Notesale: Turn your study into money

Document Preview

Extracts from the notes are below, to see the PDF you'll receive please use the links above

---

Multiple lives
Insurance and premiums
Joint and last survivor insurances
1
...

𝐴𝑥𝑦 = 1 − 𝛿𝑎𝑥𝑦
𝑑𝐴𝑥𝑦
𝑃𝑥𝑦 =
1 − 𝐴𝑥𝑦
𝑡 𝑝𝑥𝑦 = 𝑡 𝑝𝑥 𝑡 𝑝𝑦
𝑡 𝑝𝑥𝑦 = 𝑡 𝑝𝑥 + 𝑡 𝑝𝑦 − 𝑡 𝑝𝑥𝑦
2
...

𝐴𝑦 + 𝐴𝑥 = 𝐴𝑥𝑦 + 𝐴𝑥𝑦
3
...
And also, a last survivor insurance should have an
expected present value that is lower than an insurance on either life
...

1
4
...
This symbol means to treat it as a
/𝑥𝑦\

single status, not a combination of two statuses
...
𝐴 1 : 10 means a 10-year term insurance paying upon the first death of (x)
/𝑥𝑦\

and (y)
...
𝐴 1 : 10 means a pay on the last death of (x) and (y) if they both died within 10
𝑥𝑦

years
...
Calculate joint pure endowment
In order to calculate the expected present value of an endowment insurance or a
term insurance, a joint-life pure endowment may be needed to calculate
...
Two ways:
1
...
The probability of survival is calculated as a quotient of the single life
table’s 𝑙𝑥 ’s
...
Multiply the product of two single-life pure endowments, one for each life, by
1
...

6
...


- For an insurance on the joint-life status, we need to calculate
00
01
02
(𝜇𝑥+𝑡:𝑦+𝑡
+ 𝜇𝑥+𝑡:𝑦+𝑡
)𝑑𝑡
...

- For an insurance on the last survivor status, we need to calculate
01 13
02 23
𝜇𝑥+𝑡 + 𝑡𝑝𝑥𝑦
𝜇𝑦+𝑡 )𝑑𝑡
...

7
...
Calculate the EPV of the insurance in state 1
...

𝑏1 = 𝐸𝑃𝑉
b
...
𝑏2 = 𝐸𝑃𝑉

c
...
Like multiple-decrement
insurances with constant forces of decrement:

𝑏1 𝜇01 +𝑏2 𝜇02
𝜇01 +𝜇02 +𝛿

d
...
Treat EPV of the annuity in state 1 or state 2 as the
benefit amount of an insurance on the join-life status that pays upon
transition to the state
...

Contingent insurances
- An insurance on (x) if (x) dies first plus an insurance on (y) if (y) dies first is the
same as an insurance on the joint status: 𝐴1𝑥𝑦 + 𝐴𝑥𝑦1 = 𝐴𝑥𝑦
- For insurances for second deaths and the last survivor status: 𝐴2𝑥𝑦 + 𝐴𝑥𝑦2 = 𝐴𝑥𝑦
- All the A’s in either equality may be barred, and the equalities work for term
1

1

insurances as well: 𝑛𝐴𝑥𝑦 + 𝑛 𝐴𝑥𝑦 = 𝑛𝐴𝑥𝑦
2

2

- 𝑛𝐴𝑥𝑦 is not the same as 𝐴𝑥𝑦: 𝑛
𝑛
- 𝐴1𝑥𝑦 + 𝐴2𝑥𝑦 = 𝐴𝑥
- If the payments made on: 1) if (x) dies second, nothing gets paid
...
Therefore:
𝐴1𝑥𝑦 − 𝐴𝑥𝑦2 = 𝐴𝑥 − 𝐴𝑥𝑦 = 𝐴𝑥𝑦 − 𝐴𝑦



---