ESTIMATION AND GRADUATION OF CRITICAL ILLNESS INSURANCE DIAGNOSIS RATES. Howard Waters

Transcription

1 ESTIMATION AND GRADUATION OF CRITICAL ILLNESS INSURANCE DIAGNOSIS RATES Howard Waters Joint work with: Erengul Ozkok, George Streftaris, David Wilkie Heriot Watt University, Edinburgh University of Cologne, May

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

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

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 1 3 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

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 inforce at the start and end of each year policy-years of exposure Details of claims settled in claims Modelling by Erengul Ozkok + G Streftaris, H R Waters & A D Wilkie PhD thesis and papers in

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

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 All claims have a date of settlement; 82% have a date of diagnosis 2 The exposure corresponds to claims settled, not to claims diagnosed 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 1 Estimate missing dates of diagnosis as: Date of settlement median of appropriate CDD 2 Use the CDD to adjust the exposure within u years] 9

10 Claim Delay Distribution: X represents the delay; z is a vector of covariates; β is a vector of coefficients We want: 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 ) 10

11 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) =

12 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)

13 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, let: λ (j) x;θ θ be a set of covariates, including office be the diagnosis inception rate for cause j at age x with covariates θ E(u : x; θ) be the number of policies (age x, covariates θ) inforce at time u, 0 u 4 N (j) (x; θ) be 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) 13

14 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 14

15 Critical Illness diagnosis rates: Model: λ (j) x;θ = λ(j) 1 (x) + exp(λ(j) 2 (x)) exp(β zt ) where 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 15

16 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 16

17 Critical Illness diagnosis rates: MNS Cancer Inception Rates 1.7e Weighted Rates Crude Rates (CR) CR +( ) 2SE Age 17

18 Critical Illness diagnosis rates: MS/MNS Cancer Ratio of Inception Rates MS/MNS Age 18

19 Critical Illness diagnosis rates: FS Death Inception Rates 6.1e e Crude Rates (CR) Smoothed Rates CR+( )2SE Age 19

20 Critical Illness diagnosis rates: MS HA Inception Rates 1.5e 08 5e e Crude Rates (CR) Smoothed Rates CR+( )2SE Age 20

21 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 21

22 Critical Illness: Bibliography OZKOK, E. (2011) A stochastic model for Critical Illness Insurance. PhD thesis presented to Heriot Watt University. CMI WORKING PAPERS 28 (2007), 33 (2008), 43 (2010) & 50 (2011) 22