MODELLING CRITICAL ILLNESS INSURANCE DATA

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1 MODELLING CRITICAL ILLNESS INSURANCE DATA Howard Waters Joint work with: Erengul Dodd (Ozkok), George Streftaris, David Wilkie University of Piraeus, October

2 Plan: 1. Critical Illness Insurance (CI) 2. The Continuous Mortality Investigation (CMI) 3. A Markov model for Critical Illness 4. Data 5. The claim delay distribution 6. Critical Illness diagnosis rates 2

3 Critical Illness: Policy description Fixed term policy, usually ceasing at age 65 Level monthly premiums payable throughout the term A fixed sum insured payable on the diagnosis of one of a specified list of critical illnesses The policy ceases at the end of the term or on payment of the sum insured UK sales peaked in 2002 with around 1 million new policies being sold Benefit type: Full acceleration (FA): Death is included as a critical illness (88%) Stand alone (SA): Death is not included as a critical illness (12%) 3

the term or on payment of the sum insured UK sales peaked in 2002 with around 1 million new policies being sold Benefit type: Full

4 Critical Illness: Diseases covered Critical illnesses and percentage of claims (FA) in Critical Illness % of claims Critical Illness % of claims Cancer 49.0 Total & permanent disability (TPD) 2.6 Death 17.6 Coronary artery bypass graft (CABG) 2.1 Heart attack (HA) 11.6 Kidney failure (KF) 0.6 Stroke 5.4 Major organ transplant (MOT) 0.2 Multiple sclerosis (MS) 4.3 Other causes 6.6 Males 57.3 Non smokers 73.9 Females 42.7 Smokers

6 Coronary artery bypass graft (CABG) 2.1 Heart attack (HA) 11.6 Kidney failure (KF) 0.6 Stroke 5.

5 The CMI: The CMI is a research organisation established by the UK actuarial profession in 1924 It collects data from contributing UK life insuance companies on: Mortality for life insurance policyholders Mortality and Morbidity for Income Protection policyholders Mortality and Morbidity for Critical Illness policyholders Mortality for members of self administered pension schemes The data collected covers of the UK market The CMI: Analyses the data for each contributing office and reports condfidentially Analyses the aggregated data and issues regular reports see Develops models and publishes standard tables 5

Mortality for members of self administered pension schemes The data collected covers 1 3 1 2 of the UK market The CMI: Analyses the data for each contributing

6 A Markov model for CI: Healthy 0 Dead 10 λ (10) x;θ λ (1) x;θ λ (2) x;θ λ (9) x;θ Cancer 1 Heart Attack 2.. Other Causes 9 λ (j) x;θ is the diagnosis intensity for cause j at age x with covariates θ. 6

7 Data: CI data for supplied to Heriot Watt University by the CMI: Details of policies in force at the start and end of each year policy-years of exposure Details of claims settled in claims 7

the start and end of each year 18 000 000 policy-years

8 Data: Covariates in the data: Covariate Age Number of levels Continuous Sex 2 Smoker status 2 Policy duration Continuous Office 13 Benefit type Benefit amount Policy type Sales channel 2 (FA & SA) Continuous 2 (Single/Joint life) 5 (Bancassurer, Direct, IFA, Other, Unknown) 8

13 Benefit type Benefit amount Policy type Sales channel 2 (FA & SA)

9 Data: Diagnosis is the insured event and there is a delay between diagnosis and settlement E[Delay] = 176 days; SD[Delay] = 269 days Practical problems 1 Missing Dates of Diagnosis. All claims have a Date of Settlement; only 82% have a Date of Diagnosis 2 Mismatch between Exposure and Claims. Exposure relates to policies in force from 1/1/ /12/2005 Claims data corresponds to claims settled (not diagnosed) in 1/1/ /12/2005 9

All claims have a Date of Settlement; only 82% have a Date of Diagnosis 2 Mismatch between Exposure and

10 Data: Diagnosed before 1/1/99 Unknown date of diagnosis Claims settled?? Settled after 31/12/05 1/1/99 31/12/05 In force/diagnosis Time 10

11 Data: Practical problems 1 Missing Dates of Diagnosis. 2 Mismatch between Exposure and Claims. Practical solutions Construct a claim delay distribution (CDD) F (j) (u : x; θ) F (j) (u : x; θ) = Pr[claim diagnosed age x, cause j, covariates θ, will be settled within u years] 1 Estimate missing dates of diagnosis as: Date of settlement median of CDD 2 Use the CDD to adjust the exposure: Multiply the exposure by the probability that a claim diagnosed in the observation period will be settled in the observation period 11

j, covariates θ, will be settled within u years] 1 Estimate missing dates of diagnosis as: Date of settlement median of CDD 2 Use

12 Claim Delay Distribution: X represents the delay; z is a vector of covariates; β is a vector of coefficients X 3-parameter Burr f X (u) = ατ(u/s) τ u[1 + (u/s) τ ] α+1 E[X] = exp(β z T ) s = Γ(α) Γ(α 1 τ )Γ(1 + 1 τ ) exp(βzt ) α and τ are shape parameters s is a scale parameter 12

ατ(u/s) τ u[1 + (u/s) τ ] α+1 E[X] = exp(β z T ) s = Γ(α) Γ(α 1 τ )Γ(1

13 Claim Delay Distribution: Select the covariates to be retained using Bayes variable selection Covariate retained Effect on E[X] Benefit type FA/SA = 1.07 Policy type SL/JL 1.07 Benefit amount Policy duration Decreasing Decreasing Office Highest/Lowest = 2.45 Cause Highest (Stroke)/Lowest (Death) =

14 Claim delay distribution: Examples: Scenario Benefit type FA FA FA FA FA Policy type JL JL JL JL JL Benefit amount (GBP) Policy duration (years) Office Cause Cancer Cancer Cancer Death TPD E[X] (days)

50 000 50 000 50 000 Policy duration (years) 4 4 1 4 4 Office 11 11 11 11

15 Critical Illness diagnosis rates: Estimation of cause specific diagnosis rates: Our observation period is 1999 to 2005 Not all offices contribute for the whole 7 years Suppose Office 1 contributes data for 2000 to For this office: λ (j) x;θ θ is a set of covariates, including office is the diagnosis inception rate for cause j at age x with covariates θ E(u : x; θ) is the number of policies (age x, covariates θ) inforce at time u, 0 u 4 N (j) (x; θ) is the number of claims (cause j, age x, covariates θ) diagnosed and settled in N (j) (x; θ) Poisson ( λ (j) x;θ 4 u=0 E(u : x; θ) F (j) (4 u : x; θ) du) 15

For this office: λ (j) x;θ θ is a set of covariates, including office is the diagnosis inception rate for cause j at age x with covariates θ E(u : x; θ) is

16 Critical Illness diagnosis rates: Estimator/crude rate: / 4 ˆλ (j) x;θ = N (j) (x; θ) u=0 E(u : x; θ) F (j) (4 u : x; θ) du / 4 V[ˆλ (j) x;θ ] = λ(j) x;θ u=0 E(u : x; θ) F (j) (4 u : x; θ) du / ( 4 N (j) (x; θ) u=0 ) 2 E(u : x; θ) F (j) (4 u : x; θ) du 16

17 Critical Illness diagnosis rates: Model: ( λ (j) x;θ = λ(j) 1 (x) + exp λ (j) 2 (x) + β zt ) λ (j) i (x) is a polynomial function of age only, i = 1, 2 λ (j) 1 (x) 0 for each cause except death λ (j) 1 (x) 0 log linear model for λ(j) x;θ Use R to select the statistically significant covariates 17

1, 2 λ (j) 1 (x) 0 for each cause except death λ (j) 1 (x) 0 log linear

18 Critical Illness diagnosis rates: Cause Significant covariates CABG Age Sex Smoker status Cancer Age Sex Year Smoker status Death Age Sex Smoker status Heart Attack Age Sex Smoker status Kidney Failure Age MOT MS Sex Smoker status Policy duration Other causes Age Sex Office Benefit type Stroke Age Sex Smoker status TPD Age Year Policy duration 18

Sex Smoker status Kidney Failure Age MOT MS Sex Smoker status Policy duration Other

19 Critical Illness diagnosis rates: Males, non smokers Cancer MNS Cancer Inception Rates 1.7e Weighted Rates Crude Rates (CR) CR +( ) 2SE Age 19

20 Critical Illness diagnosis rates: Males Cancer: Smokers/Non smokers MS/MNS Cancer Ratio of Inception Rates MS/MNS Age 20

21 Critical Illness diagnosis rates: Females, smokers Death FS Death Inception Rates 6.1e e Crude Rates (CR) Smoothed Rates CR+( )2SE Age 21

22 Critical Illness diagnosis rates: Males, smokers Heart Attack MS HA Inception Rates 1.5e 08 5e e Crude Rates (CR) Smoothed Rates CR+( )2SE Age 22

23 Critical Illness diagnosis rates: Heart Attack λ HA x:θ = exp [ β Male β Smoker + x ( β Smoker ) x 2 ( β Smoker ) ] Females Males Age Non smoker Smoker Non smoker Smoker Values of λ HA x:θ

24 Critical Illness diagnosis rates: Males, non smokers, Office 1, 2003, Pol Durn Inception Rate Other MS MOT KF TPD CABG Stroke Death Heart Attack Cancer Age 24

25 References: OZKOK E., STREFTARIS G., WATERS H.R. and WILKIE A.D. (2012) Bayesian modelling of the time delay between diagnosis and settlement for Critical Illness Insurance using a Burr generalised-linear-type model. Insurance: Mathematics and Economics, 50, OZKOK E., STREFTARIS G., WATERS H.R. and WILKIE A.D. (2013) Modelling Critical Illness claim diagnosis rates I: Methodology. Scandinavian Actuarial Journal, DOI: / OZKOK E., STREFTARIS G., WATERS H.R. and WILKIE A.D. (2013) Modelling Critical Illness claim diagnosis rates II: Results. Scandinavian Actuarial Journal, DOI: /

Critical Illness insurance AC04 diagnosis rates

Health and care conference 2011 Dave Heeney and Dave Grimshaw, CMI Critical Illness Committee Critical Illness insurance AC04 diagnosis rates 20 May 2011 2010 The Actuarial Profession www.actuaries.org.uk