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Chapter 7

ReserveReserve

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7.1z A benefit reserve is the difference between

the expected value of future benefits and theexpected value of future premiums.

( ) ( )tV E PV of future benefits E PV of future premiums=

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7.2 Fully Continuous Benefit

Reservesz Prospective Loss at time t

zThe expected value of the prospective loss at

duration t, given that the insured survives atleast until t, or , is called the

reserveat duration t for the policy, and wecan calculate it as followed:

( )( )( )

T x tx T x tt L v P A a

=

( | ( ) )tE L T x t>

( ) ( )x x t xt x tV A A P A a+ +=

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Example1z For a fully continuous whole life insurance,

the force of mortality is constant and equal to, the force of interest is.

Calculate the t-th reserve for the policy

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Example2z For a fully continuous 20-year endowment of

(x):1. The force of mortality is 0.02

2.=0.063. The benefit premium is determined by the

equivalence principle.

Calculate the 10th terminal reserve.

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Fully Continuous Benefit Reserves:

Age at Issue x, Duration t, Unit benefit

Name Actuarial notation Prospective formula

Whole life insurance

n-year term insurance

n-year endowment

insurance

h-payment years,whole life insurance

( )xtV A

:( )x ntV A

( )

h

xtV A

( )x t x x tA P A a+ +

1 :( )x ntV A

1 1

: : :( )

0

x t n t x n x t n tA P A a t n

t n

+ +