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.