Summary of Lecture Notes – ACTSC 232, Winter 2010
Part 3 – Life Annuities
An annuity is a series of payments paid at equal intervals or continuously over a fixed term.
•
PV of the annuitydue of 1 for
n
years is
¨
a
n
= 1 +
v
+
· · ·
+
v
n

1
=
n

1
X
k
=0
v
k
=
1

v
n
d
.
•
PV of the annuityimmediate of 1 for
n
years is
a
n
=
v
+
v
2
+
· · ·
+
v
n
=
n
X
k
=1
v
k
=
1

v
n
i
.
•
PV of the continuous annuity with annual rate of 1 for
t
years is
¯
a
t
=
Z
t
0
v
s
ds
=
1

v
t
δ
.
A life annuity on (
x
) is a series of payments paid at equal intervals or continuously over a
random term when the life (
x
) is alive.
3.1 Discrete life annuities on
(
x
)
:
(a) A discrete whole life annuitydue of 1 on (
x
)
pays 1 at the beginning of each year
until (
x
) dies.
The present value of the payments in the annuity is
¨
a
K
x
+1
=
1

v
K
x
+1
d
.
The APV of the annuity is denoted by ¨
a
x
and
¨
a
x
=
E
[¨
a
K
x
+1
] =
∞
X
k
=0
v
k
k
p
x
=
1

A
x
d
.
The variance of the present value is
V ar
[¨
a
K
x
+1
] =
V ar
[
v
K
x
+1
]
d
2
=
2
A
x

(
A
x
)
2
d
2
.
1
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(b) A discrete whole life annuityimmediate of 1 on (
x
)
pays 1 at the end of each year
until (
x
) dies.
The present value of the payments in the annuity is
a
K
x
.
The APV of the annuity is denoted by
a
x
and
a
x
=
E
[
a
K
x
] =
∞
X
k
=1
v
k
k
p
x
,
where
a
0
= 0.
Note that
¨
a
x
= 1 +
a
x
.
(c) A discrete
n
year temporary life annuitydue of 1 on (
x
)
pays 1 at the beginning of
each year as long as (
x
) survives during the
n
year term.
The present value of the payments in the annuity is
¨
a
(
K
x
+1)
∧
n
=
¨
a
K
x
+1
,
0
≤
K
x
≤
n

1
¨
a
n
,
K
x
≥
n
=
1

v
(
K
x
+1)
∧
n
d
,
where
a
∧
b
= min
{
a, b
}
.
The APV of the annuity is denoted by ¨
a
x
:
n
and
¨
a
x
:
n
=
n

1
X
k
=0
v
k
k
p
x
=
1

A
x
:
n
d
.
The variance of the present value is
V ar
[¨
a
(
K
x
+1)
∧
n
] =
V ar
[
v
(
K
x
+1)
∧
n
]
d
2
=
2
A
x
:
n

(
A
x
:
n
)
2
d
2
.
Recursive formulas for ¨
a
x
and ¨
a
x
:
n
:
¨
a
x
=
1 +
vp
x
¨
a
x
+1
,
¨
a
x
=
¨
a
x
:
n
+
n
E
x
¨
a
x
+
n
,
¨
a
x
:
n
=
1 +
vp
x
¨
a
x
+1:
n

1
.
The actuarial accumulated value at time
n
of the
n
year temporary life annuitydue
of 1 on (
x
) is denoted by ¨
s
x
:
n
and
¨
s
x
:
n
=
¨
a
x
:
n
n
E
x
.
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 Summer '08
 MATTHEWTILL
 ax, Whole life insurance, APV, kx

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