life actuarial

1. Reserving Life Insurance Policy Alterations Complementary Insurance Universal Life Unit Linked
Life Contingencies II
Atuariado Vida II
Gracinda R. Guerreiro
Mestrado em Matemática Atuarial
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2. Reserving Life Insurance Policy Alterations Complementary Insurance Universal Life Unit Linked
Program
Part 1. Advanced Life Contingencies
1 Reserving Life Insurance
2 Policy Alterations
3 Complementary Insurance
4 Universal Life and Unit Linked Models
Part 2. Construction of Mortality Tables
5 Estimation of Lifetime Distributions
6 Forecasting Mortality
7 Graduation
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3. Reserving Life Insurance Policy Alterations Complementary Insurance Universal Life Unit Linked
– Part 1 –
Advanced Life Contingencies
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4. Reserving Life Insurance Policy Alterations Complementary Insurance Universal Life Unit Linked
Reserving Life Insurance
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5. Reserving Life Insurance Policy Alterations Complementary Insurance Universal Life Unit Linked
Introduction
• At the beginning of the contract (beginning of year 1), the
present value of the insurer’s liabilities is equal to the value of
the Insured’s liabilities (Actuarial Equivalence Principle).
• With this condition of equivalence the pure premiums (single
or annual) are determined.
• In the end of year k (k = 1, 2, . . .), the present value of the
Insurer’s liabilities is different from the present value of the
Insured’s liabilities.
• This equivalence would only remain unchanged over time if
the payments being made by the policyholder correspond
exactly to the risk being supported by the Insurer.
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6. Reserving Life Insurance Policy Alterations Complementary Insurance Universal Life Unit Linked
Introduction
0,00 €
50,00 €
100,00 €
150,00 €
200,00 €
250,00 €
300,00 €
350,00 €
400,00 €
450,00 €
500,00 €
30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49
Temporary Life Insurance
Level Premium Risk Premium
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7. Reserving Life Insurance Policy Alterations Complementary Insurance Universal Life Unit Linked
Introduction
• Whether the contract is paid at a single premium or at leveled
premiums, there is a mismatch between the cost of the
guarantees given by the Insurer and the amount paid by the
policyholder in each year of the contract.
• The anticipation of payments by the insured induces the need
for the insurer to set up a provision (or reserve) that will,
later on the contract, cover payments by the policyholder,
which are insufficient to cover the risk to which the insurer is
subject.
The difference between the future liabilities of the insurer and the
policyholder (immediately prior to payment of the premium) is
designated by Net Premium Reserve.
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8. Reserving Life Insurance Policy Alterations Complementary Insurance Universal Life Unit Linked
Introduction
Notation:
In the following, let:
– x: age of the insured
– n: term of the contract
– p: term of premium payments
– kVx - Net Premium Reserve in the end of year k of the
contract (immediately prior to the payment of the premium)
– An−k
x+k - present value of future liabilities of the insurer
– Pn−k
x+k - present value of future liabilities of the policyholder
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9. Reserving Life Insurance Policy Alterations Complementary Insurance Universal Life Unit Linked
Prospective Method
Prospective Method
kVx = An−k
x+k − Pn−k
x+k
Observations:
• 0Vx = 0
• nVx = 0
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10. Reserving Life Insurance Policy Alterations Complementary Insurance Universal Life Unit Linked
Prospective Method
Exercises
Solve Exercise 1.
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