function µ x = f d(x) S d (x) = Bcx. Weibull Distribution: Survival function has the form [ S d (x) = exp (x/m ) m /σ ], where m > 0 and σ > 0 are respectively location and dispersion parameters. See also Carriere (1992). Popularly known in survival analysis and reliability theory. EA Valdez (MSU) Life Insurance Policy Termination and Survivorship ERCIM / 31

15 Calibration results Time-until-withdrawal Preliminary investigation - histogram observed density time until withdrawal EA Valdez (MSU) Life Insurance Policy Termination and Survivorship ERCIM / 31

16 Calibration results Time-until-withdrawal By type of plan Plan Type Number Min Mean Median Max Std Dev PlanTypeP 27, PlanTypeT 18, PlanTypeO 19, Aggregate 65, density density density time until withdrawal time until withdrawal time until withdrawal EA Valdez (MSU) Life Insurance Policy Termination and Survivorship ERCIM / 31

17 Calibration results Time-until-withdrawal MLEs for the various duration models Parameter Log-Normal Generalized Gamma GB2 Regression coefficients β 0 (intercept) (0.0263) (0.0419) (0.0238) β 1 (PlanTypeP) (0.0071) (0.0061) (0.0054) β 2 (PlanTypeT) (0.0068) (0.0060) (0.0055) β 5 (RiskClassY) (0.0063) (0.0056) (0.0060) β 6 (Male) (0.0053) (0.0047) (0.0041) β 7 (SmokerN) (0.0079) (0.0065) (0.0063) β 8 (SmokerC) (0.0099) (0.0086) (0.0079) β 10 (Face Amount) (0.0004) * (0.0003) (0.0004) β 11 (Temp FEAmt) (0.0026) (0.0027) (0.0020) β 12 (Perm FEAmt) (0.0028) (0.0023) (0.0024) β 13 (MEFact) (0.0240) (0.0162) (0.0216) β 14 (IssAge) (0.0002) (0.0002) (0.0002) Model specific parameters σ (0.0018) (0.0130) (0.0065) m (0.0966) - γ (0.0168) γ (0.0486) Model fit statistics Number of observations 65,435 65,435 65,435 Log-likelihood -209, , ,199.5 Number of parameters Akaike information criterion 418, , , Notes: a. Face amount is re-scaled in 100,000. b. Standard errors are in parenthesis. c. An asterisk * identifies not significant at the 5% level. EA Valdez (MSU) Life Insurance Policy Termination and Survivorship ERCIM / 31

18 Calibration results Time-until-withdrawal Assessing the quality of the model fit Log-Normal, Generalized Gamma and GB2, respectively EA Valdez (MSU) Life Insurance Policy Termination and Survivorship ERCIM / 31

19 Calibration results Time-until-withdrawal Assessing the quality of the model fit PP plots of Log-Normal, Generalized Gamma and GB2, respectively EA Valdez (MSU) Life Insurance Policy Termination and Survivorship ERCIM / 31

20 Calibration results Age-at-death Observed deaths by issue age and sex Mortality status Issue Age Survive Death Total Males 30 8, , ,341 1,006 25, , ,354 > Total 40,196 2,480 42,676 Females 30 6, , , , , ,911 > Total 21,693 1,066 22,759 EA Valdez (MSU) Life Insurance Policy Termination and Survivorship ERCIM / 31

21 Calibration results Age-at-death Maximum likelihood estimation technique Maximum likelihood techniques used. While we investigated several other parametric models, it boiled down to choosing between the Gompertz and Weibull models. Our observable data, (z i, t w,i, t wd,i, δ i ), consists of the age at issue, the time of withdrawal, the time of death from withdrawal (if applicable), and a censoring variable. For an uncensored observation, the log-likelihood contribution is For a censored observation, it is log f d(z i + t w,i + t wd,i ). S d (z i + t w,i ) log S d(z i + t w,i + t wd,i ). S d (z i + t w,i ) EA Valdez (MSU) Life Insurance Policy Termination and Survivorship ERCIM / 31

22 Calibration results Age-at-death Maximum likelihood estimates Parameter Gompertz Weibull m (0.1428) (0.1811) σ (0.0975) (0.1337) σ Male (0.1161) (0.1481) Model fit statistics Number of observations 65,435 65,435 Log-likelihood -18, , Number of parameters 3 3 Akaike information criterion 36, , EA Valdez (MSU) Life Insurance Policy Termination and Survivorship ERCIM / 31

23 Calibration results Age-at-death Gompertz - Male and Gompertz - Female, respectively EA Valdez (MSU) Life Insurance Policy Termination and Survivorship ERCIM / 31

24 Calibration results Age-at-death Weibull - Male and Weibull -Female, respectively EA Valdez (MSU) Life Insurance Policy Termination and Survivorship ERCIM / 31

25 Implications of results What do all these results imply? To understand the implications of results of our models, we examined two items: The presence of mortality antiselection: this refers to whether there is greater survival rate after termination of the insurance contract. There is presence of antiselection at withdrawal in life insurance if S d w (t d t w ) > S d (t d ), for every t d t w. See Carriere (1998) and Valdez (2001). The financial cost of policy termination. EA Valdez (MSU) Life Insurance Policy Termination and Survivorship ERCIM / 31

26 Implications of results Mortality selection Mortality antiselection To interpret the previous definition: Antiselection is evidently present when survival of those terminated policies, conditional on all periods of termination, have generally better unconditional survival. Now, to look for evidence in our data, we consider a specific type of a policyholder with the following characteristics: issue age 35, permanent whole life, a non-smoker, male, face amount of 250,000, and not-so-risky with no flat extra charges. Then, we compare the conditional and unconditional survivorship curves for this policyholder for terminating in different years from issue: withdrawals for years 2, 4, 6, 8, 10, 15, 20 and 30. EA Valdez (MSU) Life Insurance Policy Termination and Survivorship ERCIM / 31

27 Implications of results Mortality selection Survival curves after policy termination for (35) survival curves survival curves duration from withdrawal duration from withdrawal survival curves duration from withdrawal survival curves duration from withdrawal For various policy terminations: years 2, 8, 20 and 30. EA Valdez (MSU) Life Insurance Policy Termination and Survivorship ERCIM / 31

28 Implications of results Financial cost The financial cost of policy termination We considered the following case for illustration: Issue age 35, male, non-smoker, permanent whole life policy, death benefit of 250,000 Two types of expenses - assumptions based on Segal (2002, NAAJ): acquisition cost: 80 plus 4.5 per 1,000 of death benefit maintenance expense: 60 plus 3.5 per 1,000 of death benefit Interest rate is 5% Time-until-withdrawal were simulated based on Generalized Gamma. Age-at-death were simulated based on Gompertz. The financial impact is the loss incurred when policy terminates: accumulated values of all past expenses incurred, plus policy reserves, reduced by the accumulated value of all past premiums paid. EA Valdez (MSU) Life Insurance Policy Termination and Survivorship ERCIM / 31

29 Implications of results Financial cost Distribution of the loss at policy termination frequency 0.0e e e 05 2e+05 1e+05 0e+00 1e+05 2e+05 loss at policy termination Summary statistics of loss at policy termination Number Min Mean Median Max Std Dev 100, ,500 1,223-3, ,000 19,065 EA Valdez (MSU) Life Insurance Policy Termination and Survivorship ERCIM / 31

30 Concluding remarks Concluding remarks We examined and modeled life insurance policy termination and survivorship: time-until-withdrawal - duration models age-at-death - survival models Our modeling aspect was driven by the observable data in our dataset. We find that: several policy characteristics do affect policy termination, but not survivorship after policy termination. The modeling results can be used for: understanding the presence of mortality selection of policy withdrawal, and predictive modeling of loss upon policy termination. EA Valdez (MSU) Life Insurance Policy Termination and Survivorship ERCIM / 31

31 Concluding remarks Acknowledgement Acknowledgement The authors wish to acknowledge financial support from the Actuarial Science program at Michigan State University, and Janet and Mark L. Goldenson Actuarial Research Center of the University of Connecticut. EA Valdez (MSU) Life Insurance Policy Termination and Survivorship ERCIM / 31

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