Ch 28: Pricing health and care contracts

Question 1

Income protection
Multi state modelling 4,4

Answer 1

• PH tracked through various stages of health
• muxij=>transition intensity from i to j @ age x
• tPx HH => prob healthy life aged x will be healthy in t years age x+t
• tPx HHbar=> prob healthy life x will remain healthy for t years age x+t

In practice MSM:
* Det prop lives in each state using duration based intensities
* Val claims outgo = # lives recieving Ben * avg SA
* Lack of detailed data avail to est all intensities
* Reduce # of states to avoid spurious accuracy

Question 2

Notation

Answer 2

add images here

Question 3

The general method 7

Answer 3

• Simplification of MSM=> converted to decrement table
• Gives exp cost of ben from claimsincepting in a particular year
• Cost is a product of 5 components:
  
  • Prob PH eligible for claim in year => survival prob
  • Claim inception rate for year
  • Value of annuity payable for duration of claim
  • Discount factor=> disc ann from claim to policy inception
  • Annual benefit amount

Question 4

Claim inception rate [idx] 4

Answer 4

• Prob claim becomes payable in year of age x to x+1
• PH becomes sick d weeks/months earlier and remains sick to ben payment
• Diff to sickness inception rate and relates to time benefit starts @ end of deferred period
• Prob sick * prob remaining sick throughout deferred period

Question 5

Disabled life annuity 3

Answer 5

• PV @ date of claim of exp claim payments to PH
• After deferred period to termination
• Allow for claim escalation, interest, Mort, Recovery

Question 6

Expected claims outgo

Answer 6

• (Ann Ben Amt) * (Disc factor) * (Prob elig claim) * (prob claim)
• 5 * 4 * 1 * 2

Question 7

Valuing disability benefits using claim inception & termination rates

Answer 7

• Formalize general method into 2 decrement tables
• 1st table
  
  • Healthy PH, decrements are falling sick and death
  • Allows for recovery and subsq sickness
  • Incept rate for disability (aq)ix dep on sickness incept rate not claim incep
• 2nd table
  
  • PH already receiving benefits, decrements are recovery and death
  • ‘Survival’ (neither recovery or dying)
  • Det dis ann payable until death/recover/termination

Question 8

Valuing disability benefits using integrals

Answer 8

• Using life table values
• Approx integral

Question 9

S(ID) Tables from CMIR12

Answer 9

• pg 138 => 1,2,4,5
• Claim inception rates (ia)x,d
  
  • central rates by age & def period
  • exp # claim inceptions over [x,x+1]
  • age sick = x-d
  • apply to all lives alive = calc prob PH eligible
• Annuity rates a ssbar x,z 2 rates given
  
  • PV current claim ann (already sick)
    
    • dep on duration of sickness
    • reserve calc
    • RHS
  • PV ann (healthy lives)
    
    • dep on deferred period
    • prem calc
    • LHS
• LHS rates => still need 1,2,4,5 mult together
• RHS rates => already incorp 1,2,3,4 so mult by 5 only

Question 10

CMIR12

Answer 10

• 2 claim inception rates
  
  • ia(x,d)= ca(x,d)/ Lx
  • ib(x,d)= cb(x,d)/Lx
• ia(x,d) consider claims incept in [x,x+1]
• ib(x,d) consider sickness incept in [x,x+1]
• Both applied to all lives and not just premium paying

Question 11

Summary of claim rates

Answer 11

Initial claim inception
* prob claim incepts[x,x+1]
* sick in [x-d,x+1-d]
* surv prob is healthy @ x-d

Central claim inception
* exp # inceptions [x,x+1] / avg # lives alive
* Surv prob is prob alive @ x+1/2

Central claim inception 2
* exp # sick inception [x,x+1] => claims [x+d,x+1+d] / avg # alive
* surv prob is prob alive @ x+1/2

Question 12

Inception rates for disability

Answer 12

• (al)x: # healthy lives aged x
• (ad)^d x+t: # healthy lives dying
• (ad)^i x+t: # healthy lives becoming sick
• (ar)x+t: # sick lives becoming healthy

(al)x+t+1= (al)x+t - (ad)d x+t - (ad)i x+t + (ar)x+t
* lives can re enter and be double counted

Question 13

Complete integrals from notes