Survival Models and Life Tables
Long Term Actuarial Mathematics (I) (LTAM1)
Don Hong
Actuarial Science Program
Middle Tennessee State University
don.hong@mtsu.edu
©Spring 2022
Don Hong
LTAM2
Survival Models and Life Tables
Survival Models and Life Tables
1. Basic Probability Review
2. Cumulative, survival, and hazard rate functions
3. Future life time random variable Tx
4. Curtate Future Lifetime Random Variable Kx
5. Life Table
Don Hong
LTAM2
Survival Models and Life Tables
1.
2.
3.
4.
5.
Basic Probability Review
Cumulative, survival, and hazard rate functions
Future life time random variable Tx
Curtate Future Lifetime Random Variable Kx
Life Table
Probability Space and Random Variable
I Definition 1. Given a set ⌦, a probability P on ⌦ is a function defined in
the collection of all (subsets) events of ⌦ such that
(i) P(;) = 0.
(ii) P(⌦) = 1.
(iii) If An , n = 1,
2, · · · , are disjoint events, then
P
1
P([1
n=1 An ) =
n=1 P(An ).
⌦ is called the sample space.
I Definition 2. A random variable X is a function from the sample space
⌦ into R. We will abbreviate random variable into r.v.
I For any r.v. X , we usually need to find its mean value for center point
measurement and its variance or standard deviation for data distribution
spread information.
Don Hong
LTAM2
Survival Models and Life Tables
1.
2.
3.
4.
5.
Basic Probability Review
Cumulative, survival, and hazard rate functions
Future life time random variable Tx
Curtate Future Lifetime Random Variable Kx
Life Table
I Many insurance concepts depend on accurate estimation
of the life span of a person. It is of interest to study the
distribution of lives’ lifespan. The life span of a person (or
any alive entity) can be modeled as a positive (r.v.) random
variable. To model the lifespan of a life, we use
age-at-death random variable X . The probability P(X > x)
is called the survival function of the newborn.
I For inanimate objects, age-at-failure is the age of an object
at the end of termination.
Don Hong
LTAM2
Survival Models and Life Tables
1.
2.
3.
4.
5.
Basic Probability Review
Cumulative, survival, and hazard rate functions
Future life time random variable Tx
Curtate Future Lifetime Random Variable Kx
Life Table
2. Cumulative, survival, and hazard rate functions
I Definition 3. The cumulative distribution function of a r.v. X is
FX (x) = P(X  x), x 2 R.
If Let X denote a newborn’s age, then corresponding to this random
variable, the function S(x) = 1 FX (x) = P(X > x) represents a
survival probability, therefore, it is called the survival function.
I The survival function satisfies the following
(i) S(x) is non-increasing, i.e. for each x1  x2 , S(x1 ) S(x2 ).
(ii) S(x) is continuous from left, i.e. limh!0 S(x + h) = S(x).
(iii) For each x  0, S(x) = 1.
(iv) limx!1 S(x) = 0.
0
S (x)
I The hazard rate function hX (x) = 1 fXF(x)(x) = S(x)
, here we assume
X
S(x) is differentiable. It is also called the force of mortality in survival
analysis, and denoted it as µ(x).
Don Hong
LTAM2
Survival Models and Life Tables
1.
2.
3.
4.
5.
Basic Probability Review
Cumulative, survival, and hazard rate functions
Future life time random variable Tx
Curtate Future Lifetime Random Variable Kx
Life Table
I Relationship Formulas:
0 (x).
(1) FX (x) + s(x) = 1, fX (x) =
s
Rx
s0 (x)
(2) µ(x) = s(x) , s(x) = e 0 µ(s)ds .
I For any random variable X , its mean value E(X ) and
variance VR(X ) are defined as follows.
1
• E(X ) = 1 xf (x)dx
R1
2
• Var (X ) = E[(X µ) ] = 1 (x µ)2 f (x)dx =
E(X 2 ) [E(X )]2 .
• For new born’s
haveR
R 1 life time r.v. X , we
1
E(X ) =e̊0 = 0 s(x)dx and E(X 2 ) = 2 0 xs(x)dx.
Don Hong
LTAM2
Survival Models and Life Tables
1.
2.
3.
4.
5.
Basic Probability Review
Cumulative, survival, and hazard rate functions
Future life time random variable Tx
Curtate Future Lifetime Random Variable Kx
Life Table
Future life time random variable Tx
I We are interested in analyzing and describing the future lifetime of an
individual. Let (x) denote a life age x for x 0. Then the random
variable describing the future lifetime, or time until death, for (x) is
denoted by Tx , Tx 0.
I The CDF of Tx is the probability that (x) dies in t years:
Fx (t) = P[Tx  t] = P[x < X  x + t|X > x] =
S(x)
S(x + t)
.
S(x)
We also denote this probability by using the notation t qx .
I Then, the probability that (x) survives over t more years is
R x+t
S(x+t)
µ(s)ds
x
=e
t px = 1
t qx = S(x) . We can have t px = e
I For very small x,
S(x) S(x+ x)
S(x)
S 0 (x) x
S(x)
Rt
0
µ(x+s)ds
.
Pr [x < X  x + x|X > x] =
⇡
= µ(x) x.
Therefore, µ(x) x can be interpreted as the probability that a newborn
who has attained age x dies between x and x + x.
Don Hong
LTAM2
Survival Models and Life Tables
1.
2.
3.
4.
5.
I Relationship Formulas:
• t px +t qx = 1, t px = s(x+t)
, t qx = 1
s(x)
Basic Probability Review
Cumulative, survival, and hazard rate functions
Future life time random variable Tx
Curtate Future Lifetime Random Variable Kx
Life Table
t px
=
s(x) s(x+t)
.
s(x)
• The probability that (x) survives over t more years but cannot make
S(x+t+u)
over the next u years is t|u qx = S(x+t) S(x)
= t px · u qx+t .
• s+t px = s px · t px+s ,
• Denote 1 px = px and 1 qx = qx . For positive integer n,
n px = px · px+1 · · · px+n 1 , where px+k = 1 px+k .
P
Pn 1
• Denote k| qx = k|1 qx . Then, 1
q
=
1,
and
k=0 k| x
k=0
Don Hong
LTAM2
k | qx
= n qx .
1.
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4.
5.
Survival Models and Life Tables
Basic Probability Review
Cumulative, survival, and hazard rate functions
Future life time random variable Tx
Curtate Future Lifetime Random Variable Kx
Life Table
I The pdf of Tx is
d
fx (t) = Fx (t) =
dt
S 0 (x + t)
=
S(x)
S 0 (x + t) S(x + t)
·
= µ(x + t) · t px .
S(x + t)
S(x)
I The expected value of Tx , or the mean value of T (x), denoted by e̊x ,
also called theRcomplete life expectancy
of (x), is
R1
R1
1
e̊x = E[Tx ] = 0 t · fx (t)dt = 0 t · t px µ(x + t)dt = 0 t px dt.
(Ex. Show this by using Integrating by Parts)
R
R
R
I E[T 2 ] = 1 t 2 fT (t)dt = 1 t 2 · t px µ(x + t)dt = 1 2t · t px dt and thus,
0
2
Var [T ] = E[T ]
0
2
2
(E[T ]) = E(T )
Don Hong
e̊2x .
LTAM2
0
Survival Models and Life Tables
1.
2.
3.
4.
5.
Basic Probability Review
Cumulative, survival, and hazard rate functions
Future life time random variable Tx
Curtate Future Lifetime Random Variable Kx
Life Table
I Example-1. Suppose that the survival function of X is
s(x) = e x (x + 1), x 0. Find E[min(X , 10)].
I Solution: fX (x) = S 0 (x) = xe x . E[min(X , 10)] =
=
Z
1
0
min(x, 10)f (x)dx =
x2
= ( 2)e (
+ x + 1)|10
0
2
This value is also equal to e̊0:10| .
x
Z
10
0
xe
10e
x
x
xdx +
Z
1
10
10e
(x + 1)|1
10 = 2
x
xdx
12e
10
.
I In general, for Tx , we have the term expectation of life:
Rn
R1
Rn
E[min(Tx , n)] =e̊x:n| = 0 t ·t px µ(x +t)dt + n n·t px µ(x +t)dt = 0 t px dt.
I Relationship formula: e̊x:m+n| =e̊x:m| + m px e̊x+m:n| .
Don Hong
LTAM2
Survival Models and Life Tables
1.
2.
3.
4.
5.
Basic Probability Review
Cumulative, survival, and hazard rate functions
Future life time random variable Tx
Curtate Future Lifetime Random Variable Kx
Life Table
Some commonly used forms for the force of mortality
I It is sometimes convenient to describe the future lifetime random
variable by explicitly modeling the force of mortality. Some of the
parametric forms that have been used are:
I Gompertz: µ(x) = Bc x , 0 < B < 1, c > 0
I Makeham: µ(x) = A + Bc x , 0 < B < 1, c > 0
I de Moivre’s: µ(x) = ! 1 x , 0  x < !
I Exponential: µx = , for 0  x  1.
I Weibull: µx = kx n , for x > 0, k > 0, n > 1.
Don Hong
LTAM2
Survival Models and Life Tables
1.
2.
3.
4.
5.
Basic Probability Review
Cumulative, survival, and hazard rate functions
Future life time random variable Tx
Curtate Future Lifetime Random Variable Kx
Life Table
I Example 2. [SOA LTAM Spring 2019 Q3] You are given: A life table
uses a Makeham’s mortality model with parameters
A = 0.00022, B = 2.7 ⇥ 10 6 , c = 1.124. 10 p50 = 0.9803. Calculate
d
( q ) at t = 10.
dt t 50
I Solution. At t = 10,
60
d
(
q
)
=
f
(10)
=
p
µ(60)
=
(0.9803)
·
(A
+
B
·
c
) = 0.003158.
t
50
50
10
50
dt
Don Hong
LTAM2
Survival Models and Life Tables
1.
2.
3.
4.
5.
Basic Probability Review
Cumulative, survival, and hazard rate functions
Future life time random variable Tx
Curtate Future Lifetime Random Variable Kx
Life Table
Curtate Future Lifetime Random Variable
I We are often interested in the integral number of years lived in the
future by an individual. This discrete random variable is called the
curtate future lifetime and is denoted by Kx = bTx c.
I We can consider the pmf (probability mass function) of Kx :
Pr [Kx = k ] = Pr [k  Tx < k + 1] = k| qx = k px qx+k .
I We can find the mean value of Kx , also called the curtate expectation of
life, denoted by ex ,
E[Kx ] = ex =
1
X
k · k px qx+k =
1
X
k · k px
k =0
and the 2nd moment and the variance of Kx :
E[K 2 ] =
1
X
k =0
k 2 · k px qx+k = 2
k=1
Don Hong
LTAM2
1
X
k px .
k =1
ex , V [Kx ] = E[K 2 ]
(ex )2 .
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Survival Models and Life Tables
I Example 3. Suppose you are given:
(
0.04,
µ=
0.05,
Calculate e̊25:25| .
I Solution.
t p25 = e
Rt
0 µ(25+t)dt
=
(
e
e
0.04t
Basic Probability Review
Cumulative, survival, and hazard rate functions
Future life time random variable Tx
Curtate Future Lifetime Random Variable Kx
Life Table
x < 40
x 40.
0  t < 15,
.
15  t  25.
,
0.04(15)
· e 0.05(t 15) ,
R 25
R 15 0.04t
R 25 0.04(15)
e̊25:25| = 0 t p25 dt = 0 e
dt + 15 e
·e 0.05(t 15) dt ⇡ 15.60.
I Example 4. [SOA MLC S2016 Q2] You are given the survival function:
S0 (x) = (1
x 1/3
) , for 0  x  60.
60
Calculate 1000µ35 .
0
(x)
I Solution. µx = SS(x)
= 3(601 x) and thus, 1000µ35 = 13.3333.
I In general, the so-called Generalized De Moivre’s Law is expressed as
S0 (x) = (1 !x )↵ , for 0  x  !.
Don Hong
LTAM2
Survival Models and Life Tables
1.
2.
3.
4.
5.
Basic Probability Review
Cumulative, survival, and hazard rate functions
Future life time random variable Tx
Curtate Future Lifetime Random Variable Kx
Life Table
5. The Life Table
I A life table is a tabular presentation of the mortality evolution of a cohort
group of lives.
I Beginning with `0 number of lives (e.g. 100,000), also called the radix of
the life table, the (Expected) number of lives who are age x is
`x = `0 · S0 (x) = `0 · x p0 .
I The (Expected) number of deaths between ages x and x + 1 is denoted
as dx = `x `x+1 and the (Expected) number of deaths between ages x
and x + n is n dx = `x `x+n .
I Conditional on survival to age x, the probability of dying within n years
is n qx = n`dxx = (`x ``xx+n ) .
I Conditional on survival to age x, the probability of living to reach age
x + n is n px = 1 n qx = `x+n
.
`x
Don Hong
LTAM2
Survival Models and Life Tables
1.
2.
3.
4.
5.
Basic Probability Review
Cumulative, survival, and hazard rate functions
Future life time random variable Tx
Curtate Future Lifetime Random Variable Kx
Life Table
I Examples of Life Table:
• Illustrative Life Table (ILT)
• Standard-Ultimate-Life-Table (SULT)
• U.S. Life Table for the total population, 2004, Center for Disease Control and Prevention (CDC):
x
`x
dx
qx
px
e̊x
0
100,000
680
0.006799
0.993201
77.84
1
99,320
48
0.000483
0.999517
77.37
2
99,272
29
0.000297
0.999703
76.41
3
99,243
22
0.000224
0.999776
75.43
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
50
93,735
413
0.004404
0.995596
30.87
51
93,323
443
0.004750
0.995250
30.01
52
92,879
475
0.005113
0.994887
29.15
53
92,404
507
0.005488
0.994512
28.30
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
97
5,926
1,370
0.231201
0.768799
3.15
98
4,556
1,133
0.248600
0.751400
2.95
99
3,423
913
0.266786
0.733214
2.76
I For more life tables, you can browse CDC Life Tables
Don Hong
LTAM2
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2.
3.
4.
5.
Survival Models and Life Tables
Basic Probability Review
Cumulative, survival, and hazard rate functions
Future life time random variable Tx
Curtate Future Lifetime Random Variable Kx
Life Table
I From a life table, we can have the following relationship formulas:
P1
• `x = k =0 dx+k : the number of survivors at age x is equal to the
number of deaths in each year of age for all the remaining years.
Pn 1
• n dx = `x `x+n = k=0 dx+k : the number of deaths within n years is
equal to the number of deaths in each year of age for the next n years.
• the probability that (x) survives the next m years but dies the following
n years after that can be derived using:
m|n qx
= m px
n+m px
Don Hong
=
n dx+m
`x
LTAM2
=
`x+m
`x+n+m
`x
.
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2.
3.
4.
5.
Survival Models and Life Tables
Basic Probability Review
Cumulative, survival, and hazard rate functions
Future life time random variable Tx
Curtate Future Lifetime Random Variable Kx
Life Table
I From the relationship formula:
Rx
`x = `0 · S(x) = `0 · e 0 µ(s)ds , we can show that the force
of mortality can be expressed in terms of life table function
`0x
as: µx = `x and with a simple change of variable, it is
easy to see that µx+t =
1
`x+t
• It follows immediately that:
d `x+t
dt
=
1
t px
·
d
dt (t px )
=
t px
µx (t).
·
d t px
dt .
I the curtate expectation of life can be expressed now as
P1
P1 `x+k
E[Kx ] = ex = k =1 k px = k=1 `x .
I The n-year temporary curtate expectation of life is
Pn
Pn `x+k
ex:n| = k=1 k px = k=1 `x .
Don Hong
LTAM2
Survival Models and Life Tables
1.
2.
3.
4.
5.
Basic Probability Review
Cumulative, survival, and hazard rate functions
Future life time random variable Tx
Curtate Future Lifetime Random Variable Kx
Life Table
I Example 5. You are given the following extract from a life table:
x
`x
94
16,208
95
10,902
96
7,212
97
4,637
98
2,893
99
1,747
100
0
Calculate (i) 2| q95 . (ii) e95 . (iii) the variance of K95 , the curtate future lifetime of (95). (iv) e95:3| .
I Solution. (i) 2| q95 = `97 `98 = 4637 2893 = 0.15997065.
`95
10902
P1
P4
`95+k
7212+4637+2893+1747 ⇡ 1.5125.
(ii) e95 =
p
=
95
k
k =1
k =1 `95 =
10902
P4
2
2
2
2 d95+k
(iii) Var (K95 ) = E(K95
)
e95
⇡ 2.0984 since E(K95
)=
k
· `
= 4.386076.
k =1
95
P3
`95+k
(iv) e95:3| =
= 1.3522.
k=1 `
95
Don Hong
LTAM2
Survival Models and Life Tables
1.
2.
3.
4.
5.
Basic Probability Review
Cumulative, survival, and hazard rate functions
Future life time random variable Tx
Curtate Future Lifetime Random Variable Kx
Life Table
I Example-6. [2020F SOA LTAM Writing Problem Q2] Consider the
following expression for the force of mortality: µx+t = ↵e (x+t) .
t
(i) Show that for ↵ > 0, > 0, t px = exp[ ↵( e 1 )].
(ii) Derive an expression for t px when = 0.
Don Hong
LTAM2
Survival Models and Life Tables
1.
2.
3.
4.
5.
Basic Probability Review
Cumulative, survival, and hazard rate functions
Future life time random variable Tx
Curtate Future Lifetime Random Variable Kx
Life Table
I Example-7. [Written Q4. LTAM 2021F]
(a) Show that ex = ex:n| + x pn ex+n .
(b) You are given: (i) Mortality follows the Standard
Ultimate Life Table (SULT). (ii)e87 = 6.56. Show that
e90 = 5.2 to the nearest 0.1. Calculate the value to the
nearest 0.01.
(c) Let H = min(3, K87 ) denote the 3-year temporary
curtate future lifetime of (87). Calculate the standard
deviation of H.
Don Hong
LTAM2