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Survival Analysis and Actuarial Life Tables
Linus Waelti ‘07
Swarthmore College, Department of Mathematics & Statistics
Actuarial life tables
World of Uncertainty
How well can
you predict my
schedule?
Pr  t  T  t  t | T  t 
lim
The future is full of uncertainty. In the course of our lives we
may encounter undesirable events that can impact us both
emotionally and financially. While extra care can be taken to
reduce the likelihood of some undesirable events such as
accidents attributed to carelessness, there are some events that
are simply inevitable. Death is a prime example. People have
little control over when they die (barring suicide) but an
untimely death presents a major risk to any family. Consider the
impact on a family’s economic wellbeing if the family loses its
sole breadwinner to an untimely death. To reduce the financial
impact of an untimely death and safeguard economic security,
families and individuals have the option to purchase life
insurance.
For many life insurances, the size and timing of premium
payments depend greatly on the time of death of the insured.
Unfortunately the insurance companies do not know precisely
when death will occur, but mathematics and statistics do offer
tools for making viable predictions.
Survival analysis is concerned with the estimation of survival
probabilities in a population. Typical questions one may
encounter in survival analysis are:
• What fraction of a population will survive past a certain time?
• Given survival to a certain age, what is the rate at which
members of a population will die?
• How do particular characteristics and circumstances increase
or decrease the odds of survival?
t 0
Life tables find applications in a multitude of fields including
engineering, biostatistics and demography. In life actuarial practices,
actuaries use life tables to build models for their insurance systems
designed to assist individuals facing uncertainty about the times of
their deaths.
Since the age we reach when we die is an uncertain figure we
call this age-at-death a random variable, denoted by X.
X has cumulative density function, c.d.f. (lifetime distribution
function) FX ( x) which tells us the probability that a newborn
dies by age x: FX ( x)  Pr  X  x 
t
qx  1 t px  Pr T ( x)  t 
The symbol t px can be interpreted as the probability a person of
age x will attain age x + t.
The symbol t qx can be interpreted as the probability a person of
age x will die within t years.
The survival function, s(x), is the complement of FX ( x) and is
defined as the probability that a newborn will attain age x:
In the special case of a newborn, we have T (0)  X and x p0  s( x)
s( x)  1 FX ( x)  Pr  X  x 
Properties of s(x)
• s(x) is non-increasing
• s(0) = 1
• s(x) tends to 0 as x approaches infinity
Note: It common practice to define a limiting age ω such that
s(x) = 0 for all x ≥ ω. For human lives, there have been so few
observations of age-at-death beyond 110 thus it is assumed that
there exists a limiting age that nobody survives beyond.
Force of Mortality
Probability Density Function (p.d.f.) and s(x)
The p.d.f. f X ( x) is the derivative of the c.d.f. so:
x

s( x)  1  FX ( x)  1   f (v)dv   f (u)du
0
x
Conditional Probability
The conditional probability that a newborn will die between the
ages x and z, given survival to age x, is
s ( x)  s ( z )
Pr( x  X  z | X  x) 
s ( x)
The probability of death within the next instant of time given
survival to age x is:
Fx ( x  x)  Fx ( x) f x ( x)x
Pr  x  X  x  x | X  x  

1  Fx ( x)
1  Fx ( x)
The force of mortality, µ(x), is the value of the conditional
p.d.f. of X at exact age x, given survival to age x:
d
 s ( x)

f X ( x)
FX ( x)
d
dx
 ( x) 


   ln s ( x) 
1  FX ( x)
s ( x)
s ( x)
dx
Finding the p.d.f. of T(x) and X
d
d
d
fT (t )  FT (t )  t qx  (1  t px )   t px
dt
dt
dt
d  s( x  t )  s( x  t )  s( x  t ) 
fT (t )   


 t px  ( x  t )



dt  s( x) 
s ( x)  s ( x  t ) 
So, for the special case of a newborn, f X ( x)  x p0  ( x  0)  s( x) ( x)
E[ z (t )]   z (t ) g (t )dt
0

 z (0)   z(t )[1  G(t )]dt
0
In the context of our life table, this theorem enables us to
express the complete-expectation-of-life in simpler terms.
The complete-expectation-of-life in actuarial notation is
denoted by e x which is equivalent to E[T(x)].
By the definition of the expected value we have:
0

  t t px  ( x  t )dt
0
Applying the theorem using z (t )  t and G(t )  1  t px yields:
ex  

0 t
Random Survivorship Group
X = x + T(x)
To make probability statements about T(x), we have the
following actuarial notation:
s( x  t )
t px  Pr  X  x  t | X  x  0   Pr T ( x)  t  
s ( x)

ex  E[T ( x)]   t fT (t )dt
Consider a person of age x. This person’s future lifetime is
described by a continuous random variable T (x). Since the
future lifetime of this person assumes he has already survived
to age x, we have T ( x)  X  x | X  x
x
If T is a continuous random variable with c.d.f. G(t) such that
G(0) = 0 and p.d.f. G′(t) = g(t), and z(t) is such that:
• it is a nonnegative, monotonic, differentiable function
• E[z(T)] exists, then

Time-until-Death, Force of Mortality
and some whacky actuarial notation
0
The Survival Function, s(x)
t
In actuarial science, a life table or mortality table is a table of
statistics that provides information on the average probability of
survival or death at different ages, the remaining life expectancy for
people of different ages and the proportion of the original birth
cohort still alive. In terms of the random variable X (age-at-death)
the life table summarizes the distribution of X. Life table are usually
constructed separately for men and for women and other
characteristics can also be used to distinguish different risks, such as
smoking-status, occupation and socio-economic class.
A Theorem of Expected Values
Life tables seem to describe the mortality experience of a group
of l0 newborns. It is common practice in actuarial life tables to
set l0  100,000
Does this mean that the lives of 100,000 newborns are observed
until all are dead? Yes, in the case of cohort life table, but no in
the more readily available period life tables. Period life tables
do not represent the mortality experience of an actual birth
cohort. Instead, the table presents what would happen to a
hypothetical cohort if it experienced throughout its entire life
the mortality conditions of particular period of time. For
example: the life table above assumes a hypothetical cohort
subject throughout its lifetime to the age-specific death rates
prevailing for the actual population in 2003. The 100,000
members therefore constitutes a random survivorship group.
The expected number of survivors to age x from the l0 newborns
is related to the survival function by: lx  l0 s( x)
The function, lx  ( x) , is interpreted as the expected density of
deaths in the age interval (x, x + dx).
The Curve of Deaths
px dt , a much simpler expression to solve.
For broader applications, this theorem is useful in that it gives
us a way to find the expected value of some continuous random
variable, T, when a well-defined p.d.f. g(t) is not readily
available.
References and Acknowledgements
Arias, E. United States Life Tables, 2003. National Vital Statistics Reports,
Vol. 54, No. 14. April 19, 2006.
Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A. and Nesbitt, C.J.
1986. Actuarial Mathematics, 1st Edition, edited by M.M. Treloar. The
Society of Actuaries, Itasca, IL.
Klein, J.P. Survival Distributions and their Characteristics. A contribution
to the Encyclopedia of
Biostatistics.<www.biostat.mcw.edu/tech/tr022.ps> Accessed 2006,
Oct 7.
Shand, K. Survival Distribution and Life Tables. Warren Center for
Actuarial Studies and Research.
<http://home.cc.umanitoba.ca/~kshand/ACT_3130_Summary_Notes.p
df>Accessed 2006, Oct 7.
4,000
Webshots. Grim Reaper.
<travel.webshots.com/photo/1190518017059728898MsbLlW>
Accessed 2006, Nov 20.
3,500
3,000
2,500
Wikipedia. Life Tables. <en.wikipedia.org/wiki/Life_tables> Accessed
2006, Oct 7.
lx  ( x) 2,000
1,500
Wikipedia. Survival Analysis. <en.wikipedia.org/wiki/Survival_analysis>
Accessed 2006, Oct 7.
1,000
500
0
0
10
20
30
40
50
-500
Age
60
70
80
90
100
 I am grateful to my fellow MATH 97 classmates and especially
Walter Stromquist for valuable comments made in the course of
preparing this poster.