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					Measuring the Effect of Mortality Improvements on the Cost of Annuities

                      M. Khalaf-Allah, S. Haberman and R. Verrall
                       Faculty of Actuarial Science and Statistics
                                CASS Business School
                                City University London



 This paper uses the mortality projection model of Sithole, Haberman and Verrall (2000)
 to investigate the effect of mortality improvements on the cost of annuities.
 Analytical results are derived using an approach which is an extension to the one
 developed by Pollard (1982). The results are illustrated using data from the Continuous
 Mortality Investigation Bureau (CMIR, 16). Simulation methods are used to consider
 the distribution of the annuity cost, as well as the more often quoted point estimates.
 The effects of the age of the pensioner at inception, the rate of interest and the gender
 of the pensioner are considered.
 Finally, a Bayesian model is developed which incorporates the estimation of the
 parameters of the mortality projection model with the simulation of the annuity cost.
 This allows us to consider the effect of parameter uncertainty on the projected
 distribution of the annuity cost.




 1. Introduction
 Over the past century mortality rates in the developed countries, including the UK, have
 improved remarkably (Charlton, 1997). There is still much uncertainty as to the
 processes that cause ageing and there is much debate as to whether there are upper
 limits to human longevity. However, it is clear that there is evidence to suggest that
 current mortality rates are close to reaching any kind of lower bound (see Thatcher,
 1999). This persistent decrease in mortality rates has become a main concern of annuity
 and pension providers, especially mortality improvements for post-retirement ages
 which have a significant financial impact as far as survival benefits are concerned.
 Under these conditions of improving mortality, the projection of future annuitants’ and
 pensioners’ mortality is essential. To ignore improvements would be to endanger the
 financial stability of the insurer selling policies providing such survival benefits, taking
 into consideration that longevity has a direct impact on the cost of survival benefits for
 both annuitants and pensioners.

 The effect of longevity risk (Olivieri and Pitacco, 2002) is even more important with
 the combination of improving mortality and falling interest rates, which have shaken
 the annuity market in recent years, especially for products offering guarantees (Ballotta
 and Haberman, 2002).

 There are a number of broad approaches to forecasting mortality rates – using models
 based on the underlying biomedical process, causal models based on economic-type
 relationships, and trend models that are extrapolative in character. We will consider
 only the last category in this paper.


                                              1
Projecting the behaviour of future mortality rates is a rather complicated process given
that there are many factors that are likely to affect future mortality rates. Some of these
can be easily measured, such as age, sex and marital status, while the effect of others
like social, economic, cultural and ethnic factors is difficult to measure or even to
model. These factors affect different people differently, which makes the process of
forecasting the future course of mortality change a challenge.

But, due to the importance of mortality projection as mentioned above, many attempts
have been made and many methods have been proposed for projecting mortality in the
future. The standard extrapolative approaches used in the literature include: a) models
based on the independent projection of age-specific mortality rates or forces of
mortality, including mortality reduction factor models; b) related models based on the
logit transformation; c) models based on graduating mortality rates with respect to age
for specific time period and then projecting the parameters; d) models based on
graduating mortality rates with respect to age and time simultaneously; e) the Lee-
carter method. Description of these methods can be found in Lee and Carter (1992),
Benjamin and Soliman (1993) and Renshaw et al (1996).
In general there is no single best method and the choice of the appropriate method will
depend on the propose of the projection and the quality and the quantity of the data
available.

This paper aims to investigate the effect of mortality improvements on the expected
costs of annuities using the mortality projection model of Sithole, Haberman and
Verrall (2000), which is of type d). This investigation is performed as an attempt to
answer questions such as: given the improvement in future mortality rates, which age
ranges will contribute the most to the expected changes in annuity values, and what is
the effect of initial age, gender, rate of interest and the level of mortality improvement
on the additional cost implied?
Analytical results are derived using an approach that is an extension to that developed
by Pollard (1982). The results are illustrated using data from the Continuous Mortality
Investigation Bureau (CMIR, 16). Simulation methods are used to consider the
distribution of the annuity cost, as well as the more often quoted point estimates. The
effects of the age of the pensioner at inception, the rate of interest and the gender of the
pensioner are considered.

Finally, a Bayesian model is developed which incorporates the estimation of the
parameters of the mortality projection model within the simulation of the annuity cost.
This allows us to consider the effect of parameter uncertainty on the projected
distribution of the annuity cost.

The paper is organized as follows. In section 2 a brief description of the mortality
projection model of Sithole et al (2000) is given. An analytical analysis of the
differences in annuity values for both male and female pensioners is performed in
section 3, To confirm the results obtained in section 3 and to have an idea regarding the
shape of the distribution of the present value of annuity payments for the different
experiences, the differences in annuity values are investigated using simulation
techniques in section 4. Section 5 deals with the estimation of the parameters of the
mortality projection model using the simulation of the distribution of annuity cost in a



                                             2
Bayesian framework. Lastly, section 6 discusses the overall conclusions and makes
some recommendations.

2. Mortality projection models of Sithole et al (2000)
These models are an application of the structure suggested by Renshaw et al (1996),
which is itself an extension of the “Gompertz-Makeham” (GM) formula used by the
Continuous Mortality Investigation Bureau (CMIB), with an age specific trend
adjustment added. The equation representing the force of mortality at age x in year t is
as follows:

                              s                 r          s                 
         µ x ,t = exp β 0 + ∑ β j L j ( x′) exp ∑  α i + ∑ γ i , j L j ( x′)t ′i 
                                                                                        (2.1)
                                                  i =1                         
                            j =1                         j =1                


where Lj is a Legendre polynomial
and x′, t′ denote the age and time variables which have been transformed linearly and
mapped on to the range [− 1,1] .

From equation (2.1), it can be seen that the first multiplicative term takes the form of a
GM(0,s) formula. The second term (the age specific term) is the product of r
expressions that are very similar to a GM(0,s), with the difference that now each
exponent is multiplied by a power of t′. The values of the parameters can be estimated
using the maximum likelihood method and the optimum values of r and s can be
obtained by comparing the improvement in the scaled deviance, resulting from
successive increases in the values of r and s, with critical values for the χ2 distribution
with one degree of freedom. The optimum values should be the ones after which the
improvement in the deviance is not statistically significant. Full details are given in
Sithole et al (2000).

One difficulty that can arise is ensuring that the fitted model also leads to projected
mortality rates that have a good shape. Thus, the model that provides the best fit to the
historic observed data is not essentially the one to be used, since the smoothness, shape
and the suitability of the model to be used for projections have to be taken into
consideration.

Sithole et al (2000) have developed models for projecting mortality improvement for
pensioners for both males and females by fitting the Renshaw et al (1996) model to
CMI data. The data used relate to male and female life office pensioners1 for the period
from 1983 to 1996. The model that provides the best fit to the data has been
determined, and then projections based on the model over a 20-year period are
considered. By using both the model and the resulting projections, the model can then
be revised in order to produce the reduction factors that can be used subsequently.



1
 i.e. members of pension schemes administered by life insurance companies who, on retirement are
compelled to annuitize: for a discussion of adverse selection in the UK market see Finkelstein and
Poterba (2002).


                                                        3
The reduction factor model recommended here is defined as a ratio of the forces of
mortality rather than mortality rates as originally suggested by the CMIB. Thus, for a
life attaining age x after t years from the base year, the formula for the projected force
of mortality at time t will be as follows:

µx,t = µx,0 RF(x,t)

where µx,0 is the value of µx in the relevant ‘base’ table.

Using the data set mentioned above, the models for the reduction factor, for life office
pensioners, that have been developed by Sithole et al (2000) are as follows:

•   For Female life office pensioners:

       RF(x,t)= exp[(-0.050651+0.000489x)t]



•   For Male life office pensioners:

       RF(x,t)= exp[(-0.078846+0.000744x)t]


When the reduction factor exceeds 1, it is set to be 1.

Table 2.1 shows the reduction factors calculated using these two models for selected
ages (x) and time (t = 10) ahead of the base year.

Table 2.1: Sithole et al (2000) reduction factors

     Age                                Reduction Factors
                                               t=10
                      Female office Pensioners      Male office Pensioners
      65                     0.828068                     0.737227
      70                     0.848564                     0.765168
      75                     0.869567                     0.794168
      80                     0.891090                     0.824268

3. Analysis of the differences in annuity values
In this section we will use the models developed by Sithole et al (2000) for projecting
mortality improvement for both male and female pensioners. To investigate how the
cost of an annuity will change when mortality improvements are allowed for, we will
use an analytical approach to describe the relationship between mortality differences
and the corresponding change in the expected present value of a life annuity at age x.

We first start by defining the cumulative hazard rate:




                                             4
                               x
                 M x + t = ∫ µ x + u du = − ln t p x                         (3.1)
                               0
Consider two bases of mortality, 1 and 2. These could represent different time periods
or one basis could allow for mortality improvements while the other basis does not, or
these could correspond to the two genders. An exact formula, explaining the difference
between the corresponding annuity values is given in equation (3.2):
                                   ∞

                                     x    ( (
                 ax2 − ax = ∫ v t t p1 exp M 1 − M x2 − 1 dt
                        1
                                             x                 ) )           (3.2)
                                   0

Using integration by parts, equation (3.2) can be written as:

                                   ∞
                 a − a = ∫ v t t px ax + t ( µ 1 + t − µ x + t )dt
                     2
                     x
                           1
                           x
                                  2 1
                                               x
                                                         2
                                                                             (3.3)
                                   0


It can also be expressed as:
                                   ∞
                 ax2 − ax = ∫ v t t p1 ax2+ t ( µ 1 + t − µ x + t )dt
                        1
                                     x            x
                                                            2
                                                                             (3.4)
                                   0
Both these equations (3.3 and 3.4) are exact, and there appears to be no theoretical
reason to prefer one to another, so that the integral could be expressed in either form. In
this study we will consider the arithmetic mean of equations (3.3) and (3.4), viz:
                                   ∞
                 ax2 − ax = ∫ ( µ 1 + t −µ x + t ) wx + t dt
                        1
                                  x
                                           2
                                                                             (3.5)
                                   0



               vt
where wx +t   = ( t p x a x +t + t p1 a x2+t ) .
                      2 1
                                    x
               2

This approach is an extension to the one developed by Pollard (1982). If the rate of
interest is equal to zero, equation (3.5) will be exactly the same as that obtained by
Pollard (1982).

The integral in equation (3.5) is not generally convenient for numerical purposes, and
hence an approximation is needed to make the calculations more convenient.
Define:
                           n

                 n   Qy = ∫ µ y + u du                                       (3.6)
                           0



In this case and for numerical evaluation purpose, we calculate nQy from:
                            l 
                   Qy = − ln y + n                                         (3.7)
                 n           l 
                             y 

Then, the mean value theorem for integrals can be used, and we can replace the integral
in (3.5) by a sum of one-year integral, leading to the following approximation:




                                                           5
                                                            (                )
                                                    ∞
                            ax2 − ax ≈ ∑ 1 Q1 + s −1 Qx2+ s wx + s +1 2
                                   1
                                            x                                                                              (3.8)
                                                   s =0


     In equation (3.8), the weights are not calculated at integer ages so that the interpolation
     formula is introduced:

                                  v s +1 2
                  wx + s +1 2 =
                                     2
                                             (   s +1 2                                    )
                                                            p x ax + s +1 2 + s +1 2 p1 ax2+ s +1 2 for each s ∈ (0, ∞ )
                                                              2 1
                                                                                      x                                    (3.9)


     In order to be able to calculate the different values of the weights, a further assumption
     is needed, which assumes that deaths are uniformly distributed over the age range
     (x + s, x + s + 1) for each choice of s. Under this assumption equation (3.9) becomes:

                                                                                                                                   
                    
                  s +1 2
                                         1 1                1
                                                                (
                                           ax + s + px + s ax + s +1                 )              1 2
                                                                                                                  (
                                                                                                        ax + s + p x + s a x2+ s +1   ) 
wx + s +1 2   =
                v
                  2 
                          2
                                  (
                     s px 1−1 2 qx + s 2
                                   2

                                             1 1 
                                                        )                             +  p 1− q
                                                                                            1
                                                                                                 (1
                                                                                        s x 1 2 x+s
                                                                                                      2       )
                                                                                                          1 2 
                                                                                                                                        
                                                                                                                                        
                                                                                                                                       
                                          1 − q x + s                                              1 − q x + s                 
                                           2                                                        2                          
                                                                                                                       (3.10)

     The proof of the above equation is given in full details in Khalaf-Allah (2001).

     Using equation (3.8), the differences in annuity values due to mortality improvements
     can be investigated in order to assess the mortality risk in a life annuity. This analysis is
     performed for life office pensioners for the two genders. In each case the total
     additional annuity cost resulting from allowing for mortality improvements in the future
     is calculated as well as highlighting the age groups that contribute most to this
     additional cost.

     It is also important to test the sensitivity of results to the different factors that are likely
     to affect the mortality risk in a life annuity. In this paper, the effects of age, gender,
     assumed interest rate and the level of mortality improvement on the mortality risk are
     investigated.

     We will consider a single life annuity with annual payment of £1 payable continuously
     each year and a 6% interest rate for discounting payable to a person aged 60 years. In
     each case the expected present value of the annuity is then compared to the expected
     present value of the annuity based on the (1991-1994) mortality tables without allowing
     for any improvement in mortality in the future. Then the sensitivity of the results
     obtained to changes in the factors affecting longevity risk is tested, by allowing age at
     inception, the assumed rate of interest and the assumed level of mortality improvement
     to vary.

     3.1 Analysis of the differences in annuity value for female pensioners

     The projected rates of mortality are produced by applying the reduction factors
     developed by Sithole et al (2000) to the q-type mortality rates from the base table
     ((1991-1994) mortality table). An adjustment is needed, as the reduction factor is



                                                                                 6
defined as a ratio of the forces of mortality rather than mortality rates. The
approximation given by Waters and Wilkie (1987) for qx as a function of µx is used.

                                0.5(µ x + µ x +1 )
               qx =                                                                               (3.11)
                                 1 + 0.5µ x +1

The projected rates can then be used to calculate the additional annuity cost resulting
from allowing for the improvement in mortality by using equation (3.8). An illustration
of these calculations and the contribution to this additional cost by age is shown in
appendix A, table 1.

The total additional cost of annuity after allowing for mortality improvement using the
log-link model for female pensioners is 0.36243967, which represents a relative
increase of 3.04%. A graphical presentation of the difference in the cost of the annuity
for a female aged 60, due to incorporating future mortality improvement using the log-
link model for female pensioners, and how it is spread over future years is given in
figure 3.1.

Figure 3.1 The difference in the cost of the annuity for a female aged 60

                                  Contributions to the differences in annuity value

                                5.00%

                                4.00%
                % Contributed




                                3.00%

                                2.00%

                                1.00%

                                0.00%
                                        1
                                            6
                                                 11
                                                      16
                                                           21
                                                                    26
                                                                         31
                                                                              36
                                                                                   41
                                                                                        46
                                                                                             51
                                                                                                  56
                                                                                                       61




                                                                         t


We can see from figure 3.1 that the percentage contributed to the additional cost
increases by age until it reaches a peak after 16 years, i.e. at age 76 where 4.73655% of
the additional cost is contributed, then it decreases, approaching zero as age increases.
From table 1 in appendix A, it can be seen that no improvement in mortality is assumed
after age 104. It can also be noticed that most of the contribution to the additional cost
has been accumulated over the first 30 years, with 90% of the contribution accumulated
by age 87.

3.2 Analysis of the differences in annuity value for male pensioners

As for the log-link model for female pensioners, the reduction factor is defined as a
ratio of the forces of mortality rather than mortality rates. Hence, the same
approximation in (3.11) for qx is used. Again using equation (3.8) the additional annuity



                                                                7
cost resulting from allowing for the improvement in mortality can be calculated. An
illustration of these calculations and the contribution to this additional cost by age is
shown in Appendix A, table 2. The total additional cost of annuity after allowing for
mortality improvement using the log-link model for male pensioners is 0.65100667,
which represents a relative increase of 6.01%. A graphical presentation of the
difference in the cost of the annuity for a male aged 60, due to incorporating future
mortality improvement using the log-link model for male pensioners, and how it is
spread over future years is given in figure 3.2.


Figure 3.2 The difference in the cost of the annuity for a male aged 60 – mortality

                              Contributions to the differences to the annuity value


                              6.00%
                              5.00%
              % Contributed




                              4.00%
                              3.00%
                              2.00%
                              1.00%
                              0.00%
                                              11
                                                   16
                                                        21
                                                             26
                                                                  31
                                                                       36
                                                                            41
                                                                                 46
                                                                                      51
                                                                                           56
                                                                                                61
                                      1
                                          6




                                                                  t


The percentage contributed to the additional cost is increasing by age until it reaches a
peak after 14 years, i.e. at age 74 where 4.99486% of the additional cost is contributed,
and then decreases approaching zero as age increases. From table 2 (Appendix A) it can
be seen that improvements in mortality are assumed until age 105 only. It can also be
seen that 90% of the additional contribution is accumulated by age 85.

3.3 Sensitivity testing

The value of annuity payments is dependent on many factors, such as age, rate of
interest and the assumed level of mortality. Incorporating mortality improvement is
essential but not enough by itself, as allowing for mortality improvement using a
certain model does not mean that annuity and pension providers are protected against
mortality risk. The future mortality improvement could prove to be more or less than
what has been assumed under the model, and so it is necessary to undertake a
sensitivity test and investigate how the performance of the model varies with changes in
with age, interest rate and the parameters of the mortality model.


3.3.1 Changes in the differences of annuity value with age
This section investigates for each model how the difference in the value of an annuity
changes with age. Since we are interested in analysing the mortality risk for pensioners,
the ages that have been considered for sensitivity testing purpose are those above age


                                                         8
60. The difference in annuity values for the two models have been calculated for the
following ages at inception: 65, 70, 75, and 80. A summary of the results obtained for
each age is shown in table 3.1.

Table 3.1: Differences in annuity values due to the allowance of future mortality
improvement for different ages as a percentage of the present value of annuity
payments when mortality improvements are not included

       Model                                 Age at inception
                             65              70            75               80
 Model for female          4.11%           2.76%         2.38%            1.88%
   pensioners
  Model for male           5.82%           5.29%          4.49%           3.52%
   pensioners

    From table 3.1 it can be seen that, as age increases, the relative differences in
annuity value decrease, reflecting the decreasing effect of mortality improvements as
age increases.
It can also be seen from the table that, for the two models, the additional cost arising
from incorporating future mortality improvements for age 80 is still important and is
more significant in the case of male pensioners than it is for female pensioners.

3.3.2 Changes in the differences of annuity value with the interest rate
This section investigates for each model how the difference in the value of an annuity
values changes with changes in the interest rate. The differences in annuity values for
the two models have been calculated at rates of interest 2% and 8% to investigate how
the differences in the value of the annuity is affected if the interest rate is changed. A
summary of the results obtained under each model is shown in table 3.2.

Table 3.2: Differences in annuity values at age 60 due to the allowance of future
mortality improvement for different rates of interest as a percentage of the present value
of annuity payments when mortality improvements are not included

       Model                                 Rate of interest
                                    2%                             8%
log-link model for                 5.17%                          2.36%
female pensioners

log-link model for                 9.95%                          4.74%
 male pensioners


From the table 3.2 it can be seen that, as the interest rate increases, the relative
differences in annuity values decrease. This effect arises because, at higher interest
rates, the effect on the expected present value of the annuity of mortality improvement
is reduced by the greater discount applied to future payments (See McCrory, 1986).




                                             9
It can also be seen from table 3.2 that the additional annuity cost is sensitive to the rate
of interest used in the calculation, with the percentage for a 2% interest rate being more
than double the percentage for a rate of interest 8%.

3.3.3 Changes in the differences of annuity value with the change in the level of
mortality improvement assumed
This section investigates the effect of changing the parameters of the mortality
projection model (in a particular way) on the differences in annuity values. Throughout
this section the difference in annuity values has been calculated using the reduction
factors as assumed previously, but we now consider the effect if the level of
improvement (expressed in terms of reduction factors) were higher or lower than
initially assumed.

Two cases will be considered, a 35% increase in the reduction factors, and a 35%
decrease in the reduction factors assumed under each of the log-link models.

3.3.3.1 Increasing the reduction factors by 35% (assuming a lower level of
mortality improvement)
Under the two models, if we define a to be the reduction factor for a life attaining age x
and after a period of t years from the base year, then

                     a = RF ( x, t ) = exp[(α + βx )t ]

a* is defined as

                                 [(             )]
                     a* = exp α * + β * x t = 1.35a

when x=55 and duration t=65.The value of α* can be found as follows:

                     α * = log a ( t )− xβ
                                      *          *
                                                                          (3.12)

If we assume also that β=β*, then (3.12) can be expressed as:

                             (
                   α * = log 1.35a
                                          65
                                               )− 55β                      (3.13)

where a = RF (55,65) .

Using equation (3.13) we can calculate the value of α * under the two models, and
hence the revised reduction factor RF * ( x, t ) for each model will be as follows:

        • For Female life office pensioners:
        RF ∗ ( x, t ) = exp[(− 0.046034006 + 0.000489 x )t ]
        • For Male life office pensioners:
        RF ∗ ( x, t ) = exp[(− 0.074229006 + 0.000744 x )t ]




                                                     10
3.3.3.2 Decreasing the reduction factors by 35% (assuming a higher level of
mortality improvement)
The calculation of the revised reduction factors ( a * ) in this case is the same as for case
(1). Hence (α * ) is given by

                          (
                α * = log 0.65a
                                  65
                                       )− 55β                                (3.14)

where a = RF (55,65) .

Using equation (3.14) the value of α * under the two log-link models can be found.
Hence the revised reduction factors RF * ( x, t ) for each model are as follows:

         • For Female life office pensioners:
         RF ∗ ( x, t ) = exp[(− 0.057278429 + 0.000489 x )t ]
         • For Male life office pensioners:
         RF ∗ ( x, t ) = exp[(− 0.085473429 + 0.000744 x )t ]

The summary of the results obtained for a life aged 60 and interest rate of 6% under
each model after using the revised reduction factors is shown in table 3.3.

Table 3.3 Differences in annuity values at age 60 due to the allowance of future
mortality improvement for different scenarios of mortality improvements as a
percentage of the present value of annuity payments when mortality improvements are
not included

       Model                          Revised Reduction Factors
                         Increasing the reduction Decreasing the reduction
                             factors by 35%             factors by 35%
 Model for female                 1.84%                      4.84%
   pensioners
 Model for male                        4.66%                       8.02%
   pensioners

It can be seen from table (3.3) that the additional cost of the annuity calculated using
the two models is sensitive to the assumed level mortality improvements.
Decreasing the reduction factors (i.e. assuming higher mortality improvements.) has a
stronger effect on the additional cost of annuity than increasing the reduction factors by
the same percentage.


4. Analysing annuity values using simulation techniques

In this section the analysis of the difference in the cost of annuities using a stochastic
simulation method will be discussed. In recent years the use of these methods has
increased, with the rapid progress in computer technology and the decline in the real
price of the hardware and the software, which have made simulation methods a cost-
effective way for representing the uncertainty associated with many actuarial problems.


                                                11
In general, simulation methods offer a very powerful tool for handling actuarial
problems, as they allow the modelling of various scenarios that provide a spread of
results and allow the computation of the likelihood of the outcomes. So we can use
simulation techniques to model a particular path that a group of persons can follow
during their lifetime, and hence we can comment on the effect of the mortality risk on a
life annuity portfolio.

By using simulation techniques, we can allow the time of death of each insured in the
portfolio to be a random value, and then perform the same valuations as those
performed in the deterministic framework, so that the results obtained from both
approaches can be compared. Also the mean of the distribution of simulated present
values can be compared with the calculations performed analytically. This will be done
for ages 60, 70 and 80 for both male and female office pensioners.

The following sections give a brief description of the simulation procedures and
methodology used. The results of the simulations are then summarised and compared
with those obtained using the analytical approach.


4.1 Methodology

For each age, for both males and females, the calculations are based on a single life
annuity with payments of £1 due at the end of each year and an interest rate of 6%. The
cost of the annuity is calculated to be the average present value of payments made to
the members of the portfolio. For each policyholder and for each year we generate a
random number from a uniform (0,1) distribution. If the q-type probability of the
policyholder is smaller than this number, we consider that the policyholder survives and
we then record that the relevant survival payment is made for that year. If the
policyholder survives for that year another random number is generated and if again
this is bigger then the q-type probability for that interval it is assumed that the
policyholder survives again and the relevant payment is recorded. This continues until
the policyholder dies. All the payments made to this policyholder during his/her
lifetime are then recorded and the present value (at outset) of the annuity payments
received, for each policyholder, is calculated. This is carried out for all n policyholders.
We now have a sum comprising all of the discounted payments made to the group of
policyholders. It is assumed that the policyholders are all of the same age, and hence
the n simulations can be regarded as applying to one single policyholder instead of one
simulation for each of n policyholders. (These two approaches should lead to the same
result.)

The number of simulations in each case has been determined such that the results
obtained are representative, as the distribution of the results is found to be sensitive to
the number of simulations performed. i.e. the number of simulations is chosen such that
the mean of the simulated distribution is close to the value obtained analytically, and it
is noted that increasing the number of simulations does not greatly affect either the
standard deviation or the coefficient of variation of the simulated distribution.

For each age and sex combination, the simulation results for the two different sets of
mortality rates are compared; the (1991-1994) life office pensioners tables without


                                            12
              allowing for any future mortality improvement and the (1991-1994) life office
              pensioners tables with mortality improvement being allowed for using the log-link
              model suggested by Sithole et al (2000).

              4.2 Simulation results for female pensioners

              4.2.1 Simulated distributions based on various mortality bases for female
              pensioners aged 60
              For female pensioners aged 60, the number of simulations that has been used is 60000.
              Table 4.1 shows the summary of the descriptive statistics of the distribution of present
              values of annuities under the two mortality bases assumed; the (1991-1994) life office
              pensioners tables without allowing for any future mortality improvement and the (1991-
              1994) life office pensioners tables with mortality improvements being allowed for using
              the log-link model suggested by Sithole et al (2000).

              Table 4.1 Summary of the descriptive statistics of the simulated distributions- female
              pensioners aged 60

Mortality basis           (1)             (2)            (3)               (4)               (5)         (1)-(5)/(5)
                    Mean value of     Standard      Coefficient of    Skewness of        Expected          Error
                          the        deviation of    variation             the        present value of
                     distribution         the                         distribution      the annuity
                                     distribution                                        obtained
                                                                                        analytically
  (1991-1994)           11.419          3.115          27.28%          -1.37854          11.40663           0.11
pensioners tables                                                    (left skewed)


    Allowing for        11.852          3.109          26.23%          -1.56035          11.76808           0.71
     mortality                                                       (left skewed)
   improvement
 (log-link model)

              From table 4.1 it can be seen that the mean value of the distribution under each case is
              close to the analytical value. A graphical presentation of the simulated distributions
              based on the two different mortality bases is shown in figure 4.1.




                                                         13
                             Figure 4.1

                             Distribution of Present value                                                                       Distribution of Present value
                                          Female Age 60                                                                                       Female Age 60
                                Based on 1991-1994 Pensioners tables                                                Based on 1991-1994 Pensioners tables, with mortality improvements
                                                                                                              being allowed for using the log-link model suggested by Renshaw et al (1996)
            2500
                                                                                                       3200




                                                                                                       2200
Frequency




            1500




                                                                                           Frequency
                                                                                                       1200

            500

                                                                                                       200


                    0    2      4     6      8     10     12    14     16   18                                  0        2       4       6       8      10     12      14      16       18
                             Present value of annuity payments                                                               Present value of annuity payments




                             It is observed that the basis allowing for mortality improvements produces a
                             distribution that is more skewed to the left than the distribution obtained by using the
                             (1991-1994) mortality tables with no margin for mortality improvement, reflecting a
                             higher expected present value of the annuity after allowing for mortality improvement.

                             4.2.2 Simulated distributions based on various mortality bases for female
                             pensioners aged 70
                             For female pensioners aged 70, the number of simulations that has been used is also
                             60,000. The same mortality bases as assumed for female pensioners aged 60 have been
                             used to produce the required simulated distributions. The descriptive statistics of the
                             distribution of the present value of annuities under the two bases of mortality are shown
                             in table 4.2.

                             Table 4.2 Summary of the descriptive statistics of the simulated distributions; female
                             pensioners aged 70

                    Mortality                  (1)                    (2)            (3)                       (4)                            (5)                       (1)-(5)/(5)
                     basis                Mean value              Standard       Coefficient              Skewness of                      Expected                       Error
                                             of the              deviation of        of                        the                          present
                                          distribution                the         variation               distribution                    value of the
                                                                 distribution                                                               annuity
                                                                                                                                           obtained
                                                                                                                                          analytically
                   (1991-1994)               8.7316                    3.5412     40.56%                       -0.64                       8.715579                          0.18
                    pensioners                                                                            (left skewed)
                      tables

                    Allowing for             9.0296                    3.6171     40.06%                       -0.71                         8.968883                        0.68
                     mortality                                                                            (left skewed)
                   improvement
                      (log-link
                       model)



                                                                                      14
                   From tables 4.1 and 4.2, it can be seen that the standard deviation for female pensioners
                   aged 70 is higher than the corresponding one for female pensioners aged 60 for the two
                   mortality bases, indicating a higher level of variability. It is also clear that the
                   coefficient of variation has increased dramatically for the two mortality bases compared
                   to the corresponding values for female pensioners aged 60. Again, this reflects a higher
                   variability in the distribution of the present value of annuity payments as age increases.

                   A graphical presentation of the simulated distributions based on the two different
                   mortality bases is shown in figure 4.2.

                   Figure 4.2


                         Distribution of Present value
                                       Female Age 70                                                          Distribution of Present value
                           Based on (1991-1994) Pensioners tables                                                                  Female Age 70
                                                                                                     Based on 1991-1994 Pensioners tables, with mortality improvements
                                                                                                 being allowed for using the log-link model suggested by Renshaw et al (1996)
            2500
                                                                                          2500
Frequency




                                                                              Frequency


            1500
                                                                                          1500




            500
                                                                                           500


                     0             5                   10           15
                                                                                                     0                     5                    10                    15
                         Present value of annuity payments
                                                                                                              Present value of annuity payments




                   In general, the two distributions show less skewness to the left than the corresponding
                   distributions for age 60 as the present value of the annuity payments is expected to
                   decrease as age increases. As before, we observe that the basis allowing for mortality
                   improvements produces a distribution that is more skewed to the left than the
                   distribution obtained by using the (1991-1994) mortality tables with no margin for
                   mortality improvement. This reflects a higher present value of the annuity after
                   allowing for mortality improvement. However, the difference is not as great as it is for
                   age 60, reflecting the decreasing effect of mortality improvements as age increases.

                   4.2.3 Simulated distributions based on various mortality bases for female
                   pensioners aged 80
                   For female pensioners aged 80 the number of simulations was increased to be 65000, in
                   order to obtain a distribution that is sufficiently stable.
                   The same mortality bases as assumed for female pensioners aged 60 and 70 were used.
                   The descriptive statistics of the distribution of the present value of annuities under the
                   three bases of mortality are shown in table 4.3.




                                                                         15
                      Table 4.3 Summary of the descriptive statistics of the simulated distributions- female
                      pensioners aged 80

 Mortality                       (1)                                   (2)              (3)                             (4)                             (5)                            (1)-(5)/(5)
  basis                    Mean value of                           Standard        Coefficient of                  Skewness of                       Expected                            Error
                          the distribution                      deviation of the    variation                           the                       present value
                                                                  distribution                                     distribution                   of the annuity
                                                                                                                                                     obtained
                                                                                                                                                   analytically
 (1991-1994)                      5.9474                                 3.3812      56.85%                             -0.17                        5.853390                              0.16
  pensioners                                                                                                       (left skewed)
     tables
 Allowing for                     6.0852                                 3.4512      56.71%                             -0.45                           5.972037                           0.19
  mortality                                                                                                        (left skewed)
improvement
   (log-link
    model)

                      From table 4.3, it can be seen that the mean value of the distribution for each case is
                      close to the analytical value. Although the variability of the distributions increases as
                      age increases, the value of the standard deviation for female pensioners aged 80 is
                      lower than the corresponding one for female pensioners aged 70. But again the
                      coefficient of variation has increased dramatically for the two mortality bases compared
                      to the corresponding values for female pensioners aged 60 and 70 reflecting a higher
                      relative variability in the distribution of the present value of annuity payments as age
                      increases. A graphical presentation of the simulated distributions based on the two
                      different mortality bases is shown in figure 4.3.

                      Figure 4.3

                          Distribution of Present value                                                                 Distribution of Present value
                                         Female Age 80
                                                                                                                                          Female Age 80
                            Based on the (1991-1994) Pensioners tables                                          Based on 1991-1994 Pensioners tables, with mortality improvements
                                                                                                            being allowed for using the log-link model suggested by Renshaw et al (1996)
               4200                                                                                  4000


               3200
                                                                                                     3000
   Frequency




                                                                                         Frequency




               2200
                                                                                                     2000

               1200
                                                                                                     1000

               200

                      0              5                     10               15
                          Present value of annuity payments                                                    0                      5                      10                     15
                                                                                                                       Present value of annuity payments


                      It is hardly noticeable from the graphical presentation that the two distributions are
                      skewed to the left. This is consistent with what was expected: the level of skewness
                      becomes less negative as age increases reflecting the fact that the present value of the
                      annuity payments is expected to decrease as age increases.


                                                                                    16
           Again it can be seen that the basis allowing for mortality improvements produces a
           distribution that is more skewed to the left than the distribution obtained by using the
           (1991-1994) mortality tables with no margin for mortality improvements. This reflects
           a higher present value of annuities after allowing for mortality improvements.
           However, the difference is not as great as it is for age 60 or even as for age 70,
           reflecting the decreasing effect of mortality improvement as age increases. Moreover,
           we note that the distribution based on the (1991-1994) mortality tables exhibits a very
           similar pattern to that obtained after allowing for mortality improvements, as after age
           80 the effect of mortality improvement is of lesser significance.


           4.3 Simulation Results for male pensioners

           4.3.1 Simulated distributions based on various mortality bases for male pensioners
           aged 60
           For male pensioners aged 60, 20000 simulations were sufficient to represent the
           population under consideration. Table 4.4 gives a summary of the descriptive statistics
           of the distribution of present values of annuities under the same mortality bases used
           before for female pensioners: the (1991-1994) life office pensioners tables without
           allowing for any future mortality improvements and the (1991-1994) life office
           pensioners tables with mortality improvement being allowed for using the log-link
           model for male pensioners.


           Table 4.4 Summary of the descriptive statistics of the simulated distributions- male
           pensioners aged 60

Mortality basis          (1)            (2)           (3)             (4)               (5)            (1)-(5)/(5)
                    Mean value      Standard      Coefficient    Skewness of        Expected             Error
                       of the      deviation of        of             the        present value of
                    distribution        the        variation     distribution      the annuity
                                   distribution                                     obtained
                                                                                   analytically
   (1991-1994)        10.349           3.331        32.19%         -1.03493         10.32564              0.23
pensioners tables                                                (left skewed)
   Allowing for       10.994           3.438        31.27%         -1.19103          10.97443             0.18
    mortality                                                    (left skewed)
  improvement
(log-link model)

           From table 4.4, it can be seen that the mean value of the distribution under each case is
           close to the analytical value. We observe that the coefficient of variation is higher for
           the two mortality bases than the corresponding values for female pensioners aged 60.
           This reflects a higher variability in the distribution of the present value of annuity
           payments in the case of male pensioners. A graphical presentation of the simulated
           distributions based on the two different mortality bases is shown in figure 4.4.




                                                      17
                        Figure 4.4

                        Distribution of Present value                                                                        Distribution of Present value
                                                                                                                                             Male Age 60
                                       Male Age 60
                                                                                                                   Based on 1991-1994 Pensioners tables, with mortality improvements
                           Based on 1991-1994 Pensioners tables                                                 being allowed for using the log-link model suggested by Renshaw et al (1996)
            900
                                                                                                        1100
            800

            700

            600




                                                                                            Frequency
Frequency




            500                                                                                         600
            400

            300
            200

            100                                                                                         100

                    0             5                  10           15                                                0                    5                    10                   15
                        Present value of annuity payments                                                                   Present value of annuity payments


                        It can be seen that the basis allowing for mortality improvements using the log-link
                        model for male pensioners produces a distribution that is more left skewed than the
                        distribution obtained by using the (1991-1994) mortality tables with no margin for
                        mortality improvements. This reflects a higher present value of annuities after allowing
                        for mortality improvements. It is also worth mentioning that the level of the skewness
                        for the two mortality bases is less than the corresponding levels for female pensioners
                        at age 60: this can be attributed to the lower mortality rates for females which lead to
                        higher expected annuity values.
                        Figure 4.4 shows that the two distributions are a bit more spread out than the
                        corresponding ones for female pensioners aged 60 (Figure 4.1).

                        4.3.2 Simulated distributions based on various mortality bases for male pensioners
                        aged 70
                        For male pensioners aged 70, the number of simulations used was increased to 60000.
                        The same mortality bases were used to produce the required simulated distribution. The
                        descriptive statistics of the distribution of present value of annuities under the two
                        mortality bases are shown in table 4.5.

                        Table 4.5 Summary of the descriptive statistics of the simulated distributions- male
                        pensioners aged 70

                  Mortality basis                  (1)                     (2)          (3)                         (4)                            (5)                             (1)-(5)/(5)
                                              Mean value               Standard      Coefficie                 Skewness of                      Expected                             Error
                                                 of the                 deviation      nt of                        the                      present value
                                              distribution                of the     variation                 distribution                  of the annuity
                                                                       distributio                                                              obtained
                                                                            n                                                                 analytically
                 (1991-1994)                         7.5036              3.5547      47.37%                         -0.35                       7.462788                                  0.55
              pensioners tables                                                                                (left skewed)
                 Allowing for                        7.9118              3.7268      47.10%                         -0.41                         7.881624                                0.38
                  mortality                                                                                    (left skewed)
                improvement
              (log-link model)




                                                                                       18
                   From table 4.5, it can be seen that the mean value of the distribution under each case is
                   close to the analytical value. Again the coefficient of variation has increased
                   dramatically for the two mortality bases as we move from age 60 to 70, reflecting a
                   higher variability in the distribution of the present value of annuity payments as age
                   increases. The coefficient of variation for the two models is also higher than the
                   corresponding values for female pensioners at age 70 confirming the conclusion
                   reached before in section 3 regarding the higher variability for the distribution of the
                   present value for male pensioners compared to female pensioners at the same age. A
                   graphical presentation of the simulated distributions based on the two different
                   mortality bases is shown in figure 4.5.

                   Figure 4.5
                   In general, the two distributions are less skewed to the left than the corresponding ones
                          Distribution of Present value                                                       Distribution of Present value
                                         Male Age 70                                                                           Male Age 70
                                                                                                     Based on 1991-1994 Pensioners tables, with mortality improvements
                             Based on 1991-1994 Pensioners tables
                                                                                                 being allowed for using the log-link model suggested by Renshaw et al (1996)

            3000
                                                                                          2500




                                                                              Frequency
            2000
Frequency




                                                                                          1500



            1000
                                                                                          500



                                                                                                     0                     5                     10                    15
                      0             5                  10           15                                        Present value of annuity payments
                          Present value of annuity payments


                   at age 60 as the present value of the annuity payments is expected to decrease as age
                   increases. The graphs for males have the same general shape as those for females at age
                   70 (Figure 4.2). We can observe that, as for females, the basis allowing for mortality
                   improvements produces a distribution that is more skewed to the left than the
                   distribution obtained by using the (1991-1994) mortality tables with no margin for
                   mortality improvements reflecting a higher present value of the annuity after allowing
                   for mortality improvements. However, the difference is not as great as it is for male
                   pensioners aged 60, reflecting the decreasing effect of mortality improvements as age
                   increases. Again, the level of the skewness for each of the mortality bases is less than
                   the corresponding one for female pensioners at age 70 as a result of the lower mortality
                   rates expected by females It is also noted that, as for the case for male pensioners aged
                   60, the two distributions are a bit more spread out than the corresponding ones for
                   female pensioners at the same age.

                   4.3.3 Simulated distributions based on various mortality bases for male pensioners
                   aged 80
                   For male pensioners aged 80, the number of simulations was increased to 65000 in
                   order to obtain a distribution that is sufficiently stable, and the same mortality bases
                   were used. The descriptive statistics of the distribution of present value of annuities
                   under the three bases of mortality are shown in table 4.6.




                                                                         19
                         Table 4.6 Summary of the descriptive statistics of the simulated distributions- male
                         pensioners aged 80

            Mortality                (1)                                (2)           (3)                    (4)                                    (5)                           (1)-(5)/(5)
             basis             Mean value of                        Standard      Coefficient          Skewness of the                           Expected                           Error
                              the distribution                     deviation of        of               distribution                          present value
                                                                        the        variation                                                  of the annuity
                                                                   distribution                                                                  obtained
                                                                                                                                               analytically
 (1991-1994)                            4.8812                       3.1901        65.35%                      0.262224                          4.762788                             0.0249
  pensioners                                                                                                  (positively
     tables                                                                                                    skewed)
 Allowing for                           5.0597                       3.3074        65.37%                      0.233297                            4.946035                           0.0230
  mortality                                                                                                   (positively
improvement                                                                                                    skewed)
   (log-link
    model)

                         From table 4.6 it can be seen that the mean value of the distribution under each case is
                         fairly close to the analytical value. Although the variability of the distributions
                         increases as age increases, the value of the standard deviation for male pensioners aged
                         80 is lower than the corresponding value for male pensioners aged 70. But if we
                         compare the values of the coefficient of variation we find that at age 80 it has increased
                         dramatically for the two mortality bases compared to the corresponding values for male
                         pensioners aged 60 and 70. This reflects a higher relative variability in the distribution
                         of the present value of annuity payments as age increases. The coefficient of variation
                         for the two models is also higher than the corresponding values for female pensioners at
                         age 80 confirming the conclusion reached before regarding the higher variability for the
                         present value distribution for male pensioners compared to female pensioners at the
                         same age. A graphical presentation of the simulated distributions based on the two
                         different mortality bases is shown in figure 4.6.

                         Figure 4.6

                        Distribution of Present value                                                                          Distribution of Present value
                                       Male Age 80
                                                                                                                                                  Male Age 80
                            Based on 1991-1994 Pensioners tables                                                       Based on 1991-1994 Pensioners tables, with mortality improvements
                                                                                                                   being allowed for using the log-link model suggested by Renshaw et al (1996)
            5500                                                                                            5500

            4500
                                                                                                            4500
Frequency




            3500
                                                                                                Frequency




                                                                                                            3500

            2500
                                                                                                            2500

            1500
                                                                                                            1500
             500
                                                                                                            500
                   0               5                    10            15
                        Present value of annuity payments                                                             0                      5                     10                      15
                                                                                                                              Present value of annuity payments




                         We note from the graphical presentation that the two distributions are positively
                         skewed. Although this feature is not observed for the case of female pensioners aged


                                                                                      20
80, it is not unexpected. Thus, we expect the distribution of the present value of annuity
payments to turn from a negatively skewed distribution to be a positively skewed one at
some age, as age increases, to reflect the fact that the present value of the annuity
payments is expected to decrease as age increases.

We observe that the basis allowing for mortality improvements produces a distribution
that is less skewed to the right than the distribution obtained by using the (1991-1994)
mortality tables with no margin for mortality improvements. This reflects a higher
present value of annuities after allowing for mortality improvement. However, the
difference is not as great as it is for age 60 or even as for age 70, reflecting the
decreasing effect of mortality improvements as age increases.

Both the distribution of present values using the log-link model to allow for mortality
improvements and the distribution based on the (1991-1994) mortality tables, which
does not allow for mortality improvements, exhibit a similar pattern. This is because
after age 80 the effect of mortality improvements is of lesser significance.


5. Analysing annuity values in a Bayesian framework
In this section we adopt a Bayesian approach that combines the estimation of the
parameters of the Sithole et al (2000) mortality projection models together with the
simulation of the annuity cost.

Although Bayesian statistical methods may be seen as the most convenient method for
the implementation and analysis of many models arising in actuarial science, it was not
until recently that it was fully used following the development of Markov Chain Monte
Carlo (MCMC) simulation methods.

A review of the basis of MCMC method is given below followed by a brief description
of the methodology used. The results obtained from this approach are then compared
with those discussed earlier in this paper.

5.1 Markov Chain Monte Carlo

MCMC methods provide a unifying framework in which many complex problems can
be analysed. Consider a vector random variable U=(U1,…,UK) with a joint distribution
f(U1,…,UK). Suppose f (U) has a complicated and analytically intractable form, and the
expected value of some integrable function h(U) is sought. Even if this calculation
cannot be performed analytically, it is still possible that the probabilistic model
associated with f(U) may be simple enough to permit independent random draws U(t) ,
t=1,…,n from it. If this is the case, then the desired expectation can be approximated
using the sample averages.
                1 n
i.e. E [h(U )] ≈ ∑ h(U ( t ) ) .
                n t =1
This procedure is called Monte Carlo integration. Unfortunately, many complicated
models will not readily permit independent random draws. In this case a MCMC
method is used to simulate realizations from a Markov chain that is constructed so that
its stationary distribution is the posterior distribution. Thus, this Markov chain has f(U)



                                            21
as its stationary distribution (see, for example, Gilks et al 1996). Various algorithms
exist for carrying out the required simulations, including Gibbs sampling which was
devised by Geman and Geman (1984) and subsequently introduced to the statistics
literature by Gelfand and Smith (1990).

5.2 Methodology

A full Bayesian model has been constructed to implement the corresponding MCMC-
Bayesian analysis which is needed to estimate the parameters of the Sithole et al (2000)
models, and hence to forecast future forces of mortality. Then, we consider a portfolio
of persons and we allow the time of death for each person to be a random variable and
then calculate the present value of annuity payments for each person and obtain the
mean value of the simulated distribution of annuity payments. This can be carried out in
the same manner as mentioned earlier in section 4, except for the fact that it is
performed in a Bayesian framework.
In line with the calculations performed in section 4, for each age for both male and
female life office pensioners, we will consider a single life annuity with payments of £1
due at the end of each year and an interest rate of 6 percent.
To carry on the MCMC-Bayesian analysis, a specialized software package that
performs Bayesian inference using Gibbs sampling known as WinBugs was used. A
useful review of actuarial application of MCMC methods is given by Scollnik (2001).


5.2.1 Estimation of the parameters of Sithole et al (2000) mortality projection
models
Following the Bayesian framework, we treat all model parameters as unknowns and we
specify prior information via probability density functions. In this section, totally
uninformative priors for the parameters have been used.

Let θ denote the parameter vector θ=(β0,β1,β2,β3,α1,γ11), where β0,β1,β2,β3,α1 and γ11
are the 6 parameters of the Sithole et al model with prior probability p(θ), and y denote
the data vector which in this case will be the CMI life office pensioners’ experience. y
will represent the female office pensioners’ experience when we estimate the
parameters for female pensioners and it will represent the male office pensioners’
experience when we estimate the parameters for male pensioners.

In order to make a probability statement about θ given y, we must begin with a model
providing a joint probability distribution for θ and y which can be expressed as the
product of the prior p(θ ) and the sampling distribution p( y | θ ) . Following Bayes’
theorem the posterior density p(θ | y ) is equivalent to:

                    p (θ | y ) ∝ p (θ ) p ( y | θ )               (5.1)


The descriptive statistics of the parameters values for both male and female office
pensioners is shown in table (5.1).

Table 5.1 Summary of the descriptive statistics of the parameters of Sithole et al(2000)-
male and female life office pensioners


                                                      22
       Male pensioners Model                        Female Pensioners model
Parameter       Estimate       Standard       Parameter       Estimate       Standard
                 (mean)        deviation                       (mean)        deviation
     β0           -2.535       0.003822            β0           -3.177       0.005547
     β1           1.822         0.01069            β1           1.838         0.01157
     β2          -0.2639       0.009513            β2          -0.0263        0.01379
     β3         -0.04892        0.01169            β3         -0.03614        0.01524
     α1          -0.1257       0.003409            α1         -0.08306       0.007964
     γ11        0.09621         0.00982            γ11        0.05582         0.01725

It can be seen from table 5.1 that the parameter values using Bayesian inferences are
very close to the maximum likelihood estimates obtained by Sithole et al (2000).

5.2.2 Analysing annuity values
After estimating the parameter values we use simulation techniques to model a
particular path that a group of people can follow during their life time in the same
manner as described earlier in section 4, but in the context of a Bayesian analysis.

The resulting distributions of the present value of annuity payments can then be
compared with the corresponding ones in section 4, as well as comparing the mean
value to the present value of annuity payments obtained analytically. This will be
implemented for ages 60, 70 and 80 for both male and female life office pensioners.
A summary of the descriptive statistics of the simulated distribution of annuity
payments in Bayesian framework for both male and female life office pensioners
is shown below in table 5.2.

Table 5.2 Summary of the descriptive statistics of the simulated distributions of annuity
payments male and female life office pensioners

 Age            Male pensioners Model                  Female Pensioners model
           Mean Prediction        Value            Mean    Prediction      Value
           value     error       obtained          value      error       obtained
                                analytically                            analytically
  60       11.000    3.508        10.974           11.800     3.161        11.768
  70        7.960    3.718         7.882           9.087      3.508        8.969
  80        5.011    3.230         4.946           5.940      3.302        5.970

It can be seen from table 5.2 that the mean value of the distribution is close to the
corresponding analytical value as well as being close to the corresponding value,
obtained in section 4 for all ages examined for both genders. It is also obvious that the
coefficient of variation increases dramatically with age for both males and females,
reflecting a higher level of variability as age increase. For males it is 31.9%, 46.7% and
64.5% for ages 60, 70 and 80 respectively, and for females it is 26.6%, 38.6% and
55.6% for ages 60, 70 and 80 respectively.


5.2.3 Measuring the effect of parameter uncertainty
In this section, the distribution of the present value of annuity payments is calculated


                                            23
using parameter values that have been estimated within the model itself. This means
that unlike the case in section 4, we have allowed each of the parameters to be a
random variable with a distribution rather than a fixed value that is the point estimate of
this random variable. Hence, the variability due to parameter uncertainty is included in
the prediction error of the mean value of the present value of annuity payments.
As a measure of parameter uncertainty, we can fix the value of each of the parameters
to be the maximum likelihood estimates obtained by Sithole et al (2000), and carry out
a similar simulation process as in section 4 in order to obtain a distribution of the
present value of annuity payments in a Bayesian framework. We expect the prediction
error to decrease reflecting the contribution of parameter uncertainty to the overall
variability.

The prediction error of the mean value of the simulated distribution of annuity
payments in a Bayesian framework for both male and female life office pensioners is
shown below in table 5.3. Comparison of tables 5.2 and 5.3 shows the change in
prediction error which can be regarded as a measure of the effect of parameter
uncertainty.

Table 5.3 Prediction error of the mean value of the simulated distribution of annuity
payments in Bayesian framework where all the parameters are fixed to a certain value -
male and female life office pensioners

     Age             Male pensioners Model                Female Pensioners model
                    Prediction    Change as a            Prediction    Change as a
                      error            %                   error            %
      60              3.431          2.19%                 3.106          1.74%
      70              3.700          0.48%                 3.500          0.23%
      80              3.229         0.031%                 3.289          0.39%


6. Summary and Conclusions

In this paper, the theory of Pollard (1982) has been extended to describe the
relationship between mortality differences and the corresponding change in the
expected present value of a life annuity at age x. As a result of this extension, equation
(3.8) has been developed providing a simple and effective tool for calculating the
difference in annuity values resulting from using two different sets of mortality rates.

Equation (3.8) is used in section 3 to analyse the differences in the present value of
annuity payments after allowing for mortality improvements using the log-link models
proposed by Sithole et al (2000). From the results obtained, it is concluded that the
difference in annuity values, as a percentage of the present value of annuity payments
before allowing for mortality improvements, for males is higher than the corresponding
values for females at the same age. This implies a higher relative impact on annuity
values from future mortality improvements for males than for females under these two
models.

From the analysis in sections 3 and 4, we note that, for a starting age of 60, the age
range that contributes the most to the additional cost associated with allowing for future
mortality improvements is ages 73- 80. It may seem reasonable initially to believe that


                                            24
the years that contribute most to the difference in the annuity values would be those
immediately following the inception of the annuity. However it has been demonstrated
that the main contribution comes from ages that are 13-20 years on average from the
annuity inception.

Incorporating future mortality improvements for older ages (80 and above) is also
important, reflecting the high level of the contributions made by these ages to the
additional cost of the annuity. It is seen in section 3 that, even for an age at inception of
80, the percentage increase in the annuity value due to the allowance for mortality
improvement is still significant for both male and female office pensioners models,
ranging from 1.88% increase in the value of the annuity obtained by using the log-link
model for female pensioners to a 3.52% increase in the value of the annuity for male
pensioners. This reflects the importance of mortality improvements for ages 80 and
above.

As mentioned in section 3, the present value of annuity payments, and hence the
difference in annuity values, is sensitive to the rate of interest used in valuations. This
means that the lower the rate of interest, the higher the effect of the longevity risk on
the present value of annuity payments. In other words, the effect on the present value of
annuity payments of living longer than average is increased by the lower discounting
applied to the future payments in a low interest environment.

As was also mentioned in section 3, the present value of annuity payments, and hence
the difference in annuity values, is sensitive to the level of mortality improvement
assumed in the model. It is also noted that decreasing the reduction factors (i.e.
assuming higher mortality improvements) has a stronger effect on the additional cost of
annuity than increasing the reduction factors by the same percentage.

The above results are also confirmed using a simulation-based approach. Using the
simulated distributions, we can draw some conclusions regarding the mortality risk in
the context of a life annuity portfolio:

•   Generally, there is less variation in the distribution of the present value of annuity
    payments for female pensioners than in the corresponding one for male pensioners,
    and for both genders the coefficient of variation increases with age.
•   As age at inception increases, the effect of mortality improvement decreases and the
    shape of all the distributions become similar to each other.
•   For younger ages, at inception, the shape of the distribution of present value of
    annuity payments is negatively skewed. With increasing age, the distribution
    becomes less negatively skewed until, at some point, it becomes positively skewed,
    reflecting the fact that the present value of annuity payments is expected to be lower
    for older ages. This has been shown for males at age 80.

Finally, a full Bayesian model is constructed to implement the corresponding MCMC-
Bayesian analysis needed to estimate the parameters of the Sithole et al (2000) models.
This is then used to forecast future mortality rates and simulate the annuity cost. An
approach for measuring the effect of parameter uncertainty is presented and
implemented.



                                             25
In general, Bayesian simulation techniques have the advantage of providing a better
assessment for the risk under question. This is because they give the whole distribution
of the present value of annuity payments while the deterministic approach provides
only the mean value of the distribution. For example, the deterministic approach does
not provide any information regarding the dispersion of the distribution. MCMC
methods also provide a very flexible framework, in the sense that they allow the
modelling of different scenarios at the same time, within which different models, or
different aspects of particular models, can be investigated.




                                           26
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