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/
1/15
c
2008.
Miguel A. Arcones. All rights reserved.
Manual for SOA Exam MLC.
Chapter 3. Life tables.
Section 3.3. Continuous computations.
Continuous computations
Although, a life table does not show values of `x for non integers
numbers, we will assume that `x is known for each x ≥ 0. In the
next section, we will discuss how to estimate `x for non integers.
Knowing `x , for each x ≥ 0, we can get
`x
,
`0
d
`0
µ(x) = − (log(`x )) = − x ,
dx
`x
Z ∞
`x
◦
e0 =
dx,
`
Z0 ∞ 0
`x+t
◦
ex =
dt.
`x
0
s(x) =
2/15
c
2008.
Miguel A. Arcones. All rights reserved.
Manual for SOA Exam MLC.
Chapter 3. Life tables.
Section 3.3. Continuous computations.
Definition 1
Tx is the expected number of years lived beyond age x by the
cohort group with l0 members.
TheoremR1
(i) Tx =
(ii)
∞
0 `x+t
◦
dt and e x = E [T (x)] =
2
E [(T (x)) ] =
2
Tx
`x .
R∞
x
Ty dy
.
`x
Notice that the expected number of years lived beyond age x by an
◦
individual alive at age x is e x . The expected number of individuals
◦
alive at age x is `x . Hence, Tx = `x e x .
3/15
c
2008.
Miguel A. Arcones. All rights reserved.
Manual for SOA Exam MLC.
Chapter 3. Life tables.
Section 3.3. Continuous computations.
Proof.
(i) We have that
Tx = `0 E [(X − x)I (X > x)] = `0 P{X > x}E [X − x|X > x]
Z ∞
Z ∞
=`x E [T (x)] = `x
`x+t dt.
t px dt =
0
(ii) Using that Tx =
Z
0
∞
2
0
R∞
`x+t dt =
Z
`t dt, we get that
Z
∞Z t
`t dt dy = 2
x
y
∞
Z
(t − x)`t dt = 2
=2
x
∞Z ∞
Z
Ty dy = 2
x
R∞
x
`t dy dt
x
x
∞
u`x+u du.
0
So,
2
R∞
x
Ty dy
=
`x
Z
∞
2u · u px du = E [(T (x))2 ].
0
4/15
c
2008.
Miguel A. Arcones. All rights reserved.
Manual for SOA Exam MLC.
Chapter 3. Life tables.
Section 3.3. Continuous computations.
Example 1
2
◦
Suppose `x = `0 1 − ωx 2 , for 0 ≤ x ≤ ω. Find e x , E [(T (x))2 ]
and Var(T (x)) using Theorem 1.
5/15
c
2008.
Miguel A. Arcones. All rights reserved.
Manual for SOA Exam MLC.
Chapter 3. Life tables.
Section 3.3. Continuous computations.
Example 1
2
◦
Suppose `x = `0 1 − ωx 2 , for 0 ≤ x ≤ ω. Find e x , E [(T (x))2 ]
and Var(T (x)) using Theorem 1.
Solution:
We have that
Z ∞
Z
Tx =
`x+t dt =
ω−x
(x + t)2
`0 1 −
dt
ω2
0
0
2
2
3ω (ω − x) − ω 3 + x 3
3ω t − (x + t)3 ω−x
= `0
=
=`0
3ω 2
3ω 2
0
3
2ω − 3ω 2 x + x 3
=`0
,0 ≤ x ≤ ω
3ω 2
Hence,
◦
ex =
2ω 3 − 3ω 2 x + x 3
2ω 2 − ωx − x 2
(ω − x)(2ω + x)
Tx
=
=
=
.
`x
3(ω 2 − x 2 )
3(ω + x)
3(ω + x)
6/15
c
2008.
Miguel A. Arcones. All rights reserved.
Manual for SOA Exam MLC.
Chapter 3. Life tables.
Section 3.3. Continuous computations.
Example 1
2
◦
Suppose `x = `0 1 − ωx 2 , for 0 ≤ x ≤ ω. Find e x , E [(T (x))2 ]
and Var(T (x)) using Theorem 1.
Solution:
We also have that
Z ∞
Z ω 3
2ω − 3ω 2 y + y 3
Ty dy =
`0
dy
3ω 2
x
x
3
8ω y − 6ω 2 y 2 + y 4 ω
=
12ω 2
x
8ω 4 − 6ω 4 + ω 4 − 8ω 3 x + 6ω 2 x 2 − x 4
12ω 2
3ω 4 − 8ω 3 x + 6ω 2 x 2 − x 4
=
.
12ω 2
=
7/15
c
2008.
Miguel A. Arcones. All rights reserved.
Manual for SOA Exam MLC.
Chapter 3. Life tables.
Section 3.3. Continuous computations.
Example 1
2
◦
Suppose `x = `0 1 − ωx 2 , for 0 ≤ x ≤ ω. Find e x , E [(T (x))2 ]
and Var(T (x)) using Theorem 1.
Solution:
R∞
Ty dy
3ω 4 − 8ω 3 x + 6ω 2 x 2 − x 4
=
`x
6(ω 2 − x 2 )
3ω 3 − 5ω 2 x + ωx 2 + x 3
(ω − x)2 (3ω + x)
=
=
,
6(ω + x)
6(ω + x)
(ω − x)(2ω + x) 2
(ω − x)2 (3ω + x)
Var(T (x)) =
−
6(ω + x)
3(ω + x)
2
(ω − x)
2
=
3(ω
+
x)(3ω
+
x)
−
2(2ω
+
x)
18(ω + x)2
(ω − x)2
=
ω 2 + 4ωx + x 2 .
2
18(ω + x)
2
E [(T (x)) ] =
2
x
8/15
c
2008.
Miguel A. Arcones. All rights reserved.
Manual for SOA Exam MLC.
Chapter 3. Life tables.
Section 3.3. Continuous computations.
Definition 2
n Lx
is the expected number of years lived between age x and
age x + n by the `x survivors at age x.
Theorem 2
◦
Z
n Lx = Tx − Tx+n = `x e x:n| =
n
`x+t dt.
0
9/15
c
2008.
Miguel A. Arcones. All rights reserved.
Manual for SOA Exam MLC.
Chapter 3. Life tables.
Section 3.3. Continuous computations.
Definition 2
n Lx
is the expected number of years lived between age x and
age x + n by the `x survivors at age x.
Theorem 2
◦
Z
n Lx = Tx − Tx+n = `x e x:n| =
n
`x+t dt.
0
Proof: (i) Since the deceased at age x do not live between age x
and age x + n, n Lx is the expected number of years lived between
age x and age x + n by the initial `0 lives. So,
n Lx
= Tx − Tx+n .
10/15
c
2008.
Miguel A. Arcones. All rights reserved.
Manual for SOA Exam MLC.
Chapter 3. Life tables.
Section 3.3. Continuous computations.
Definition 2
n Lx
is the expected number of years lived between age x and
age x + n by the `x survivors at age x.
Theorem 2
◦
Z
n Lx = Tx − Tx+n = `x e x:n| =
n
`x+t dt.
0
Proof: (i) Since the deceased at age x do not live between age x
and age x + n, n Lx is the expected number of years lived between
age x and age x + n by the initial `0 lives. So,
n Lx
= 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,
n Lx
◦
= `x e x:n| .
11/15
c
2008.
Miguel A. Arcones. All rights reserved.
Manual for SOA Exam MLC.
Chapter 3. Life tables.
Section 3.3. Continuous computations.
Definition 2
n Lx
is the expected number of years lived between age x and
age x + n by the `x survivors at age x.
Theorem 2
◦
Z
n Lx = Tx − Tx+n = `x e x:n| =
n
`x+t dt.
0
Proof: (i) Since the deceased at age x do not live between age x
and age x + n, n Lx is the expected number of years lived between
age x and age x + n by the initial `0 lives. So,
n Lx
= 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,
n Lx
◦
(iii) `x e x:n| = `x
Rn
0 t px
dt =
◦
= `x e x:n| .
Rn
0
c
2008.
Miguel A. Arcones. All rights reserved.
`x+t dt.
Manual for SOA Exam MLC.
12/15
Chapter 3. Life tables.
Section 3.3. Continuous computations.
R1
We will abbreviate Lx = 1 Lx , i.e. Lx = 0 `x+t dt is the expected
number of years lived between age x and age x + 1 by the `x
survivors at age x. Previous equation display implies that
Z 1
Lx =
`x+t dt = Tx − Tx+1 .
0
13/15
c
2008.
Miguel A. Arcones. All rights reserved.
Manual for SOA Exam MLC.
Chapter 3. Life tables.
Section 3.3. Continuous computations.
Corollary 1
(i) Tx =
◦
(ii) e x =
◦
P∞
k=x Lk .
P∞
k=x Lk
.
`x
P
(iii) e x:n| =
x+n−1
k=x
`x
Lk
.
14/15
c
2008.
Miguel A. Arcones. All rights reserved.
Manual for SOA Exam MLC.
Chapter 3. Life tables.
Section 3.3. Continuous computations.
Corollary 1
(i) Tx =
◦
(ii) e x =
P∞
k=x Lk .
P∞
k=x Lk
.
`x
P
x+n−1
◦
L
(iii) e x:n| = k=x`x k .
Proof:
(i)
Z ∞
Z
Tx =
`x+t dt =
0
=
∞
X
∞
`t dt =
x
∞ Z
X
k=x
k+1
`t dt =
k
∞ Z
X
k=x
1
`k+t dt
0
Lk .
k=x
◦
(ii) e x =
(iii)
Rn
◦
ex =
0
Tx
`x
=
P∞
k=x
`x
`x+t dt
=
`x
Lk
.
R x+n
x
`t dt
`x
Px+n−1 R k+1
=
c
2008.
Miguel A. Arcones. All rights reserved.
k=x
k
`t dt
`x
Manual for SOA Exam MLC.
Px+n−1
=
k=x
`x
Lk
.
15/15