VIEWS: 6 PAGES: 30 POSTED ON: 4/25/2011 Public Domain
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 27 28 References Ballotta, L. and Haberman S. (2002). Valuation of Guaranteed Annuity Conversion Options. Under review. Charlton, J. (1997). Trends in all-cause mortality, 1841-1994. The Health of Adult Britain, 1841-1994. Office for national statistics, vol. 1, Decennial Supplement No. 12. Chapter 4. HMSO, London Benjamin, B and Soliman, A (1993). Mortality on the Move. Distributed by Actuarial Education Service, printed by City University London. Continuous Mortality Investigation Bureau (1998) Continuous Mortality Investigation Reports. CMIR, 16, The Institute of Actuaries and the Faculty of Actuaries, United Kingdom. Finkelstein, A. and Poteba, J. (2002). Selection Effects in the United Kingdom Individual Annuity Market. The Economic Journal, vol. 112, pp 28-50. Gelfand, A.E. and Smith, A.F.M. (1990) Sampling-based Approximations to calculating Marginal Densities. J of the American Statistical Association, vol. 85, pp 398-409. Geman, S. and Geman, D. (1994) Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images. IEEE Transactions on Pattern Analysis and Machine Intelligence., vol. 6, pp721-41. Gilks, W. R., Richardson, S. and Spiegelhalter, D.J. (1996) Markov Chain Monte Carlo Methods in Practice. Chapman and Hall Khalaf-Allah, M (2001) Mortality Changes and the Cost of life Annuities. MSc dissertation, City University. Lee R. and Carter L. (1992). Modelling and Forecasting US Mortality. Journal of American Statistical Association, vol. 87, pp 659-671. McCrory, R.T (1986) Mortality Risk in Life Annuities. Transactions of the Society of Actuaries, vol. 36, pp.309-338. Olivieri, A. and Pitaco, E. (2002). Inference about Mortality improvement in Life Annuity Portfolios. Presented to 27th international Congress of Actuaries, Cancun, Mexico. Pollard, J.H, (1982) The Expectation of Life And Its Relationship to Mortality. Journal of the Institute of Actuaries, vol. 109, pp.225-240. Renshaw, A., Haberman, S. and Hatzopoulos, P. (1996) Modelling of Recent Mortality Trends in UK Male Assured Lives. British Actuarial Journal, vol. 2, pp 449-477. Scollnik, D.P.M. (2001) Actuarial Modelling with MCMC and BUGS. North American Actuarial Journal, 5 (2), pp 96-124. 29 Sithole, T.Z., Haberman, S. and Verrall, R.J. (2000) An Investigation into Parametric Models For Mortality Projections, With Applications to Immediate Annuitants’ and Life Office Pensioners’ Data. Insurance Mathematics and Economics, vol.27, pp.285-312. Thatcher, A.R. (1999). The Long Term Pattern of Adult Mortality and Highest Attended Age. Journal of Royal Statistical Society, Series A, vol. 162, pp 5-43. Waters, H.R and Wilkie A.D. (1987). A Short Note On The Construction Of Life Tables And Multiple Decrement Tables. Journal of the Institute of Actuaries, vol. 114, pp 569- 580. 30