Manual for SOA Exam MLC.

Chapter 3. Life tables.
Manual for SOA Exam MLC.
Chapter 3. Life tables.
Section 3.3. Continuous computations.
c○2008. Miguel A. Arcones. All rights reserved.
Extract from:
”Arcones’ Manual for SOA Exam MLC. Fall 2009 Edition”,
available at http://www.actexmadriver.com/
c○2008. Miguel A. Arcones. All rights reserved. Manual for SOA Exam MLC.
1/15

Continuous computations
Chapter 3. Life tables. Section 3.3. Continuous computations.
Although, a life table does not show values of ℓx for non integers
numbers, we will assume that ℓx is known for each x ≥ 0. In the
next section, we will discuss how to estimate ℓx for non integers.
Knowing ℓx, for each x ≥ 0, we can get
s(x) = ℓx
,
ℓ0
µ(x) = − d
dx (log(ℓx)) = − ℓ′ x
,
� ∞
◦ ℓx
e0 =
0 ℓ0
� ∞
◦ ℓx+t
ex =
0 ℓx
dx,
dt.
c○2008. Miguel A. Arcones. All rights reserved. Manual for SOA Exam MLC.
ℓx
2/15

Definition 1
Chapter 3. Life tables. Section 3.3. Continuous computations.
Tx is the expected number of years lived beyond age x by the
cohort group with l0 members.
Theorem 1
(i) Tx = � ∞
(ii)
0 ℓx+t dt and ◦ ex = E[T (x)] = Tx
ℓx .
E[(T (x)) 2 ] = 2 � ∞
x Ty dy
Notice that the expected number of years lived beyond age x by an
individual alive at age x is ◦ ex. The expected number of individuals
◦
alive at age x is ℓx. Hence, Tx = ℓxex.
c○2008. Miguel A. Arcones. All rights reserved. Manual for SOA Exam MLC.
ℓx
.
3/15

Proof.
(i) We have that
Chapter 3. Life tables. Section 3.3. Continuous computations.
Tx = ℓ0E[(X − x)I (X > x)] = ℓ0P{X > x}E[X − x|X > x]
=ℓxE[T (x)] = ℓx
� ∞
0
tpx dt =
� ∞
ℓx+t dt.
0
(ii) Using that Tx = � ∞
0 ℓx+t dt = � ∞
x ℓt dt, we get that
So,
2
=2
� ∞
x
� ∞
x
Ty dy = 2
� ∞ � ∞
x
(t − x)ℓt dt = 2
2 � ∞
x Ty dy
ℓx
=
� ∞
0
ℓt dt dy = 2
y
� ∞
uℓx+u du.
0
� ∞ � t
2u · upx du = E[(T (x)) 2 ].
c○2008. Miguel A. Arcones. All rights reserved. Manual for SOA Exam MLC.
x
x
ℓt dy dt
4/15

Example 1 �
Suppose ℓx = ℓ0
Chapter 3. Life tables. Section 3.3. Continuous computations.
1 − x2
ω 2
and Var(T (x)) using Theorem 1.
�
, for 0 ≤ x ≤ ω. Find ◦ ex, E[(T (x)) 2 ]
c○2008. Miguel A. Arcones. All rights reserved. Manual for SOA Exam MLC.
5/15

Example 1 �
Suppose ℓx = ℓ0
Chapter 3. Life tables. Section 3.3. Continuous computations.
1 − x2
ω 2
and Var(T (x)) using Theorem 1.
Solution:
We have that
� ∞ � ω−x
Tx = ℓx+t dt =
0
�
, for 0 ≤ x ≤ ω. Find ◦ ex, E[(T (x)) 2 ]
ℓ0
0
ω−x
�
1 −
�
(x + t)2
dt
�
3ω2t − (x + t) 3
=ℓ0
3ω2 � � �
��� 3ω2 (ω − x) − ω3 + x 3
= ℓ0
0
3ω2 �
=
� �
2ω3 − 3ω2x + x 3
=ℓ0
, 0 ≤ x ≤ ω
Hence,
◦
ex = Tx
ℓx
3ω 2
= 2ω3 − 3ω 2 x + x 3
3(ω 2 − x 2 )
ω 2
= 2ω2 − ωx − x 2
3(ω + x)
c○2008. Miguel A. Arcones. All rights reserved. Manual for SOA Exam MLC.
= (ω − x)(2ω + x)
.
3(ω + x)
6/15

Example 1 �
Suppose ℓx = ℓ0
Chapter 3. Life tables. Section 3.3. Continuous computations.
1 − x2
ω 2
and Var(T (x)) using Theorem 1.
�
, for 0 ≤ x ≤ ω. Find ◦ ex, E[(T (x)) 2 ]
Solution:
We also have that
� ∞ � ω �
2ω3 − 3ω2y + y 3
Ty dy = ℓ0
x
x
3ω2 �
dy
�
8ω3y − 6ω2y 2 + y 4
=
12ω2 � �
��� ω
x
= 8ω4 − 6ω4 + ω4 − 8ω3x + 6ω2x 2 − x 4
12ω2 = 3ω4 − 8ω3x + 6ω2x 2 − x 4
12ω2 .
c○2008. Miguel A. Arcones. All rights reserved. Manual for SOA Exam MLC.
7/15

Example 1 �
Suppose ℓx = ℓ0
Chapter 3. Life tables. Section 3.3. Continuous computations.
1 − x2
ω 2
and Var(T (x)) using Theorem 1.
Solution:
E[(T (x)) 2 ] = 2 � ∞
x Ty dy
�
, for 0 ≤ x ≤ ω. Find ◦ ex, E[(T (x)) 2 ]
ℓx
= 3ω4 − 8ω 3 x + 6ω 2 x 2 − x 4
6(ω 2 − x 2 )
= 3ω3 − 5ω2x + ωx 2 + x 3
=
6(ω + x)
(ω − x)2 (3ω + x)
,
6(ω + x)
Var(T (x)) = (ω − x)2 �
(3ω + x) (ω − x)(2ω + x)
−
6(ω + x)
3(ω + x)
= (ω − x)2
18(ω + x) 2
� 2
3(ω + x)(3ω + x) − 2(2ω + x) �
= (ω − x)2
18(ω + x) 2
� 2 2
ω + 4ωx + x � .
c○2008. Miguel A. Arcones. All rights reserved. Manual for SOA Exam MLC.
� 2
8/15

Definition 2
Chapter 3. Life tables. Section 3.3. Continuous computations.
nLx is the expected number of years lived between age x and
age x + n by the ℓx survivors at age x.
Theorem 2
◦
nLx = Tx − Tx+n = ℓxe
x:n| =
� n
ℓx+t dt.
0
c○2008. Miguel A. Arcones. All rights reserved. Manual for SOA Exam MLC.
9/15

Definition 2
Chapter 3. Life tables. Section 3.3. Continuous computations.
nLx is the expected number of years lived between age x and
age x + n by the ℓx survivors at age x.
Theorem 2
◦
nLx = Tx − Tx+n = ℓxe
x:n| =
� n
ℓx+t dt.
0
Proof: (i) Since the deceased at age x do not live between age x
and age x + n, nLx is the expected number of years lived between
age x and age x + n by the initial ℓ0 lives. So,
nLx = Tx − Tx+n.
c○2008. Miguel A. Arcones. All rights reserved. Manual for SOA Exam MLC.
10/15

Definition 2
Chapter 3. Life tables. Section 3.3. Continuous computations.
nLx is the expected number of years lived between age x and
age x + n by the ℓx survivors at age x.
Theorem 2
◦
nLx = Tx − Tx+n = ℓxe
x:n| =
� n
ℓx+t dt.
0
Proof: (i) Since the deceased at age x do not live between age x
and age x + n, nLx is the expected number of years lived between
age x and age x + n by the initial ℓ0 lives. So,
nLx = Tx − Tx+n.
(ii) Since ◦ e x:n| is the expected number of years lived between age x
and age x + n by a live aged x,
◦
nLx = ℓxe
x:n|.
c○2008. Miguel A. Arcones. All rights reserved. Manual for SOA Exam MLC.
11/15

Definition 2
Chapter 3. Life tables. Section 3.3. Continuous computations.
nLx is the expected number of years lived between age x and
age x + n by the ℓx survivors at age x.
Theorem 2
◦
nLx = Tx − Tx+n = ℓxe
x:n| =
� n
ℓx+t dt.
0
Proof: (i) Since the deceased at age x do not live between age x
and age x + n, nLx is the expected number of years lived between
age x and age x + n by the initial ℓ0 lives. So,
nLx = Tx − Tx+n.
(ii) Since ◦ e x:n| is the expected number of years lived between age x
and age x + n by a live aged x,
◦
nLx = ℓxe
x:n|.
◦ � n
(iii) ℓxe
x:n| = ℓx 0 tpx dt = � n
0 ℓx+t dt.
c○2008. Miguel A. Arcones. All rights reserved. Manual for SOA Exam MLC.
12/15

Chapter 3. Life tables. Section 3.3. Continuous computations.
We will abbreviate Lx = 1Lx, i.e. Lx = � 1
0 ℓx+t dt is the expected
number of years lived between age x and age x + 1 by the ℓx
survivors at age x. Previous equation display implies that
Lx =
� 1
0
ℓx+t dt = Tx − Tx+1.
c○2008. Miguel A. Arcones. All rights reserved. Manual for SOA Exam MLC.
13/15

Corollary 1
(i) Tx = �∞ (ii) ◦ ex =
(iii) ◦ e x:n| =
k=x Lk.
P∞ k=x Lk
. ℓx Px+n−1 k=x
Lk
ℓx
Chapter 3. Life tables. Section 3.3. Continuous computations.
.
c○2008. Miguel A. Arcones. All rights reserved. Manual for SOA Exam MLC.
14/15

Corollary 1
(i) Tx = �∞ (ii) ◦ ex =
(iii) ◦ ex:n| =
Proof:
(i)
=
Tx =
k=x Lk.
P∞ k=x Lk
. ℓx Px+n−1 k=x
Lk
ℓx
� ∞
0
∞�
Lk.
k=x
(ii) ◦ ex = Tx
ℓx =
(iii)
◦
ex =
� n
0 ℓx+t dt
ℓx
Chapter 3. Life tables. Section 3.3. Continuous computations.
.
ℓx+t dt =
P ∞
k=x Lk
ℓx
=
� ∞
.
x
� x+n
x
ℓt dt
ℓx
ℓt dt =
=
∞�
k=x
� k+1
k
� x+n−1
k=x
ℓt dt =
� k+1
k
ℓt dt
c○2008. Miguel A. Arcones. All rights reserved. Manual for SOA Exam MLC.
ℓx
∞�
� 1
ℓk+t dt
k=x
0
=
� x+n−1
k=x
ℓx
Lk
.
15/15