Modeling and Forecasting Mortality Using the Lee-Carter Model for Indian Population Based on Decade-wise Data

The stochastic mortality model given by Lee and Carter (1992) has been used in literature for fitting and forecasting the human mortality. We have modeled mortality rates of Indian population using the Lee-Carter (LC) model based on decade-wise data available separately for Indian female and male populations in the form of life tables for the period 1901-2011. The Singular Value Decomposition (SVD) approach is used for estimation of the parameters of the LC model. Forecasted values of time dependent parameter of the LC model are obtained for next five decades using best fitted auto regressive integrated moving average (ARIMA) model. Forecasted values of life expectancy at different ages with 95% confidence intervals are also reported for the next five decades. As an application, using forecasted mortality rates for the next five decades, net single premium for whole life and term insurance, actuarial present values of life annuities for some selected ages are also evaluated.


Introduction
Mortality modeling and its forecasting have been a major research area for the practitioners in actuarial science and demography.During the last century many countries have experienced considerable improvements in mortality rates.Dramatic decline in mortality brings very serious financial exposures for insurers The rest of the paper is organized as follows: In Section 2, we discuss the data used in this paper.In Section 3, we have presented some results about changes in mortality pattern in India during last century.The LC model and modeling procedure is discussed in Section 4. Section 5 gives the results of fitted model.Section 6 deals with results on the mortality forecasting for India.In Section 7, as an application of study, we have presented some actuarial calculation.Conclusions are reported in Section 8.

Data Description
Our study is based on data obtained from abridged life tables of Indian population for 11 decades from 1901-1911 to 2001-2011.These life tables are taken from the websites www.lifetable.de and www.who.org.in.Life tables of Indian female and male populations are separately available for the decades 1901-1911 to 1971-1981 for the ages 0, 1-4 and 5-80 (quinquennial) and 80+ whereas for the decades 1981-1991 to 2001-2011 these life tables are available for the ages 0, 1-4 and 5-100 (quinquennial) and 100+.To maintain uniformity in our data, we have converted the life tables of the decades 1981-91 to 2001-11 for the ages 0, 1-4 and 5-80 (quinquennial) and 80+.From these life tables, we have derived decade-wise age group specific central death rates as (1) where, denotes the number of deaths in the age group during decade .denotes the average number of persons in the age group during decade .
is also known as central rate of mortality or central mortality rate for age group during decade .Data on decade-wise age group specific central death rates for Indian female and male populations are reported in Table 1 and Table 2 respectively.
Using data on life tables we have obtained life expectancies (LE) for female and male populations for all decades.Results obtained on LE are reported in Table 3 for some selected age groups.In India, Life expectancy at birth has been increased from 23.23 years to 67.30 years for female and from 24.76 years to 63.81 years for male during last century.During decades 1901-11, 1911-21 and 1921-31, life expectancy of Indian female and male at age group 20-24 was higher than life expectancy at birth.This is due to high rate of infant mortality during these decades.
To obtain estimated parameters ̂ and ̂ , we applied singular value decomposition on matrix , where that is Applying SVD to the matrix , we achieve the decomposition , where , are the singular values in increasing order with and as the corresponding left and right singular vectors.The approximation to the first term of gives the estimates ̂ and ̂ The LC model is popular due to its simplicity for the parameter estimation by SVD.The proportion of variation explained by the LC model is ∑ .

Fitting of the LC Model to Indian Mortality Data
This section presents the results of estimation of parameters in LC model.Estimated values of age dependent parameters and are reported in Table 4 and estimated values of time dependent parameter is reported in Table 5 for female and male populations in India based on decade-wise life tables (1901-11 to 2001-11).For SVD analysis, we have used MATLAB program.
From SVD analysis, we found that 98.16% and 96.47% variation explained by fitted LC model for Indian female and male mortality data respectively.In Figure 2, we have plotted observed and fitted age group specific central death rates for three decades, 1901-11, 1951-61 and 2001-11.We observed that the fitted mortality rates are very close to observed (actual) mortality rates except for lower and higher ages for the decade 2001-11.

Forecasting
Forecasting is the main aim behind the stochastic modeling.One of the noteworthy property of the LC model is that, once it is fitted (i.e.once values of ̂ ̂ and ̂ are found), only the mortality index ( ) over time needs to be forecasted for future time points.Lee and Carter (1992) fitted autoregressive integrated moving average (ARIMA) (0,1,0) (i.e.random walk with drift) for modeling mortality index for US population and also suggested to use the appropriate ARIMA models for different populations.We have considered some possible choices of ARIMA models for modeling mortality index .Table 7 includes values of Akaike Information Criteria (AIC) and Bayesian Information Criteria (BIC) for considered models.According to AIC and BIC, ARIMA(1,2,0) and ARIMA(0,2,0) are best fitted models for mortality index estimated by SVD for Indian female and male populations respectively.Values in Table 7 are obtained by the R 'forecast' package.(see Hyndman, and Khandakar, ( 2008)).We have used best fitted ARIMA models for forecasting future values of mortality index, and other corresponding quantities of life table.
In the Table 8, we have given the forecasted values of mortality index, along with its 95% confidence intervals (CI) for next five decades.These values are forecasted one by one for each next decade and then by adding the forecasted value in the previous values of and again by using best fitted model, next decade's value for was forecasted and so on.At each forecasting stage we got the same model with different parameters.We observed that in next five decades mortality are expected to decline for both female and male population in India.This is due to decreasing nature of .We have forecasted values of age specific death rate, by using estimated parameters ̂ ̂ and forecasted values of mortality index .Table 9 reveals the forecasted age specific central death rates in terms of deaths per 100,000 for the five decades, 2011-21, 2021-31, 2031-41, 2041-51 and 2051-61.We observed that infant mortality rate will decline from 42 to 15 per thousand for female and 44 to 18 for male from period 2011-21 to 2051-61.By the decade 2051-61, deaths rates for age groups between 1 to 49 years for female and age groups 5 to 34 years for male expected to be lower than one per thousand.The age specific death rates will continue to be lower for female population as compare to the male population for all five decades.Table 10 presents the LE along with 95% CI at birth and for the age groups 20-24, 40-44 and 60-64.
Let be the effective rate of interest, which is interest earned on 1 unit invested in one period (generally one period is one year) and is the present value or discounted value of unit 1 payable after one period.Hence, .The NSP for whole life insurance payable for one unit benefit payable at the end of year of death of person aged years, denoted by and its expression is given by, ∑ The NSP for -year term life insurance payable for one unit benefit payable at the end of year of death of person aged years if death occur within year from the policy issue, denoted by ̅| and its expression is given by, The APV of whole life annuity due for one unit payment at the beginning of each year throughout the remaining lifetime of an individual now aged x denoted by ̈ and its expression is given by, The APV of -year temporary life annuity due for one unit payment at the beginning of each year for the next years or till survivor whichever occur first for an individual now aged x denoted by ̈ ̅| and its expression is given by,

Conclusions
We modeled the central mortality rates of Indian population by using the LC model estimated by SVD, approach based on decade-wise mortality data from 1901-11 to 2001-11.We observed the following.
i) The general pattern of mortality ( ̂ ) for both female and male populations shown high infant mortality, an accidental hump around ages 20 years and nearly exponential increase at older ages.ii) The sensitivity of mortality ( ̂ ) has shown mortality decline at high rate for ages 25-34 years for female and for ages 15-24 years for male population than other ages.iii) Mortality index ( ̂ ) has shown decreasing trend.iv) Improvement in female mortality is larger than male mortality.v) Mortality improvement and its impact on actuarial quantities are observed at all ages for Indian female and male populations.

Figure 1 :
Figure 1: Pattern of age group specific central death rates for Indian Population

Table 1 :
Decade-wise age group specific central death rates for Indian female population during 1901-1911 to 2001-2011

Table 2 :
Decade-wise age group specific central death rates for Indian male population during1901-1911 to 2001-2011

Table 3 :
Life expectancy (in years) of Indians during 1901-2011 at selected age groups

Parameter Estimation by Singular Value Decomposition (SVD
Lee and Carter (1992))estimated the parameters of the LC model given in equation (1) by the SVD method.The estimated parameter vector ̂ is determined as the average over time of the logarithm of the central death rates as

71 1971-81 1981-91 1991-01 2001-11
Chiang (1984)22)d age group specific central death rates for three decadesFrom the values of actual and model based LE, we observed a good fit.To obtain model based LE, we have used definitions for construction of abridged life table given byGreenwood (1922)andChiang (1984).

Table 6 :
Observed and fitted life expectancy at some selected age groups using the LC model

Table 7 :
Fitted ARIMA models for estimated mortality index

Table 8 :
Forecasted values of mortality index with 95% CI

Table 9 :
Forecasted values of age specific central death rates in terms of deaths per 100,000 for Indian population

Table 10 :
Forecasted values of life expectancy at different age groups with 95% confidence intervals

Table 11
presents the values of and ̅̅̅̅ | calculated at 6.75% effective rate of interest per annum.(as per current repo rate on 29.02.2016 of Reserve Bank of India).In the Table 12, we have reported the APV of life annuities, ̈ and ̈ ̅̅̅̅ | calculated at 6.75% effective rate of interest per annum (as per repo rate on 29.02.2016 of Reserve Bank of India).These values of NSP of life insurances or APV of life annuities may be useful for finding total cost of one time premium for person of age and 60.

Table 11 :
Forecasted values of NSP of whole life insurance, and for 20-year term life insurance, ̅̅̅̅ | at some selected ages

Table 12 :
Forecasted values of APV of whole life annuity, ̈ and for 20-year temporary annuity, ̈ ̅̅̅̅ | at some selected ages