The Steps Using Lee Carter Model Economics Essay In this section the BRP will be assessed for the 20 years by using the Lee-Carter model to estimate population. Since the Central Statistics Office (CSO) has not yet published the data for 2012, there will be the need to estimate the demography for the Island of Mauritius for the year 2012, thereafter forecasting the demography will be done as from year 2013 to 2031. In order to be able to assess BRP for the Island of Mauritius, there are two important elements which are essential to be able to estimate the BRP cost: 1) The estimated population who will be eligible to the BRP 2) The estimated future cost of the BRP per eligible. RESEARCH METHODOLOGY 4.1 The estimated population who will be eligible to the BRP To be able to forecast the population level at a particular time, the estimated number of death is one of the most determining factor. The method used to estimate the population at a particular year is through the use of Lee-carter model and Least Square method by estimating the regression equation. Estimating Death Forecasting deaths observations over time are stochastic data and it is one of the major elements which are required to predict the costs of BRP in the future. Lee-Carter model is one of the most widely used in both academic and practical applications. This essay is an example of a student’s work Disclaimer This essay has been submitted to us by a student in order to help you with your studies. This is not an example of the work written by our professional essay writers. Essay Writing Service Dissertation Writing Service Who wrote this essay Place an Order The model under Lee and Carter is an equation for the natural logarithm of the mortality rate for the year t of a specific group age x. ln(rm,t) = ax + bxkt + єx,t (1) where, rm,t -the mortality rate which is subject to changes over time ax and bx are parameters єx,t – the error term, which cannot be predicted. Kt -the time indicator of mortality level Assumptions under Lee-Carter model: 1) The parameters a and b in equation (1) are different across different age group x. However, the parameters are assumed to remain fixed over time. 2) The authors proposed to impose some constraints under its model: Across age group x: ∑ b = 1 Across time: ∑K = 0 Steps using Lee-Carter Model 1)Calculate ax In accordance with assumption (1), where the parameter is fixed over time but changes across ages. Therefore, an average of the all ax over time is appropriate. In addition to the first assumption, the second assumption will also be used. Since across time, ∑ K = 0 Letting n denote the number of years used, the formula becomes as The values of ln (mx,t) computed are shown in Appendix 1 and the values of The values of ax obtained for the Island of Mauritius in Appendix 2 Figure 1. 2) Calculate Kt Once ax for each age group is found, Kt can now be computed. Given ∑b(across ages) =1, Kt (for each year) can be calculated, denoted by the equation below: The values of Kt obtained for the year 2002 to 2011 are shown in Appendix 2 Figure 2. 3)Calculate bx bx in equation (1) basically is the slope of the difference between the logarithm of the mortality rate and a over time(t) at a specific age group (x) and the values of k obtained in Appendix 2 Figure 2 ). The calculated slopes, b, for each x are calculated across time are shown in Appendix 2 Figure 1. 4) In this step, Lee-carter proposed to re-estimate Kt in such a manner that the death rate observed in reality is aligned with the parameters a and b obtained for each x. Therefore, since Kt should varies over time and be re-estimated for each age group (x) in such a way to align the equation (1) with the real life data, the following re-estimated Kt has been produced in the Appendix 3. The computation has been made using the formula: DRx,t = eax+bxKt , where DR is the death rate obtained at t at a specified x, ax and bx are the data in Appendix 2 Figure 1. Lee-carter used ARIMA model to estimate Kt, with εt ∼ N(0, σ2), and the random walk with drift in such a way to get rid of outliers in its model. However to simplify the estimation, linear regression will be used to estimate forecasted Kt without random drift.That is, the assumptions made to be able to estimate Kt using linear regression are: 1) The data set 2002-2011 are reliable information without outliers and it represents an appropriate sample to be able to forecast the future Kt values with minimum variances. 2) There is a linearity relationship between dependent and independent variables in the equation Kt. 3) There is no serial correlation; constant variance of errors and normality of the error distribution Steps using Linear Least Squares method To estimate Kt linear regression equations in the form: ŷ = b0 + b1x, where, By fitting these equation in this situation, ŷ is the estimated Kt. x represent the time period ( t = 1,2,3,4,5…) ys’ in equation b1 and b0 are the re-estimated Kt values which has been computed. Therefore, to minimize the error term Least Squares Method has been used to be able to predict with a degree of accuracy. Since the analysis will be about estimating the BRP for the next 20 years, the age group which fits this particular situation will affect only those who have the age group 40+ as at 2011. Based on the simple linear equation from Appendix 4, the estimation of Kt can now be done for the future. Therefore, death can now be estimated by using the following equation: Dx,t = P(t-1)* eax+bxkt where , Dx,t is the actual death for a year t for a specific age group x This essay is an example of a student’s work Disclaimer This essay has been submitted to us by a student in order to help you with your studies. This is not an example of the work written by our professional essay writers. Essay Writing Service Dissertation Writing Service Who wrote this essay Place an Order P(t-1) is the population amount in the range of x of the year before t ax and bx are the parameters defined under Lee-carter which is assumed to remained fixed. Kt are the estimated values based on the linear equations obtained. Forecasting Population Now that death can be estimated, population can also be estimated for the years 2012-2031. Assumptions to forecast the population. Upon request to obtain a detailed information for each age (as from 40 years old) for population 2011, it has been noticed that only the age group (85+) population has not been provided in details. Therefore, to remediate to this lack of information for the 85+ population in the digest of demographic statistics, the proportion obtained from a population census undertaken in 2011 has been used to have a quite a proxy of the population 85+. 1) The Appendix 5 shows that the census population is overly stated, this might probably due to a double counting or fraud?. Thus, it will assume that with or without double counting issue, the error can be remediate by using the proportion of the census to estimate the 85+ population as at 2011. 2) Since the census shows data at latest 104+, to estimate the population, it has been assumed that the resident population will die by latest 104 years old. Age Pop_ 2011 Death 2011/12 Pop_ 2012 Death 2012/13 Pop. 2013 Death 2013/14 Pop. 2014 40 17408 (61) 41 19036 (61) 17347 (57) 42 17415 (61) 18975 (62) 17290 (56) 43 19528 (61) 17354 (57) 18913 (62) 17234 44 18568 (61) 19467 (64) 17297 (56) 18851 45 20491 (103) 18507 (92) 19403 (95) 17241 46 19851 (100) 20388 (101) 18415 (90) 19308 47 20135 (101) 19751 (98) 20287 (100) 18325 48 19388 (97) 20034 (99) 19653 (96) 20187 49 19062 (96) 19291 (96) 19934 (98) 19557 50 17513 (135) 18966 (144) 19195 (145) 19837 Death transition The table above is an extract of the results based on the computation. The ‘not available’ data are the non-required computation to assess the sustainability of BRP. Forecasting population at a particular time t, the death rate transition was computed. To be able to forecast that death rate transition, the formula (previously mentioned) was used. This in fact implies that for each death rate transition, the population (before the death rate transition column) should be used. To understand better, the example for population 2011 at age 40 will be taken as illustration. Given that: (1) Population ageing 40 years old (Pt-1) as at 2011 is 17408, (2) The values of forecasted Kt can be known by using the linear regression equation in Appendix 4 (i.e, K2012 is known), and Given that a and b parameters are also known (fixed over time but only changes over age group), The formula: Dx,t = P(t-1)* eax+bxkt can be applied. By inserting the values known values, the forecasted death transition 2011- 2012 is now known. Therefore, the population ageing 40 years old in 2011 ‘will be’ 41 years old in 2012. By repeating this process across time and ages, the forecasted population will be as shown in Appendix 6 (i) and (ii). 4.2 The estimated future cost of the BRP per eligible. As history has shown, there has been considerable development – the ‘ Mauritian miracle’ Subramanian and Roy(2001) -in the Mauritian economy. Now, Mauritius has developed and increased its economic stability considerably which has proved its resilience through the recent and prevailing euro crisis. Thus, to assess the future evolution of BRP, the period 2000 to 2012 has been taken to assess the future cost of BRP per eligible. Before analyzing the figures obtained, it has been found in the government budget 2012 and 2013 that BRP has been increased to compensate rising cost of living. Unfortunately, prior to 2012, a detailed view of the government budgets have not been accessible online. Upon a talk to the Ministry of social security, it was said that BRP increases according to cost of living. Therefore, to be on the safe side, by using of the data available below, a deduction and an assumption. Year Average increase in BRP Inflation rate 1999 Not computed 6.9 2000 0.071593915 4.2 2001 0.050163755 5.4 2002 0.07308347 6.4 2003 0.052570224 3.9 2004 0.052704972 4.7 2005 0.074312453 4.9 2006 0.040718445 8.9 2007 0.087059747 8.8 2008 0.089910874 9.7 2009 0.051027859 2.5 2010 0.034997676 2.9 2011 0.032063789 6.5 2012 0.065387079 3.9 2013 0.042999717 4.7i i It is the headline inflation rate estimated by the Bank of Mauritius (April 2012) The table above shows average increase in BRP and inflation rate which has been computed and extracted respectively. Based on the budget 2012 and 2013, it has been notice that BRP , in a particular year, has generally been increased almost the same inflation rate of the previous year. For example, for the year 2003 and 2004, BRP has been over or under adjusted which has almost been net off. It can be noticed that generally BRP has been adjusted quite fairly in relation with inflation rate. The reasons not to account for the inflation rate immediately could be because of political or economic stability reasons. However, there are large discrepancies for the average increase in BRP for the year 2005 in comparison with inflation rate in the year 2004. There may be several reasons for such a large discrepancies. One of the reasons could be because of the introduction of income test in BRP, which had been introduced in August 2004 to July 2005 and had allowed the government to more than compensating the eligible pensioners at that time. This essay is an example of a student’s work Disclaimer This essay has been submitted to us by a student in order to help you with your studies. This is not an example of the work written by our professional essay writers. Essay Writing Service Dissertation Writing Service Who wrote this essay Place an Order Therefore, to be able to estimate the cost of the BRP per individual in the future, forecasting or estimating inflation rate is required. In estimating the BRP costs, it will also be assumed, besides the assumptions made previously, that the BRP (as from 2015) will commonly increase, in both scenarios below, by ‘identical’ percentage as the average inflation rate. For the sake of analysis, 2 scenarios (A and B) will be recorded, (i) BRP per individual will increase at a constant rate of 5.62% (as from 2015). To calculate average inflation rate or cost of living, the time period (1999-2013) has been taken. For 2014, BRP will increase by 4.7%. It has been assumed that the government will be able this degree of macro-economic management. In other words, in this scenario, it has been assumed that inflation rate in the future will follow very close the inflation rate of this century. (ii) BRP per individual will increase at 8.4% (as from 2014 [i] ). The 8.4% has been estimated by averaging inflation rate for the period 1975-2012, where only all the factual data are taken into account. if only the data (1975-2012) available are taken. It is assumed that inflation rate will follow the average of the overall data which are only factual (‘already occur’ data) concerning inflation rate. The outcome scenarios A and B are shown in Appendix 7 (i). Scenarios A and B are assumed to follow two different paths: Scenario A assumes that the country is now able to manage their inflation rates as compared to 15 years ago. Therefore, the average used was for the period 1999-2013. On the other hand, Scenario B assumes that the cost of living will be increasing accordingly to the average of all past data available. In the literature review, the World Bank (2004) proposed three options to the sustainability of the universal pension- Targeting options, reducing benefit level and raising the eligibility age. However, based on based experience about corruption and incentives to distort information on targeting and because of political reasons for not reducing poverty level, increasing the eligibility age is most probable to happen. Therefore, it will be important to assess the impact of increase the BRP eligibility age. Therefore, under these 2 scenarios described, there will be another 3 scenarios (1,2 and 3) where BRP eligible age will be either 60 (not change) or extended, not progressively but immediately, the eligible age to 65 years old and 70 years old respectively. The population structure outcomes were summed from the Appendix 6 and the outcome scenarios 1, 2 and 3 are shown in Appendix 7 (ii). Scenario A Scenario 1 Scenario 3 Scenario 2 Scenario B Scenario 3 Scenario 1 Scenario 2 ANALYSIS In the analysis section, there will be 6 scenarios which will be assessed: A1, A2, A3, B1, B2 and B3 (which are the outcomes shown in Appendix 8). In Scenario A1 and A2, the outcomes show what will be the estimated costs of BRP in the future up to 2031 if the basic retirement pension system is left unchanged. Obviously scenario B shows a more pessimism situation if it is compared to scenario A (if the “% change” in the Appendix, which represents the % change in total costs, is being analyzed). The reason for that lies in the fact made in the assumption for scenarios A and B concerning the BRP increase per eligible. Scenario A Combination Outcomes In the analyzing the possible scenarios in scenario A, it can be concluded that if the government had decided to increase the eligibility age to 65 and 75 years old as from 2013, the costs of the BRP would have dropped down significantly by 31.06 and 56.90 percent respectively. In nominal values, the government would make a huge savings if the eligibility age rises. 1) If the government had introduced a rise in eligibility age to 65 as from 2013, the costs, which would be saved, would start at Rs 217.4 million and would gradually increased up to Rs750.5 million in 2031, which would represent a rise of almost 3.5 times in the savings made in one year. 2) If the government was to introduce a rise in eligibility age to 70 as from 2013, the costs saved which would start at Rs 355.0 million (in 2013) and would gradually increased to Rs 1.5 billion for the year 2031. This would represent an increase in the savings nominal amount which would more than quadruple as from 2013 to 2031. This essay is an example of a student’s work Disclaimer This essay has been submitted to us by a student in order to help you with your studies. This is not an example of the work written by our professional essay writers. Essay Writing Service Dissertation Writing Service Who wrote this essay Place an Order Scenario B Combination Outcomes Since BRP per individual for 2013 is already known, it cannot be assumed that for the year 2013 BRP per eligible will increased by the assumed, computed percentage. Thinking logically, BRP costs for the year 2013 (if the government had increased the eligible age in 2013) would have the same impact as the outcomes with scenario B. However, in nominal values, the savings that government would have made if the BRP eligibility age had increased to 65 and 70 years old would be more impactful than the outcomes with scenario A. 1) If the eligibility age to BRP was at minimum 65 as from the year 2013, the government would have saved approximately Rs 217.4 million in the year 2013 and the savings per year over time would gradually increase up to Rs 1.209 billion for the year ended 2031. The transition change from 2013 to 2031 represents a savings which is almost 5.5 times (compared to 3.5 times which is the outcome from scenario A). 2) If the eligibility had increased to 70 years old in 2013, the government would have saved approximately Rs 355.0 million for the year ended 2013 and the savings made would gradually increased up to Rs 2.44 billion for the year 2031 only. This transition change in the costs, from 2013 to 2031, shows that BRP savings with scenario B would approximately increase by 6.9 times( as compared to 4 times with scenario A) To assess the overall impact on BRP costs (as from 2013 to 2031) as a result of changes in the eligibility age as from 2013, the Implicit Pension Debt (IPD) will be used to assess the costs needed today to be able to cover the costs of the estimated future BRP costs. Therefore, in analyzing the IPD in the different scenarios in scenarios A and B, the impact of a change in the IPD in the scenarios 1, 2 and 3 can be assessed to know the real significance of that change in terms of the value today. To calculate the IPD, one major question should be asked: “What is the present value of the payment stream that the pension scheme will have to pay current participants and their survivors for the contributions made up to the current date, provided the rules of the scheme stay the same?” Robert Holzmann et al. (2004) Since the government’s, now the Ministry of Social Security, duty (since 1950) to provide the payment of BRP to the eligible concerns, the BRP can be considered almost as a riskless financial asset. Therefore, a ‘riskless’ discount rate should be used- the repo rate in the Bank of Mauritius will be used to assess the impact of changes in eligible age in BRP. However, the problem which arises is which discount rate should be used to assess the impact. Accordingly to the repo rates on the BoM website, there are several rates which range around 5 to 8 percent for the year 2006- 2013. Since the repo rate in Mauritius can change several times in one year and changes over time, it is difficult and almost impossible to forecast the repo rate up to 2031. Therefore, since the repo rate obtained on the website of the BoM was around the range 5 to 8 percent, the discount rate used to calculate the IPD will be 5, 6, 7 and 8 respectively. To obtain the outcome shown in the table below, the total costs in Appendix 8 has been discounted back to the year 2012 to obtain the IPD for each scenario (A1, A2…). The formula used to discount back the cash outflows of the government to the year 2012 is as follows. Where, C is the cash outflows at t T is the time as from 2012 (T =0, 1,2,…) ri is the repo-rate. Thereafter, to make an appropriate assessment of the impact of changing BRP eligibility age, a ratio of IPD to GDP (at market price) 2012 is used. It should be noted that the GDP 2012 is amounted to Rs 344.550 billion and is only an estimate because the official revised amount has not yet been published. Scenario A1 A2 A3 B1 B2 B3 Discount rate (%) IPD as a % of GDP (Market Prices) as at 2012 5 4.93% 3.45% 2.23% 6.51% This essay is an example of a student’s work Disclaimer This essay has been submitted to us by a student in order to help you with your studies. This is not an example of the work written by our professional essay writers. Essay Writing Service Dissertation Writing Service Who wrote this essay Place an Order 4.56% 2.95% 6 4.45% 3.11% 2.01% 5.83% 4.08% 2.64% 7 4.04% 2.82% 1.83% 5.25% 3.67% 2.37% 8 3.67% 2.56% 1.66% 4.74% 3.31% 2.14% The table shows the overall impact of increasing BRP eligibility age through the use of IPD. There are 4 scenarios which are assumed. The assumptions are that the yearly average of repo rate (set up by the Bank of Mauritius) up to 2031 will be 5, 6, 7 and 8 percent in the different scenarios. Outcomes under scenario A In the scenarios with the assumption that BRP per eligible increases by 4.7% in 2014 and by 5.62% as from 2015 to 2031 (scenario A in appendix 7), there is a general decrease in the IPDs calculated as the BRP eligible age increased to 65 and 70 ( for example, A2 and A3 respectively). In the assumption that the repo rate is 5 percent for the year 2013-2031, it can be found that the overall impact on the IPD as at 2012 has decreased by 1.48% and 2.70% ( if the government had increased the BRP eligible age to either 65 or 70 respectively. In deeper analysis, it can be noticed that the impact of a 5 years increased (to 65 years old) is more significant than increasing the eligible age to 70. The reason for making such analysis is because IPD reduction, with a 65 years old retirement, is 1.48% as compared to 2.70% which represents less than the double of 1.48%. The same conclusion can also be made with the other discount rates- 6, 7and 8- where scenario A2 is more impactful than scenario A3. Outcomes under scenario B Under the assumption that BRP per individual will increase as from 2014 by 8.4%, it can be found that the overall impact of increasing the eligible age to 65 and 70 reduced the IPD ratio considerably. For example under the discount rate 5 percent, the IPD ratio got reduced from a large gap between scenarios A1 and B1 to almost no gap between scenarios A3 and B3. The same analysis can be concluded in other discount rates which had been used in the table. Exactly as the same outcomes estimated under scenario A can be found in scenario B. It can be analyzed that if the discount rate was at 5 percent as from 2013 to 2031, the estimated IPD ratio would reduced to 4.56 or 2.95 percent if the government had changed the eligible age to 65 or 70 respectively as from 2013. In other discounts rates, the same conclusion can be found and it can be notified that the significance of increase eligible age to 65 is a more important than increasing the eligible age to 70. The table also shows that if the government wanted to finance the BRP for the citizens of the island of Mauritius starting from the year 2012 to year 2031, the government should finance either 4.93, 4.45, 4.04, 3.45, 3.11 percent or any other figures on the table (depending on the circumstances) out of the GDP in 2012. Thus, it can be deduced that the higher is the value of IPD, the least sustainable or the most expensive it is for the government to finance it. It can be found that the worst case for the government is the costs of BRP in 2012 would be 6.51 percent of annual GDP to be able to finance the BRP as from 2012 to 2031. However, in the best scenario, the government would need to finance only 1.66% of the annual GDP (2012) to finance the same overall BRP up to 2031. CONCLUSION 5.1 Evaluation The results of the last section shows that the impact on IPD shows that increasing the actual eligible age from 60 to 65 is more significant than to an increase in the eligible age to 70. It has also been found that as the BRP age eligibility increases, the impact on IPD ratio between the scenarios A and B decreases. That is, the increase differential between scenarios A and B in the BRP is not enough to offset the increase in the age eligibility to maintain or increase the gap. Therefore, it can be said that even though the BRP per individual increases by 8.4% in the BRP per individual over the years, the costs implications for such an increase can be minimized by increasing the age eligibility to BRP. 5.2 Limitations of the Analysis One important limitation of the scenario analysis is that the discount factor used is not realistic in practice. In fact, the repo rate used are 5, 6, 7 and 8 which are assumed not to changed over time and will remain constant yearly up to 2031. However, if we questioned about how the repo rate is set up in the real life, it would be obvious that this assumption is not relevant. In the case of Mauritius the repo rate, which is set up by the Bank of Mauritius, can change several times during a year (with very low probably for it not to change in 2 consecutive years and almost impossible for it to remain constant till year 2031). A second limitation from which the analysis had been made is that the inflation rate will most probably not be constant for the year 2013 to 2031. Though the majority, if not all, of the governments’ main economic agenda is to maintain low inflation rate in their respective countries, it does not mean that the governments have the absolute power to determine the inflation rate at a specific and desired percentage. Therefore, it would not be realistic to say that the BRP per individual would raise at a constant rate for the 19 consecutive years. Last and not least, the BRP costs outcomes assumed that there is no migration involves for those 40 years old or above as at year 2011. However, in practice, migration also involve people who are above 40 and even above 60 though they are less pronounce among elderly people. Going further back to the beginning of the section- Research Methodology under the Lee-carter model , there are several assumptions which have been made to be able to use the simple linear regression model to estimate the forecasted value Kt. The first assumption: The data set 2002-2011 are reliable information without outliers and it represents an appropriate sample to be able to forecast the future Kt values with minimum variances, is quite unrealistic because such a small time period is usually not appropriate to make a good sample representative. The second assumption about the linearity relationship cannot be guessed with ease with a small time sample. Finally, the last assumption that there is constant variance of errors, normality of the error distribution and no serial correlation has been assumed instead of being tested, which otherwise require more time to be compute all these years. In addition to these assumptions put forward under the Lee Carter model, forecasting the number of death over time will never considered as accurate. In fact, no matter how lengthy and complex the procedures used to calculate the forecasted mortality are, the forecasted mortality obtained in the Research Methodology section will always be an estimation. Therefore, the question of how accurate are the data are is questionable. In the options proposed to replace universal pension, put forward by the World Bank reports, it has been discussed previously that targeting and reducing benefit level is unlikely to happen based on past experiences and for political reasons. However, based on the past experience of France concerning increasing the retirement age, there was a ‘huge strike’ concerning this issue. The question which arises is that does increasing eligibility age will disrupt the serenity in the Mauritius? As the population of Mauritius is aging, the power of electoral votes will move much more in the hands of the elderly. Therefore, if the government of Mauritius is planning to make an important reform in the pension system , it is now that the government should react. Otherwise, it may be harsher to make reforms in the future. 5.3 Future Works To improve the evaluation of this project, the sample of years can be increased much more to make a more accurate analysis with testings instead of assumptions. Besides, there are factors such as inflation rate and migration rate for those concerned can be assessed and thereafter forecasted to address or at least take in consideration the factors which can affect the analysis.