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Class Notes
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Canada
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University of Waterloo
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Actuarial Science
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ACTSC 232
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Jun Cai
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Lecture

Description

Summary of Lecture Notes { ACTSC 232, Winter 2010
Part 4 { Bene▯t Premiums
▯ The loss random variable L for an insurance or annuity is de▯ned as
L = The PV of bene▯ts ▯ The PV of premiums.
▯ The expectation of the loss random variable is
E[L] = The APV of bene▯ts ▯ The APV of premiums.
▯ Equivalence Principle (EP): Set premiums such that E[L] = 0 or
The APV of bene▯ts = The APV of premiums.
▯ Percentile Principle (PP): Set premiums such that PrfL > 0g = ▯.
Note that we assume the equivalence principle throughout this courses unless otherwise
stated.
4.1 Fully continuous bene▯t premiums { In an insurance, the bene▯ts of the insurance form
a continuous life insurance while the premiums of the insurance form a continuous life
annuity.
(a) A fully continuous whole life insurance of 1 on (x) with an annual premium rate of
P:
L = v Tx▯ P ▯ ;
Tx
A▯x ▯ Ax 1 ▯ ▯▯ax
P = = = ;
▯x 1 ▯ Ax ▯x
▯!2 ▯! 2h i
V ar[L] = 1 + P V ar(v ) = 1 + P 2A ▯ (A ) 2
▯ ▯ x x
2Ax▯ (A x 2
= 2 :
(▯▯x)
(b) A fully continuous h-Payment whole life insurance of 1 on (x) with an annual pre-
mium rate of P:
L = v x ▯ P ▯T ^h;
x
A▯x
P▯ = :
▯x:h
1 (c) A fully continuous n-year term insurance of 1 on (x) with an annual premium rate
of P:
8
< v x ▯ P ▯ ; T ▯ n
L = Tx x
: 0 ▯ P ▯n; T x n
A▯x:
P▯ = :
▯ x:n
(d) A fully continuous n-year endowment insurance of 1 on (x) with an annual premium
rate of P:
Tx^n ▯
L = v ▯ P ▯Tx^n;
▯ ▯
▯ Ax:n ▯ Ax:n 1 ▯ ▯▯x:n
P = ▯ = 1 ▯ A = ▯ ;
x:n ! x:n x:n !
P 2 P 2h i
V ar[L] = 1 + V ar(vTx^n) = 1 + 2A▯x:n (A▯x:n2
▯ ▯
2 2
Ax:n▯ (A▯x:n
= 2 :
(▯▯x:n)
(e) A fully continuous n-year pure endowment of 1 on (x) with an annual premium rate
of P:
8
< 0 ▯ P ▯ ; Tx▯ n
L = Tx
: v ▯ P a▯n; T x n
A x:n nE x
P = = :
▯x:n ▯x:n
(f) A fully continuous n-year deferred whole life annuity of 1 on (x) with an annual
premium rate of P:
8
< 0 ▯ P ▯ ; T ▯ n
L = Tx x
: v a▯ ▯ P a▯n; Tx> n
Tx▯n
nj▯x nE x▯x+n ▯x▯ a▯x:n
P▯ = = = :
▯x:n ▯x:n ▯x:n
(g) A fully continuous h-Payment, n-year endowment insurance of 1 on (x)(h < n) with
an annual premium rate of P:
Tx^n ▯
L = v ▯ P ▯ x ^h
▯
▯ A x:n
P = ▯ :
x:h
4.2 Fully discrete bene▯t premiums { In an insurance, the bene▯ts of the insurance form a
discrete life insurance while the premiums of the insurance form a discrete life annuity-
due.
2 (a) A fully discrete whole life insurance of 1 on (x) with an annual premium of P:
K x1
L = v ▯ P Kx+1;
A dA 1 ▯ d
P = x = x = x;
x 1 ▯ Ax x
▯ ▯ 2 ▯ ▯ 2
P K x1 P h2 2i
V ar[L] = 1 + V ar(v ) = 1 + Ax▯ (A x
d d
2A x (A )x 2
= :
(dx)2
(b) A fully discrete h-Payment whole life insurance of 1 on (x) with an annual premium
of P:
L = v Kx+1 ▯ P (K +1)^h
x
Ax
P = :
x:h
(c) A fully discrete n-year term insurance of 1 on (x) with an annual premium of P:
8
< vKx+1 ▯ P Kx+1; K x n ▯ 1
L =
: 0 ▯ P n; Kx> n ▯ 1
1
Ax:n
P = :
x:n
(d) A fully discrete n-year endowment insurance of 1 on (x) with an annual premium of
P:
(Kx+1)^n
L = v ▯ P (Kx+1)^n;
A x:n dA xn 1 ▯ daxn

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