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Attempts to provide robust estimates of prior distributions.
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INTRODUCTION
Mortality is the measure of the intensity of death in a certain population which translates
to the measure of the expected or mean frequency of death occurrences within a particular span
of time across populations. It may definitely seem unusual to measure or determine when will an
individual die, or when will someone reach the end of his or her life. But in an economical
perspective, becoming aware of mortality models is fundamental in quantifying longevity risks
and providing the basis of pricing and reserving (Chen, Cox, & Yan, 2011). But, despite the
enormous amount of data they provide, these models are subject to inadequacies and
inconsistencies because of the uncertainty, which is endogenous in the model. This uncertainty
may be from the model itself or from the parameters they contain. To address this problem,
statisticians estimate these parameters to provide a more reliable and generalizable information.
In Bayesian analysis, one method that can be used to determine estimates is the fractile method
of estimation.
A fractile (or sometimes called quantile) is a measure of location, which comes in the
form of medians, quartiles, deciles or percentiles. These measures of location provide us with an
idea on the percentage of values in our data set that are less than or equal to it. In fractile
estimation, these fractiles are subjectively estimated from a given prior density and from there
one chooses the parameter of a given functional form of a density that closely matches these
fractiles. Though this may be a more robust way of point estimation, two problems arise
immediately: the choice of quantiles for estimation, and the possible non-uniqueness of a
solution due to the larger dimension of the quantiles to be used than that of the parameters to be
estimated.
With all these in mind, the group is inclined into attaining the following:
1. To apply the concepts of Bayesian Point Estimation of Priors on mortality data
2. To determine the proper laws of mortality that lie beneath the mortality data; and
3. To come up with solutions which are both unique and robust, if such may even exist
BACKGROUND OF THE STUDY
Life tables are published tables which primarily consist of the basic mortality functions
such as the expected number of survivors up to age , ; expected number of deaths between
ages and , ; and the probability that a person aged will die in years, , to name
a few. (Actuarial Mathematics, 2nd ed., pp. 58-59) Such tables, especially in Actuarial Science,
may also include joint and marginal actuarial functions such as life insurances and annuities, to
name a few. Most life tables are based on true proportions from a population – be it for those
living in a time period (usually measured per year) or for those living in a certain generation (or,
in the context of demography and actuarial science, cohort) – estimates from previous studies
and models, or simulated from specified models. The last kind of life table is called the
Illustrative Life Table, which is used in the SOA exams starting from the Mathematics of Life
Contingencies (MLC) Exam.
Just as the Table of Random Numbers is simulated from a specific statistical distribution,
then the Illustrative Life Table can be thought of as simulated from a certain distribution, better
known as a law of mortality. The Free Dictionary (Farlex, 2009) defines this as “a mathematical
relation between the numbers living at different ages, so that from a given large number of
persons alive at one age, it can be computed what number is likely to survive a given number of
years.” Thus, a life table, in general, may fit according to an underlying law of mortality.
The SOA Exam MLC syllabus, patterned on the two official review materials – Actuarial
Mathematics (Bowers, Gerber, Hickman, Jones, & Nesbitt, 1997) and Actuarial Mathematics for
Life Contingent Risks (Hardy, Dickson, & Waters, 2009) – includes 4 laws of mortality integral
to the development of Demography and Actuarial Science: the de Moivre, Gompertz, Makeham,
and Weibull laws.
The de Moivre law of mortality was developed by Abraham de Moivre in 1724. It is a
one-parameter distribution based on the assumption of a uniformly distributed force of mortality,
i.e.
, )( )
The single parameter is known as the limiting age, not necessarily reflective of that
given in a life table.
The Gompertz law of mortality was developed by Benjamin Gompertz in 1825. The law
asserts that death is an age-exponential age-dependent component, i.e.
{
( )}
The parameter is known as the aging hazard; if the parameter is unity, then this law
of mortality becomes the constant force of mortality (CFM) assumption.
The Makeham law of mortality is an update of the Gompertz law, developed by William
Makeham in 1860. It is similar to the earlier law of mortality but it asserts that the law must
contain another age-independent component, i.e.
{
( )}
The added parameter is also known as the accident hazard, taking into consideration
those not accounted for by the aging hazard.
The Weibull law of mortality was developed by Waloddi Weibull in 1939. It is a two-
parameter distribution commonly used in survival analysis, in general.
{
}
METHODOLOGY
After the original life table data is tabulated on a spreadsheet application, a proper
transformation must be employed so as to attain probabilities which behave like a cumulative
density function (CDF); the laws of mortality are expressed as the probability that a newborn
will die in years; however, the life table usually gives cohort probabilities, i.e. the probability
that a person aged will die in one year, or . A useful function for in terms of the cohort
probabilities is given by:
∏( )
The next procedure would be the interpolation of the quantiles; the group has decided to
use the three quartiles as it is commonly used in the fractile method. Although there are proper
interpolation methods for fractional ages discussed in Actuarial Science, linear interpolation will
be employed instead, as it is the simplest.
The next procedure would be the fractile method itself. The quartiles will be substituted
into the laws of mortality and will form a system of at most three equations. As established
earlier, the method usually results in non-unique solutions; the group has agreed that all possible
solutions using the quartiles will be obtained.
The last step would be the assessment of the solutions; if they tend to cluster more
closely, then it can be said that the life table may follow the law of mortality used in estimation.
SOLUTIONS
Interpolation
1st Quartile:
Age
(
)
2nd
Quartile (Median):
Age
(
)
3rd
Quartile:
Age
(
)
Estimation
De Moivre
Gompertz
Eq. 1:
{
( )}
*( )+
Eq. 2:
{
( )}
*( )+
Eq. 3:
{
( )}
*( )+
Equating Eqs. 1 and 2 yields Eq. 4:
Equating Eqs. 1 and 3 yields Eq. 5:
Equating Eqs. 2 and 3 yeilds Eq. 6:
This can be treated as a system of “linear” equations, with
[
] [ ] [
]
However, the coefficient matrix is singular, and thus there is no unique solution. However, this
can be solved using the Newton-Raphson Method, an iterative algorithm that solves for the
zeroes of an equation. Plugging Eqs. 4-6 into the software, the results are as follows:
Equation
Equation 4 1 -
1.0814 0.000137656
Equation 5 1 -
1.0846 0.000113903
Equation 6 1 -
1.0889 0.0000827
Makeham
Equation 1:
0.25= { ( )
( )} { ( )
( )}
Solve for A:
( )
Equation 2:
0.50= { ( )
( )} { ( )
( )}
Solve for A:
( )
Equation 3:
0.75= { ( )
( )} { ( )
( )}
Solve for A:
( )
On Equation 1 and 2, we eliminate A (By elimination method):
( )
( )
Hence we obtain Equation 4:
or
( )
We do the same technique on Equations 1 and 3:
( )
( )
We obtain a simplified equation, which we will name as Equation 5:
( )
Since Equations 4 and 5 include the variable B, we can express the two equations as follows:
and
Hence,
Cancelling ln c and combining similar terms, we get:
Further simplification yields the following result:
Solving for c, we get a unique solution c=1.09714. This value satisfies the requirement on
Gompertz (and also on Makeham) that c>1. Using this value, we are able to get the estimates for
B by substituting c to either one of the equations.
( )
( ) ( )
REMARK: The value of the other B using the second equation is approximately equal to the
previously stated value (4.733707414x ). The slight discrepancy in their
values may be accounted to the c value obtained due to rounding errors. The value c obtained is
not the exact value because of the limited memory of the computer used in calculating its value.
In calculating the value of A, the first B value will be used.
Now that B value is already known, we can now calculate the value of A:
( )
( )
( )
The values are consistent with the definition and constraint that A>-2B. Hence, we say that the
estimates conform to their expected values.
Weibull
Equation 1: 0.25 = 1 - ( )
= 1 - ( )
( )
( )
( )
( ) ( )
Equation 2: 0.5 = 1 - ( )
= 1 - ( )
( )
( )
( )
( ) ( )
Equation 3: 0.75 = 1 – ( )
= 1 – ( )
( )
( )
( )
( ) ( )
From Equations 1 and 2:
( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( )
( ) ( )
( )( )
( ) ( )
( ) ( )
n = 4.550872599
Substituting in equation 1:
( ) ( )
From Equations 1 and 3:
( ) ( )
( ) ( )
(( ))
( ) ( )
( )
(( )) (( ))
( ) ( ) ( )
( )
(( ))
( )
( ) ( )
(( )) ( )
(( ))( )
( ) ( )
(( )) ( )
n = 5.060007323
Substituting in equation 1:
( ) ( )
From Equations 2 and 3:
( ) ( )
( ) ( )
(( ))
( ) ( )
( )
(( )) (( ))
( ) ( ) ( )
( )
(( ))
( )
( ) ( )
(( )) ( )
(( ))( )
( ) ( )
(( )) ( )
n = 5.85805057
Substituting in equation 2:
( ) ( )
With 3 equations and 2 unknowns it is possible to have multiple values for these 2
parameters. The values of n are around 5 and the values of k are both very small. A small
discrepancy between the values of a parameter suggests that the data could follow the Weibull
distribution.
CONCLUSION
It is clear that the fractile method yields non-unique solutions, and often the method is
very tedious and sometimes requires trial-and-error, or even possible computational methods (as
illustrated in the estimation of the mortality parameters of the Gompertz law). However, no
matter how tedious the process is, the results are numerically stable, with the exception of the de
Moivre law which has a wider range of possible values.
The group recommends that for the next study related to this one, computational methods
must be heavily employed. Also, some other goodness-of-fit indicators must be used as well,
especially if the discrepancies between possible prior point estimates are infinitesimal.
Bibliography Bowers, N. U., Gerber, H. U., Hickman, J. C., Jones, D. A., & Nesbitt, C. J. (1997). Actuarial Mathematics,
2nd ed. Society of Actuaries.
Chen, H., Cox, S. H., & Yan, Z. (2011). A Family of Mortality Jump Models with Parameter Uncertainty:
Application to Hedging Longevity Risk in Life Settlements.
Farlex. (2009). Retrieved April 2013, from The Free Dicitonary by Farlex:
http://www.thefreedictionary.com/
Hardy, M. R., Dickson, D. M., & Waters, H. R. (2009). Actuarial Mathematics for Life Contingent Risks.
Cambridge: Cambridge University Press.