Acturial maths

FM 2002 ACTUARIAL MATHEMATICS I 1

CHAPTER 1


Introduction
 Insurance policy: Two types of RISK
 life insurance: the variability in the claim
is only the time at which the claim is
made, since the amount of the claim is
specified by the policy.
 Other types of insurance (such as auto or
casualty): there is variability in both the
time and amount of the claim.


Why a Survival Model is Necessary
 Annuity base: to find present value, use discount
factor.

Discounting Process

 Life insurance:


Actuarial Present Value
 Present value
n
Pv
 Actuarial Present Value n
Pv P (E )

In life insurance: Random Event is


Survival Function
 The basic of a survival model is a
random lifetime variable and its
corresponding distribution.
 X: Life time of a newborn.
 X: Cts, X  [0,  ]
f X ( x)
 Probability density function
 Probability distribution F ( x )
X


Survival Distribution
x

F X ( x )  P ( X  x )  P (new born dies before age x )   f X ( s ) ds
0


Survival Function
 The survival function sX(x) is defined as the
probability that a newborn survives to age x.
Since this is the event that X>x we have

s X ( x )  P ( X  x )  1  P ( X  x )  1  FX ( x )


 Suppose for example that sX(75)=0.12,
so that FX (75)=0.88. MEANS?


2
 x 
If s ( x )   1   , for 0  x  100, find F X (75), f X (75)
 100 

2
 x 
F X ( x )= 1 - s ( x )  1   1  
 100 
2
 75 
F X (7 5)= 1 -  1   1  0 .2 5
2

 100 
F X (7 5)= 0 .9 3 7 5
d 2  x 
f X ( x )= FX ( x)  1  
dx 100  100 
2  75 
f X (7 5)= 1  
100  100 
f X (7 5)= 0 .0 0 5

DeMoivre Law
 Historical Model

x
s( x)  1  , for 0  x  

  is the limiting age by which all have died.
 X : uniform distribution on the interval (0,)


The Future Lifetime of Age (x)

 Life insurance is usually issued on a person who has
already attained a certain age x.
 Life age (x).
 Future life time of age (x): T(x)

age
x X
(today) (death)

x years of past life T years of future life


W e are given that X  x , so
T  X x
S ince T is a function of X , its density function f T ( x ) ( t )
and distribution function FT ( x ) ( t ) shoud be re lated to X .



 This gives the conditional probability that a newborn will die
between the ages x and x+t. OR
 age (x) dies before reaching to age x+t, OR
 age (x) dies within next t years.

t
p x  the probability of survival to age x  t give n survival to age x.
= P[T ( x )  t ]

t
q x  the probability of death before age x  t giv en survival to age x.
= P[T ( x )  t ]  1  t p x

Remarks
T he sym bol, t q x can be interpreted as the p robability that ( x ) w ill die w ithin t years;
that is t q x is the distribution function of T ( x ).

T he sym bol, t p x can be interpreted as the p robability that ( x) w ill survive another t years;
that is t p x is the survival function of T ( x).

W hen t  1 the prefix is om itted and one j ust w rites p x and q x respectively.
p x = P [T ( x )  1], the probability that ( x) survive s another year.
q x  P [T ( x )  1], the probability that ( x) w ill die w ithin next ye ar.


Probability Concept

t
qx  1 t px


Special symbol
 (x) will survive t years and die within the
following u years: i.e. (x) will die between ages
x+t and x+t+u.

t |u
q x  P [t  T ( x )  t  u ]  P [ x  t  X  x  t  u ]

t |u
qx  t u
qx  t qx  t
px  t u
px

t |u
qx  t
px u
q x t

Prove that
t |u
qx  t
p x u q xt


Compute tpx for the DeMoivre law of mortality.
Conclude that under the DeMoivre law T(x) has
the uniform distribution on the interval (0,ω-x)


Force of Mortality
 Consider s( x)  s( x  t)
t
q x  P[ x  X  x  t | X  x ] 
s(x)

 Now take t=Δx

FX ( x   x )  FX ( x )
x
q x  P[ x  X  x   x | X  x ] 
1  FX ( x )

 FX ( x   x )  FX ( x )   x  f X ( x)x
=  
 x   1  FX ( x )  1  FX ( x )

Force of Mortality
f X ( x)
 For each age x, it gives the value of th e conditional p.d.f.
1  FX ( x )
of X at exact age x, given survival to that a g e.

 Force of mortality f X ( x)
  ( x)
1  FX ( x )
f X ( x)x
x
qx    ( x) ( x)
1  FX ( x )

 This means the probability that x dies in the "next
instant" delta(x).


Force of Mortality
f X ( x)  s ( x )
 ( x)  
1  FX ( x ) s( x)

 In Actuarial Science µ(x) is called the force of
mortality. In reliability theory, the study of the
survival probabilities of manufactured part
system, µ(x) is called the failure rate or hazard
rate or, more fully, the hazard rate function.


What is the meaning of qx = 0.008 versus the
meaning of µ(x) = 0.008?


 Prove that  x 
s ( x )  exp     ( t ) dt 
 0 

ds( x)
 s ( x ) dx
 ( x)    
s( x) s(x)
x
ds( x)
 s( x)
    (t ) d (t )
0

x
 x 
ln ( s ( x ))     ( t ) d ( t )  s ( x )  ex p     ( t ) d ( t ) 
0  0 


Example
2
If  ( x )  for 0  x  100, find s ( x ), F X ( x ), f X ( x )
100  x
Let x  40, find FT ( x ) ( t ) and f T ( x ) ( t )


Properties of µ(x)
 t 
s ( x )  exp     ( t ) dt 
 0 

 ( x)  0 x



  ( x ) dx  
0


Show that density function of T(x) can be
written in the following form
fT ( x ) (t )  t p x  ( x  t )


If the force of mortality is constant then the life random
variable X has an exponential distribution. Further T(x) is
also exponentially distributed.


Find the force of mortality for DeMoivers’ law.


Complete Expectation of Life
 The expected value of T(x), (E(T(x)) is known as
the complete expectation of life at age x.
0
C omplete expectation life= e x  E ( T ( x))
 Prove 
0
e x  E (T ( x ))   t
p x dt
0



Show that E (T ( x ) )  2  t t p x d t
2

0

Remark:
Var (T ( x ))  E (T ( x ) )  ( E (T ( x )))
2 2


If X follows DeMoivres’ law, compute E(T(x))

 x
s( x)  , 0  x 

s( x  t)   ( x  t)
px   , 0  t  -x
 x
t
s( x)
x
0
 x
e x  E (T ( x ))   t
pxdt 
2
0


Life Tables
 In practice the survival distribution is estimated by
compiling mortality data in the form of a life table.
Here is the conceptual model behind the entries in
the table. Imagine that at time 0 there are l0
newborns. Here l0 is called the radix of the life table
and is usually taken to be some large number such as
100,000. Denote by lx the number of these original
newborns who are still alive at age x. Similarly ndx
denotes the number of persons alive at age x who die
before reaching age x + n.


 Show d x  lx  lx n
n

 Since ndx is the number alive at age x who die
by age x + n, this is simply the number alive at
age x, which is lx, minus the number alive at age
x + n, which is lx+n.


 Consider a group of newborns l0 .
 Each newborn's age-at-death has a distribution
specified by survival function s(x).
 L(x) - random number of survivors at age x.
 Each newborn is viewed as a Bernoulli trial: survive –
success, death – fail.
 Hence L(x) has a binomial distribution: n= l0 ,
p=P(success) =s(x).
 lx means expected # of survivors:


Basic Relationships
s( x)  s( x  n) lx  lx n dx
s( x  n) lx n qx    n

n
px   n
s( x) lx lx
s(x) lx
l x  l x 1 dx
l x 1 qx  
px  lx lx
lx

lx n  lx n m d x n
n |m
qx   m

lx lx  s ( x ) 
lx
 ( x)  
s( x) lx
l x  n  l x  n 1 d xn
n|
qx  
lx lx


Curtate Future Lifetime
 A discrete random variable associated with the
future lifetime is the number of future years
completed by (x) prior to death. It is called the
curtate future lifetime of (x), denoted by K(x), is
defined by the relation:

K ( x )  T ( x ) 

 Here [ ] denote the greatest integer function.


The curtate future lifetime of (x), K(x) is a discrete
random variable with density:

 The curtate lifetime, K(x), represents the number of
complete future years lived by (x).


Given the following portion of a life Table, find
the distribution of K for x=90.


How to find fractional part???

 Three approaches:

 Uniform Distribution of Deaths in the Year of
the Death (UDD)
 Constant Force of Mortality
 Balducci Assumption


Uniform Distribution of Deaths in the Year of the Death
(UDD)
 The number alive at age x + t, where x is an integer and 0
< t < 1, is given by:

 The UDD assumption means that the age at death of
those who will die at curtate age x is uniformly
distributed between the ages x and x + 1. In terms of the
survival function the UDD assumption means:

Where x is an integer and 0<t<1.

S h o w th at t q x  tq x

t
p x  1  tq x

qx
 (x  t) 
1  tq x


Consider Previous Example
 The ideas here will be introduced in the context
of previous Exercise , a 3-year, discrete survival
model for 90-year-old. The UDD linearly
interpolates among these 4 points to obtain the
complete graph of lx for all x between 90 and 93.


 Now let T be the complete future lifetime of a
90-year-old from previous problem where we
have extended the life table to a continuous
model via the UDD assumption.

l90  t  l90  t
  
l90  t
f T ( x ) ( t )  t p 90  (90  t )   
l90  l90  t  l90


Curtate Life Expectancy
 E(K) is known as the complete life expectancy and is
denoted by ex
  x 1   x 1

ex  E [ K ]   kf K ( x ) ( k )   k k pxqxk
k 0 k 0
x

0 x x
lxt
 lxt dt
ex   p x dt   dt  0
t
0 0
lx lx
  x 1   x 1
l x  k 1 l x  1  l x  2  ...  l 1
ex   k 1
px   lx

lx
k 0 k 0


With UDD Assumption
 Let T=K+S, then S is uniformly distributed
over [0, 1).

 Find the complete and curtate life expectancies at age 90
for the survival model


Constant Force of Mortality
 The assumption of a constant force of mortality in
each year of age means that
μ(x+t)=μ(x), for each integer age x and 0<t<1

px  ( px )
t
t

s ( x  t )  s ( x ) exp(   t ), w here  =- ln p x


Balducci Assumption
1 1 t t
 
s(x  t) s(x) s ( x  1)

Find expressions for tqx and μ(x+t), under this
assumption


The Expected Number of Years Lived by (x)

0
e x : n  T he expected num ber of years lived by ( x )
before age x  n
e x :n  T he expected num ber of com plete years lived by ( x )
before age x  n

T if T  n K if K  n  1
 
T   
K 
 n if T  n  n if K  n


0
2
If  ( x )  , fo r 0  x  1 0 0, co m p u te e 5 0:2 5
100  x


Select Mortality and the Underwriting Process

 (x) may pass the medical test to buy insurance
policy.
 Survival function is actually dependent on two
variables.
 The age at the selection (application for insurance)
 The amount of time passed after the time of
selection
 A life table which takes this effect into account is
called a select table.


Notations
 q[x]+i denotes the probability that a person dies between years x +
i and x + i + 1 given that selection occurred at age x.

 q25 - Probability that an insured 25-old will die in the
next year.
 q25 values for individuals underwritten at ages 0, 1, 2,
...,24, 25 are respectively denoted by q[0]+25, q[1]+24, …, q[25].


 A select mortality table is based on this idea. As one
might expect, after a certain period of time the effect of
selection on mortality is negligible. The length of time
until the selection effect becomes negligible is called the
select period. The Society of Actuaries uses a 15 year
select period in its mortality tables. The Institute of
Actuaries in UK uses a 2 year select period. The
implication of the select period of 15 years in
computations is that for each j≥0


Aggregate Table
 A life table in which the survival functions are
tabulated for attained ages only is called an aggregate
table. Generally, a select life table contains a final
column which constitutes an aggregate table. The
whole table is then referred to as a select and
ultimate table and the last column is usually called an
ultimate table.


 Consider:
 3-year select period
 85%, 90%, 95% and 100% of general mortality in policy
year 1,2,3 and 4, respectively.

 With a 3-year select period an individual underwritten at
age 21 would be subject to mortality rates

at age 21, 22 and 23.


 You are given the following extract from a 3
year select and ultimate mortality table.

Assume that the ultimate table follows
DeMoivre’s law and that d[x]=d[x]+1=d[x]+2 for
all x. Find 1000( 2|2q[71] )


2|2
q [ 71]  P robability of age 71 survies tw o years and
w ill die the follow ing 2 years.


Acturial maths

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