Asian Economic and Financial Review A MODEL FOR ESTIMATING THE DISTRIBUTION OF FUTURE POPULATION. Ben David Nissim.

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1 Asian Economic and Financial Review journal homeage: htt:// A MODEL FOR ESTIMATING THE DISTRIBUTION OF FUTURE POPULATION Ben David Nissim Deartment of Economics and Management, Emek Yezreel Academic College, Israel Garyn-Tal Sharon Deartment of Economics and Management, Emek Yezreel Academic College, Israel ABSTRACT In most counties, statistical authorities collect data on the number of deaths in each age grou. These data enables the calculation of life exectancy as well as the calculation of death and survival robabilities for each age grou. In this aer, we develo an easy-to-use tool to estimate the distribution of future survivors for each cohort. Such a distribution defines the robabilities for the number of survivors at a given future time. Many institutions can benefit from the estimation of the distribution of future survivors (e.g., ension funds, geriatric institutions and medical authorities in general. Thus, our aer contributes not only to the literature on the rojection of mortality rates, but it also has significant ractical imlications. 4 AESS Publications. All Rights Reserved. Keywords: Distribution, Future survivors, Life exectancy, Cohort, Mortality rates, Projection. Contribution/ Originality This study uses new methodology for estimating future distribution of survivors. Estimating the robability of surviving individuals in time T via the continuous normal distribution solution requires a multile-integral calculation. We suggest a comact model that allows doing the calculation via a single integral over the density distribution function.. INTRODUCTION In most counties, statistical authorities collect data on the number of deaths in each age grou, which enables the calculation of death robabilities (and survival robabilities as well as the calculation of life exectancy for each age grou. However, a large degree of uncertainty is associated with these future survival robabilities. Booth and Tickle (8 reviewed the history of mortality rojections and identified three main rojection methodologies. The first methodology builds eidemiological models in order to 55

2 exlain articular causes of death or known risk factors (see, Alderson and Ashwood, 985. The second methodology defines a target life exectancy and then determines a ath to that life exectancy. Such an aroach allows exert oinion to be factored into the statistical rocess (see Pollard, 987; Olshansky, 988. The third methodology is extraolation - a term given to the forecasting of future mortality based on ast atterns (see Lee and Carter, 99; Milevsky and Promislow, ; Brouhns et al., ; Dahl, 4; Biffis, 5; Renshaw and Haberman, 6; Schrager, 6; Cairns et al., 9. Many institutions would be interested in and benefit from the estimation of the distribution of future oulation. Such a distribution defines the robabilities for the number of survivors for each cohort at any given time in the future. First of all, ension funds would benefit from forecasting the distribution of the exected number of survivors in cohorts assed retirement age. When a death of an insured individual occurs, ension funds usually ay a given comensation to the family in addition to the continuation of the ost-death ension ayments (ayment level might be different relative to the re-death ayments. Funds can use ast records and documentations of death robabilities in order to evaluate the number of exected deaths. However, there is a high robability that the actual number of deaths would be much larger than the anticiated number (calculated by the multilication of the number of eole in a cohort by their robability of dying. This means that even if ension funds oerate with an exected actuarial balance, they might end u extremely unbalanced due to volatility in the exected forthcoming deaths. If ension funds could forecast the distribution of the number of deaths for each cohort, it would imrove their forecasting tools (by using the confidence interval of the exected number of deaths instead of solely using the exected value and thus will enable them to take stes to reduce their oerational risks. Insurance comanies would also find it useful to evaluate the distribution of the future exected number of survivors for each cohort since they are interested in estimating the robabilities associated with their commitments, i.e., ucoming death comensation ayments. Although death robabilities enable the estimation of exected ayments, they do not fully describe the comlete risks. For examle, if the robability of dying for a given cohort is %, then the exected number of deaths for, individuals (from this cohort is. However, there is some robability that more (and even much more than individuals will die, causing substantial losses to the insurance comany. The same considerations hold for deaths caused by car accidents (drivers' insurance risks, earthquake insurance risks as well as health insurance risks. Additionally, medical authorities would be interested in the same information. For examle, medical authorities evaluate death robabilities caused by eidemic diseases based on ast records. In articular, they are interested in the robability that the actual number of deaths would be much larger than the exected number of deaths. Moreover, being exosed to a disease might create a given risk of death in all future eriods. Here again, the medical authority would benefit from taking into consideration the distribution of the future deaths in each time eriod. 56

3 A variety of aroaches have been roosed for modeling randomness in the aggregation of mortality rates over time. Lee and Carter (99 worked with discrete time models and focused on eriodical alication of stochastic mortality and its statistical analysis (see also Renshaw and Haberman, 6. Other researchers develoed continuous time frameworks (see Dahl, 4; Schrager, 6. In this aer, we first assume that in each future eriod an agent faces a given likelihood to survive. We assume that future annual survival robability is defined according to the reort of the realized death robabilities ublished by the statistical authorities. Then, we resent a model that enables the rediction of future survival distribution for each cohort.. THEORY/CALCULATION.. Estimating the Distribution of Future Survivors for Each Cohort... The Bernoulli Distribution and the Binomial Distribution Let us assume that individuals were born in eriod. According to statistical authorities, their survival robability during eriod is. That robability can be found in mortality rates tables ublished by the authorities. We can refer to the survival distribution at age as a Bernoulli distribution. Each agent faces a "successful trial" with robability (survival and a failure with a robability (death. Using the Newton Binomial formula, we can calculate the robability rob ( that j agents will survive at age t=, for j =,,,. As, the binomial j distribution function is exressed in terms of the standard normal distribution function. Using the same idea, the distribution of the number of survivors in eriod t deends on the realized number of survivors in eriod t-,. Thus, given the distribution of the number of survivors in time t-, we can calculate the distribution of the number of survivors in the following eriod. Each individual faces a survival robability t during eriod t, and, given that individuals survived eriod t-, any j surviving agents face a survival robability j N rob t ( t- in time t. The surviving agents face in time t+ a new individual survival robability t and, given that individuals survived eriod t, a new Bernoulli trial with survival robability rob t N. ( j t 57

4 ... An Examle: The Binomial Distribution In order to simlify the resentation, let us assume that we estimate the distribution of survival of a nonhuman secies with survival robabilities of and at years and, resectively. Assuming that only N= objects were born in year zero, table below shows the calculations of the future survival distribution of the objects born in year, according to the Bernoulli distribution. Table--Panel-A. The survival distribution in year and in year (given the number of survivors in year Number of Year survivors in survival year robability ( ( ( ( Number of Year survival survivors in Probability given the year number of survivors in year ( ( ( ( ( ( ( ( ( ( Notice that the survival robability in year (table - anel B is the roduct of the robability of j survivors in year, j=,,,, multilied by the robability of k survivors in year, such that k is lower than or equal to j. For examle, the robability that j= objects survive year and k= object survives year is: where: ( ( 58

5 59 ( is the robability that eole survive year and ( is the robability that erson survives year given that eole had survived year. Based on table, the overall survival distributions in years and are resented in table. The year survival robabilities are similar to those resented in anel A of table. The year survival Table- Panel-B. Survival distribution in year Number of survivors in year Number of survivors in year Year overall survival robability given the number of survivors in year ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( robabilities are calculated as follows. For each ossible number of survivors k (k=,,, in eriod t=, we take into consideration all ossible number of survivors j in the receding eriod (time, k,...n j k j, and then we sum these robabilities of the different scenarios leading to k survivors in time : ( ( ( N k j j k j k. To exlain table, consider year survival robability for one survival:

6 5 ( ( ( ( ( ( ( Prob of one survivor Prob of survivors Prob of survivors in eriod in eriod in eriod... An Examle: The Binomial Distribution A Numerical Solution Let us now continue the binomial distribution examle from section... As an examle, given 96. and, the survival distribution in years and are resented in table. Table-. The overall Survival distribution in years and Number of survivals Year survival robability Year survival robability ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( Table-. The survival distribution in years and given that 96. and.. The Normal Distribution In order to ease the calculations, we can use the rincile that as, the binomial distribution function is exressed in terms of the standard normal distribution function. Suose is the number of surviving individuals at the end of time t-, t is the robability of surviving the ucoming eriod t and is a random variable reresenting the number Number of survivors Year survival robability Year survival robability 6.4E

7 of surviving individuals at the end of the eriod t. Then and as,. We define: the PDF (density distribution function for the normal distribution. initial number of individuals in time. the required number of surviving individuals in time T. a random variable, normally distributed, reresents the number of surviving individuals in time t. the realized number of surviving individuals in time t. - the robability of surviving in time t. The imlementation of these definitions yields that reresents the number of surviving individuals in time and is normally distributed: ( In general, the normal distribution for deends on the realized number of surviving individuals in time t-. For t=,,,t, this distribution of given is: ( The PDF of is: while the PDF of is: For simlicity, we start with a simle examle, the same as the examle discussed in sections Let us describe the robability for surviving individual in time. We substitute T= and and get that: ( ( Thus, the robability for surviving individual in time is: Note that equation ( includes two terms: the external integral reresents the robability for the number of survivors in time, and the internal integral reresents the robability that only one individual will survive in eriod. Thus, we can further subdivide equation ( into: 5

8 Given that. 96 and, we use equation ( for calculating the survival distributions in year, and the results are as follows: the robability for survivor in eriod is.6449, the robability for survivors in eriod is.4744, and the robability for survivors in eriod is These numbers are similar in magnitude to those resented in Table. The differences can be exlained by the small samle of T= as: individuals. Generalizing equation (, we can exress the robability of surviving individuals in time we get: Placing the PDFs of and of, as defined by equations (6-(9, into equation (, Note that after the internal integration over in equation (, we get an exression that deends on. Then, when integrating the external integration over, it does the integration also over the that remained after the internal integration. Now let us generalize the examle above to T time eriods and survivors in time T. We get that the robability of survivors in time T is: Let us exlain the underlying intuition of equation (4. The first art of equation (4 is: is the realized number of surviving individuals in time. We would like to be lower than or equal to (since is the initial number of individuals in time, we cannot have more surviving individuals in time than the initial number of individuals in the time, but greater than or equal to (since is the required number of surviving individuals in time T, we cannot have less surviving individuals in time. The second art of equation (4 is: We know now how many individuals survived time : the answer is. Now we would like to consider the ossible values for - the realized number of surviving individuals in time. We would like to be lower than or equal to (we cannot have more surviving individuals in time than the number of surviving 5

9 individuals in the time, the receding eriod, but greater than or equal to (since is the required number of surviving individuals in time T, T>, we cannot have less surviving individuals in time. In general, we can exlain what haens in time t. Part number t of equation (4 is: We know now how many individuals survived time t-: the answer is. Now we would like to consider the ossible values for - the realized number of surviving individuals in time t. We would like to be lower than or equal to (we cannot have more surviving individuals in time t than the number of surviving individuals in the time t-, but greater than or equal to (since is the required number of surviving individuals in time T, T>t, we cannot have less surviving individuals in time t. The last art of equation (4 is: We know now how many individuals survived time T-: the answer is. We also know that - the realized number of surviving individuals in time T should equal to. Thus, the limits of the last integral of equation (4 are: and. The result is the estimation of the robability of surviving individuals in time T.... Another Examle We start with individuals in time. Let us describe the robability for surviving individual in time. We substitute, T= and and get that: (, (, (, ( Thus, given that, the robability for surviving individual in time is: Let us exlain the underlying intuition of equation (5. The first art of equation (5 is: is the realized number of surviving individuals in time. We would like to be lower than or equal to but greater than or equal to. Exlanation: notice that since we start with, the initial number of individuals in time, we cannot have more surviving individuals in time. On the other hand, since is the required number of surviving individuals in time, we cannot have less than surviving individuals in time. The second art of equation (5 is: In time t=, - the number of individuals that survived eriod - is given and known. Now we would like to consider the ossible values for - the realized number of surviving individuals in time. We would like to be lower than or equal to but greater than or equal to. Exlanation: since is the realized number of surviving individuals in time, we cannot have more surviving individuals in 5

10 time. On the other hand, since is the required number of surviving individuals in time, we cannot have less surviving individuals in time. The exlanation for the third art of equation (5 is similar. The fourth and last art of equation (5 is: We know that - the realized number of surviving individuals in time t should be equal to. Thus, the limits of this last art of equation (6 are: and. The result is the estimation of surviving individuals in time. In equation (6 we further decomosed equation (5 based on the following different scenarios:. (this is the first line in the decomosition of equation (5.. (this is the second line in the decomosition of equation (5.. (this is the third line in the decomosition of equation (5. 4. (this is the fourth line in the decomosition of equation (5... Normal Distribution The Comact Solution Equation (4 in section. reresents the continuous normal distribution solution. It estimates the robability of surviving individuals in time T via T integrals, since it considers and accounts for every realization regarding the number of surviving individuals that could occur. The first integral goes over the ossible values for the realized number of surviving individuals in time ; The second integral goes over the ossible values for the realized number of surviving individuals in time, given the realized number of survivors in eriod ; Integral number t goes over the ossible values for the realized number of surviving individuals in time t, given the realized number of survivors in eriod t-; And the last integral, integral number T, estimates the robability for exactly, the required number of surviving individuals in time T, given the realized number of survivors in eriod T-. Let us reconsider the role of the first T- integrals in equation (4. These T- integrals go over all ossible values for the realized number of surviving individuals in times,,,,t-. The result 54

11 of these T- integrals is exlicit and simle: it is the calculation of the exected number of surviving individuals in time T-. In order to avoid solving the roblem via the multile integral solution as in equation (4, we suggest another way to exress the average exected number of individuals in time T-,. We define the robability to survive from time and until time T- as: Thus, the exected number of individuals at time T- is: We can now exress the robability of surviving individuals in time T as: where: ( Using equations (8 and (9 instead of using the multile integrals in equation (4 enable us to calculate the survival robability distribution at time T in a much more comact way.... A Real-Life Examle We used the mortality rates tables ublished by the United States Social Security Administration (SSA to get the mortality rates for individuals born in 96 for each year between 96 and 7 (the survival rate is minus the mortality rate. Table A. in aendix A resents art of the SSA data. According to the data in table A., the robability of individuals born in 96 to survive 96 is.9766, the robability of those individuals, born in 96, to survive 96 is.9986, while the robability of those individuals to survive 96 is and so on. We can now calculate the robability of individuals that were born in 96 to survive until 6 (included as the multilication of the annual survival rates resented in table A. for the time eriod The calculated robability of individuals born in 96 to survive until 6 (included is Thus, among every individuals born in 96, will survive till the end of 6, on average. The robability of those individuals surviving 7 is also given in table A. and is equal to The distribution of surviving individuals in 7 is calculated via the following equation (.The robability for surviving individuals is - for: ( 55

12 Probability We rogram the solution defined by the integral in equation ( with its characteristics in ( via the Wolfram Mathematica software. The Mathematica code is rovided in aendix B. Table C. in aendix C reorts the distribution of surviving individuals in 7. The distribution of surviving individuals in 7 is also described in figure. We can also forecast the estimated distribution of future survivors in any eriod in the future. For examle, we can forecast the distribution of survivors in 5 for the same grou of individuals born in 96. Let us go back to the SSA mortality tables which reorts mortality rates only until the year 7. The cohort born in 96 was 47 years old in 7. However, we can use the mortality rates data collected by the statistical agencies for 7 for eole at ages 48, 49, 5, 5, 5, 5, 54 and 55, and attribute them to the cohort born in 96 and use it as estimation for the 8, 9,,,,, 4, 5 mortality robabilities, resectively. For examle, the mortality rate for the cohort at the age of 48 in 7 is.46. Thus, we can estimate the forecasted 8 mortality robability for the cohort born in 96 as.46 and thus we can estimate the forecasted survival robability (that equals minus the forecasted mortality robability as We continue to make these estimations until we reach the 5 mortality robability forecast. These forecasts are resented in table D. in aendix D. Figure-. The distribution of surviving individuals in 7, for individuals born in 96 9 Number of surviving individuals We showed before that for eole born in 96, the calculated robability of surviving until 6 (included was.8995 and thus the exected number of survivors in 6 was We can now calculate the robability of individuals born in 96 to survive until 4 (included by multilying the calculated robability of surviving until 6 by the robability of surviving 7 (given in the SSA table as well as in table D. and by the forecasted robabilities of surviving 8-4 (included, as reorted in table D.. The estimated robability to survive until 4 (included is.8547 and thus the exected number of survivors in 4 is (out of the original individuals born in 96. Using our technique and the forecasted survival robability for 5 (which is.99 as reorted in table D., we can calculate the exected distribution of future survivors in 5. The forecasted distribution of future surviving individuals in 5 is again calculated via equation (, for: 56

13 Probability ( The forecasted distribution of the future survivors in 5 is described in figure. Figure-. The forecasted distribution of surviving individuals in 5, for individuals born in Number of surviving individuals. RESULTS AND DISCUSSION In most counties, the statistical authorities collect data on the number of deaths in each age grou. That enables the calculation of life exectancy as well as the calculation of death and survival robabilities for each age grou. However, many institutions (e.g., ension funds, geriatric institutions and the medical authorities in general would benefit from estimating the future distribution of survivors as well. In this aer, we develo a tool that can be used to estimate the future distribution of survivors for each cohort. Such a distribution defines the robability for the number of survivors at a given future time. Assuming individuals were born in eriod, and assuming that their robability to survive time t is defined as t, we can refer to the survival distribution at time t as a Bernoulli distribution. In the Bernoulli distribution, each agent faces a "successful trial" with robability t (survival and a failure with the robability (death. Using the Newton Binomial formula we can calculate the robability that individuals will survive at any time T>t. As, the binomial distribution function is exressed in terms of the standard normal distribution function. t Estimating the robability of surviving individuals in time T via the continuous normal distribution solution requires a multile-integral calculation. Alternatively, we suggest a comact model in which the average exected number of individuals in time T-,, is estimated by multilying the robability to survive from time until time T- by the initial number of individuals in time,. That allows us to exress the robability of surviving individuals in time T via a single integral over the density distribution function of the normal distribution, with the integral boundaries of and. For examle, our tool enables us to calculate the whole distribution of surviving robabilities in 7. Using the mortality rates tables ublished by the SSA, we can calculate the robability of 57

14 individuals born in 96 to survive until 6 (included. The robability of individuals born in 96 to survive until 6 (included is Thus, among every individuals born in 96, will survive till the end of 6, on average. We can also tell that, on average, out of the original individuals, about 888 of them will survive till the end of 7. But, there might be a given robability that the realized number of survivals will be much larger or much lower than the exected number. Thus, relying solely or mainly on the exected value of survivals, many institutions such as ension funds, geriatric institutions and the medical authorities in general, may end u extremely unbalanced. If such a comany could estimate the distribution of the exected future number of deaths for each cohort, it could exand its forecasting tools and thus reduce its oerational risks by using the confidence interval of the exected number of deaths instead of solely using the exected value. We can also forecast the estimated distribution of future survivors in any eriod in the future. For examle, we can forecast the distribution of survivors in 5 for the same grou of individuals born in 96. Naturally, the SSA mortality tables do not go into the future. However, we can aly the current mortality rates data and use it as an estimate for the future mortality robabilities for the cohort born in 96. Continuing our examle, the estimated robability for the cohort born in 96 to survive until 4 (included is.8547 and thus the exected number of survivors in 4 is (out of the original individuals born in 96. Using our technique and the forecasted survival robability for 5 (.99, we can calculate the exected forecasted distribution of future survivors in 5. Thus, our aer contributes not only to the literature on the rojection of mortality rates, but it also has significant ractical imlications because it enables the authorities, as well as other relevant institutions, to be better reared for the ucoming future and to better handle unexected changes associated with it. REFERENCES Alderson, M. and F. Ashwood, 985. Projection of mortality rates for the elderly. Poulation Trends, 4: 9. Biffis, E., 5. Affine rocesses for dynamic mortality and actuarial valuations. Insurance. Mathematics and Economics, 7: Booth, H. and L. Tickle, 8. Mortality modelling and forecasting: A review of methods. Annals of Actuarial Science, : 4. Brouhns, N., M. Denuit and J.K. Vermunt,. A oisson log-bilinear regression aroach to the construction of rojected life tables. Insurance Mathematics and Economics, : 7 9. Cairns, A.J.G., D. Blake, K. Dowd, G.D. Coughlan, D. Estein, A. Ong and I. Balevich, 9. A quantitative comarison of stochastic mortality models using data from England & wales and the United States. North American Actuarial Journal, : 5. 58

15 Dahl, M., 4. Stochastic mortality in life insurance: Market reserves and mortality-linked insurance contracts. Insurance Mathematics and Economics, 5: 6. Lee, R.D. and L.R. Carter, 99. Modeling and forecasting U.S. Mortality. Journal of the American Statistical Association, 87: Milevsky, M.A. and S.D. Promislow,. Mortality derivatives and the otion to annuitise. Insurance Mathematics and Economics, 9: Olshansky, S.J., 988. On forecasting mortality. The Milbank Quarterly, 66(: Pollard, J.H., 987. Projection of age-secific mortality rates. Poulation Bulletin of the United Nations, : Renshaw, A.E. and S. Haberman, 6. A cohort-based extension to the lee-carter model for mortality reduction factors. Insurance Mathematics and Economics, 8: Schrager, D.F., 6. Affine stochastic mortality insurance. Mathematics and Economics, 8: Aendix-A: Table-A.. SSA data: mortality and survival rates for individuals born in 96, for each year between 96 and 7 Year Age at that year Mortality rates Survival rates Aendix-B: The Mathematica code for the distribution of surviving individuals in 7, for individuals born in 96: strm=oenwrite [" fill in the required outut ath "] num= For[m=,m<num,m++,Write[strm,Integrate[/Sqrt[Pin(-]Ex[-(x-n^/(n(- ], {x, m-.5, m+.5},assumtions {n=89.947,=.99579}]]] Aendix-C: Table-C.. The distribution of surviving individuals in 7, for individuals born in 96. Number of surviving individuals Probability Continue 59

16 Aendix-D: Table-D.. Forecasted mortality and survival robabilities for the cohort born in 96. The cohort born at the year Age at the year 7 Mortality rates Survival rates An estimation for the mortality and survival robabilities for the cohort born in 96 for the year