•
MARVA HOUSTON MOORE*. Statistical Analysis of Changes in Mortality
Patterns with Reference to Actuarial Functions. (Under the
direction of NORMAN L. JOHNSON.)
Methods of analyzing mortality data to determine whether a
particular life table should be used to calculate actuarial functions,
and to determine whether two mortality experiences could be merged
are outlined.
Ways in which hypotheses might be established and
tested, in ftccord with specific practical requirements are illustrated.
The hypotheses are formulated in terms of sets of annuity values or
linear functions of annuity values .. Methods of approximating these
it
pJI .
•
•
functions are used to illustrate that hypotheses about them involving
annuity values at many ages may be replaced by hypotheses involving
values at only a few ages.
The asymptotic distributions of estimates
of these functions are derived, and used in constructing test
procedures.
•
*This research was supported in part by the U.S. Army Research
Office, under contract #DAAG29-77-C-0035 .
ACKNOWLEDGEMENTS
•
With sincere appreciation, I acknowledge the tremendous
assistance given by my adviser, Professor N.L. Johnson.
His ideas,
patience, and encouragement were extremely helpful.
I would also like to thank Professors R.J. Carroll, I.M.
Chakravarti, W. Hoeffding,G.D. Simons,C.M. Suchindran, and E.J.
Wegman for the work they did while serving on my committee.
The speedy typing job done by Ms. Joyce Hill was greatly
appreciated.
The support and cooperation of my husband and daughter were
invaluable.
TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS
iii
LIST OF TABLES
vii
LIST OF FIGURES
ix
INTRODUCTION
x
Chapter
I.
II.
ACTUARIAL BACKGROUND
1
1.1 Life Tables . .
1.2 Makeham-Gompertz Law of Mortality
1.3 Compound Interest . .
1.4 Actuarial Functions
1.5 Special Notation
1
3
12
FOR ACTUARIAL FUNCTIONS
14
APPROXIM~TIONS
2.1
Approximating Annuity Values and
Differences Between Annuity Values
from Two Life Tables
3
6
14
"
2.1.1
2.1.2
2.2
Approximating a Linear Function of Annuity
Functions and Linear Function of the
Difference Between Annuity Functions from
Two Life Tables
....
.
2.2.1
2.2.2
2.2.3
2.2.4
.
Functional Form of the Annuity
Function
..
.
.
Lagrangian Interpolation of Annuity
Values and Differences Between
Annuity Values from Two Life
Tables
..
.
General Description of n-point .
Integration en-ages) Method
Approximations Using the n-ages
Method . . . . .
....
The n-th Derivative of the Annuity
Function Assuming the MakehamGompertz Law
Examples.
"
.
14
22
33
33
35
37
46
v
Page
Chapter
III.
ASYMPTOTIC DISTRIBUTIONS OF ESTIMATES OF
ACTUARIAL FUNCTIONS .
. ..
.
3.1
Asymptotic Joint Distribution of Annuity
Function Estimates for m-Ages
3.1.1
3.1.2
3.1.3
3.1.4
3.2
IV.
Mean of Estimate of Annuity
Function
.
..
Variance of Estimate of Annuity
Function .
....
Covariance of Estimates of Annuity
Functions at Two Ages
Distribution
Asymptotic Distribution of the Estimate of a
Linear Function of Annuity Functions for
m-Ages.
.
..
..
. .
49
50
50
50
54
57
63
TESTeS) OF ADEQUACY OF A STANDARD LIFE TABLE
65
4.1
4.2
Formulation of Problem..
.
.
.
Some Notes on General Procedure for
Testing Hypothesis that a Parameter is
Within a Given Interval . .
Testes) of Hypotheses Concerning
Annuity Values
.
66
4.3.1
4.3.2
4.3.3
79
84
4.3
4.3.4
4.4
Test Procedure
Properties of the Test
Effect of a Change in Interest
Alone .
Examples
Testes) of Hypotheses Concerning Linear
Functions of Annuity Values
4.4 .1
4.4 .2
V.
. ..
Test Procedure
Example .
.
TESTeS) OF POSSIBILITY OF MERGING TWO SURVIVAL
EXPERIENCES
....
5.1
5.2
5.3
Formulation of Problem
Testes) of Hypotheses Concerning the
Difference Between Annuity Values
.
from Two Populations.
Testes) of Hypotheses Concerning Linear
Functions of the Differences Between
Annuity Values from Two Populations
76
79
85
86
96
96
98
102
102
104
106
vi
....
..
Chapter
Page
APPENDIX .
108
BIBLIOGRAPHY
115
LIST OF TABLES
Table
2.1
.
Page
Maximum Absolute Error over Ages 20-80
Made Interpolating
2.2
A67 a - C58 a
[
x
x
1
29
A67
_C58ax) .
( ax
30
Maximum Absolute Error over Ages 30-65
Made Interpolating
2.5
28
Maximum Absolute Error over Ages 20-65
Made Interpolating
2.4
...
Maximum Absolute Error over Ages 20-70
Made Interpolating
2.3
A67
_C58 ) . . . .
ax
ax
[
[A67 ax -C58axl . . . .
.
.
31
Maximum Absolute Error over Ages 35-65
Made Interpolating
A67 ax _C58ax) .
(
32
8
..
2.6
Approximate Value of
l~! Dn~x Makeham-
Gompert z Law (A = .002, B = .0000357,
c=1.0996589) - i = .06
',' .
44
8
2.7
Approximate Value of
~
Dna
Makehamn!
x
Gompertz Law (A = .002, B = .0000357,
.. .
c =1.0996589) - i = .10
45
\1
8
2.8
Maximum Approximate Value of Il~! Dnaxl
Makeham-Gompertz Law (A = .002, B = .0000357,
.
..
c =1.0996589) - i = .06 . . .
45
8
2.9
Maximum Approximate Value of
ll~!
Dnaxl
Makeham-Gompertz Law (A = .002, B = .0000357,
c =1.0996589) - i = .10
46
2.10
Values of
47
2.11
Five-Ages Approximation for
i\
y
b.
1
from Model Office Data •
y**
i\a
y=y* y y
I
jY**I
i\,
y=y* y
48
viii
Table
3.1
Page
Error Made Approximating
A67
a X::t::t-Xr
rrn-:::1
••.••
A67 a
by
x
53
n Times Variance of Annuity Function Estimate
Using U.K. Assured Lives: Durations 2 and
over Experience Assuming Total U.S. Population
1969-71 Life Table with w* =100 - i = .04
..
55
n Times Variance of Annuity Function Estimate
Using U.K. Assured Lives: Durations 2 and
over Experience Assuming Total U.S. Population
1969-71 Life Table with w* = 100 - i = .08
56
4.1
(Untitled)
87
4.2
(Untitled)
88
4.3
(Untitled)
92
4.4
(Untitled)
94
4.4
(Untitled)
100
A.l
Standard Ordinary Issues for Nineteen Companies
Medical and Non-Medical Combined Male and
Female Lives Combined Premiums at 5% Interest
Rate by Age-at-Issue
.
. . . .
111
Standard Ordinary Issues for Nineteen Companies
Sum Assured and Premiums at 5% Interest By
Attained Age . .
•.
112
Grouped Values of A 5% Interest Rate
($1,000,000 units)
....
113
Smoothed Values of A By Attained Age 5%
Interest Rate ($1,000,000 units)
114
3.2
3.3
A.2
A.3
A.4
LIST OF FIGURES
FIGURE
Page
1
Annuity Values (A67 Life Table)
17
2
Immediate Annuity at 6% Interest Rate
18
3
Immediate Annuity at 10% Interest Rate
19
4
Residual Plots (6% Interest Rate)
20
5
Residual Plots (10% Interest Rate)
21
6
Absolute Error Made Interpolating
A67 (.06)
a
.
.
24
x
7
Absolute Error Made Interpolating
C58
..
a
x
Using Five Ages
25
8
Bounds for Power Function Example 1
9
Bounds for Power Function W.R.T.
e
89
91
10
Bounds for Power Function Example 2
93
11
Bounds for Power Function Example 3
95
12
Power Function
101
INTRODUCTION
We outline and apply methods of analyzing mortality data with
the aim of detecting whether changes in the mortality pattern should
cause a life insurance company to consider replacement of a standard
life table for calculation of actuarial functions by one more in
accord with current experience.
We also outline a method for
determining whether two mortality experiences are sufficiently alike
to be merged.
Practical interests will dictate the criteria for determining
whether a life table should be changed and whether experiences
should be merged.
Therefore, it is impossible to establish completely
general criteria.
However, using certain actuarial functions that
are commonly used we illustrate how hypotheses might be established
and tested, in accord with specific requirements.
The hypotheses formulated concern annuity values at every
integer-valued age over a wide age interval or a linear function of
the annuity values for a number of different ages.
The emphasis here
is on a practical approach so we explore the possibility of replacing
these hypotheses by hypotheses concerning annuity values at a
relatively few ages.
We have concentrated on types of changes in mortality patterns
that are likely to occur in practice and the methods used may not be
applicable to other, theoretically possible, changes in mortality.
CHAPTER I
ACTUARIAL BAC KGROUND
The notation introduced in this chapter will be used throughout
without further explanation.
It will be used for the purpose given
here exclusively unless it is obvious from the context in which it
is used that it is being used for some other purpose.
Most of the notation introduced in this chapter is standard
actuarial notation, however, Section 1.5 gives some special notation
we will use.
1.1
Life Tables
The life table is a device for exhibiting a mortality distribu-
tion.
There are two forms of the life table.
The cohort life table
is intended to show the mortality experienced by a group ("cohort ")
of individuals born about the same time, as it progresses through
life.
Current life tables use the observed death rates in a given
population during a relatively short period of time to project the
life span of individuals in a hypothetical cohort assumed to experience these rates throughout life.
In this section, we will only be
concerned with current life tables which may be either complete or
abridged.
(The complete life table uses only one year age intervals,
•
whereas, the abridged life table deals with age intervals greater
than one year.)
2
A life table typically has at least the four columns:
(i)
age interval,
tx
(ii)
x to
x + t;
=the number living at the beginning of the age interval;
tdx =the number dying during the age interval;
(iii)
tqx =the proportion of persons alive at the beginning of age
(iv)
interval, dying during the interval.
Probabilities of death and survival may be expressed in terms
of
x
t
x
and tdx'
Let
Px
be the probability that an individual aged
will survive to attain age
x +1,
Px =
then
t X+1
--:r;-
The probability that an individual aged
age
x +n
is denoted by
p,
n x
x
will survive to attain
x
will die before attaining
and
t x+n
nPx =
--:r;-
The probability that an individual aged
age
x +1
is denoted by
qx'
and
=1
- Px --
d
qx
where
d
x +1.
Let
x
die within
x
rx '
is the number of deaths during the age interval
nqx
n
be the probability that an individual aged
years,
x
to
x
will
then
nqx =1 - nPx =
t x -tx+n
--:t.....--x
The probability that an individual aged
and die in the
(n +1) -th
x
will survive for
year is denoted by n Iqx'
d
x+n
= --r- .
x
and
n
years
,"
3
1.2
Makeham-Gompertz Law of Mortality
When proposing an analytic function that would closely reproduce
the typical
.f.
x
curve, Gompertz (1825) stated, "It is possible that
death may be the consequence of two generally coexisting causes:
the
one, chance, without previous disposition to death or deterioration;
the other, a deterioration, or increased inability to withstand
destruction."
However, his law of mortality only takes account of
the second cause.
Makeham (1860) combined the two causes additively
and derived the Makeham-Gompertz law of mortality,
]J
x
= A + Bc
x
,
is the force of mortaZity (a measure of the mortality at
where
the precise moment of attaining age
where
D.f.x
x).
denotes the derivative of
an expression for
.f.
x
.f.<x
with respect to
x,
may be derived from the expression for
so
]l
x
•
Compound Interest
1.3
Suppose that on an investment of 1, one earns a constant amount
of interest,
of one year is
i,
each year.
(1 + i)
Then the value of the
1
at the end
and the total at the end of t years is
1 + it
for
t
2::
O.
Interest earned in this manner is simpZe interest.
At the end of one year simple interest earned during that year
is not reinvested to earn additional interest during the next year.
If it is reinvested, the interest is said to be compounded.
earns compound interest at a rate of
of
1
i
If one
per year on his investment
then the value of his investment at the end of the first year
4
is
(1 + i)
at the end of the second year is
and at the end of
t
(1 + i) + i (1 +i)
= (1 + i) 2
years is
(1 + i) t
.
In our work we will be concerned with compound interest functions
and will generally be dealing with amounts that will be invested at
the present time in order to yield a given amount(s) at some time(s)
in the future.
Therefore, we must explain the discount function.
one expects an investment to be worth
1
If
at the end of a year then
its value at the beginning of that year must be
v =
1
1 + i '
since this amount will accumulate to 1 at the end of the year.
invests
1/ (l + i) 2
If one
now at the end of one year it is worth
1
1
+ io
(1 + i) 2
(1 + i) 2
1
= 1 +i ,
and at the end of two years it is worth
1
l+i +
.
J.°
1
1
1+i =
In general, to have an investment worth
in
t
years, one
after
t
years is now
1
must invest
t
v =
now.
1
t 2::0
In otherwords, the present value of
1/(1 +i)t.
v
t
1
is called the discount function.
The effective rate of discount,
d,
during the t-th year is the
ratio of the amount of interest earned during the year to the value
of the investment at the end of the year
d = v
t-1
v
_·v
t-1
t
i
= 1 +i
5
Suppose that on an investment of
year convertible
1
m times per year is
the rate of interest per
i (m).
Then the effective
annual rate of interest is
. (m)/)m
m -1
. - (1
1-
+1
The force of interest is the rate of interest per year convertible
infinitely often during the year,
i (00),
and is denoted by
o.
Clearly,
and
v =e
-0
(1. 2)
A series of payments made at equal intervals of time is called
an annuity.
If the payments are going to be made (with probability
one) for a fixed period of time only, the annuity is called an annuity
certain.
The present value of an annuity certain providing payments of 1
at the end of each year for
n
n
a
ill
years is
t
= LV.
t=l
This kind of annuity, with the first payment made at the end of the
first year is called an immediate annuity.
If the first payment
were made at the beginning of the year, it would be an annuity-due.
The notation for an annuity-due is distinguished from that of the
corresponding immediate annuity by a double dot over the
present value for the annuity-due would be
n-l
a
=
11]
Lvt
t=O
Clearly,
va
ffi
= a
ii1
a.
The
6
Frequently, a payment or a series of payments is made contigent
upon the survival of an individual and this is the type of monetary
functions we are interested in analyzing.
1.4
Actuarial Functions
Suppose an insurer promises to pay an individual, now aged
the sum of
1
at the end of
n
years if he is then alive.
net single premium the individual must pay at age
x
x,
The
for this
n-year pure endoUJrnent of 1 is
E
n x
(A net premium does not include any provision for the expense involved
in the transaction; it is just the present value of the payment to be
made .)
A series of annual payments of
year if an individual now aged
x
1
beginning at the end of one
is alive then and continuing
throughout his lifetime is called a life annuity.
Its present value
is
w-x-l
a
x
=
=
L E
t=l t x
w-x-l t
L
t=l
v
P
t x
w is the youngest age at which the probability of surviving beyond
that age is negligible.
Suppose the force of mortality is increased by a constant amount,
say
8,
at every age, then the present value of the life annuity at
interest rate
i
on the changed mortality basis may be expressed as
follows (using (1.1))
4It
7
(i)
a
w-x-l t
L
=
x
v tP
t=l
x
w-x-l
L
=
vtexp[_
Jt
t=l
.
(]Jx+n + 8)dn]
0
It
w-x-l
= L\' (ve -8 ) t exp [ - ]Jx+ndn]
t= 1
Clearly,
0
is equal to the present value of a life annuity on
the original mortality basis, with interest rate
i
l
=(1+i)e
8
-l
Life annuities are sometimes payable more than once a year.
The present value of a life annuity of
to a life now aged
x
1
payable
m times a year
is
1 m(w-x)
m
L
E
t=l tim x
When m becomes infinite, the resulting annuity is called a continuous
annuity.
Its present value is
00
a = lim.!..
L
E
m~ m t=l tim x
x
=
fOOtE dt
J
o
x
Usually the survival function has no simple mathematical expression
and it is necessary to use approximations of
approximation is
ax·d:ax +.1.2
a.
x
The usual practical
8
The life annuity function is the basic function we will use in
our analysis.
However, certain special annuities, for which payments
are limited to a maximum number of years, or the first payment is
deferred, will also be needed.
The n-yeap tempopapy life annuity which provides payments of
at the end of each year for
n
years if an individual now aged
I
x
survives, has the present value,
n
ax:i'il =
The n-yeap defepped life
n
t~IV
t
tPx
annuity~
a life annuity with the first
payments omitted has the present value,
w-x-l
nlax =
An
Vttpx
n-yeap defepped m-yeap tempopapy life annuity providing
annual-payments of
x +n +I
L
t=n+l
I
to an individual now aged
x
when he is
years old and continuing for m-years if he survives has the
present value
a
nlm x
=
If the first payment of an annuity is made at the end of a
payment period it is called an immediate Ufe annuity. The annuities
mentioned above are all- immediate annuities.
An
payments at the beginning of the payment period.
annuity-due provides
The following
relationships exist between immediate annuities and annuities-due:
a
x
=I
+a
(1. 3)
x
ax:1i1 = 1 +ax;n-l\
njax
= n_Ilax
'
9
and
Annuities are payable if an individual survives; payments
contingent on death rather than survival are provided by insurances.
Usually the insurer pays the specified amount upon the death of
the insured, however, values of insurances have traditionally been
calculated by assuming that the specified amount is paid at the
end of the year of death.
So, we will also make that assumption
since the probabilities of death and survival to the end of a year
can easily be calculated from life table data.
Suppose an insurer promises to pay an amount of
of the year of death of the insured who is now aged
insurance is called whole life insurance.
e
single premium at age
Ax =
x
1
x,
at the end
such an
The present value or net
for such an insurance is
w-x-l t+l
w-x-l
1
I vt+ld
I v t ,qx = r t=O
x+t
t=O
x
Each term in this expression is the probability that death will occur
in a particular year multiplied by the present value of a payment of
1
at the end of that year.
It can be shown that the convenient relationship
(1.4) .
A = 1 - da ,
x
x
exists between the present value of a whole life insurance and an
annuity-due.
Generally, insurance is not purchased by a single premium
payment but rather by a series of periodic premium payments.
The
present value of the sequence of net premiums must be equal to the
present value of the insurance.
Let
P
x
denote the premiums paid
10
once a year for whole life insurance with present value
xax
A ,
x
then
= Ax
P
or
P
x
Since A = I -dii
x
=A
x
lax
x
P
x
1
--d
li
x
=
When one buys an insurance policy, he agrees to pay a set
premium each year and in return the insurer promises to pay a certain
amount in the future.
It is important for the insurer to have a
measure of the amount which he should have on hand at any time to
assure payment of the future benefits assuming that all future
premium payments will be made as they are due.
is such a measure.
The policy reserve
It is the difference between the present value
of the benefits to be paid in the future, and future premium payments.
For a whole life insurance of
denote the reserve at the end of
V
t x
=A
x+t
=I
since
- (P
- P
issued at age
1
t
x
let
tVx
years.
a-
x x+t
x
+ d)a
x+t
,
Ax+ t = I - dax+ t .
The aggregate reserve on all the whole life insurance policies
issued by a company is
w-l n
w-l n
I
I
t S tV = I
I t Sx (Ax+ t - Px' ax+t )
x=O t=O
x
x x=O t=O
w-l
=
I
n
I
x=o t=O
(tSx Ax +t - tGx a x +t )'
~
11
where
5
t x
= total face value of all the whole life policies issued t
years ago to persons then aged
x,
pI = the annual premium per unit amount actually being paid for a
x
whole life policy issued
t
years ago on a life then aged
x,
tGx = total gross annual premium for all whole life policies
issued
t
years ago on lives then aged
x,
and
n = the number of years since the first whole life policy was
issued.
Let the total sum assured for persons now aged
L
(x,t):x+t=y
t 5x
=5
y,
y
and the total gross premiums for persons now aged
L
G
y,
= G
(x,t):x+t=y t x
y
then we may express the aggregate reserve as
w-l
n
L L
x=o t=O
w-l
t5
x
tV
x
=
L (5
y=O
A
y Y
-G
a) .
YY
Using (1.4) we obtain
w-l
n
L L
w-l
5
x=O t=O t x
tVx =
L [S
y=O
y
- (dS + G ) a
Y
]
Y Y
We should note that
on mortality.
Sand G are actual amounts but
y
y
Throughout we will let
A
Y
=dS
Y
+G
ay
depends
.
Y
Using (1.3) we express the aggregate reserve as
w-l n
w-l
l: L tSx tVx = l: [(S - A ) - A a ]
x=O t=O
y=O
Y Y
YY
w-l
w-l
=
L (S
y=O
Y
-A ) -
Y
LA a
,
y=O Y Y
(1.5)
12
The aggregate reserve depends on the
function
\
Ly
A a
y
a 's
y
only through the linear
Y
1.5 .Specia1 Notation
Throughout we will use the non-conventional notation explained
in this section.
Our discussion is generally limited to integer valued ages,
therefore, if we let
denotes integer-valued
x
denote age,
x
x*
~x ~x**
in the interval
or
x E[X*, x**]
[x*, x**].
Frequently, it will be necessary to indicate the interest rate
at which an actuarial function is calculated.
We indicate this using
a superscript with the principal symbol for the actuarial function.
For example,
(i)
= the
ax
present value of a life annuity at interest rate
a life aged
i
for
x.
We use a prefixed superscript with the principal symbol for an
actuarial function to indicate the mortality basis used to determine
the value of the function.
For example,
Sax = the present value of a life annuity for a life aged· x
based
on a standard life table.
"Standard life table" means some established life table used by a
company.
C
a
x
=
the present value of a life annuity for a life aged
x
based
on the current mortality experience of the population of
concern.
When it is necessary to distinguish between current mortality experiences for two populations, primes and double primes are placed on the C.
e
13
In our examples we have used several established life tables
as mortality bases.
The following abbreviations are used to indicate
the particular life table used:
A67 = A1967-70 Life Table (A1967-70 (1975), p. 15)
M69 = Life Table for U.S. Males 1969-71 (U.S. Life Tables
(1975), p. 8)
C58 = The Commissioners 1958 Standard Ordinary Mortality Table
(e.g., Jordan (1975), pp. 342-344)
MG = Life Table Following the Makeham-Gompertz Law with
A = .002,
B = .0000357,
c =1.0996589.
As an example,
M69a (·04) =the present value of a life annuity for a life aged
x
x,
at 4% interest, based on the life table for U.S. Males
1969-71.
CHAPTER II
APPROXIMATIONS FOR ACTUARIAL FUNCTIONS
If one could establish that all the values of an actuarial
function, or the differences between the values of this function
from two life tables can be approximated from the values at a few
ages with sufficient accuracy, it would be possible (for practical
purposes) to replace hypotheses relating to differences between the
values from two life tables at all ages (in a certain range) by
hypotheses relating to differences at just a few ages.
We are particularly interested in hypotheses concerning life
annuities and linear functions of life annuities.
Therefore, in
this chapter we examine methods of approximating these functions.
2.1
2.1.1
Approximating Annuity Values and Differences Between Annuity
Values from Two Life Tables
Functional Form of the Annuity Function
Usually there is no mathematically simple and well-defined
functional relationship between age and the annuity value
over the "range of relevant ages".
(x and a )
x
(This range will vary from
company to company but for most cases the interval [20,80] will cover
the great majority of ages of concern to the company.
We have, in
fact, confined our investigations to the age interval [15, 96].)
From a study of graphs of the annuity value against age we have
observed that
15
(i)
the annuity function is a "smooth" function of
(ii)
x;
for a given mortality basis the functional forms of the annuity
functions for interest rates greater than 5% and not
very different from each other are very similar;
(iii)
over the range of relevant ages (in practice) it can be
assumed that annuity values from different mortality bases
are similar type functions of
x.
To illustrate these points we have used annuity values based on
the A1967-70 life table, (A67), the life table for U.S. males 1969-71,
(M69), and a life table following the Makeham-Gompertz law with
A = .002,
B = .0000357, and
c
=1.0996589,
(MG).
The A67 life table
is based upon data supplied the Institute of Actuaries and the Faculty
of Actuaries by forty-four insurance companies transacting business
in the United Kingdom; the data relate to the experience during
1967-70 of male lives accepted for whole life and endowment insurances.
The life table for U.S. males
1969~7l
is a current life table for
the United States male population based on age-specific mortality rates
for the period 1969-71.
It has been found empirically that the
Makeham-Gompertz law gives a survival function (probability that a
new life will survive to attain age
x) that follows the usual
mortality pattern closely when the parameters are within the following
ranges:
.001 <A < .003
10- 6 <B < 10- 3
1. 08 < c < 1.12
[See Jordan (1975) p. 23-24J
The parameters we have used are within
e
16
the above ranges but were not chosen to fit any particular population.
We have chosen to use these three mortality bases because
they are from such varied sources (a population of assured lives,
a national population that includes relatively few (if any) of these
same assured lives, and an "artificial" population).
"
1 sows
h
F19ure
grap h s
0f
A67 ax aga1nst
.
of 0, 2, 4, 6, 8, and 10 percent,
x
for interest rates
Figure 2 shows graphs of
A67 (.06) M69 (.06)
MG (.06)
a
against x, and Figure 3
ax
' and
ax
'
x
M69 (.1)
A67 (.1)
MG (.1) against x
shows graphs of
a
,
a
ax
' and
x
x
of these graphs illustrate (i) "
each
Figure 1 suggests (ii) since for our purposes we may ignore an
interest rate of 0%.
The annuity at 0% interest rate has been
included merely because it is just the curtate expectation of life.
Figures 2 and 3 suggest (iii).
Using the method of least squares
piecewise straight lines were fitted to the annuity values for 6% and
10% based on the three life tables over the age intervals [15, 25],
[26, 36], [37, 54], and [55, 80] and an exponential function,
x
ax =a8,
was fitted to the values over the interval [81, 96].
Figures 4 and 5 exhibit the residual plots obtained.
The residual
plots have the same form and the magnitude of the residual at each
age,
x,
is about the same for each life table (the greatest
difference is .007 atx =46) over the intervals [IS, 25], [26, 36],
and [37, 54].
This indicates that, in these intervals,. the annuity
function is the same type of function of
x
for each of the life
tables and the difference between the fitted annuity values is a
very good estimate of the difference between the actual annuity values
at the same age.
Over the interval [55, 75] all three residual plots
17
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34.28
42.85
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22
have generally the same form but over [76, 80] the MG differs from
the A67 and M69.
The residual plots over the interval [81, 96]
2
indicate that the model
a =aSx+yx
may be more appropriate for
x
the MG and A67 tables, but, in each case, there is a different
pattern for the M69 table.
The interval [76, 96] covers more
advanced ages, which are usually of relatively minor importance in
practice.
Since the residual plots have the same form for all
three life tables up to about age 76 there should be very few cases
(in practice) where it is unsafe to assume (iii).
2.1.2
Lagrangian Interpolation of Annuity Values and Differences
Between Annuity Values from Two Life Tables
Let
given.
m values of the function
f(x), f(x.),
l.
i = 1,2, ... ,m,
Using Lagrange's m-point interpolation formula
approximated by the polynomial through
m
I
i=l
l.
is
(x., f(x.)) (i =1,2, ... ,m),
l.
m
f (x . )
f(x)
be
nj =1 Lrex - xJ. ) / (x.
l.
l.
- x .)l
J
J
.
jli
Clearly, the approximation for the difference between two functions
obtained using Lagrange's interpolation formula is simply the
difference between the approximations of the individual functions
obtained using Lagrange's interpolation formula.
Therefore, if we
can interpolate the individual functions accurately, we can interpolate the difference between the two functions accurately.
Usually we are more interested in differences between annuity
values than in annuities themselves.
...
However, since
in practice, it
can be assumed that annuity values from different mortality bases
are similar type functions of
it follows that:
x
over the relevant range of ages,
23
(i)
the error made using Lagrange's interpolation formula
to approximate annuity values from two life tables of the kind that
might be encountered in practice will usually have the same sign.
Therefore, the absolute value of the error made approximating the
difference between annuity values from two such life tables is less
than the maximum of the absolute value of the error made approximating
the two annuity values; and
(ii)
to check that Lagrange's interpolation formula gives an
accurate approximation for the difference between annuity values from
two life tables we need only illustrate that i t gives an accurate
approximation of the annuity function based on a "typical" life table.
Figure 6 shows graphs of the absolute value of the error obtained
using the four, five and six point Lagrangian interpolation formulas
to approximate
A67 (.06)
a
x
for
X E:
[20, 80]
against
x.
In each
case equispaced ages over the interval [20, 80] with 20 and 80 being
two of the ages were used as pivotal points for the interpolation.
From Figure 6 we see that the maximum absolute errors obtained using
the four, five, and six point formulas were .1507 at age 72, .0213
at age 25, and .0126 at age 76 respectively.
Clearly, the five and
. p01ntormu
.
fl
.
.
S1X
as '
g1ve very accurate approx1mat10ns
0f
A67 a (.06) .
x
A question of considerable importance to us is how few ages may be
used.
Since the absolute error using the five point formula is so
much smaller than that using the four point formula at every age
except the pivotal ages for the four point formula, it seems
desirable to use five ages.
Figure 7 further illustrates how accurately Lagrange's five
point interpolation formula approximates annuity functions.
It shows
4It
24
o
.02&
.1l4
.0116
.05:7
()
o
o
0
(J
oca
o
o
o
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- - - - '- - - - - - - - · - · - - - · - - - - . - ••
~-----
__a
•
•
~._.
j
_ _
e
ABSOLUTE ERROR MADE INTERPOLATING
o
L/)
~
N
e
e
C58 a
x
USING FIVE AGES
"1"----------------------:--------------.. . . .- -----.. . . .----.
U)
~
tIl
o
~i
0+
o ++ +
1
i-.oo
i-.035
...0
~J
+
o
o
+
+
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o
db
~
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0
or
o
~
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N
o
o
D~D
....
Q
tt
o
I1
o
I
I
10
,
•
+t>
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20
~~~I>t>~~t>
I
30
o
"lI
-~~+#i#
11-0
,\.\,
,;
Atr
~
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SO
Figure 7
,-
A U1:lr:t. A~
.
U - 1-.10
zv..
Utr
X
AGE
.
~
+
I
I
A+
0+
~+++ ~.
f
A~-·08
.A A. t'{]
~A
o
..
~+ i-.06
~
o
co
~~
60
70
80
90
100
26
the absolute value of the error obtained using the formula to approximate
for
C58 (0)
a
,
x
x €[20, 80]
C58 (.035)
ax
'
C58 (.06)
a
,
x
graphed against
x.
C58 (.08)
ax
'
C58 (.1)
ax
and
The maximum absolute errors
were .0847, .0683, .0433, .0267 and .0174 for 0,
percent interest rates respectively.
3.5, 6, 8 and 10
In each case except for the
case of 10% interest, the maximum absolute error is attained at age
75.
In the case of 10% interest, it is attained at age 25.
For our
purposes, interest rates of less than 5% are not of major importance,
since in practice such low interest rates for actuarial functions
have been rare recently.
Therefore, large errors for interest rates
less than 5% will not cause us to judge the approximations inadequate.
In order to give some idea of the magnitude of the absolute
value of the error made when approximating the difference between
annuity values from two life tables using Lagrange's formula we
include tables showing the maximum absolute error made approximating
(A67 a x _C58 a x )
over several
age'1nterva 1s.
Using Lagrange's three, four, five, and six point interpolation
formulas to interpolate
(A67 a
x
_ C58 a )
x
it seems reasonable to
believe that the difference at each age can be approximated with
sufficient accuracy from the differences at a few ages.
The
differences were interpolated using equispaced ages over the interval
of ages [20, 80] with 20 and 80 being two of the ages.
Interest rates
of 0, 2, 4, 6, 8 and 10 percent were used.
Define the absolute error as the absolute value of the interpolated value of the difference of the annuity values based on the
two life tables minus the actual value of the difference of the
annuity values based on the two life tables.
27
Tables 2.1, 2.2, 2.3, 2.4, and 2.5 show the maximum absolute
errors, over age intervals [20, 80], [20, 70], [20, 65], [30, 65],
and [35, 65] respectively, that were made interpolating
for
x
£
(A67 a
e
C58)
ax
x -
[20, 80] using Lagrange's three, four, five and six point
formulas.
We observe that the greatest absolute error (.1502) occurs at
age 73 using the three-point formula for 0% interest rate.
In fact,
the greatest absolute errors in the age intervals, [20, 70], [20, 65],
[30, 65], and [35, 65] occur using the three-point formula for 0%
interest rate, in the case of ages 20-70 it is .138 at age 70 and in
the other intervals .1172 at age 65.
These errors seem fairly large
but they occur at points that are, for our purposes, of relatively
minor importance.
The 0% interest rate case is of importance simply
because it is the curtate expectation of life, a 0% interest rate
for actuarial functions is highly unrealistic.
We observe that the
greatest absolute errorS obtained using the four, five, and six
point formulas are much smaller .0718 at 2% interest, .0427 at 4%
interest, and .0492 at 0% interest respectively - all at age 75.
(See Table 2.1).
Since there is such a reduction in the greatest absolute error
when using more than three ages we will focus on the four, five, and
six point formulas.
If we concentrate on the age interval where a person is most
likely to have accumulated enough cash to purchase an insurance
policy or an annuity (ages 35-65) we observe that the greatest
absolute error using the four, five, and six point formulas are
particularly small, .0274 at age 35, .0088 at age 41, and .0064 at
,.
28
TABLE 2.1
Maximum Absolute Error over Ages 20-80
Made Interpolating
i
.00
.02
.04
e
.06
.08
x
- C58 a
x
J
6 Pt.
5 Pt.
4 Pt.
3 Pt.
Ierror I
.0492
75
.0415
74
.0668
74
.1502
73
Ierror I
.0432
75
.0418
75
.0718
75
.1051
73, 74
Ierror I
.0394
75
.0427
75
.0682
74
.0556
75
Ierror I
.0368
75
.0419
75
.0563
75
.0679
32
Ierror I
.0347
75
.0394
75
.0447
75
.0660
33
.0326
75
.0359
75
.0343
75
.0622
66
ages
ages
ages
ages
ages
Ierror I
.10
(A67 a
ages
29
TABLE 2.2
Maximum Absolute Error over Ages 20-70
. Made Interpolating
i
.00
.02
.04
.06
.08
.10
(A67 a
x
- C58)
a
x
6 Pt.
5 Pt.
4 Pt.
3 Pt.
Ierror I
.0278
24
.0215
24
.0339
32
.1380
70
Ierror I
.0186
25
.0194
25
.0431
70
.0854
70
Ierror I
.0123
24, 25
.0172
25
.0390
70
.0540
33
Ierror I
.0085
25
.0144
25
.0289
70
.0670
33
.0084
70
.0126
70
.0182
70
.0647
32
.0076
70
.0107
70
.0098
26
.0622
66
ages
ages
ages
ages
Ierror I
ages
Ierror I
ages
e
30
31
TABLE 2.4
Maximum Absolute Error over Ages 30-65
Made Interpolating
[A67 a
C58)
ax
x -
6 Pt.
5 Pt.
4 Pt.
3 Pt.
Jerror J
ages
.0091
30
.0089
30
.0339
32
.1172
65
Ierror J
.0064
30
.0097
30
.0300
32
.0617
65
Ierror I
.0045
30
.0097
30
.0192
32
.0540
33
Ierror I
ages
.0032
30
.0086
30
.0092
33,34
.0670
33
.08
lerrorJ
ages
.0025
30
.0070
30
.0647
32
.0647
32
.10
lerrorJ
ages
.0020
65
.0052
30
.0066
30
.0612
65
i
.00
.02
.04
.06
ages
ages
e
32
TABLE 2.5
Maximum Absolute Error over Ages 35-65
Made Interpolating
i
.02
...
e
.04
.06
x
x
J
6 Pt.
5 Pt.
4 pt.
3 Pt.
.0064
35
.0088
41
.0274
35
.1172
65
Ierror I
.0048
35
.0041
41, 55, 56
.0239
35
.0617
65
Ierror I
.0036
35,36
.0018
55,56
.0161
35
.0515
35
.0027
35
.0007
37~45
.0089
50
.0666
35
Ierror I
.00
(A67 a . - C58 a
ages
ages
ages
Ierror I
ages
55,56
.08
.10'
lerrorJ
ages
.0023
65
.0007
45
.0032
50, 51, 65
.0656
35
Ierror I
.0020
65
.0006
55,56
.0012
46,47
54, 55
.0612
65
ages
33
age 35
respectively each at 0% interest.
(See Table 2.5).
In the age interval [35, 65] we notice that for each of the
interest rates 2, 4, 6, 8, and 10%, the maximum absolute error is
smallest using the five-point formula.
It is smallest using the
six-point formula in the case of 0% interest, however, there is only
a difference of .0024 between the maximum absolute errors using the
six and five point formulas.
(See Table 2.5).
In the age intervals
[20, 80], [20, 70], [20, 65], and [30,65] for each of the interest
rates used the smallest maximum absolute error is given by either
the five-point or the six-point formula.
and 2.4).
(See Tables 2.1, 2.2, 2.3,
The difference of the maximum absolute errors using the
six and five point formulas is at most .0059.
So, there is not a
great reduction in the maximum absolute error obtained using the
six-point formula rather than the five-point formula.
Using the five-point formula the absolute error for each of
the interest rates remains below .01 in the age interval
(30, 70)
and for interest rates 4, 6, 8, and 10 percent, the absolute error
of interpolation remains below .002 in the age interval [35, 65].
Therefore, it seems satisfactory to use as few as five ages for the
approximations.
2.2
Approximating a Linear Function of Annuity Functions and Linear
Function of the Difference Between Annuity Functions From Two
Life Tables
2.2.1' General Description of n-point Intergation (n-ages) Method
The n-point integration method given by Beard (1947) is a method
r
of approximate evaluation of functions of the form
f (y)¢(y)dy
b
34
or
C
Lf(y)ljJ(y) ,
b
(n)
by a sum of relatively few
terms.
It is assumed that
(i)
the first
n derivatives of
fey)
exist and are
cpntinuous on an interval containing y =h,
(ii)
the moments of
mr =
ljJ(y),
r
C
(y - h{ ljJCY)d0 ¢(y)dy ,
b
exist for
b
r =0,1, ... ,n -1 .
We want to determine weights,
c
b. ,
1
fC
I
such that
JC
f(y)ljJ(y)dy = nL b.f(y.) ljJ(y)dy +R
ljJ(y)dy,
. 111
n
b
1=
b
b
where the
small.
bi and Yi
If
fey)
fey) =
Yo
between
c
R
n
I
¢(y)dy=
L
'1
j=O
J.
.,
J.
'1
t;(y)
b.
1
1. =
b obtain
Substituting in (2.1) we find
.
fCf(n)(C())
(y_h)JljJCy)dy+
i y (y_h)nljJCy)dy
b
- . nL1b.1 ~-lf(j)(h)
I
. 0
J.
1=
RnfCljJ(Y)d Y is
f(n)(y)
.
o (y - h) n ,
(y _h)J +
n!
n~l f(j)(h)
n-lf(j)(h)JC
L
. 0
and
is expanded by Taylor's theorem, we
hand y.
J=
b
are independent of fey)
(2.1)
b
(y. -h)
1
j
+
1
JfC ljJ(y)dy
(Yi-h)n (n)
I
f
(t;. (y.))
n.
1 1
t;. (y. )
betweenh and y and
n.
1
between
h and y ..
1
b
(2.2)
If the
are determined from the equations
=mr ,
r
=0, 1, ... , n
- 1
(2.3)
35
the terms of (2.2) involving
f (j) (h) ,
j = 0,1, ... , n - 1
will
(2.4)
~.(y.)
and
between
(~(y)
between
2.2.2
Approximations Using the n-ages Method
h and y
hand y.).
1 1 1
We may use the n-point integration method described above to
express a linear function of annuity functions, say
terms of annuity functions at selected ages.
Ly¢(y)ay
in
From (2.1) notice that
n
LyHy)a y = i~lbiaYiLy¢(Y) +RnLycjJ(y) ,
where the
~(y)
b.
1
between
are determined from (2.3) by letting the integral
hand y
between
~. (y.)
and
1
1
denotes the n-th derivative with respect to
If
IRnLycjJ(y)
I
y
b.
1
hand y.,
1
where
n
D
x.
is sufficiently small then
L cjJ(y)a
The
(2.5)
Y
~
n
L b.a L cjJ(y)
i=l
1
Yi Y
.
(2.7)
obtained are independent of the annuity function,
therefore, a linear function of the difference between annuity
functions from two life tables, say
LycjJ(y) (a'y - a")
y
may be approximated
tit
36
by the difference of the approximations for Ly¢(y)a~ and Iy¢(y)a~ ,
n
L b.(a' -all)L ¢(y),
i=l 1 Yi
Yi Y
if
I (R'n
- R")
n Ly ¢ (y)
I,
where
R'n and R"n
(2.8)
are the
R'
n s
remainder terms obtained by using (2.5) to express
in the
Ly¢(y)a~ and
\!..y ¢ (y) a"y respectively,
is sufficiently small.
.
Usually, we will be interested in linear functions of the
difference between annuity functions from two life tables, say
Ly ¢ (y) (a~ - a~)
¢(y) ~o
where
V y and Ly¢(Y)
,
is not necessarily very small.
Nevertheless, in order to study relative accuracy, it is convenient
to suppose that the
.
'
¢(y)
have been standardized so that Ly¢(Y) =1.
If the annuity values are known for every age in the interval
over which
is summed, then (2.5) may be used to assess the
y
accuracy of the approximations, (2.7) and (2.8).
In general, all the necessary annuity values will not be known so
to evaluate the accuracy of the approximations, some reasonable
assumptions about the bounds for
Suppose
V n ~y ~
B,
y
and
IRnl
will have to be made.
is summed over the interval
LY¢ (y)
¢(y)
~O
Without any loss of generality, we may
= 1.
make the restriction that
[n,B],
h
(n, B) • With this restriction,
and the
R
n
y.'s
1
all be in the interval
involves unknown parameters only
through the n-th derivative of the annuity values in the interval
(n,B) .
It is obvious that a very crude upper bound for
~
I-~
[I
I
IRn I
IR I max.
Dna I
Iy -h In¢(y) +
lb. (y. _h)n 1Jl
n
x y=n
. 1 1 1
n<x< B n.
1=
is given by
,
37
and
Rn
n
is the remainder obtained by approximating Ly¢(y)ayby
1: b.a
. Therefore, to make assumptions about bounds for the
~ y.
~
. 1
~=
absolute value of the remainder term we need only investigate
In Section 2,1.1 we noted that over the range of relevant ages
(in practice) it can be assumed that annuity values from different
mortality bases are similar type functions of
ax
against
x,
x.
From graphs of
such as those in Figures 2 and 3, it appears
reasonable to take values of
Makeham-Gompertz law with
n
Da
x
A = .002,
Therefore, we will investigate
of the order of those for the
B = .0000357,
and
c =1.0996589.
assuming the Makeham-Gompertz
law.
2.2.3
The n-th Derivative of the Annuity Function Assuming the
Makeham-Gompertz Law
If we assume the Makeham-Gompertz law
I
(lJ
x
=A
+ Bc
x
),
then
x lJ dy =Ax + (BcX/fu c) - B/fu c
Oy
x
x
c
= -fu s -fu g -1 ,
Consequently,
(2.9)
38
ax
The usual practical approximation for
ax :;: ax +..L2
Hence,
is
(Jordan (1975, p. 49)).
n
. Dn. D a x =;= a x
and we will use this approximation for
Recall that,
using this and (2.9) it is easily seen that
_
xfoo t t t
cx(ct_l)
Da = (m c) (mg)c
v s (c -l)g
dt
x
0
(2.10)
Throughout this section, let
. . foo t t t
. x ( t 1)
1.=(lng)J c Jx vs (c _l)Jgc c - dt
J
.0
where
In s= -A and m g= -B/lnc.
Clearly,
(2.12)
Dr. = (m c) (j 1. + I. 1)'
J
From (2.10) we see that
(2.11)
J
J+
(2 .13)
we will use (2.12) and the following lemma.
To approximate
Lemma 2.1.
where
a = Bcx/ln c= _cXmg
Let
-0 = In v,
andh = (A +o)/ln c
then
1(_l)r(~)aj+h-rf(r
I. =ea(m c)-l
J
r=O
where
-h, a)
(2.14 )
f(r -h, a)
is an incomplete gamma function,
oo -t a-I
(f (a, x) = x e t
dt).
f
Proof:
-0 =m v
. . foo
>v
I. =(mg)JcJx
(c
J
0
Let
Y =c
t
-1
then
t
t'
=e
-ot
,
-l)Jexp[cx(c
therefore, from (2.11) we obtain
t
-l)(mg)- (A+o)t]dt.
t = [m(y +l)]/m c,
dt =dy/[(y +l)m c],
and
39
I j = (-a) j (in c) -1
Jooy j (y +1) -h-1 e -aydy.
(2.15)
o
Using formulas given by Erde1yi (1954, pp.129, 137) we observe that
(2.17)
Applying the following formula given by Abramovitz and Stegun (1974,
p. 262):
'(P
~[x
n
dX
-a
n -a-n
rea, x)] = (-1) x
rea +n, x),
n =0,1, ...
and Leibnitz's formula for differentiation of a product,
n
D (uv) =
n [ rn ) Dn-ruDr v ,
L
r=O
to evaluate the derivative in (2.17) we determine that
. - ·hl
- e -aYdy=
yJ(y+l)
OO.
Jo
. +r (0) a h- r
Lj(_1)J
; e a
r(r
-h,a)
(2.18)
r=O
. Substituting (2.18) into(2.l5) we obtain (2.14).
Theopem 2.Z Assume Makeham-Gompertz Law of Mortality applies
then
Dna
x
where
$~j),
~ ea(fu c)n-l .I1 $0)
n
t..
(_l{[j)aj+h-rr(r -h, a)
r
J=
r='0
a Stirling number of the second kind, is the number of
ways of partitioning a set of n elements into
a, h, and
x
non-empty subsets,
are as in Lemma 2.1.
-n n
T = (in c) D aFrom the expressions for
n
x
in (2.13) and Dro in (2.12) it follows that
Pr>oof:
Da
r(r -h, a)
j
Let
J
40
n
I
T =
n
(2.19)
. I.
b
j=l n,J J
with
b
. =0
n, J
j =0
if
or
n =j - 1,
(2.20)
and
b
(2.21 )
1 . =jb . +b . 1
n+,J
n,J
n,J-
We must determine the exact form of the
b
..
n, J
From (2.20) and
(2.21) it is clear that
b . . =b. 1 . 1
J,J
J- ,J-
and
b
= b
n,l
However, (2.13) implies that
b
1,
.
n-1,1
1 =1,
therefore,
bj,j =b j _1 ,j_1 = ... =
and
bn,l = bn-1,1 = ... = b , 1 = 1 .
1
To solve the difference equation, (2.21), with
may get a special solution of (2.21) for
b
is determined from
or
where
H.
J
b
n+1,2
b. 1 . =0
J- ,J
n, j
= 2b ' +b
n,2
n,l
= 2b n, 2 + l.
A special solution of this is
b
n,2
=-1
'
then
n
b n ,2 =2 H2 -1
2
b2, 2 = 2 H2 - 1 = 1 => H2 =
so, in general,
b
n,2
n 1
= 2 - -l.
Similarly, it can be determined that
1
2'
j
fixed, we
then add to it
b. . =1.
J,J
Let
.n
J H.
J
j = 2,
41
Assume
(2.22)
where
If
j
is a positive integer then
~jg(X) =
=
f [~J
i [~J(-l)rEj-rg(X)
(_l)j-r Er g (x)
r=O
r=O·
and
j
E g(x) = g (x + jk)
where
k
is any constant.
Taking
then
Let
b' . = (j -l)!b .,
n,J
n,J
if (2.21) holds then
b'
. =jb' . +(j -l)b' .
n+1,J
n,J
n,J-1
Under the assumption (2.22)
b' . =~j-1(1)n-1
n, J
1
= j (-1) r [j 11 (j _ r) n .. 1
r=O
)
I
.
ten,
sJ.nce
h
~
(_1)r+1 = (_1)r-1
1
b'
. =jb' . +jI (_1)r-1 r [j-1)(j _r)n-1 .
n+1,J
n,J r=l
r
Recalling that,
and letting
r' =r -1, we obtain
k = 1,
42
-2
b
=jb ' . + j L
(_l)r I (j -1) [ .J-,2) (j -1 _rl)n- 1
'n+1,j
n,J r'=O
r
=jb ' . +(j -l)b ' . 1 .
n,J
n,J-
.
Therefore,
does satisfy
bI
• =j b I
•
+ (j - 1) b I • 1
n+l,J
n,J
n,J• = (j - 1) ! b
. the assumption (2.22) satisfies (2.21).
n,J
n,J
Also, the assumption (2.22) satisfies the conditions b . . =1, since
J,J
n n
L'I x =n!, and b 1 =1, since L'l°(l)n =1. Therefore, substituting (2.22)
n,
and since
bI
in (2.19) we see that
(2.23)
Recalling that,
implies that
and using the fact shown in Freeman (1960, p. 124) that
it follows from (2.23) that
Tn
=
=
.
Since,
43
i t follows that
(2.24)
Substituting the expression in Lemma 2.1 for
into (2.24) the
I.
J
desired result is obtained.
n
In order to examine the behavior and magnitude of
I wrote a program to evaluate the approximation of
Theorem 2.1.
To evaluate
r(r -h, a)
D a Inl
x
Dna
given in
x
the program uses a sub-
routine, written by Vernon Chinchilli, based on an algorithm given by
Fullerton (1972).
From studying the values calculated by the program, it seems
reasonable to assume that,
(i)
(ii)
IDnax In! I
IDna In! I
x
n
increases, for
decreases as
n
increases for all
the points of inflection of
Da
Tables 2.6 and 2.7 illustrate these points.
for four ages (x) and
n =1,2, ... ,7
x =1,2, ... ,100
not found.
A = .002,
Da,
x
i = .10,
increases for
1
~x ~
SO
Values are only given
and a contradiction was
B = .0000357,
ID2aso/2!
the Makeham-Gompertz law and
and
c = 1. 09965S9,
SO < x < 100 .
I < ID3aso/3! I
From
but
n~3. For the same.parameters in
IDnaso/n!1 > IDn+lasO/(n+l) I I for
the point of inflection of
x
and decreases for
Table 2.7 it can be seen that
x except
in these tables, however, the
values were calculated for
Using
n ~ 3;
decreases as
i = .06, our calculations suggest that
Da
x
is
x =74
but
so even at the points of inflection
decreases as
that
n
increases in some cases.
Tables 2.S and 2.9 suggest
44
(i)
...
.
IDna In!
x
I
is very small for
n ;::5;
I
x;::: 94
is attained at
n ;:::4;
for
(ii)
max IDna In!
x
(iii)
max IDna In!
x<80
x
I
is substantially smaller than
max IDna In!
x<96
x
I
for
n ~4.
TABLE 2.6
8
Approximate Value of
Makeham-Gompertz Law
x
n
e
..
(A
= .002,
i = .06
40
20
!2Dna
n!
x
B = .0000357, c
=1. 0996589)
60
80
1
-4 403 733.316
-11 587 057.628
-23 072 482.543
-26 145 128.907
2
-115 317.831
-250 085.041
-268 774.084
181 144.786
3
-1 791.940
-2 312.232
3 252.705
9 102.033
4
-15.475
15.374
125.199
-67.506
5
.006
.779
.665
-3.907
6
.002
.010
-.028
.009
7
+.000
+.000
-.001
.001
45
TABLE 2.7
8
~ Dna
Approximate Value of
Makeham-Gompertz Law
nl
x
(A = .002, B = .0000357, c =1.0996589)
i
= .10
Maximum Approximate Value of
Makeham-Gompertz Law
108
1
nl Dn",.
"'x
(A = .002, B = .0000357,
C
I
=1.0996589)
i = .06
-
n
max
Attained
at . x of
x <96
max
Attained
at x of
x <80
1
27 302 662.042
74
27 302 662.042
74
2
405 527.404
95
303 503.491
52
3
9 560.164
77
9 560.164
77
4
909.475
94
130.848
63
5
80.949
94
3.936
79
6
8.461
94
.055
69
7
.821
94
(.001)
-
46
TABLE 2.9
It
8
I~Dna
n!
xI
Maximum Approximate Value of
Makeham-Gompertz Law
(A = .002, B = .0000357, c = 1. 0996589)
i = .10
.
n
max
x <96
Attained
at x of;
max
x <80
Attained
at x of
1
19 191 783.851
80
19 179 089.526
79
2
514 018.266
95
253 781.863
62
3
44 416.096
95
8 163.503
79
4
6 263.386
95
133.355
69
5
852.184
95
3.022
79
6
101.159
95
.061
73
7
10.982
96
.003
79
Examples
2.2 A
Table 2.10 gives the values of
b.
1
for the n-ages approximation
of
y**
L
ep(y)f(y)
where
y=y*
and the
A'S
y
!
y**
L A
y y=y* y
ep(y) = A
are given in Table A.4 of the Appendix,
selected values of
yare
y., i =1,2, ... ,5.
1
n =5.
The
47
TABLE 2.10
Values of
[y*, y**]
= [1,
96]
[y*, y**]
b.
1
= [20,
96]
[y*, y**]
= [20,
96]
i
y.
b.
y.
b.
y.
b.
1
20
.1729800
25
.0824503
30
.448767
2
35
-.0604667
40
.260893
40
-1.22522
3
50
.6140280
55
.326062
50
2.71882
4
65
.0542104
70
.233794
60
-2.05944
5
80
.2192480
85
.0968005
70
1.11707
1
1
1
1
1
1
Using the values in Table 2.10 we approximate
y**
I
Aa
y=y* y y
/
y**
I
A ,
y=y* y
(2.25)
5
by
I
. 1
1=
(i)
b.a
1
y.1
Table 2.11 illustrates that
the absolute values of the errors made using the n-ages
approximation for (2.25) for two mortality bases of the kind
that might be encountered in practice are of the same order
of magnitude when the same selected ages are used;
(ii)
the absolute value of the error made using the n-ages
approximation varies with the set of selected ages used, but
appears to be reasonably small, provided the selected ages
do cover fairly well the range of summation.
TABLE 2.11
co
o::t
y**
Five-Ages Approximation for
(1)
(2)
L
Aa
y=y* y y
(3)
!y**
LA, A from Model Office Data
y=y* y
Y
(4 )
(5)
Actual
Approximation
L/\y)
L
[ 5 b.a )
i=l 1. Yi
(6)
(7)
Error
IError I as
(5) - (4)
% of Actual
y*, y**
Selected Ages
1,90
20,35,50,65,80
C58
11.2024
11.2163
+.0139
.12
1, 96
20, 35, 50, 65, 80
M69
11.1192
11.1084
- .0108
.10
20, 96
25,40,55, 70,85
M69
10.796181
10.796186
.000005
20, 96
25,40,55,70,85
MG
11.966779
11.967511
-.000732
20,96
30,40,50,60, 70
M69
10.796181
10.840785
.044604
.41
20,96
30,40,50,60,70
MG
11.966779
11.954446
-.012333
.10
Life Table
(LyAya y /
e
e
~
.00005
.006
e
CHAPTER III
ASYMPTOTIC DISTRIBUTIONS
OF ESTIMATES OF ACTUARIAL FUNCTIONS
The life annuity, temporary life annuity, deferred life annuity,
and deferred temporary life annuity are all linear functions of
Vttpx'
therefore, all results given in this chapter are applicable
to each of these annuities.
To simplify notation the results are
given for the immediate life annuity only.
The other annuity
functions (and estimates thereof) can be obtained simply by varying
the limits of summation.
The present value of an immediate life annuity,
a ,
x
can be
estimated unbiasedly by
ax
where
=
w-x-I t
L v t x
t=l
P
is an estimate of the probability that a life aged
will survive to age
x
+ t.
x
Throughout, our calculations are
conditional on the actual observed numbers exposed to risk each year.
Under this condition, we estimate
t Px
by
x+t-l
=
where
k
A
Pk
to age
nPk
k=x
is an estimate of the probability of surviving from age
k
+
I,
Pk .
50
Let n = 4c n ' where for any age
x
n
x
x,
we let
- be the effective number of lives exposed to risk at age
x
(the "initial exposed to risk"),
Sx - be the number of lives that survive from age
age
x out of the nx-I alive at age
Assume each of the
between ages
age
x
x
to age
n
and
x +1
x
(x -1) to
(x -1) .
lives has the same probability of dying
x +1
then the probability of surviving from
is estimated by
A
/
Px =s x+l n x·
3.1
Asymptotic Joint Distribution of Annuity Function Estimates
for m-Ages
3.1.1
Mean of Estimate of Annuity Function
We have
w-x-l
t
Ea = L V EtP .
x
t=l
x
A
\'
A
Since
it follows that
A
EPx = Px .
The
f'
Pk
for all
k
are independent, therefore,
and
w-x-l
=
= a
3.1.2
I
t=l
v
t
P
t x
x
Variance of Estimate of Annuity Function
The variance of the estimate of
a,
x
51
(3.1)
the
Recall that,
are independent, and
EPk =Pk' Let t <u then
. tAU A
t+u A· A
tAU A
Cov(v tPx' v upx) =E(v
t Pxup)
x -Ev t PxEv uPx
=v
t+u
A
u_tpx+tVar(tpx)
Substituting in (3.1) we observe that
w-x-l 2t
t u
Var(a ) = I v var(tp ) + 21 I v + t P var(tp),
x
t=l
x
t<u
u- x+t
x
Since
(3.2)
s x+ l~ Bin(nx , Px ),
Var (p ) = P q In .
x
x x x
and
(3.3)
Suppose that
nk
+00
with
nk/n ,
k
+c
k , k' >0.
It follows that
w-x-l x+t-l 2t 2
nVar(a ) +
I (v tpxqk/ckPk)
I k=x
x
t=l
w-x-2 w-x-l
+2
I
I
u=t+l
t=l
As
n
k
gets large
(3.4)
52
Px -.t
- x+l l.tx
Substituting
and
tPx =.tx+t/.tx
in (3.4) we obtain the
computational formula for the asymptotic value of
~
nVar(a ) =
L
t=l
(vt.t 1.t)2
x+t x
w-x-2
L
+ 2
x
x+t-l
w-x-l
x
nVar(a),
t=l
L [(-; - -;+l)/ck~+l]
k=x
x+t-l
(vt+u.t
+
.tx+tl.t;)
I [(-;
I
x u
u=t+l
k=x
w-x-l
-~+l)/ck~+l]'
(3.5)
Making the same substitutions in (3.3) and (3.2) we obtain the
A
following computational formula for the variance of
Var(a:x )
"':rV2tff ~~+l (~ -~+l)
=
w-x-2 w-x-l
+2
I
t=l
I
u=t+l
+
a :
nk~+l)/nk~J
x
- C./.x+/.t/}
~
n
(~+l(~-~+l)
k=x
{X+t_l
(vt+u.tx+i.tx+t)
+
'\~+l)/nk~J -
(.t X +t l .tx
)2}
(3.6).
From (3.6), it can easily be seen that if the number exposed to
risk,
then
n ,
k
approaches zero at any age greater than or equal to
Var(a)
x
possible that
approaches infinity.
n
k
In practice, it is quite
will be zero for some ages less than
we want to calculate the variance of
presented by
w*,
n =0,
k
x,
w -1.
If
to avoid the problems
we will have to determine a practical value, say
beyond which calculations will be discontinued.
If the company writes off the business on persons older than a
particular age, then take this age as
as a practical rule
w* =100.
w*.
Otherwise, one might take,
(See Table 3.1 for example of size
error made.).
Obviously, if the company writes off the business on persons
older than a given age any differences in the values of monetary
53
functions calculated by using
w*
in place of
importance to the particular company.
are just approximating
a
If
w*
by
x
ware not of
is taken as 100, we
The error made by this
approximation,
ax - a X:~~-XI
9~ = v
99-x
99 -xpx a 99 ,
is negligible, except perhaps for the extreme old ages where great
accuracy is not required.
is written off or not.
made using
w* =100
This error is applicable whether business
Table 3.1 gives for selected ages the error
rather than
w equal to the largest age given
by the A67 life table in calculating
table.
Both
v
99-x
and
a
x
based on the A67 life
decrease as
x
decreases.
Table 3.1 clearly indicates that the error is very small in this case,
even for
x
as large as 80.
TABLE 3.1
Error Made Approximating A67 a
b
x
Y
A67
ax: 99-xl
i = .08
i = .04
error
a
.7771
4.7946
7.9570
60
50
ax: 99-xl
error
1.2777
.5856
.6921
.0057
4.0571
4.0545
.0026
7.9549
.0021
6.2550
6.2544
.0006
11. 5512
11.5501
.0011
8.3822
8.3820
.0002
15.0030
15.0023
.0007
10.0626
10.0625
.0001
x
a
98
1.3848
x: 99-xl
.6077
80
4.8003
70
x
We may assume
a
x
n >0 i f k :<;;w* -1 since it is very unlikely
k
that there will not be at least one exposed to risk for each k :<;;w* -1.
54
I have written a Fortran program to evaluate
its asymptotic value.
Using the
nVar
(ax )
and
lx given by the total United
States population 1969-71 life table
(U.S~
Life Tables (1975, p. 6))
for ages less than 100, and the exposed to risk,
n,
for assured
x
lives 1967-70 of durations 2 and over contributed by forty-four
companies in the United Kingdom to the continuous mortality
investigation conducted by the Institute and Faculty of Actuaries
(JMIC (1974), pp. 160-163) the values in Tables 3.2 and 3.3 were
calculated by this program.
(See pages 55 and 56 for Tables 3.2 and 3.3.)
Tables 3.2 and 3.3 show the actual and asymptotic values for
nVar(~)
x
using interest rates of 4 and 8 percent respectively.
Columns 2, 3, and 4 in each table show the actual value of
if each value of
n
x
nVar(~ )
x
reported by the forty-four companies combined
had been divided by 100, 10, and 1 respectively, so that they might
be thought of as representing respectively a small, a large, and
an extremely large company.
of
nVar(a)
x
ratio
nk/n
Column 5 shows the asymptotic value
if we assume that
n
increases without limit, the
remaining constant and equal to those in the JMIC;
column 6 shows the value if
nk/n =lk/I~,
where
the total U.S. population 1969-71 life table.
~
is given by
The figures in
these columns show that the asymptotic values are sensitive to the
values of
nVar(a)
x
nk/n
3.1.3
nk/n,
but they are reasonably good approximate values of
even for fairly small values of
n
i f we assume that
does have the values actually observed.
Covariance of Estimates of Annuity Functions at Two Ages
Let
x. =x. +r.
J
1
To determine the covariance of
A
a
x.1
and
A
a
x.
J
55
TABLE 3.2
n Times Variance of Annuity Function Estimate
Using U.K. Assured Lives: Durations 2 and over Experience
Assuming Total U.S. Population 1969-71 Life Table with w* =100
i
Actual
Age
ex)
e
..
e
= .04
n =
x
A67
n 1100
x
n
A67
x
=
n 110
x
Asymptotic
n
A67
x
=
n
x
n In from
x
n x In from
JMIC
U.S. ' 69-71
10
165903.83
165786.56
165774.84
165773.58
555.31
15
13785.17
13784.11
13784.01
13784.00
734.96
20
639.79
639.27
639.22
639.22
858.23
25
619.96
619.18
619.10
619.09
978.93
30
769.93
768.76
768.65
768.63
1184.64
35
1017.57
1015.81
1015.64
1015.62
1469.27
40
1384.74
1382.07
1381.81
1381. 78
1820.04
45
1929.09
1924.99
1924.60
1924.55
2230.71
50
2757.33
2750.93
2750.31
2750.24
2704.24
55
4060.84
4050.52
4049.52
4049.42
3265.68
60
6307.28
6289.90
6288.23
6288.04
3967.87
65
10009.71
9978.87
9975.89
9975.56
4969.78
70
13493.04
13436.75
13431.34
13430.74
6595.51
75
16448.42
16339.47
16329.07
16327.92
9625.44
80
20776.34
20527.28
20503.74
20501.14
16166.81
85
29494.42
28755.27
28686.4 7
28678.88
31608.12
90
56615.16
53457.23
53168.82
53137.06
53137.06
95
117555.74
106126.10
105066.63
104949.84
104949.84
56
;~
TABLE 3.3
n Times Variance of Annuity Function Estimate
Using U.K. Assured Lives: Durations 2 and over Experience
Assuming Total U.S. Population 1969-71 Life Table with w* =100
i = .08
Actual
Age
ex)
nx
A67
=
n 1100
x
n
x
A67
=
n 110
x
Asymptotic
n
A67
x
=
n
x
n x In from
n In from
x
JMIC
U.S. ' 69-71
10
46821.89
46792.60
46789.67
46789.36
61.64
15
4075.78
4075.60
4075.58
4075.58
99.40
20
97.93
97.92
97.92
97.92
121.43
25
68.18
68.18
68.18
68.18
136.89
30
78.69
78.68
78.67
78.67
175.39
35
108.00
107.97
107.97
107.97
242.03
40
162.48
162.41
162.40
162.40
339.62
45
262.80
262.64
262.63
262.62
473.21
50
455.80
455.43
455.40
455.39
651.12
55
842.08
841.23
841.15
841.14
891.17
60
1723.05
1720.98
1720.78
1720.76
1221. 52
65
3619.72
3614.49
3613.98
3613 .92
1730.88
70
5870.47
5857.69
5856.45
5856.31
2616.66
75
8210.97
8179.40
8176.34
8176.00
4409.25
80
11685.16
11594.25
11585.53
11584.57
8627.64
85
18433043
18091.75
18059.57
18056.02
19806.82
90
39756.20
37873.45
37700.18
37681.09
37681.09
95
96509.75
87622.78
86797.03
86705.98
86705.98
e
e
57
it is convenient to use the fact that
~
r
A
A
A
p a
a X. = ax .:r,
:=1 + v
r x. x.
1
1
A
The covariance of
A
Cov(a
A
x.
, a
and
x.1
a
,
x.
J
A
x.
J
1
Since the
a
) = Cov(a
x.:r
v
+
r
1
A
A
p a
r x. X.'
J
1
A
a
are independent
Pk's
J
1
A
ax. ).
J
A
and
x. :"FI
a
1
x.
, aX. )
=
r
V
J
1
A
are independent
J
and
Cov (a
x.
A
A
Cov( p a , a ).
r x. x.
X.
J
1
J
Recall that
Cov (XY, Y) =E (X) Var (Y) ,
when
A
X and Y are independent.
a
are independent,
x.
J
therefore,
A
r
A
A
Cov(a , a ) = v p Var(a ) ,
x.
x.
rx.
x.
J
1
nCov(a ,
x.
1
J
1
ax. ) =v r rx.
p [nVar(a )]
x.
J
(3.7)
J
1
and its asymptotic value
r-----J
, a )
x.
X.
nCov(a
J
1
3.1.4
= v
r
r---J
[nVar(a )]
r x.
x.
A
p
1
(3.8)
J
Distribution
For a fixed interest rate the estimated annuity function,
is a linear function of the estimates
A
a ,
X
t =1,2, ... ,w-x-l. To
determine the asymptotic joint distribution of the
A
a
x.1 , i =1,2, ... ,m
we first determine the asymptotic distribution of the set of variables
..,
t=1,2, ... ,w-x. -1
1
and
i =1, 2, . . . , m.
Using the central limit theroem it can be shown that
v1l
X
(pX -px )/;pq
rv
x x
N(O, 1) asymptotically.
(3.9)
58
This fact and the following three lemmas will be useful in determining the asymptotic distribution of the
Lemma 3.l
.
If
.
t
Px. 's.
1
Xn ~L X
and
Yn -p+ 0 then
X.
n +Y n -,-)-L X
(See Rao (1965), pp. 122-123.)
Throughout, we wi 11 suppose as before that
constant, that is,
Lemma 3.2
nk/nkl +c
In (tPx -tPx)
with mean zero and variance
k >0
k, '
and
11 =
Let
L
Sk = Iiik(Pk - Pk),
t l
n+k=x-
L
= In
(Pk + ski
nk/nk I,
lim n/n =c .
k
n+oo
(tpxqk/ckPk)'
then
In (tPx - tPx)
l
l
= Ii1(XTT P - xTT Pk)
l k=x k k=x
with
is asymptotically normally distributed
x+t-l
2
k=x
P1:'oof:
n +00
Ii\) - x+t-l
nk=x PkJ
59
x+t-l
L ~Px 2kl (Pk ~)] is a 1inear func t ion of the
k=x
'"
sk'
k =x, x +1, .. . ,x +t -1 which are independent
The function
random variables
and using (3.9) we see that each is asymptotically distributed
x+ l' ... ,Ex+ t - 1)
(£ , £
x
I.
t-variate normal distribution, therefore,
i
x+t-l
I
I
[X+t-l
IRn I s;
+
I
i! ..
+
Ie
;.
{l: l:
vn k<k'
I II
[tPx I£k£k'
[t p) £kEk' Ek " I
.~~k~<:~ll
=R'/1il
n
(PkP k ,pk"/ckc k 'C k " )]
I
)
and so
Since the
",2
ES
k
i
I
)]
'
ER
\
/
We have
!Ck 1
£k I
k=x
I~
I
R.
I/ /(PkPk'
!eke: '
'
I
I
]
qx/(Pkck)]
asymptotically.
We now consider the remainder term
II
2
k~X ~PxEk/(Pk~)]A.JN'O, k~X [tPx
I
I
has an asymptotic joint
Ek'S
2
n
s; E (R ') 2
n
In
are independent,
does not depend on
n,
(3.10)
E IE
I s; ~,
and since
it follows that
E (R ' ) 2 < B
n
where
B does not depend on
~n
n.
s;
(B/n)~
s;
elm
(3 .11)
Hence from (3.10) and (3.11)
60
Using the Markov inequality
Pr. [ IRn I
; : 0]
~ E IR
n
~ 0- 1
I/o
vtRn2
we obtain
Pr [ IR
n
as
So,
I ; : 0]
~ C/ olii -+0
n -+00.
R converges to zero in probability.
n
x+t-l
Since
I tPx£k/{PklEk'") has an asymptotically normal distri.
k~
bution and
R
converges to zero in probability it follows from
n
Lemma 3.1 that
Iii
(p
- p ) /"\-.I
txt x
N.(O
'
aSyniptotically.
Lemma 3.3
i
=1 , 2, ... , m
Let
and
n·1 t
~
=III
(t
PX. - t PX. ) ,
1
be the vector of
arbitrary fixed vector of
bit'S
t
1
nit's.
t =1,2, ...
Any arbitrary linear combination of the
=1,2, ... , w - x.1
Let
,w-x.-l,
nit's,
1
~'n
b
- 1,
be an
i =1,2, ... ,m.
is
asymptotically normally distributed.
m
=
\'
L.
i=l
w-x.-l x.+t-l
1
1
I
t=l
m
w-x.-l
1
Ct)
)] +
b.tR
1
,
[bit t p €k/ (PklC:
k
X.
. 1 t 1
1
n
1
1=
=
k=x.
I
1
I
I
61
m
L
i=l
W-x. -1 x.+t-l
1
1
L
L
k=x.1
t=l
[bit
m
=
L
i=l
is a linear function of the independent asymptotically normal
variables
Therefore, it is asymptotically normally distributed with mean
zero and variance
W_2!
klx
1
m
i~l
It can be shown using the same method as in Lemma 3.2 that
m
w-x.-1
L
i=l
1
b. R(it)
lt n
L
t=l
converges to zero in probability.
It follows from Lemma 3.1 that
distributed.
Whatever be
Theopem 3.Z
b'n
is asymptotically normally
b' .
The set of random variables
t =1,2, .•. ,w -x. -1
1
and
i =1,2, ... ,m
A
p
x.
1
has an asymptotically mu1tit
normal distribution.
Froof:
•
Let
~
be as in Lemma 3.3.
cally multivariate normal.
such that
~'~
Assume
~
is not asymptoti--
Then there exists at least one vector
is not asymptotically normal.
Lemma 3.3, therefore,
n
But this contradicts
is asymptotically multivariate normal.
b
62
It follows that the set of random variables t P ,t =1,2, .. " w - x. -1·
and
x.l.
has an asymptotically multivariate normal
i =1,2, •.. ,m
l.
distribution.
To show that the set of functions
(axl ,a , ... ,ax )
has an
2
m
asymptotic multivariate normal distribution we use the following
X
lemma.
Lemma 3.4 A set of random variables each of which is a linear
function of a set of asymptotically multinormal variables, itself
has an asymptotically multinormal distribution.
Proof:
(Kendall and Stuart (1969, p. 350), case
x' = (xl' x 2 ""'xn )
].l = 0).
Let
be a set of asymptotically multinormal variables.
Then by the continuity theorem for characteristic functions
E[exp(it'x)]
where
exp[t'].l - ~ t'Vt]
l.
E[exp(is'Ax)]
Therefore, the set of
exp[t'].l -~ t'Vt]
is the characteristic function of a
n-variate normal distribution.
s.
-+
Suppose
real and
where
A is any fixed
exp[s'A].l
-+
t' =s'A
m xn
-t s'AVA's]
m linear functions of
x
defined by
has an asymptotically multinormal distribution with mean
variance-covariance matrix
Theorem 3.2
matrix then
AlJ
y =Ax
and
AVA'.
The annuity function estimates,
w-x. -1
l.
t
\' v
a x. =
L
t=l
l.
"
A
t
Px.
i
=1,2, ... ,m
l.
•
have an asymptotic joint multinormal distribution.
~.
63
Proof:
A
The random variables
ax.'
i =1,2, ... ,m
are linear
1
functions of the set of asymptotically multinormal variables
"-
tPx.'
1
t =1,2, ... ,w -x. -1
1
and
i =1,2, ... ,m,
a'"
the annuity function estimates
i =1,2, ... ,m
,
x.1
therefore, by Lemma 3.4
have an asymptotic
joint mu1tinorma1 distribution.
3.2
Asymptotic Distribution of the Estimate of a Linear Function
of Annuity Functions for m-Ages
We may estimate an arbitrary linear combination of the annuity
functions
a
x.1
,
i =1,2, ... ,m,
say
m
L cp (x.)
a
1 X.
. 1
1=
1
by
m
L cp (x.)
i =
. 1
1 X.
1=
1
w-x. -1
m
I
i=l
t
1
L
t=l
ep(x.)
1
A
(3.12)
v tP
Xi
The mean of this estimate is
E[
I
1=
m
a
cP (x . )
1
1 X.
.
1
)
J = . L1 CP(x.)E(a
1
X.
1=
1
m
=
L ep(x.)a
1 X.
. 1
1=
(3.13)
1
The variance of the estimate is
var(
Let
var[
I ep(x.)a
) = I cp2(x.)var(a )
. 1
1
x.1
1 X.1
1=
. 1
1=
1
m
\L. cp 2 (x.)Var(aA
I cp(x.)a 1) = 1=. 1
. 1
1=
1
1'<'J
Using (3.7) we see that
r =X. -x ..
J
Lep(x.)cp(x.)Cov(a ,a )
1
J
X.1 X.J
+2L
X.
1
x.1
) +2 L L.ep(x.)ep(x.)v r p Var(a'" )
'<'
1
J
r x.1
x J.
1 J
\
\
(3.14)
nvar(
I cp(x.)a
. 1
1=
1
m
X.
1
) =
L cp2(x.)nVar(a
)
1
x.
. 1
1=
1
+2I Iep(x.)ep(x.)v
'<'
1}
1
J
r
p nVar(a ) .
r x.1
x J.
(3.15)
64
Its asymptotic value may be found by substituting the asymptotic
values of
nVar(a )
x.1
and
nVar(a
x.
) for
nVar(a
X.
J
respectively.
)
and
nVar(a )
x.
1
J
Theorem 3.3 The estimate of a linear function of annuity
functions for m-ages
m
m
I cHx.)a
= I
. 1
1 X.
1=
i=l
1
w-x.-l
I
1
t=l
¢(x.)v
t
1
is asymptotically normally distributed.
FPoof:
Let
nit
=
v'TI(tPx. - tPx)
1
v'TI(
I ¢ (x .)a. - I ¢ (x . ) a
. 1
1=
1
X.1
. 1
1=
1
is a linear combination of the
1
I
J =l' =1
X.
1
nit's
w-x. -1
t
1
I
t=l
1
Xi
I
¢(x.)a
1
X.
1
1
Thus, the estimate
m
. 1
- tPx.)
and it follows from Lemma 3.3
that it is asymptotically normally distributed.
1=
A
¢(x.)v vh(tP
is asymptotically normally distributed .
CHAPTER IV
TEST(S) OF ADEQUACY OF A STANDARD LIFE TABLE
Life insurance companies are naturally keenly interested in
changes in mortality patterns, as well as fluctuations in interest
rates, since the annuity and insurance payments they make depend on
survival and interest rates.
Of course, in order to receive one of
these payments from an insurance company one must have paid the
company for it.
premium.
The amount that one PaYs the company is called the
The company wants to be sure not to set the premiums so
low as to incur a loss nor so high as to be uncompetitive.
So, they
would not want to use a life table that differs too much from the
mortality experience of their potential policy holders, for calcula-.
ting premiums.
Also, in order to estimate the needed policy reserve, one needs
a life table that is appropriate to the current population of policy
holders.
A question of major importance to the insurer is whether the
total estimated policy reserve for a group of policies is too large
or too small.
If this reserve is larger than it need be, an excess
amount of money is being held in low risk-low return investments
that could be yielding a greater return for the company.
If this
reserve is smaller than it should be, the company may not be able
to meet all future claims.
Therefore, the company will certainly
66
be interested in how its total policy reserves relate to those
appropriate to the current population of policy holders.
In this chapter we establish methods for determining whether
the life table being used by an insurance company is adequate in this
respect.
That is, whether the actuarial functions calculated using
a "standard" life table are "sufficiently close" to those based on
the "current" (Le., recent) mortality experience of lives insured
by the company.
While these methods are analogous to decision
procedures, since no attempt is made to actually determine losses at
risk the problem is really an incompletely specified decision theory
problem.
4.1
Formulation of Problem
The practical interests of the company will dictate the criteria
for determining whether the life table is adequate, that is, which
actuarial functions should be sufficiently close and the definition
of sufficiently close.
Therefore, no attempt will be made here to
establish criteria that can be used in all cases, but rather, several
possible criteria will be presented, in the hope that the procedure
used will illustrate how hypotheses might be established and tested
in accord with the requirements of a company.
The decision to be made is whether to replace a "standard" life
table by one representing more nearly "current" mortality experience.
In establishing criteria for determining whether the life
table should be changed, we must specify which actuarial functions
are of interest.
Obviously, if all of the actuarial functions that
are calculated by the company using the life table have values
67
sufficiently close to their values based on the "current" mortality
experience of the lives insured by the company there would be no
need to change life tables.
However, the company might be willing
to base its decision on a less stringent criterion where only a
subset of the functions that might be calculated using the life
table will be required to have values sufficiently close to their
values based on the "current" experience of the lives involved.
Among the many actuarial functions that are based on the life
table, and are commonly used, are
(a)
present value of an immediate life annuity of 1 at age
x,
(b)
x
present value of a whole life insurance policy of 1 for a life
aged
(c)
a ,
x,
annual premium for a whole life insurance of 1 for a life
aged x,
(d)
A ,
x
P ,
x
reserve at the end of
t
years for an existing whole life
insurance policy of 1 issued on a life aged
x,
tVx'
and
(e)
total reserve for a group of whole life insurance policies,
\' \' txt
S Vx'
L.xL.t
where
tSx
is the total face value of all whole
life insurance policies issued
t
years ago to persons then
aged, x.
Since an insurance company uses a life table in two major ways,
(i) in the calculation of premiums for new annuities and insurances,
and
(ii) in the valuation of net liabilities (required "reserves")
in respect of business already in force, the above functions are
68
likely to be of particular interest.
We note in passing that,
depending on the different classes of business involved, a greater
variety of functions, ranging from temporary annuities and endowment
insurances to complicated special contingent annuities and insurances,
may be needed.
However, our discussion will not be directly concerned
with these.
Of course, in both (i) and (ii) mortality is not the only
important factor; rate of interest is also relevant.
In the present
work, we do not emphasize possible variation in the latter.
We will
use, for the most part, a rate of 6% though we will also try to indicate how to determine ranges of rates of interest over which any
particular conclusion may be expected to remain valid.
Using the above five functions, we could list thirty-one
5
(=2 -1)
possible criteria that might be used for determining whether
to continue to use the standard life table by requiring that for all
the functions in various subsets of the above five, values based on
the standard life table be sufficiently close to current values
estimated from mortality experience for all ages in the "relevant
range".
We use the term "relevant range" to mean a range including
present ages of most policyholders.
It will vary somewhat from one
company to another, but the range 20-80 will usually be ample.
it has been determined which functions must be sufficiently
Once
close~
"sufficiently close" must (at last!) be defined.
In Section 1.4 each of the above functions were expressed as a
function
of the present values of life annuities.
In defining
sufficiently close we will find these relationships helpful since they
will allow US to define sufficiently close for each of the functions
~
69
in terms of conditions on present values of immediate annuities or
a linear function of them.
We define the values of a function based on the standard life
table as sufficiently close to the current values of the function for
the population if the absolute values of the error made by using the
standard life table to determine the values of the function are less
than or equal to given quantities.
Determination of these quantities
naturally depend on practical requirements.
Special cases are
obtained by choosing special values for these upper limits.
Among
the special cases is the case where a limit is set on the proportional error in the value of the function at each age in the relevant
range.
This is accomplished by requiring the upper limit for the
absolute value of the error made by using the standard life table to
be a certain proportion of the true (i. e., "currenti ') value of the
function.
In this case, the fact that
IC ~S I ~ £
will be used.
(a)
(4.1)
==>
Accordingly, we define sufficiently close as follows:
The present value of life annuities are sufficiently close if
£
We could have
£
x
= £ 'rf
x,
x
in the reI evant range.
> 0, 'rf x
but this is often not very reasonable.
Among the possible choices for
£
X
is
£ =£
x
C
a . in this case we will
x'
be setting a limit on the proportional error and using (4.1) the
lower and upper limits given by (4.2) for
S
a [-£/ (1 + £) ]
x
(b)
and
S
a [£/ (1 - £) ] ,
x
C
S
ax - ax
are
respectively.
the present value of whole life insurance policies are
sufficiently close if
(4.2)
70
(4.3)
since
A =1 -da
x
(4.3) is equivalent to the condition
x
sx
> 0, V x
in the relevant range.
sx might possibly be taken as sx =sCAx
(4.4 )
In this case it follows
from (4.1) that (4.3) and (4.4) are equivalent to
(c)
Annual premiums for whole life insurance policies are
sufficiently close if
, Cp
Since
p
x
x
_ Sp
= (l/ax )
I<
>0
x - nx ,nx
-d
Vx
'
in the relevant range. (4.5)
(4.5) is equivalent to the condition
Ic~x - S~x I ~ nx '
which implies that
S..
a
S..
I -n a -<~
C..
x x
a
x
~ I
S..
+nx a x
or
S..a
x
S..
1 +n a
x
8..
- 1
~
a
C
a
x
~
x
x
S..
- 1
1 -n a
x x
or
M
-nx
S.. 2
ax
HO(Px.) : 1 + n Sa
x x
vx
-
in the relevant range.
Among the possible choices for
(i)
nx
<
nx '
nx =nSpx =n ( _1_
- d) V x in the relevant range
S ..
ax
>0,
71
or
n = n/C"a
x
x
(ii)
for the older ages,
C
nx =n I a" x
(using
at the younger ages might be excessive) might be
nCPx
.
use d t 0 approxlmate
clearly
,Cp _Sp I
x
x
: :; nx
is equivalent to
HO(px):
(d)
c
I
SIS"
ax - ax :::;n ax
Consider reserves at the end of
Let
life insurance policies.
p'
x
t
years for existing whole
denote the annual premium per unit
amount actually being paid for a policy issued
life then aged
x.
C"
nx =nI ax'
For t hose ages were
h
t
years ago on a
If
t V >0, V x
x
in the relevant range,
t
~l,
(4.6)
then
ICa _Sa I:::;
Y
since
Y
t
v I(P' +d)
x
x
where
y =X +t,
tVx = 1 - (PxI + d) a
x+t .
Let
vy =
min
[tVxI(P'x +d)]
(x, t) : y=x+t
and let us consider reserves at the end of
t
years for whole life
insurance policies as being sufficiently close if
(4.7)
tVx
might possibly be chosen as
in this case, we may use
(4.1) to see that (4.6) implies that
StV 1[(1 +v)(P ' +d)] :::;C a _Sa :::;v StV 1[(1 -v)(P' +d)]
x
x
Y
Y
x
x
and define the reserves as being sufficiently close if
-V
72
HO(YX) ; vyI
$;
Ca - Sa
Y
Y
$;
v"
(4.8)
Y
where
S
=
max
- v tY /[ (1 + v)(P I +d)]
x
x
(x, t); y=x+t
v" =
min
v ~ Y 1[(1 - v) (P I +d)] ..
(x,t):y=x+t·
x
x
VI
Y
and
Y
Vy
in the relevant range.
The reserve for an existing policy should be distinguished from
the poZicy pesepve vaZues which would apply to future policies
issued at premiums calculated from the current mortality.
The policy
reserve which would apply to future whole life policies issued at
age
x,
after
t
years, would be
= 1 - C..
a
x+t
I Ca"
x
with ~Yx provides information
x
on magnitude of changes in calculated policy values which serve as
In this case, comparison of
bases for surrender values.
t
If it is desired to test statistically
Cy
for difference between
Cy
t x
and
~Yx we may use the statistic
A
s.·a
= s··x+t
a
x
C..
a
x+t
~a
x
The method of statistical differentials (see e.g., Johnson and Kotz
(1969), pp. 28-29) may be used to approximate the mean and variance
1\
A
of the ratio
COO
a
x+t
IC a..
x
as
73
A
A
A
C
C
E (C,.a t I a, , ) ,. [C..a t I a,,) {I + C..a -2 Var (C a.. )
x+
x
x+
x
x
x
-
c..
[ a x+t
-1
C..
ax )
Cov
[AC..
a + '
x t
and
A
Var [C..
.a
A
x+
C
tI "
a ) '.. [C..a
x
A
I C" )2{C a.. -2 t Var (C..a t
x+ t ax
x+
x+
-2 (Ca:x+t
1
/'\
J + C..ax-2
A
Coo)a
Cov [C..a
x
Var
(c..ax )
A
x+ t'
Ca.. }}
x
Since a more or less uniform increase (decrease) in mortality tends to
C..
and
a , it is likely that the
decrease (increase) both
x
ratios
Ii.x+t lax
will be less sensitive to change than are the
individual annuity values.
Hence, it is not likely that a require-
ment of the policy reserves which apply to future whole life policies
being suffiaiently alose to the standard values in order to retain
the standard life table will be critical.
(e)
The total reserve for a group of whole life insurance policies
is sufficiently close if
(4.9)
It is shown in Section 1.4 that
where
A = dS + G ,
Y
Y Y
S = total sum assured for all whole life policies on lives
y
now aged
G
y
= total
y,
gross annual premiums for all whole life policies
on lives now aged
y.
""""
.
74
Using this we may express (4.9) as
(4.10)
since
for
If
Ly Ay
\' Sand
!..y y
~
Ly Ay ' ~'Ly
~'
are
are fixed.
Three of the possible choices
Ay Cay,
and
~' Lx Lt
tSx
~vx
.
~=~'Ly AyCa y then using (4.1) we see that (4.9) is "equivalent
to
Usually, the proportional error in the total reserve is of greater
interest. If
~
=
~'LxLt tSx
lVx
then using (4.1) we may express
(4.9) as
Clearly,
so that if
,Sa
y
_Ca
y
I ~~/L y Ay
,V Y in the relevant range
then
Hence, in some cases it may be desired to tighten the condition given
in (4.9) by considering
(4.12)
The face value and gross annual premium are fixed for each policy
at the time it is issued so the
A'S are all fixed.
y
However, the
75
I
11
I
,e
close only if
1
I
HO(A)
holds for a particular set of
for certain other sets of
1
I
company might choose to define the total reserve as being sufficiently
A'S.
and also
That is, the total reserve is
y
4
Ay 's ,
sufficiently close if
I
H
n
-
O(A) - AEA
where
H
O(A) ,
A = {A: A is a vector of
Ay'S
for which
HO(A)
justify retaining the standard life table}, holds.
must hold to
Whatever the special form of the hypothesis to be tested, we
ultimately want to make one of two decisions,
VO: Continue to use standard life table
or
VI:
Discontinue using standard life table.
Consider each of
HO(AX)'
HO(ax)'
HO(px)' HO(vx)' and HO(A) as
hypotheses and denote the complement of these as
Hl(PX)'
HI (Vx)'
and
respectively.
f = {T: T = A, ax,
AX, px,
or
Hl(~X)'
Hl(AX)'
Let
Vx} .
The null hypothesis,
HO =
Q*
HO(T) ,
is to be tested against the alternative
HI =
where
f*
}-d,*
denotes the subset of
HI (T) ,
f
that contains the subscripts
used in denoting the hypotheses that correspond to the conditions
that must hold in order for the desired actuarial functions to be
sufficiently close.
Let all the
HO(A)'S
be of the form given in (4.12) or suppose
76
that
At r* .
If a condition
for a given age,
x,
H (T)'
T E r*
O
is not required to hold
in order to retain the standard life table,
for simplicity consider the lower limit given by
ca - Sa
x
x
as
_00
and the upper limit as
a
C
a
r ux = {yu : Y =upper limit for
u
Let
and define
+00,
C
r-ex={Y.e.: Yt = lower limit for
x
-
S
a
for
HO(T)
x
HO(T)' T Er*}
given by
S
given by H (T)' T Ef*}.
a
O
x
x
x* =minimum age in the relevant range of ages and
age in the relevant range of ages.
Then we may state
x** = maximum
HO =
~*
HO(T)
as
HO:
and
C
S
max Y.e. ::; a x - a x ::;
Y.e.Ef-ex
HI = Tld.* HI (T)
HI :
as
C
S
ax - ax < max Y.e.
Y.e. Ef.e.x
If
A Ef*
and
(4.13)
or
ca
S
x - ax >
for some x* ::;x ::;x**.
HO(A) is a condition on a linear function of the
difference between the annuity values, then it may be found that the
conditions given by
HO(A)
are also given by some other
HO(T)'
T Er*
or the conditions given by some other HO(T)' T Er* can be altered so
that HO(A) is also satisfied.
In this case, the hypothesis to be
tested has the form given by (4.13).
Consequently, we need only con-
sider tests of hypotheses concerning annuity values and for the case
where it is just desired that the total reserve be sufficiently close,
tests concerning linear functions of annuity values.
4.2
Some Notes on General Procedure for Testing Hypothesis that a
Parameter is Within a Given Interval
Let
and
~
8 denote a parameter,
8
0
an estimate of the parameter.
a fixed value of that parameter,
The test criterion for
e
77
against
HI: 8 - 80 < yJ!..
is reject
CJ!..
and
e
H if
O
or
8 - 80 > yu '
is greater than
C
u
or less than
CJ!..
where
Cuare such that
(4.14 )
Let
s
denote the estimate of the asymptotic value of
n
(8.D.(8)
denotes standard deviation of
Z =In
(8
-8)/s
§),
and suppose that
n
has an asymptotic standard normal distribution.
ZJ!..
= v'il(CJ!..
-8 0 -Yu)/sn
z
u
= !n(C u
-8
0
In 8.D.(8),
-Y )/s
u
Let
n
and
6
= lJ1(yu
(4.15)
-Yo)/s
.{..
n
From (4.14) we see that to find the critical region for the test of
HO we must find
zJ!.. and Zu
such that
<I>(zu) - <I>(zJ!..)
=1 -
a'
(4.16)
and
(4.17)
where
~(z)
is less than
denotes the probability that a standard normal variable
z.
Because of the symmetry of the normal distribution
we may take
that is,
z.{..0 + zu =-6 '
(4.18)
78
therefore, if we find
zl and Zu
such that (4.16) and (4.18) hold
the conditions (4.16) and (4.17) will be satisfied.
The critical
points are given by the equations
Cl =8 0 +Yu +zlsn /lll
=8 0 +Yl + (zl +6) s
(4.19)
i
ln
(4.20)
u =8 0 +Yu +z u s n1m
(4.21)
C
C
u
=8 +Y o + (z
0
~
u
+1'I)s lin .
(4.22)
n
Of course, we could interchange the roles of
Y
u
and
Yl
in all of
the above equations after (4.14).
Let
tion.
<P(z) =Pr[Z ::;z]
where
Z has a standard normal distribu-
The power function for the test of H is given by
O
seel =.CDCC;n-elj
+1
_(DC~: -elJ
(4.23)
and has the following properties:
(4.24 )
S(8 0 +Yl -E) =S(8 0 +Yu +E) ,
S(8) > ex '
m~nS(8)
Y
8 > 80 +Yu
or
(4.25 )
8 < 80 +Yl
(4.26)
= S(8 0 + (Yu +Yl)/2) > 0 ,
sup
S(8)
80+Yl::;8::;8 0+y
u
= ex' >
which is unaffected by
n,
(4.27)
(4.28)
(4.29)
All of these properties follow from the properties of the standard
normal distribution.
79
4.3
Test(s)of Hypotheses Concerning Annuity Values
4.3.1
Test Procedure
Suppose that the null hypothesis formulated from the criteria set
. I
by the company for retaining the standard life table is
Ho.. Ya -<C ax -ax
S <y
- ux V x*:s:x :s:x**
and the alternative hypothesis is
Ca _Sa > y
for some x* :s:x :s:x** .
x
x
ux
or
The interval
[x*, x**] wiLl generally contain a considerable
number of integer ages.
Usually there will not exist a mathematically
simple and well-defined parametric relationship between
over this interval.
expression for
a
x
and
x
Even if such a relationship did exist an
H in terms of the parameters of the function would
O
be very complicated and it would be very difficult to establish a
simple test criterion for a test of a hypothesis of the above type.
In Chapter II it was established that the difference between
annuity values from any two life tables that are likely to be encountered in practice at any age in the interval,
[x*, x**],
can be
well approximated from the differences between the annuity values at
a
f~w
ages.
We therefore replace the above hypothesis by a simpler
hypothesis, of the form
HO,:YL. :s:Ca
.{.AJ
x
_Sa
j
x
:s:y'
j
ux
, j =1,2, ... ,m
with xl <x 2 <",<x
m
j
against the alternative
HI' :
C
(4.30)
S
a x. - a X. < Y.{.A.
~ •.
J
J
J
or
C
a
S
x.
J
- a
x.
J
>y ,
ux.
for some x.,
J
J
j =1,2, ..• ,m
First, we will give a procedure for determining values for
x. ,
J
Y1..J '
and
Y~x. '
J
then a procedure for testing
against HI"
80
Let
[x!, x'.']
J
be any subinterval of
J
[x*, x**].
In Section
2.1.1 we established that in practice it can be assumed that annuity
values from different mortality bases are similar type functions of x.
Therefore, if the standard life table has given reasonably representative annuity values in the past, it is reasonable to assume that
S
a
S
=f. (x,
J
x
x! s;x s;x'.'
J
J
e),
(4.31)
and
Ca
That is,
S
a
C
a
and
x
=f . (x, Ce),
x! s; x s; x '.' .
J
J
J
x
are the same type of function of
x
possibly different parameters, in the interval
Let
(x, Sa)
x
{'
ct. +
J
and
{'
{'
S.x and
ct.
J
J
(x, Ca ) for
x
method of least squares.
variables.
+
with
[x!, X'.'].
J
('
X
x,
J
S.x be straight lines fitted to
J
E [x!, x~], respectively, using the
Note that
Jty
ct.
J
and
{'
S.
J
are not random
They depend on the actual (unknown) values of
C
a.
x
Because of the assumption (4.31) we may use
x
8
= [\
+C~xJ -[\ +S~xJ
to estimate
S
a
x
, .
x! s; x s; X'.'
J
J
Obviously,
for any
x!s;x.s;x '.'
J
J
J
iff
Yo..
~.
J
+ <5
x
- <5
x.
J
s; <5 s; Y
x
ux.
J
therefore, to be sure that
like to choose
+
0 -0
x
x.
J
H
O
'rJ x! s; x s; x '0'
J
J
is true when
HOI
is true we would
81
(4.32)
and
y'ux.
=
J
min (y - cS )
ux
x
x!:-=:;x:-=:;x'.'
J
+ cS
J
(4.33)
x.
J
However, we do not know the actual values of
obtain
cS
x
a"'- ,
x
(x,
cS
x
.
a
x
so we cannot
Using the observed data for as many ages in
as is convenient to calculate the usual estimate of
[x! , x'.']
J
"'-
or
C
J
C
a ,
x
we may use the method of least squares to fit a straight line to
~
A
ax ),
thus obtaining an estimate of
C
(3. ,
J
say
C
(3 ..
J
We then
calculate
y"be.
J
=
~
max
x! :-=:;x:-=:;X'.' ~.ex
J
[~.J
(4.34 )
J
and
y"ux.
J
(4.35)
82
[x!, x'.'],
J
j =1,2, ... ,m
J
such that for each interval there is a
(x, Sa )
reasonably good straight line fit to
keeping
for
x
X E::
[x!, x'.']
J
J
m as small as possible.
From Chapter II it appears that it would be sufficient
to take m equal to five or at most six.
need to have
Of course, we will usually
m ~3.
We may, without any loss of generality, choose
closest to the midpoint of
A
calculate
a
lx. and y'ux.
J
J
J
J
for which it is convenient to
J
In doing so, we decrease the possibility of choosing
x.
y'
[x!, x'.']
as the age
x.
as unrealistic values (e.g., values that would make it
J
almost certain that
will be accepted).
HO' we will apply the union-intersection
principle of test construction given by Roy (1957, pp. 5-12).
To construct a test of
be the hypothesis
Let
H
. y'
OJ·
and
a
x.
J
x.
J
: .:; y'
ux.
then
m
H '
HO' =
j=l oJ
n
lx.
J
and
H '
Oj
x.
S
- a
>y'
ux.
x.
J
J
J
J
m
HI' = UBI'
j =1 J
denote the acceptance region for
region for
Ca
or
<y'
J
J
S
- a
x.
be the related alternative hypothesis
H
lj
A.
C
a
J
S
Let
<
lx.-
H
Oj
and
A!
J
the rejection
then applying the union-intersection principle the
nA.
j=l
m
acceptance region for HO'
m
for HO' is A' = U A! .
j=l J
Let
is
A=
S(a ) = Pr(rejecting
x.
J
and the rejection region
J
./ax. )
H
OJ
J
83
then
L(a
x.
J
Let
~
Ho.1
a )
J x.
) = Pr(accepting
=I
denote the vector
J
- S(a
(a
) .
x.
J
xl
, a
x2
, ... ,a ) ,
xm
L(a) = Pr(accepting
HOII~),
S(g) = Pr(rejecting
Holla) ,
and
also let
S(ax , a x ) = Pr (rej ecting H ' and HOkia , a ).
x
x
OJ
k
k
j
j
Using Bonferroni inequalities we obtain the following bounds for the
operating characteristic function for the overall test of
against
HII
m
1 -
HOI
I S(ax. )
. 1
J=
m
$;L(i!) $;1-IS(a
j=l
J
xj
(4.38)
)+IIS(a,a)
j<k
xj
xk
Correspondingly, we have the following bounds for the power function
of the test
m
m
(4.39)
I S(ax ) - I IS(ax ' a ) $;S(§;) $; I S(a ) .
. 1
X
x.
j=l
j
j<k
j
k
J=
We wish to construct a test of
HOI
maximum value of the power function, when
exceed a designated value, say
a.
J
against
HOI
HII
such that the
is true, does not
To obtain such a test we
construct tests of size
j =1,2, ... ,m.
a =a/m for testing H
against Hlj ,
Oj
j
In this case the upper bound given by (4.39) is a.
The hypothesis
e =C ax. '
H
Oj
is the hypothesis that a parameter,
minus a fixed value,
S
e = a
x. , is within a given interval
J
J '" '"
[Y.e.' Yu] = [ykj' y~j]' If we let a I =a.J' e= a x.' and s n denote the
'" J
estimate of the asymptotic value of In S.D.(a ) obtained by subx.
o
J
84
stituting Px for px in the expression for the asymptotic value of
(ax. ),
In S.D.
the procedure given in Section 4.2 gives us a test
]
with the desired features.
4.3.2
Properties of the Test
Since the rejection region for H ' is the union of the
O
rejection regions for H '; j =1,2, ... ,m, it is clear that
Oj
S(~)
=1, i f Sea
x. ) =1 for at least one of j =1, ... ,m,
(4.40)
]
and
S(~) ~
max Sea
l~jsm
)
(4.41)
xj
Using (4.39) and (4.41) we obtain the following bounds for the power
function for the test of HO' against HI'
m
max (S(a
I<
_]. <
s.m
x ].
),
L Seax . )
. 1
- I IS(a ,a )) ~ S(~) ~min(l,
x ].
xk
] ]. <k
]=
m
L Sea
j=l
xj
)).
(4.42)
Clearly, these bounds cannot be worse than those obtained using only
the Bonferroni inequalities.
S
ax
Let
~l
+ Y..em
- E)
denote the vector
and
~u
denote the vector
m
S
(a
S
S
+y 1 +E,
ax +y 2 +E, ... , a +y +E). It follows from the
xl
u
2 u
xm urn
sYmmetric properties of the multinormal distribution that
. SC~l)
From (4.25) we know that
S(~) ~aj
= S(~u)
Sea
x
=a/m
(4.43)
.
) >a.
when
]
j
when
HI'
HI]'
is true, therefore,
(4.44 )
is true.
m
Since the upper bound for
S(~)
given by (4.39) is
since by (4.27) we know that when
unaffected by n,
H
Oj
L S(ax
j=l
is true supS(a
xj
) and
j
) =a. is
]
it follows that
m
supS(~):o;
I a. = a
j=l ]
when
H '
O
is true.
(4.45)
~.
85
From (4.29) we know that
Sea
L.J S(ax
This implies that
x. ) -+0
J
) -+ 0
as
as
n
when
-+00
n -+00 when
j
H I
O
is true.
is true,
therefore,
S (!!J -+ 0
as
n -+00
when
H I
O
is true.
(4.46)
Nandi (1965) showed that under certain conditions the consistency of
a test constructed using the union intersection principle follows
from that of the component tests.
Sea
x. ) -+1
that
J
4.3.3
as
S(a)-+l
n -+00 when
H
lj
Similarly, by (4.28) we know that
is true, then it follows from (4.41)
is true.
when
Hence the test is consistent.
Effect of a Change in Interest Alone
Recall that,
with respect to
v =1/(1 +i),
therefore, the derivative of
a
x
i,
d~ [w-I-\l + i) -ttP 1
d
-:-r- a
=
d1 X
t=l
1
xJ
w-x-l
=-v
L
t=l
w-x-l
=-v t=O
L
tv
t
t
la
P
t x
x
Expressing this in differential form,
w-x-l
da = -v L
di
x
t=O tlax
It follows that for "small" changes in
w-x-1
/:'a
x
=i= -v
L
t=O t
i,
IaxM
therefore,
(i I) . a (i)
ax
'T
x
and
Ca(i ' ) _Sa(i ' )
x
x
+ Ca(i)
x
_Sa(i)
x
i
I
-
i w- ~ -1 [C ( i ) _ S (i) .J (4 47 )
1+ i
t~O
tlax
tlax
'
.
86
.\i,
where
i
is the original interest rate and
i'
the new interest
rate.
To determine whether conclusions about the differences between
annuity values on two mortality bases at an interest rate
valid for the interest rate
substitute
bounds for
bounds of
~
tlax
i'
when
"
1
'I
i
are
is small, we may
-1
C (i) in (4.47)
' and use this to determine
tl a x
Ca(i') _Sa(i') in view of the conclusions about the
x
for
x
C (i) S (i)
a
- a
.
x
x
Clearly for
I1"
"
-1
very small the two
differences will have approximately the same bounds.
If
Ii' -il
is large it is best to test the hypotheses for both interest rates,
i'
and i.
4.3.4
Exampl es
Example 1
Suppose a company has been using the M69 life table and the
null hypothesis formulated from the criteria set by the company for
retaining this life table is
-.05 <_C a (·06) _M69 a (·06) -<.05 V 25
x
x
~x ~
55,
Suppose that the current experience is that given by JMIC (1974) for
assured lives 1967-70 of durations 2 and over, ages 20 and over,
with the exposed to risk and deaths divided by 100 and rounded to
the nearest whole number.
C
a
x
Using this experience's data, we estimate
in the usual manner, by
A
a
x
By observing the plot of (x, M69a (.06))
x
given in Figure 2, we
decide that there is a reasonably good straight line fit to
87
[25 , 35] , [36 , 44] , [45 , 55] ,
(x, M69ax(·06))· over the age l'ntervals
[56, 64], and [65, 75].
Table 4.1 shows the values of
Y'd.., y"
-LA
UX • '
j
J
y'
be . '
and
J
y'
ux .
J
we obtained using (4.34), (4.35), (4.36), and (4.37) respectively.
TABLE 4.1
j
x!-x'.'
J
J
x.
y"
J
a.
y"
ux.
J
y'
a.
y~x.
J
J
J
1
25-35
30
.0180
-.0180
-.0180
.0180
2
36-44
40
-.0035
.0035
-.0035
.0035
3
45-55
50
-.0408
.0408
-.0408
.0408
4
56-64
60
.0209
-.0209
-.0209
.0209
5
65-75
70
.0951
-.0951
-.0951
.0951
We obtain an estimate of the asymptotic value of
say
by substituting
asymptotic value of
square root.
nVar(a
px for
x.
)
px
In S. D.
(ax. ),
in the expression for the
given by (3.4) and extracting its
J
Then the statistic
z = In (ax.
J
-a
x.
J
)/s
n
has an asymptotic standard normal distribution and the procedure
given in Section 4.2 is used to determine the critical regions for
testing
,
<C (.06) M69 (.06) <_y'
H ': y -LA.
0._
ax.
a X.
ux.
OJ
J
J
J
J
against
H .: Ca (·06) _M69 a (·06) <y'
IJ
x.
J
j =1,2, .•• ,m.
x.
J
be.
J
J
88
ax. ,6
Table 4.2 gives the values of
Z, zl and zu'
values for
and
C ,
and
z
(see 4.15), the critical
J
for tests at the 1% significance level.
u
'"a
and the critical values for
x. ,
Co
.{.
J
The values of
were determined using tables for the normal distribution
u
given by Kelley (1948, pp. 37-137) so
exactly when
zl +zu does not equal
-6
but we used the values closest to
(z ) - <I> (z I) =1 - a.
u
J
the solution that the table had.
<I>
TABLE 4.2
a. = .01
J
j
x.
6
J
zl
z
u
C
1
C
A
a
x.
J
1
30
1.1006
-3.4316
2.3378
14.2683
14.4570
14.97
2
40
.1364
-2.6521
2.5121
12.9192
13 .1842
13.79
3
50
.9736
-3.2905
2.3455
10.9693
11 .4417
12.01
4
60
.2874
-2.7478
2.4573
8.5827
9.3398
9.61
5
70
.7944
-3.1559
2.3575
5.9376
7.2578
6.81
All of the estimates,
A
ax ,
except
j
region for the test of
A
u
A.
HOj
against
Hlj ,
A
ax
are in the critical
5
therefore, the vector
1\
(a , a , ... ,a ) is in the rejection region for the test of H '
O
xl
x2
x5
We would conclude that the M69 life table should not be retained.
Figure 8 shows the upper and lower bounds for the section of
m
the power surface, for the overall test of H =
O j=l HOJ' against
m
HI =
HI' given by (4.42), where the true annuity value is the
j=l J
.
same distance from Ma6 9
+y,
at every age. Th at 1S,
x.
ux.
J
J
M69
a X. ==
a x. + y'ux. +£ 'if j =1,2, ... ,5 or because of (4.43)
n
U
J
J
J
e
89
BOUNDS FOR POWER FUNCTION EXAMPLE 1
O-+---...---t-----r--~--_r_--_-__t--
-0.2
~o.
12
~o.oq.
0.0+
__"'"--_r_-~
0.12
0.2
£
Figure 8
In Section 1.4, it was shown that if there is a constant increase
in the force of mortality, say
8,
at every age, the annuity values
90
obtained using this mortality base are equivalent to the annuity
values on the original mortality base, at a new interest rate
i' =(1 +i)e 8 -1 .
Therefore,
M69 (i ')
a
x.
=
(M69+8) (i)
a
J
where
8=
x.
J
tn (1 + i ') - tn (1 + i) and
(M69 + 8)
denotes the life
table where the force of mortality at each age is
that for the M69 life table.
determine the values of
8 greater than
Hence, it is not difficult to
(M69+8)a(i).
x.
Figure 9 shows the upper
J
and lower bounds for the section of the power surface where
a
x.
J
=(M69+8) a x.(i)
J
Example 2
Suppose that for the same experience as in Example 1 the
company had set criteria that resulted in choosing
as are given in Table 4.3.
~
Ylx. and
Y~x.
J
J
Table 4.3 also gives the values of
"a ,
x.
C and Cu' for tests at the
t
Using the values of a
given in Table 4.2
x.
and the critical values for
J
1% significance level.
A
J
we see that the conclusion in this case is the same as that in
Example 1.
Figure 10 shows the upper and lower bounds for the section of
M69
the power surface where a = a
+ y'
+ E, V j
or
x.
x.
ux.
J
J
J
M69
,
a =
a x. + Yo..
- E, v J for the overall test in this example.
x.
-l-A.
\.1
J
J
•
J
If we make the condition that
Y~x.
=-Ylx.'
J
for the experience
J
used in Examples 1 and 2 the smallest approximate value of
that lead to acceptance of
H ' j =1,2, ... ,5
Oj
Yhx.
J
are .53, .62, .6,
e
91
BOUNDS FOR POWER FUNCTION W.R. T.
e
.
00
o
\0
0
•
~
e
~I
\oJ
<Xl
'<t
C
.
N
o
o -t---.,..----1I---r----;----t-----1----r---r----t----1
o
0.0002
0.0006
0.0004
e
Figure 9
0.0008
0.001
92
TABLE 4.3
ct.=
J
.01
x.
J
y'
a.
Y~x.
1
30
-.02
.02
1. 2229
14.2663
14.4587
2
40
-.02
.02
.7794
12.9098
13.1927
3
50
-.02
.02
.4772
10.9831
11.2243
4
60
-,01
,02
.2063
8.5892
9.3436
5
70
-.02
.03
.2088
5.9825
7.2242
j
J
C-e.
J
C
u
e
93
BOUNDS FOR POWER FUNCTION EXAMPLE 2
--1
/
co
C>
.
1.0
c>
c>
c>
+--r----t---..,...---'-t'---r---t---...,....---+----r----l
o
0.0+
0.12
0.08
e:
Figure 10
0.16
0.2
94
.27, and 0 respectively.
Example 3
Suppose that for an experience that is the same as that in
Examples 1 and 2 except that the exposed to risk and deaths for
each age have been divided by 10 and the criteria set for retaining
Y~x.
the M69 life table are such that the
Table 4.4 gives the values of
C-e.
of
and
A
a
C ,
u
are as those in Example 2.
J
and the critical values for
6
for tests at the 1% significance level.
x. are the same as those given in Table 4.2.
J
reject
but do not reject
HOI' H02 ' and H03
5
x. ,
The values
J
In this case we
H
04
and
n
we do reject the overall hypothesis
A
a
H '
05
However,
and conclude that
H '
HO =
j=l OJ
the M69 life table should not be retained.
e
TABLE 4.4
=
Q',.
J
.01
j
x.
6
C-e.
1
30
.3867
11 .7810
14.6341
2
40
.2465
12.6309
13.4732
3
50
.1509
10.5191
11.8886
4
60
.0652
7.7835
10.1510
5
70
.0660
4.6504
8.5535
J
C
u
Figure 11 shows the upper and lower bounds for the section of
the power surfac e where
"If j
a
x.
J
= Sa
x.
J
+ y'
ux.
J
+E V j
for the overall test in this example.
or a
x.
J
=
sa
x.
J
+ y'
lx. J
E
9S
BOUNDS FOR POWER FUNCTION EXM1PLE 3
·e
,
. . . -,-----------------r-------------..,
.
ex:>
o
1.0
0
,.....
~I
'-'
c::Q.
cI"-
0
C\l
o
O-+---t----t-----t---T--..,---;---t---__t----t----1
o
0.1
0.2
e:
Figure 11
0.3
0.5
O.lI-
The estimates of the asymptotic values of
nCov(a, it )
x.
x
J
k
needed to calculate the bivariate normal probabilities required by
(4.42) were obtained by substituting
for the asymptotic value of
Px
nCov(ax ,
j
for
ax )
px
in the expression
given by (3.8).
A
k
Fortran program, using an International Mathematical Statistical
Library subroutine to calculate the value of
96
written to evaluate the bounds given by (4.42).
to approximate
(:3(~)
No attempt was made
since the bounds given by (4.42) are relatively
narrow even for smaller samples, see Figure 11 which is based on a
sample of size
n =16,678.
Recall "that
n =I n
and that we are
x x
considering applications by insurance companies so data on all the
policy holders would be available.
Therefore,
n
of 16,678 is
indeed "small".
4.4
Testes) of Hypotheses Concerning Linear Functions of Annuity
Values
4.4.1
Test Procedure
Suppose it is desired that the total reserve for a group of
whole life insurance policies be sufficiently close in order to
retain the standard life table, then the null hypothesis of concern
is of the form
y**
H (1): YR..:OS; I A (Ca - Sa )
O /\
y=y* Y Y
Y
:os;
(4.48)
y .
u
Clearly, if
C
S
Ho.· YR..y -< a y - a y -<yuy Vy*<_y<_·y**,
is valid for sufficiently small
be valid.
y
uy
and large
However, if the only criterion for retaining the standard
life table is that the total reserve be sufficiently close, there is
no need to appeal to the multiple hypotheses given by
HO'
We will consider the hypothesis fora single fixed set
of
A'S
y ,
which may be regarded as representing the company's
business at a fixed point in time.
Therefore, tests of the hypothesis,
(4.48), might be based on the test statistic
y**
I
A
a,
y=y* y y
(4.49)
97
which is a linear function of the estimated annuity functions,
y* -::.y -::.y**,
A
a ,
y
Applying Theorem 3.3 we see that this statistic is
asymptotically normally distributed.
Therefore, tests of (4.48) may
be performed using the procedure described in Section 4.2.
The interval
[y*, y**]
will usually contain a considerable
number of integer ages, and as a result calculation of the estimate
of the asymptotic value of
nVar(L y Ayay ) obtained by substituting
in the expression for the asymptotic value involves some
quite lengthy summations.
With electronic computers this is not a
difficult task, but it can be advantageous to have a test involving
fewer calculations.
In Section 2.2 it was shown that the n-ages method may be used
to approximate linear functions of the annuity values with reasonable
accuracy for mortality bases that may be
~ncountered
in practice.
Therefore, we will replace hypotheses of the form (4.48) by
hypotheses concerning annuity functions at a few selected ages.
Clearly, (4.48) is equivalent to the hypothesis
y**
) /y**
HO: Y1 -: . I A (Ca - Sa
I A -::. Y~
y=y* Y Y
Y
y=y* Y
y**
I A
/ y=y* Y
against the alternative
where
Yl=Yt
y**
and
Y' =Y
I A
u
u / y=y* y
or
Using (2.8) to approximate
these hypotheses by
(4.50)
Iyy
A rCay
_Sa)
y
and we want to test this
y**
!y**
A(Ca _Sa)
L A ~y'.
y=y* Y Y
Y y=y* Y u
L
II
A
yy
(4.51)
we may replace
· 98
(4.52)
against
HI :
The
b.
J
I
m b.
j =1 J
(C a
Yj
S J < Y1
Y
- a
or
j
I
m b.
j=l J
(C a
Yj
S J >y'
- a
Yj
u
are independent of the annuity function, so tests of these
m
I
hypotheses may be based on the statistic
b.a
.
We know from
j =1 J Yj
Theorem 3.3 that this statistic has an asymptotically normal distribution, so a test of (4.52) may be constructed using the procedure
described in Section 4.2.
We will now construct a test of (4.52) such that (asymptotically)
the maximum value of the power function when
greater than a designated value a.
m C
m S
put e = I b. a , eO = I b. a ,
j=l J Yj
j=l J Yj
and let
s
n
Px for px
A
the asymptotic value of
4.4.2
In the equations of Section 4.2
m
8 = L bJ.ay , Y{)=Y~ Yu=Yu' ,
j=l
j
.{.. .{..
be the asymptotic value of
substituting
HO is true is not
In S.D.(8) obtained by
in the expression for the square root of
nVar(8).
Example
Suppose that a company has been using the M69 life table and
the criterion set by the company for retaining the life table is
that the life table will be retained if the absolute error made in
calculating the total reserve for the whole life insurance policies
is not greater than .1% of its true value.
formulated from this criterion is
The null hypothesis
99
Using (4,11) we see that this is equivalent to the hypothesis
-,00099Lx Lt tSx
M6~VX
- - - - - - - - - - $;
L/'y
where
y
denotes attained age.
Suppose that the range of relevant attained ages is
[20, 96],
the reserve is calculated at an interest rate of 5%, and the sum
assured by attained age and values of
A
y
are as given in the
Appendix, Tables A.2 and A.4 respectively, except that they are zero
for attained ages less than 20.
Using (1.5) to evaluate
LxI t tSx M6~vx it is clear that the null hypothesis is
-.2926
e
$;
96
I A (C a - M69)
a
y=20 y
y
Y
1I
96
A
$;
y=20 Y
.3254.
In Section 2.2.4 it was shown that the error made approximating
96
M69
using the five-ages approximation with selected ages,
a
I A
Y
y=20 y
25, 40, 55, 70 and 85 is very small (.000005) . Since the absolute
value of the errors made using the n-ages approximation for
I Aa
y y y
IIAy y
for any two mortality bases that might be encountered
in practice are of the same order of magnitude when the same selected
ages are used, from (4.52) we see that we may test
H : -.2926
O
against
I
$;
I
b.[C a (·05) _M69 a (·05))
j =1]
Yj
Yj
H:
b.(C a (·05) _M69 a (·05)) <-.2926
1 j=l]
Yj
Yj
or
$;
.3254.
100
y.,
where the values of
and the values of
J
b.
are 25, 40, 55, 70, and 85
j =1,2, ... ,5
are given in Table 2.10.
J
Using these values
we can calculate
10.796186
and if the experience is as in the examples of Section 4.3.4, we
obtain the value of the statistic,
5
8=
I
b.a
=11.4381
j =1 J Yj
In S.D.(8),
The asymptotic value of
is calculated by
s ,
n
extracting the square root of the quantity obtained by substituting
A
P
x
for
p
5
I
nVar(
x
b.a
j =1 J Yj
in the expression for the asymptotic value of
)
obtained from (3.15).
The statistic
5
z =In I
b.(a -a )/s
j=l J Yj
Yj
n
has an asymptotic standard normal distribution.
given in Example 1 of Section 4.3.4
in Example 3 of Section 4.3.4
of
~
For the experience
n =166,784 and for that given
n =16,678; Table 4.5 gives the values
(see 4.15), the critical values of
z, zl
and
5
I
C and
l
significance level for these two cases.
critical values for
b.a
j =1 J Yj
,
C,
u
Z ,
u
and the
for test at the 5%
TABLE 4.5
zu
166,784
7.0687
-8.7135
1.6449
C
l
10.3598
16,678
2.2358
-3.8807
1.6449
10.0487
n
~
zl
C
u
11.2654
11.5764
101
The hypothesis would be rejected, for
n
n =1C>6,784
were 16,678, at the S% significance level.
of the power function,
a=
but accepted i f
Figure 12 shows graphs
for n =166,784 and n =16,678,
8(8),
where
5
~ b .M69 ( . 05)
L
.... 3254 + £.
a
j =1 J
Yj
POWER FUNCTION
.
00
o
N
o
.
o
-0.31
-0.01
0.29
f).
E
Figure 12
S9
0.89
1.19
CHAPTER V
TEST(S) OF POSSIBILITY OF
MERGING TWO SURVIVAL EXPERIENCES
There are many circumstances which may give rise to consideration of merging several survival experiences.
Among the more
obvious situations where the possibility of merging two survival experiences may be considered are
(i)
(ii)
the merger of two companies, and
a desire to eliminate discrepancies in the premium rates
charged two groups.
Merging experiences gives rise to the same concerns as when considering whether a standard life table is adequate.
In particular,
whether the actuarial functions calculated will be appropriate to
the "current" population of policy holders.
In this chapter we outline methods that may be used when it has
been determined that two experiences will be merged if they are
tlsufficientlyalike tl .
That is, if the actuarial functions based on
the two mortality experiences are "sufficiently close".
5.1
Formulation of Problem
First, the functions that must be sufficiently close have to be
determined, and the expression "sufficiently close" must be defined.
These depend on the practical requirements of the company.
103
Proceeding analogously as in Chapter IV, we define the values
of a function based on the mortality experience of one population as
being sufficiently close to the values of the function for another
population i f the absolute values of the difference between the
functions from the two mortality bases are not greater than given
quantities.
We consider the same five actuarial functions used in Chapter IV,
now, denoting the "current" mortality bases in the two populations by
Then we define sufficiently close for these five functions
C' and C".
by replacing
C with
C'
and
S
with
C" (or vice versa) in the
definitions given in Chapter IV.
We have to choose one of two decisions
va:
Merge the two experiences
VI:
Do not merge the two experiences.
or
The hypothesis to be tested is formulated from the criterion
set by the company in exactly the same manner as in Chapter IV and
takes the form of either
(5.1)
again~t
or
or
C'
a x
C"
a >Y
x ux
for some x*
~x ~x**,
Y**
C': C"
HO: YJ!..~ L A ( a - a) ~ Y
y=y* Y Y
Y
u
against
y**
A
(C'a
Clla
<YJ!..
or
L A (CIa _C"a) >Yu •
)
HI: L
y
y
y=y* y
y
y
y=y* y
y**
(5.2)
104
5.2
Test(s) of Hypotheses Concerning the Difference Between Annuity
Values from Two Populations
Suppose that the hypothesis formulated from the criteria set by
the company for merging two experiences is given by (5.1).
We re-
place this hypothesis by the simpler hypothesis
,
HO': Yo••
s
-t.JI..
J
C'
a x. J
C"
a x. sy'ux.
J
J
against the alternative
HI':
C'
a x. -
C"
J
or
a x. <y'be.
J
a
J
m, X y'
The values for
CI
j'
-ex.'
J
x.
J
and
-
C"
a
y'
ux.
x.
J
for some x y j=l, 2, ... , m.
>y'
ux.
J
may be determined using the
J
procedure given in Section 4.3.1, replacing
e
C with
C'
and
S
(or vice versa). However, we do not know the actual values
C"
C' a
or
a , so using the observed data we may calculate
of
x
x
/"-..
C-<'
C"
and
respectively. Fitting
their usual estimates,
a
a
x
x
~
straight lines to (x,
a ) and (x, a) using the method of least
x
/".... x
............ ~
.~
CI
C"
cor-Band C"
squares, we obtain estimates of
Band
B,
B respec-
with
C"
0
tively.
let
Then
y"
f'c~
S= Band
in Chapter IV.
1
I
are given by (4.34) and (4.35) if we
~~J
:L C"
B=
B when we replqce
Having done this then
The test of HOI
I
by
C
and
y'
be.
J
and
C'
~
this case we let
S
by
C"
' are given by
Yux.
J
is constructed in exactly the same manner as
the corresponding test was constructed in Chapter IV.
I
I
and y"
ux.
(4.36) and (4.37) respectively.
I
I
I
.ex.
be the hypothesis
I
y'
-ex.J S
C' a
x.
J
C"a
x.
J
sy'
ux.
J
However, in
105
and
H
lj
be the related alternative hypothesis
C'
HI]':
a x. ]
m
HO ,=
n
C"
,
ax . <Yo..
C'
or
a
.{..A.
]
]
C" a
x.
]
x.
]
>y'
ux.
]
m
HI' = U HI'
The procedure given in Section
j =1
J
j=l
Now H
is the
4.2 may be used to construct a test of H
Then
H ]'
O
and
Oj
hypothesis that a parameter)
8=
C'
a
C"
x.
]
80 (=0)
is within a given interval,
At:'a
8=
~
a
and denote by
x.
x.
Oj
a,
x.
minus a fixed value,
J
[Y~'Yu]
=
[Ykx.,y~x.]' We put
J
J
the estimate of the asymptotic
]
J
value of In S. D. (§),
where
n
is the total exposed to risk in the
"'-
two experiences, obtained by substituting
pression for the asymptotic value of
Px
for
Px
in the ex-
vn S. D. (8) .
Usually, an individual in one experience will not be in the
other, but there might be some overlap.
However, the number of in-
dividuals common to the two experiences will probably be relatively
small, and data for them could be eliminated with little loss of
information.
We will therefore assume the two experiences to be
independent.
In this case
n Var(8)
=n
Yare
c0
a
x.
-
/'... J
=n
Var
C'
a
x. +
]
where
n
Let
k
n'
n'/n
k
and
k
in
-+
is the sum
C'
c'>O
k
and
n
k
C"
C~
a
x.
)
J
n Var
/'..
C"
a
x.
,
J
of the exposed to risk in the two experiences.
denote the initial numbers exposed to risk at age
respectively.
We suppose that as
n
increases
Then (3.4) may be used to obtain the
106
asymptotic values of
n VarC
6'a
x.
)
and
n VarC
Z
= !iiC8-8)/s
..
n
has an asymptotic standard normal distribution since
c'it'
a
The
J
J
statistic
""'a"x. ).
C"
"""'a"
C'
J
are independent asymptotic normal variables.
x.
and
x.
J
5.3
TestCs)of Hypotheses Concerning Linear Functions of the
Differences Between Annuity Values from Two Populations
The test procedure in this case is completely analogous to that
given in Section 4.4.1.
Of course, as in Section 5.2, we are making
the assumption that an individual does not appear in both of the
experiences.
In this case we test the hypothesis
•
:;:; mIb.c C' a
j=l J
-
C"
Yj
a ) :;:; y'
u
Yj
against
C" a
r
y.
J
I
and
y'
u
) < y'
-e
m
or
C'
lb. C a
j=l J
C" a
Yj
y.
J
) > y'
u '
are as in Section 4.4.1.
Using the procedure described in Section 4.2 we construct a test of
m
C'
C"
this hypothesis by putting 8= I b.C a - a ), 8 =0, Y-e=Yl, Yu=Y~'
0
j=l J
Yj
Yj
m
C~ C"ir'
8= lb. C a - a ) and letting s n be the asymptotic value of
j=l J A
Yj
Yj
InS.D.(8) obtained by substituting p for
in the expression
A
x
for the square root of the asymptotic value of
Section 5.2 we take
experiences.
n
n Var(8).
As in
as the sum of the exposed to risk in the two
107
1\
n Var (8) =
=
nvar[ I
b.
j =1 J
./"....
C'
m
- Lb.
a
Yj j=l J
nvar[ h. c\:y. ]
j =1 J
+
c~y. ]
J
[
~
m
C"
n Var Lb.
a
J
j=l J
Yj
]
.
The expression given by (3.15) may be used to obtain the asymptotic
value of
"
nVar(8),
C<'
a
C0
) and n Var(L.b.
a ) and so of
y.
y.
J J
J J
J
J
if we assume that the numbers exposed to risk get large
n VarC2.b.
in the same manner described in Section 5.2.
•
APPENDIX
MODEL OFFICE DATA
Description and Source of Data in Tables A.l-A.4
Age at Issue -
Age last birthday of person when the insurance
policy was issued.
Sum Assured
(t C)
The data were contributed to the Society of Actuaries
by nineteen insurance companies.
pp. 38-48).
(See
SAT~W
(1977),
The sum assured for each age at issue
group is a rough average of the amount of standard
ordinary insurance issued by the nineteen companies
from 1960-74 and exposed to risk between the 1974
and 1975 anniversaries.
This includes the insurance
issued with, as well as, without a medical examination on male and female lives.
The data for non-
medical issues at ages 50 and. over has been consider. ed as for ages SO-54.
It has beeh assUmed that all
policies issued at ages 70 and over were really on
persons aged 70-90.
Premium (P.) x
Premium per unit amount insured.
Premiums used are
based on the life table for total U.S. population
1969-71 given in U.S. Life Tables (1975, p. 6).
The
premiums are calculated at a 5% interest rate and the
premium for the middle age of each age at issue
109
group is assumed to be applicable to each age in the
group.
tLx - Number of persons in the age interval
(x-x+t) .in
the stationary population given by the life table
for the total U.S. population 1969-71.
(See U.S.
Life Tables (1975, p. 6),
tSy - The total sum assured on persons now aged
y+t.
to
y
This has been calculated by assuming that the
proportion of the sum assured issued to persons at
ages
y+t
x
x
x+t
to
now held by persons aged
y
to
is the same as the proportion of people aged
to
x+t
in a stationary population that are
living at ages
y
to
y+t.
It has also been assumed
that all policies on persons over age 96 have been
written off by the company.
G - The total gross annual premium on
t y
L
\ (P.
t Y x<y
L
A - Grouped Value of
t y
where
i
d= l+i
C / L ).
x txt x
A for attained age group
y-(y+t)
is the effective rate of discount.
110
A - Smoothed Value of
A for attained age
y
obtain
1.
To
A:
y
I
Plot 19t Cd tSy+tGy)
age interval
2.
y.
at the midpoint of each
y-Cy+t).
Free handedly draw a smooth curve through the
plotted points.
3.
•
Read the values of
A
y
from the smooth curve.
III
TABLE A.1
Standard Ordinary Issues for Nineteen Companies
Medical and Non-Medical Combined
Male and Female Lives Combined
Premiums at 5% Interest Rate
By Age-at~Issue
Age
at Issue
0
1
2-4
5-9
10-14
15-19
20-24
25-29
30-34
35-39
40-44
45-49
SO-54
55-59
60~64
65-69
70-90
Exact Age
(x~x+t)
0~1
1-2
2-5
5-10
10-15
15-20
20-25
25-30
30-35
35-40
40~45
45-50
SO-55
55-60
60-65
65-70
70-91
tCx
($1000000 units)
p.
x
tLx
237.87
109.40
177 .33
255.27
350.33
1251.73
2917.13
3079.33
2629.40
2188.27
1671. 73
1109.27
634.67
283.73
100.40
25.87
6.13
.00324
.00240
.00255
.00301
.00381
;00478
.00588
.00730
.00925
.01182
.01515
.01947
.02509
.03239
.04198
.05491
.11668
98283
97937
293288
487781
486880
485069
481813
478267
474562
469696
462558
451806
435805
412350
379531
335762
715311
e
•
112
TABLE A.2
Standard Ordinary Issues for Nineteen Companies
Sum Assured and Premiums a~ 5% Interest
By Attained Age
Attained
age
(y-y+t)
1-2
2-5
5-10
10-15
15-20
20-25
25-30
30-35
35-40
40-45
45-50
50-55
55-60
60-65
65-70
70-75
75-96
S
t
1
2
2
2
3
6
9
12
14
15
16
15
14
13
10
17
237 032
037 447
020. 353
271 420
611 998
837 793
705 209
708 741
211 630
181 063
484 301
005 899
744 974
752 977
140 427
987 338
934 109
Y
513.9
642.4
781. 8
328.2
581.1
279.7
121.4
255.8
061.6
846.6
105.5
139.2
472.0
599.1
967.2
931.5
166.5
3
5
6
7
13
30
52
76
100
123
139
147
143
130
110
180
768
086
885
641
945
835
760
826
357
669
067
541
098
849
988
496
775
021.95
269.62
077.77
043.20
915.29
742.11
220.37
818.15
539.33
272.89
436.34
710.37
440.20
460.09
814.49
299.23
678.40
113
TABLE A.3
Grouped Values of A
5% Interest Rate
($1,000,000 units)
Attained Age
(y-y+t)
1-2
2-5
5-10
10-15
15-20
20-25
25-30
30-35
35-40
40-45
45-50
50-55
55-60
60-65
65-70
70-75
75-96
19
A
t Y
12.1
52.5
102.1
114.8
132.3
196.5
349.9
515.0
657.6
775.7
860.1
901.4
896.6
846.1
756.5
633.5
1034.8
A
t Y
.637
2.763
5.374
6.042
6.963
10.342
18.416
27.105
34.611
40.826
45.268
47.442
47.189
44.532
39.816
33.342
54.463
e
114
TABLE A.4
Smoothed Values of A
By Attained Age
5% Interest Rate
($1,000,000 units)
e
y
A
1
2
3
4
5
6
7
8
9
10
0;40
0.64
0.80
0.89
0.95
1.00
1.00
1.02
1.04
1.10
1.12
1.15
1.20
1.20
1.25
1.30
1.38
1.48
1.59
1. 70
1.85
2.00
2.25
2.58
11
12
13
14
15
16
17
18
19
20
21
22
23
24
y
y
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
Ay
y
2.85
3.20
3.50
3.80
4.20
4.58
4.89
5.30
5.62
5.95
6.20
6.58
6.85
7.10
7.35
7.60
7.80
8.08
8.25
8.48
8.68
8.85
9.00
9.10
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
A
y
9.20
9.30
9.40
9.50
9.55
9.60
9.60
9.55
9.52
9.40
9.35
9.20
9.10
8.99
8.82
8.65
8.50
8.25
8.08
7.80
7.59
7.38
7.15
6.85
y
A
Y
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
6.50
6.18
5.80
5.47
5.19
4.88
4.57
4.20
3.82
3.58
3.20
3.00
2.78
2.50
2.25
2.00
1.80
1.50
1.20
1.00
0.79
0.58
0.30
0.00
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Rao, C. R. (1965), Linear Statistical Inference and Its
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