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Markov Chain Applications – Part 1 (Feb 14)
Disability Insurance
1
...
Premiums : Level premiums, waived when insured is disabled; group disability
insurance provided by an employer may cost less because of less anti-selection and nonpayment of premium risk
...
Periods: There is a waiting period (aka elimination period)
...
Benefit period (the maximum
length of time over which benefits are paid) is selected by policyholder
...
Disabilities are separate if a sufficient interval of time separates them
...
Level of benefits: Benefits are 50%-70% of salary to account for tax-free nature and to
encourage return to work
...
Continuous annuities – continuous sojourn annuity as an annuity that pays to a life in state i
only while in state i
...
EPV of a continuous sojourn annuity paying 1 per year for a maximum of n years to
𝑛

𝑖𝑖

someone currently in state i is 𝑎𝑥: 𝑛 = ∫0 𝑣 𝑡 𝑡 𝑝𝑥𝑖𝑖 𝑑𝑡, this annuity ends when state i is
exited and does not resume upon reentry to state i
...
If the elimination period is length w, the EPV of payments of 1 per year past the
𝑖𝑖

𝑖𝑖

elimination period using 𝑎𝑥: 𝑛 − 𝑎𝑥: 𝑤
c
...
Integrate an interest factor to time t, times the density
01
function of becoming disabled at time t ( 𝑡𝑝𝑥00 𝜇𝑥+𝑡
), times an appropriate continuous
sojourn annuity beginning at time t
...
If the maximum benefit period is k, then EPV of the insurance is
𝑛−(𝑘+𝑤)

𝐸𝑃𝑉 = ∫
0

11

11

01
𝑣 𝑡 𝑡 𝑝𝑥00 𝜇𝑥+𝑡
(𝑎𝑥+𝑡: 𝑘+𝑤 − 𝑎𝑥+𝑡: 𝑤 ) 𝑑𝑡
𝑛−𝑤

+∫
𝑛−(𝑘+𝑤)

11

11

01
𝑣 𝑡 𝑡 𝑝𝑥00 𝜇𝑥+𝑡
(𝑎𝑥+𝑡: 𝑛−(𝑥+𝑡) − 𝑎𝑥+𝑡: 𝑤 ) 𝑑𝑡

e
...
When calculate the premium, assume that it is payable only in state 0, not payable in
state 1 even when no disability benefit is being paid due to the elimination period or
maximum benefit period
...
Features (US & Canada)
a
...

b
...

LTC may have a limited benefit period, or provide lifetime benefits
...
Benefits periods separated by
separation period count as separate periods with separate waiting periods and
maximum benefit periods
...
Benefit level and premium level: benefit may be fixed amount per day or
reimbursement for nursing home/in-home expenses up to a specified maximum
...
Two LTC model
Active
0

Impaired
1

Dead
2

Active ≥ 5
ADLs
0

Cognitive
Impairment
3

Impaired 4
ADLs
1

Severly Impaired
≤3 ADLs
2

Dead
4



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