Summary of Lecture Notes – ACTSC 232, Winter 2010
Part 4 – Benefit Premiums
•
The loss random variable
L
for an insurance or annuity is defined as
L
= The PV of benefits

The PV of premiums.
•
The expectation of the loss random variable is
E
[
L
] = The APV of benefits

The APV of premiums.
•
Equivalence Principle
(EP): Set premiums such that
E
[
L
] = 0 or
The APV of benefits
= The APV of premiums.
•
Percentile Principle
(PP): Set premiums such that Pr
{
L >
0
}
=
α
.
Note that we assume the equivalence principle throughout this courses unless otherwise
stated.
4.1
Fully continuous benefit premiums – In an insurance, the benefits 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
T
x

¯
P
¯
a
T
x
,
¯
P
=
¯
A
x
¯
a
x
=
δ
¯
A
x
1

¯
A
x
=
1

δ
¯
a
x
¯
a
x
,
V ar
[
L
]
=
1 +
¯
P
δ
!
2
V ar
(
v
T
) =
1 +
¯
P
δ
!
2
h
2
¯
A
x

(
¯
A
x
)
2
i
=
2
¯
A
x

(
¯
A
x
)
2
(
δ
¯
a
x
)
2
.
(b) A fully continuous
h
Payment whole life insurance of 1 on (
x
) with an annual pre
mium rate of
¯
P
:
L
=
v
T
x

¯
P
¯
a
T
x
∧
h
,
¯
P
=
¯
A
x
¯
a
x
:
h
.
1
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(c) A fully continuous
n
year term insurance of 1 on (
x
) with an annual premium rate
of
¯
P
:
L
=
v
T
x

¯
P
¯
a
T
x
,
T
x
≤
n
0

¯
P
¯
a
n
,
T
x
> n
¯
P
=
¯
A
1
x
:
n
¯
a
x
:
n
.
(d) A fully continuous
n
year endowment insurance of 1 on (
x
) with an annual premium
rate of
¯
P
:
L
=
v
T
x
∧
n

¯
P
¯
a
T
x
∧
n
,
¯
P
=
¯
A
x
:
n
¯
a
x
:
n
=
δ
¯
A
x
:
n
1

¯
A
x
:
n
=
1

δ
¯
a
x
:
n
¯
a
x
:
n
,
V ar
[
L
]
=
1 +
¯
P
δ
!
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 Summer '08
 MATTHEWTILL
 Annuity, ax, Endowment policy

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