MODELLING CRITICAL ILLNESS INSURANCE DATA Howard Waters Joint work - - PowerPoint PPT Presentation

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: Healthy Cancer 1 Heart Attack 2 . . . . . . Other Causes

9 ✲ ◗◗◗◗◗◗ ◗ s ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❫ λ(1)

x;θ

λ(2)

x;θ

λ(9)

x;θ

❄ λ(10)

x;θ

Dead

10 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:

✲ Time

Claims settled In force/diagnosis 1/1/99 31/12/05

s ❝ s ❝ ❝ s s s ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂✂ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✂ ✂ ✂ ✂ ✂

?

s

?✂

✂ ✂ ✂ ✂ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂✂ ❝ ❄

Diagnosed before 1/1/99

❄

Unknown date of diagnosis

❄

Settled after 31/12/05

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

fX(u) = ατ(u/s)τ u[1 + (u/s)τ]α+1 E[X] = exp(β zT ) → 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

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 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

→ N (j)(x; θ) ∼ Poisson

• λ(j)

x;θ

4

u=0 E(u : x; θ) F (j)(4 − u : x; θ) du

• 15

Critical Illness diagnosis rates: Estimator/crude rate:

ˆ λ(j)

x;θ

= N (j)(x; θ) 4

u=0

E(u : x; θ) F (j)(4 − u : x; θ) du

V[ˆ

λ(j)

x;θ]

= λ(j)

x;θ

4

u=0

E(u : x; θ) F (j)(4 − u : x; θ) du ≈ N (j)(x; θ) 4

u=0

E(u : x; θ) F (j)(4 − u : x; θ) du 2

16

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

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

20 30 40 50 60 70 80

MNS − Cancer

Age Inception Rates 1.7e−05 0.00012 0.00091 0.0067 0.05 Weighted Rates Crude Rates (CR) CR +(−) 2SE 1999 2005

19

Critical Illness diagnosis rates: Males – Cancer: Smokers/Non–smokers

20 30 40 50 60 70 80 0.8 1.0 1.2 1.4 1.6 1.8

MS/MNS − Cancer

Age Ratio of Inception Rates MS/MNS

20

Critical Illness diagnosis rates: Females, smokers – Death

20 30 40 50 60 70 80

FS − Death

Age Inception Rates 6.1e−06 9.6e−05 0.0015 0.024 0.37 Crude Rates (CR) Smoothed Rates CR+(−)2SE

21

Critical Illness diagnosis rates: Males, smokers – Heart Attack

20 30 40 50 60 70 80

MS − HA

Age Inception Rates 1.5e−08 5e−07 1.7e−05 0.00055 0.018 Crude Rates (CR) Smoothed Rates CR+(−)2SE

22

Critical Illness diagnosis rates: Heart Attack

λHA

x:θ

= exp [−24.56 + 1.915 βMale + 2.041 βSmoker + x (0.4877 + 0.0003 βSmoker) −x2 (0.0036 + 0.0003 βSmoker)

• 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 λHA

x:θ × 100 000

23

Critical Illness diagnosis rates: Males, non–smokers, Office 1, 2003, Pol Durn 3

Age Inception Rate

0.003 0.006 0.009 0.012 0.015 20 30 40 50 60 Other MS MOT KF TPD CABG Stroke Death Heart Attack Cancer

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