MODELLING CRITICAL ILLNESS INSURANCE DATA Howard Waters Joint work with: Erengul Dodd (Ozkok), George Streftaris, David Wilkie University of Piraeus, October 2014 1
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
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
Critical Illness: Diseases covered Critical illnesses and percentage of claims (FA) in 1999 – 2005 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 26.1 4
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 ≈ 1 3 → 1 2 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 www.actuaries.org.uk/knowledge/cmi Develops models and publishes standard tables 5
A Markov model for CI: λ (1) ✲ x ; θ Healthy Cancer ◗◗◗◗◗◗ ❏ 0 1 ❏ λ (2) ❏ x ; θ ❏ ◗ s ❏ Heart Attack ❏ λ (10) λ (9) 2 ❏ x ; θ x ; θ . . ❏ . . ❏ . . ❄ ❏ ❫ ❏ Dead Other Causes 10 9 λ ( j ) x ; θ is the diagnosis intensity for cause j at age x with covariates θ . 6
Data: CI data for 1999 – 2005 supplied to Heriot–Watt University by the CMI: • Details of policies in force at the start and end of each year → 18 000 000 policy-years of exposure • Details of claims settled in 1999 – 2005 → 19 000 claims 7
Data: Covariates in the data: Covariate Number of levels Age Continuous Sex 2 Smoker status 2 Policy duration Continuous Office 13 Benefit type 2 (FA & SA) Benefit amount Continuous Policy type 2 (Single/Joint life) Sales channel 5 (Bancassurer, Direct, IFA, Other, Unknown) 8
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 1/1/1999 → 31/12/2005 Claims data corresponds to claims settled (not diagnosed) in 1/1/1999 → 31/12/2005 9
Data: Unknown date of diagnosis Diagnosed before 1/1/99 Settled after 31/12/05 Claims settled ❄ ❄ ❄ ❝ s s s ❝ ✂✂ ✁ ✂ ✡ ✂✂ ✂ ✁ ✡ ✂ ✂ ✁ ✡ ✂ ✂ ✂ ✁ ✡ ✂ ? ✂ ✂ ✁ ✡ ✂ ✲ Time ✂ ✁ ✡ ✂ ✂ ✂ ✂ ✁ ✡ ✂ ✂ ✂ ✁ ✡ ✂ ✂ ✂ ✁ ✡ ✂ ✂ ✂ ✁ ✡ ✂ ✂ ✁ ✡ ✂ ? ❝ s s s ❝ 1/1/99 31/12/05 In force/diagnosis 10
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
Claim Delay Distribution: X represents the delay; z is a vector of covariates; β is a vector of coefficients X ∼ 3-parameter Burr ατ ( u/s ) τ f X ( u ) = u [1 + ( u/s ) τ ] α +1 exp( β z T ) E [ X ] = Γ( α ) τ ) exp( βz T ) → s = Γ( α − 1 τ )Γ(1 + 1 α and τ are shape parameters s is a scale parameter 12
Claim Delay Distribution: Select the covariates to be retained using Bayes variable selection Effect on E [ X ] Covariate retained Benefit type FA/SA = 1.07 Policy type SL/JL 1.07 Benefit amount Decreasing Policy duration Decreasing Office Highest/Lowest = 2.45 Cause Highest (Stroke)/Lowest (Death) = 2.13 13
Claim delay distribution: Examples: Scenario 1 2 3 4 5 Benefit type FA FA FA FA FA Policy type JL JL JL JL JL Benefit amount (GBP) 50 000 250 000 50 000 50 000 50 000 Policy duration (years) 4 4 1 4 4 Office 11 11 11 11 11 Cause Cancer Cancer Cancer Death TPD E [ X ] (days) 174 158 196 109 211 14
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 2003. For this office: θ is a set of covariates, including office λ ( j ) x ; θ 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 2000 – 2003 � 4 � � λ ( j ) → N ( j ) ( x ; θ ) ∼ Poisson u =0 E ( u : x ; θ ) F ( j ) (4 − u : x ; θ ) du x ; θ 15
Critical Illness diagnosis rates: Estimator/crude rate: � � 4 λ ( j ) ˆ N ( j ) ( x ; θ ) E ( u : x ; θ ) F ( j ) (4 − u : x ; θ ) du = x ; θ u =0 � � 4 λ ( j ) λ ( j ) V [ˆ E ( u : x ; θ ) F ( j ) (4 − u : x ; θ ) du x ; θ ] = x ; θ u =0 � �� 4 � 2 N ( j ) ( x ; θ ) E ( u : x ; θ ) F ( j ) (4 − u : x ; θ ) du ≈ u =0 16
Critical Illness diagnosis rates: Model: � 2 ( x ) + β z T � λ ( j ) x ; θ = λ ( j ) λ ( j ) 1 ( x ) + exp 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
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
Critical Illness diagnosis rates: Males, non–smokers – Cancer MNS − Cancer 0.05 Weighted Rates Crude Rates (CR) CR +(−) 2SE 1999 2005 0.0067 Inception Rates 0.00091 0.00012 1.7e−05 20 30 40 50 60 70 80 Age 19
Critical Illness diagnosis rates: Males – Cancer: Smokers/Non–smokers MS/MNS − Cancer MS/MNS 1.8 1.6 Ratio of Inception Rates 1.4 1.2 1.0 0.8 20 30 40 50 60 70 80 Age 20
Critical Illness diagnosis rates: Females, smokers – Death FS − Death 0.37 Crude Rates (CR) Smoothed Rates CR+(−)2SE 0.024 Inception Rates 0.0015 9.6e−05 6.1e−06 20 30 40 50 60 70 80 Age 21
Critical Illness diagnosis rates: Males, smokers – Heart Attack MS − HA 0.018 Crude Rates (CR) Smoothed Rates CR+(−)2SE 0.00055 Inception Rates 1.7e−05 5e−07 1.5e−08 20 30 40 50 60 70 80 Age 22
Critical Illness diagnosis rates: Heart Attack λ HA = exp [ − 24 . 56 + 1 . 915 β Male + 2 . 041 β Smoker + x (0 . 4877 + 0 . 0003 β Smoker ) x : θ − x 2 (0 . 0036 + 0 . 0003 β Smoker ) � Females Males Age Non–smoker Smoker Non–smoker Smoker 50 11 40 73 268 60 27 73 186 493 70 34 62 232 418 80 21 24 141 163 Values of λ HA x : θ × 100 000 23
Critical Illness diagnosis rates: Males, non–smokers, Office 1, 2003, Pol Durn 3 0.015 0.012 Other 0.009 MS MOT Inception Rate KF TPD CABG Stroke Death 0.006 Heart Attack Cancer 0.003 0 20 30 40 50 60 Age 24
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, 266–279. OZKOK E., STREFTARIS G., WATERS H.R. and WILKIE A.D. (2013) Modelling Critical Illness claim diagnosis rates I: Methodology. Scandinavian Actuarial Journal, DOI:10.1080/03461238.2012.728537 OZKOK E., STREFTARIS G., WATERS H.R. and WILKIE A.D. (2013) Modelling Critical Illness claim diagnosis rates II: Results. Scandinavian Actuarial Journal, DOI:10.1080/03461238.2012.728538 25
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