Notesale: Turn your study into money

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Gross Premiums & Expense Premiums
Prewords
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2
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3
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4
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5
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6
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7
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Gross future loss
•

•
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𝑔𝑟𝑜𝑠𝑠 𝑓𝑢𝑡𝑢𝑟𝑒 𝑙𝑜𝑠𝑠 = 𝑝𝑟𝑒𝑠𝑒𝑛𝑡 𝑣𝑎𝑙𝑢𝑒 𝑎𝑡 𝑖𝑠𝑠𝑢𝑒 𝑜𝑓 𝑏𝑒𝑛𝑒𝑓𝑖𝑡𝑠 + 𝑒𝑥𝑝𝑒𝑛𝑠𝑒𝑠 − 𝑝𝑟𝑒𝑠𝑒𝑛𝑡 𝑣𝑙𝑎𝑢𝑒 𝑎𝑡 𝑖𝑠𝑠𝑢𝑒 𝑜𝑓 𝑔𝑟𝑜𝑠𝑠 𝑝𝑟𝑒𝑚𝑖𝑢𝑚𝑠

Under the equivalence principle: 𝐸𝑃𝑉 𝑜𝑓 𝑏𝑒𝑛𝑒𝑓𝑖𝑡𝑠 𝑎𝑛𝑑 𝑒𝑥𝑝𝑒𝑛𝑠𝑒𝑠 = 𝐸𝑃𝑉 𝑜𝑓 𝑝𝑟𝑒𝑚𝑖𝑢𝑚𝑠
Example:
For a fully discrete whole life insurance of 1000 on (45), premiums of 22 are payable annually
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𝑣 = 0
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Initial expenses are 50% of first year premium plus 100
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Renewal expenses are 5% of renewal premiums plus 4, and are paid at the beginning of
each year
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2
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9631
𝑎̈ 31 =
= 17
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04
𝑔
31
0𝐿 = 1000(0
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45(22) + 96 + [0
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94742) = 84
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An insurance for the face amount + settlement expenses
2
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An annuity-due for the coverage period of the renewal per-policy and per-1000 expense + the
excess of first year over renewal expenses
• Makeham’s Law:
𝜇𝑥 = 𝐴 + 𝐵𝑐 𝑥 𝑐 > 1, 𝐵 > 0, 𝐴 ≥ −𝐵
𝑡 𝑝𝑥

=𝑒

𝐵𝑐 𝑥
[− 𝑙𝑛𝑐 (𝑐 𝑡 −1)] −𝐴𝑡

𝑒

Expense premium: (on an equivalence basis)
• The expense premium is the excess of the gross premium over the net premium
𝐸𝑃𝑉 𝑜𝑓 𝑒𝑥𝑝𝑒𝑛𝑠𝑒 𝑝𝑟𝑒𝑚𝑖𝑢𝑚 = 𝐸𝑃𝑉 𝑜𝑓 𝑒𝑥𝑝𝑒𝑛𝑠𝑒
- Example:
For a fully discrete 20-payment whole life policy on (45) with face amount 1000, you are given
I
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II
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05
III
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00
Renewal
5%
0
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Expenses are payable at the beginning of the year
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𝐴45 = 0
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8162, 𝑎̈ 45: 20 = 12
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3004
151
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7172
20𝑝45 =
12
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3004 − 11
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5832
Method2: 𝑃𝑒 =

14
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05(12
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4]+0
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8162+4
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9391

= 2

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