B. Survival Models and Life Tables
RANDOM VARIABLES
LIFE TABLES
Tx ∼ future lifetime of (x)
lx − expected number of survivors at (x)
TEMPORARY LIFE EXPECTANCY
Z n
◦
ex:n = E[Wx ] =
t px dt
0
Kx ∼ number of completed future years by
(x) prior to death.
n dx
− expected number of deaths between
ages x and x + n
Kx = bTx c
ex:n = E[Kx ∧ n] =
E[Wx2 ] = 2
k px
Out of ω births, one dies every year until they
are all dead.
Future lifetime is uniformly distributed from 0
to ω − x.
k=1
n
Z
n
X
DE MOIVRE’S LAW
`x = k (ω − x)
t t px dt
0
Wx ∼ future lifetime or n years whichever is
less.
Wx = Tx ∧ n
FORCE OF MORTALITY
Intuitively, µx+t dt is the probability that a life
age x + t will die in the next instant.
CUMULATIVE DISTRIBUTION
Probability of (x) dying before age x + t.
Fx (t) = Pr(Tx ≤ t)
d
t qx
◦
d `
− dt
x+t
`x+t
∗
n px
d
p
dt t x
= (n px )k
= t px µx+t
= −t px µx+t
Probability of a (x) attaining age x + t.
ACTUARIAL NOTATION
PROBABILITY DENSITY FUNCTION
◦
◦
= Sx (t)
t+u px
t qx
= t px · u px+t
n px
Age doesn’t matter.
ex =
Future lifetime is distributed exponentially
with mean 1/µ.
t qx
= 1 − t px
t|u qx
t| qx
t|u qx
= Sx (t) − Sx (t + u)
= t px − t+u px
= t+u qx − t qx
= t px · u qx+t
◦
for all x
= e−nµ = (px )n
1
µ
1
µ2
Recursion
ex = px (1 + ex+1 )
◦
ex:n = ex (1 − n px )
ex:n = ex (1 − n px )
Discrete Constant Force − Kx ∼ geometric
fKx (k) = pk · q
ex = E[Kx ] =
COMPLETE EXPECTATION OF LIFE
Average future lifetime of (x).
Z ∞
◦
ex = E[Tx ] =
t px dt
Z ∞0
E[Tx2 ] = 2
t t px dt
Var[Kx ] =
p
q2
=
m
ω−x
ω−x−n
ω−x
ω−x
2
half way to omega
= n px (n) + n qx n
2
◦
ex:n
a(x) =
1
2
Discrete DML
=
µx =
`x+t = `x e−µt
k=1
For integral x, ex is the expected number of
future birthdays.
=
1
ω−x
MODIFIED DML
Sx (t) = e−µt
◦
= µx
n
ω−x
=
◦
k| qx
Var[Tx ] =
− probability that (x) will survive t years
and die within the following u years.
is the PDF for Kx
n px
fx (t) = Fx0 (t) = −Sx0 (t) = t px µx+t
CURTATE EXPECTATION OF LIFE
∞
X
ex = E[Kx ] =
k px
n qx
ω−x−t
ω−x
1
ω−x
qx =
CONSTANT FORCE
ex =
− probability that (x) dies within t year
t qx = Fx (t)
◦
n|m qx
Intuitively, the probability that (x) dies at age
x + t.
t px − probability that (x) will attain age
x + t.
t px
Sx (t) =
ex:m+n = ex:m +m px ex+m:n
◦
◦
ex ≈ px 1+ ex+1 + qx 12
µx+t = µ
Sx (t) = Pr(Tx > t) = 1 − Fx (t)
◦
ex = ex:n +n px ex+n
If µ∗x+t = k µx+t :
d
q
dt t x
SURVIVAL DISTRIBUTION
fx (t)
Sx (t)
=
= dtp
t x
Rn
n px = exp − 0 µx+t dt
µx+t =
◦
1
ω−x
µx =
Recursion
p
q
a
ω−x
`x = (ω − x)a
a
Sx (t) = ω−x−t
ω−x
a
ω−x−n
n px =
ω−x
◦
ex =
ω−x
a+1
FRACTIONAL AGES
Uniform Distribution of Deaths (UDD)
- Use linear interpolation.
◦
ex = ex +
1
2
◦
ex:1 = px + qx ( 12 )
0
c
2013
The Infinite Actuary
GOMPERTZ LAW
µx = Bcx
c > 1, B > 0
i
B
cx (ct − 1)
t px = exp − ln c
h
MAKEHAM LAW
µx = A + Bcx
c > 1, B > 0, A ≥ −B
h i
B
cx (ct − 1)
t px = exp(−At) · exp − ln c
c
2013
The Infinite Actuary