Notesale: Turn your study into money

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Multiple lives
Contingent probabilities (recall)
Notation:
1
𝑡 𝑞𝑥𝑦 : the probability that (x) dies first by time t before (y)
- To express the probability as an integral, integrate the probability of staying in 0 to
time s followed by transfer to state 2:
𝑡
1
𝑡 𝑞𝑥𝑦

=∫

00 02
𝑠 𝑝𝑥𝑦 𝜇𝑥+𝑠:𝑦+𝑠 𝑑𝑠

0

2
𝑡 𝑞𝑥𝑦 :

the probability that (x) dies second by time t
- To express the probability as an integral, use a double integral for the two
transitions:
𝑡
2
𝑡 𝑞𝑥𝑦

=∫
0

𝑡−𝑠
00 01
𝑠𝑝𝑥𝑦 𝜇𝑥+𝑠:𝑦+𝑠

∫

11
𝑢𝑝𝑥+𝑠

13
𝜇𝑥+𝑠+𝑢
𝑑𝑢 𝑑𝑠

0

Annuities
- Recall: 𝑎𝑥 + 𝑎𝑦 = 𝑎𝑥𝑦 + 𝑎𝑥𝑦
- 3 ways to calculate expected present values of annuities
1
...

The first technique is calculating annuities for each life, then cancel or augment the
payments for the joint life status when both are alive
...
Three-annuity technique: calculate 2 single life annuities and 1 joint life annuity
...
An annuity payable when only the first life is alive
...

b
...
This may require
calculating an annuity payable when the second life is alive and subtracting
from it an annuity payable when both lives are alive
...
An annuity payable when both lives are alive
...
Reversionary annuity technique: calculate a full annuity and cancel out the part that
is not paid
...
A reversionary annuity is an annuity that makes regular payments to one
status after another status has failed
...
IAN symbol: 𝑎𝑥|𝑦 , for the expected present value of the annuity
c
...
If an annuity is paid to (𝑧2 ) after (𝑧1 ) expires, where (𝑧1 ) and (𝑧2 ) can be any
combination of lives and certain statuses, then 𝑎̈ 𝑧1 |𝑧2 = 𝑎̈ 𝑧2 − 𝑎̈ 𝑧1 :𝑧2
e
...
For example, 𝑎𝑥: 20 |𝑦 would pay 1 per year
to (y) only after the later of 20 years and the death of (x)
...
The common shock model allows for both lives dying at the same time
...
For Markov chain, the probability of the joint status failing by time t is 𝑡𝑝𝑥𝑦
=
01
02
03
𝑡 𝑝𝑥𝑦 + 𝑡 𝑝𝑥𝑦 + 𝑡 𝑝𝑥𝑦
...

3
...
𝜇 03 is constant
...
𝜇𝑥+𝑡
= 𝜇𝑥+𝑡:𝑦+𝑡
+ 𝜇 03
23
01
c

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