Mathematics of Life Contingencies MATH
3281
Life annuities contracts
Edward Furman
Department of Mathematics and Statistics
York University
February 13, 2012
Edward Furman
Mathematics of Life Contingencies MATH 3281
1 / 23
Definition 0.1 (Life annuity.)
Life annuity is a series of payments made continuously or at
equal intervals while a given life survives.
Unlike in the context of life insurances, we have a series of
payments.
Discrete payments can be made not only at the end of the
year, but also at the beginning of the year.
Question.
Given a life status (u) what is the expected future lifetime? If
one dollar is payed at the beginning of every year while (u) is
alive, what would the random present value of the contract be?
What if the payment was made at the end of the year?
Recall that, e.g., K (u) : Ω → Nu ⊆ [0, ∞].
Edward Furman
Mathematics of Life Contingencies MATH 3281
2 / 23
Solution.
The expected future lifetime is E[K (u)] full years.
The random present value.. if the payments are made at the
beginning of each year is aK (u)+1 = 1 + v + · · · + v K (u)+1−1 .
Also, the random present value of the payments if they are
made at the end of each year is aK (u) = v + · · · + v K (u) .
Once again, recall that we find the price by taking expectation
operator (under the identity utility principle).
Net premium for an annuity contract
Fix a life status (u) (be careful with joint life statuses). The net
premium for a continuously payable annuity is
Z
Z
at t pu µ(u + t)dt = −
at d t pu := au .
E[aT (u) ] =
Ru
Ru
Edward Furman
Mathematics of Life Contingencies MATH 3281
3 / 23
Net premium for an annuity contract. cont.
The net premium for an annuity due payable annually is
X ..
X ..
..
..
ak +1 ∆k pu := au .
ak +1 k pu qu+k = −
E[aK (u)+1 ] =
k ∈Nu
k ∈Nu
Last but not least, the net premium for an annuity certain
payable annually is
X
X
ak ∆k pu := au .
ak k pu qu+k = −
E[aK (u) ] =
k ∈Nu
k ∈Nu
Example 0.1
Whole life annuity. Let (u) = (x). Then Ru = [0, ∞) and
Nu = {0, 1, . . .}. Thus
Edward Furman
Mathematics of Life Contingencies MATH 3281
4 / 23
Cont.
..
..
ax = E[aK (x)+1 ] =
∞
X
..
ak +1 k | qx .
k =0
Further
ax = E[aK (x) ] =
∞
X
ak k | qx .
k =0
And
ax = E[aT (x) ] =
Z
∞
0
at t px µ(x + t)dt.
Recall that.
..
an =
1 − vn
1 − Vn
1 − vn
, an =
, an =
, n ≥ 0.
d
i
δ
Edward Furman
Mathematics of Life Contingencies MATH 3281
5 / 23
Proposition 0.1
Fix a life status (u), and let i, d , δ be the effective and discount
interests and the force of interest, respectively. Then,
au =
1 − (1 + i)Au ..
1 − Au
1 − Au
, au =
, au =
.
δ
i
d
Proof.
We clearly have that
h
au = E aT (u)
i
"
1 − v T (u)
=E
δ
#
=
1 − Au
.
δ
..
The same can be done for au . For the last one
"
#
"
#
1 − v K (u)
1 − (1 + i)v K (u)+1
au = E
=E
,
i
i
as required.
Edward Furman
Mathematics of Life Contingencies MATH 3281
6 / 23
Proposition 0.2
Fix (u). The variance of the random present value can be given
by
2 A − (A )2
u
u
,
Var[aT (u) ] =
δ2
and
2 A − (A )2
..
u
u
,
Var[aK (u)+1 ] =
d2
as well as
(1 + i)2 2 Au − (Au )2
Var[aK (u) ] =
.
i2
Proof.
The proof is straightforward and similar to the proof of the
previous proposition. It is thus omitted.
Edward Furman
Mathematics of Life Contingencies MATH 3281
7 / 23
Proposition 0.3
Consider a whole life annuity due. Then
..
ax =
∞
X
v k k px =
∞
X
k Ex .
k =0
k =0
Proof.
We have that
..
ax
∞ X
k
∞
X
X
..
v i P[K (x) = k]
ak +1 k px qx+k =
=
=
k =0
∞
X
i=0
=
∞
X
k =0 i=0
∞
X
!
P[K (x) = k] v i =
k =i
P[K (x) > i − 1]v i =
∞
X
P[K (x) ≥ i]v i
i=0
∞
X
i px v
i
,
i=0
i=0
as required.
Edward Furman
Mathematics of Life Contingencies MATH 3281
8 / 23
Proposition 0.4
Let again (u) = (x). Then
..
..
ax = 1 + Ex ax+1 .
..
Here a∞ = 0.
Remark.
..
..
Note that a∞ 6= a∞ = 1/d .
Proof.
We have that
..
ax = 1 +
∞
X
k +1 px v
k +1
k =0
= 1 + vpx
∞
X
v k k px+1 ,
k =0
as required.
Edward Furman
Mathematics of Life Contingencies MATH 3281
9 / 23
Example 0.2 (n year term life annuity.)
Let Ru = [0, ∞) and Nu = {0, 1, . . . , ∞}. And let (u) = (x : n ).
In the discrete case, K (x : n ) is K (x) if K (x) < n and
K (x : n ) = n if K (x) ≥ n. The above is in fact the number of
payments for the discretely payable annuity due.
..
E[aK (x:n )+1 ] =
∞
n−1
X
X
..
..
..
an k | qx := ax:n .
ak +1 k | qx +
k =n
k =0
(Recall the endowment insurance case.) Recall, however that
the p.m.f. is

 k px · qx+k , k ≤ n − 2
p ,
k = n−1
k px:n · q(x:n )+k =
 n−1 x
0,
k ≥n
Thus
..
E[aK (x:n )+1 ] =
n−2
X
..
..
ak +1 k | qx + an n−1 px
Edward Furman
k =0
Mathematics of Life Contingencies MATH 3281
10 / 23
Example 0.2 (cont.)
Therefore
n−2
∞
X
X
..
..
ak +1 k | qx + an
..
E[aK (x:n )+1 ] =
k =0
k | qx
k =n−1
n−1
∞
X
..
.. X
ak +1 k | qx + an
k | qx
=
k =0
n−1
X
=
k =n
..
..
ak +1 k | qx + an n px
k =0
Also
E[aT (x:n ) ] =
Z
n
0
at t px µ(x + t)dt + an · n px .
Edward Furman
Mathematics of Life Contingencies MATH 3281
11 / 23
Proposition 0.5
For (u) = (x : n ), we have that
..
ax:n =
n−1
X
k Ex .
k =0
Proof.
..
ax:n
=
=
n−1 X
k
X
k =0 i=0
n−1 X
n−1
X
v i k | qx +
v i k | qx +
i=0 k =i
=
n−1
X
i=0
vi
n−1
X
i=0
n−1
X
v i n px
v i n px
i=0
n−1
X
k =i
k | qx + n px
!
=
n−1
X
i px v
i
,
i=0
as required.
Edward Furman
Mathematics of Life Contingencies MATH 3281
12 / 23
Proposition 0.6
For a general (u) (be careful with the joint life statuses), we
have that
..
..
..
au = −ab+1 b+1 pu + aa a pu +
b
X
k pu v
k
.
k =a
Proof.
Write the annuity as, for a and b being any in {0, 1, . . . , ∞}.
..
au = −
b
X
..
ak +1 ∆k pu .
k =a
..
..
au = − ak +1
b
b+1 X
..
−
k +1 pu ∆a
k pu
k +1
a
k =a
..
..
= −ab+2 b+1 pu + aa+1 a pu +
Edward Furman
b
X
!
k +1 pu v
k +1
.
=aContingencies MATH 3281
Mathematics ofkLife
13 / 23
Proof.
Further
..
..
..
au = −ab+2 b+1 pu + aa+1 a pu +
b+1
X
k pu v
k
k =a+1
..
..
= −ab+1 b+1 pu + aa a pu +
b
X
k pu v
k
k =a
Edward Furman
Mathematics of Life Contingencies MATH 3281
14 / 23
Example 0.3 (Deferred annuities.)
We are sometimes interested in an annuity that starts to pay
one dollar after n time units only. We thus have, e.g.,
∞
X
..
..
..
k
au − au:n := n au =
k pu v .
k =n
Also,
n
..
ax
=
=
∞ ∞
∞
X
X
..
..
.. X ..
−
a
a
q
=
ak +1 k qx − an
x
n
k qx
k
k +1
k =n
∞ X
k =n
..
..
ak +n+1 − an
k =0
k +n qx =
k =n
∞ n
X
v (1
k =0
− v k +1 )
d
∞
X
..
..
= vn
ak +1 n+k px · qx+n+k = n Ex · ax+n .
k +n
qx
k =0
Edward Furman
Mathematics of Life Contingencies MATH 3281
15 / 23
Example (cont.)
For the continuous case, we have that (make sure you can
prove that),
n |ax = n Ex ax+n ,
as well as
n |ax
=
Z
∞
t Ex dt.
n
Note that we can obtain, say the discretely payable deferred
whole life annuity as an expectation of the r.v.
( ..
..
aK (x)+1 − aK (x)+1 ≡ 0, K (x) < n
∗
..
..
K (x) :=
.
K (x) ≥ n
aK (x)+1 − an ,
Edward Furman
Mathematics of Life Contingencies MATH 3281
16 / 23
Example 0.4 (n year certain and then life annuity)
Consider an r.v.
∗
K (x) :=
(
..
an ,
K (x) < n
.
aK (x)+1 , K (x) ≥ n
..
Then taking expectation, we have that
..
ax:n
..
= an n qx +
∞
∞
X
X
..
..
..
ak +1 k | qx = an n qx −
ak +1 ∆k px
k =n
..
= an n qx −
..
k =n
ak +1
∞
∞ X
k +1
−
k +1 px v
k px
n
k =n
= an + v n n px +
..
∞
X
k +1 px v
k +1 ..
!
..
an + n| ax .
k =n
In a similar fashion ax:n = an + n| ax = an + n Ex ax+n .
Edward Furman
Mathematics of Life Contingencies MATH 3281
17 / 23
Proposition 0.7
The variance of the n year certain and whole life annuity is
equal to the variance of the n years deferred annuity.
Proof.
Note that if we denote by K ∗ (x) the random present value of
the deferred annuity and by k ∗∗ (x) the random present value of
the n year certain whole life annuity, then
k ∗∗ (x) = an + K ∗ (x).
..
Thus
Var[k ∗∗ (x)] = Var[an ] + Var[K ∗ (x)] + Cov[an , K ∗ (x)],
..
..
as required.
Edward Furman
Mathematics of Life Contingencies MATH 3281
18 / 23
Recall.
We know that
sn := (1 + i)n an ,
for n ≥ 0. The above is an accumulated present value of an
annuity at time n instead of time 0. Similarly, we have that
sx:n := (n Ex )−1 ax:n ,
that represents the actuarial accumulated value at the end of
the term of an n-year temporary life annuity of 1 per year
payable continuously while (x) survives. The benefit is
available at age x + n if x survives till then
Edward Furman
Mathematics of Life Contingencies MATH 3281
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Summary of discrete life annuities of 1 per annum payable at the
beginning of each year (due) or at the end of each year (immediate)
Edward Furman
Mathematics of Life Contingencies MATH 3281
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Useful relations between specific discrete annuities and insurances.
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Mathematics of Life Contingencies MATH 3281
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Summary of continuous life annuities of 1 per annum payable
continuously.
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Mathematics of Life Contingencies MATH 3281
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Typical distributions for the present value rv T .
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Mathematics of Life Contingencies MATH 3281
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