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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
+ +