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Title: multiple lives 1
Description: This note allows master's students to learn multiple lives. It is as straightforward as possible. This is part 1, there are part 2 and part 3.
Description: This note allows master's students to learn multiple lives. It is as straightforward as possible. This is part 1, there are part 2 and part 3.
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Multiple Lives
Markov chain model
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•
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01
23
02
If the two lives are independent: 𝜇𝑥+𝑡:𝑦+𝑡
= 𝜇𝑦+𝑡
, 𝜇𝑥+𝑡:𝑦+𝑡
= 𝜇13
𝑥+𝑡
Notation:
Actuarial notation
Markov chain notation
00
𝑡 𝑝𝑥𝑦
𝑡𝑝𝑥𝑦
0∙
01
02
03
𝑡 𝑝𝑥𝑦 or 𝑡𝑝𝑥𝑦 + 𝑡 𝑝𝑥𝑦 + 𝑡 𝑝𝑥𝑦
0∙
𝑝 : the probability of transitioning from state 0 to
𝑡 𝑞𝑥𝑦
any other state
...
With the assumption of independence, 𝑡 𝑝𝑥𝑦 = 𝑡 𝑝𝑥 𝑡𝑝𝑦 , but 𝑡𝑞𝑥𝑦 ≠ 𝑡𝑞𝑥 𝑡𝑞𝑦 , this generalises to any
number of lives
...
Under independence, define 𝜇𝑥𝑦 as the force of leaving state 0
...
If mortality is uniformly distributed, 𝜇𝑥 = 𝜔−𝑥
𝛼
4
...
𝜇𝑥+𝑡:𝑦+𝑡 = 𝜇𝑥+𝑡 + 𝜇𝑦+𝑡
6
...
Notation
𝑥𝑦 : the last survivor status
...
𝑞𝑥𝑦 : the probability that they both die in one year
...
If (x) dies first and (y) dies second, then 𝑡𝑝𝑥 = 𝑡𝑝𝑥𝑦 , 𝑡𝑝𝑦 = 𝑡𝑝𝑥𝑦
𝑡 𝑝𝑥𝑦 + 𝑡 𝑝𝑥𝑦 = 𝑡 𝑝𝑥 + 𝑡 𝑝𝑦
𝑡 𝑝𝑥𝑦 = 𝑡 𝑝𝑥 + 𝑡 𝑝𝑦 − 𝑡 𝑝𝑥𝑦
3
...
In the more general Markov chain situation, when the lives are not independent, calculating last
survivor probabilities is more complicated
...
When the two lives are independent, the force of mortality notation is 𝜇𝑥𝑦 (𝑡) not 𝜇𝑥+𝑡:𝑦+𝑡 , because
the force of mortality for the last survivor status is not independent of the starting ages xy
Title: multiple lives 1
Description: This note allows master's students to learn multiple lives. It is as straightforward as possible. This is part 1, there are part 2 and part 3.
Description: This note allows master's students to learn multiple lives. It is as straightforward as possible. This is part 1, there are part 2 and part 3.