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Risks
, Volume 8 (3) – Jul 1, 2020

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risks Article Retiree Mortality Forecasting: A Partial Age-Range or a Full Age-Range Model? 1, 2 Han Lin Shang * and Steven Haberman Department of Actuarial Studies and Business Analytics, Macquarie University, Balaclava Rd, Macquarie Park, NSW 2109, Australia Cass Business School, City, University of London, London EC1Y 8TZ, UK; S.Haberman@city.ac.uk * Correspondence: hanlin.shang@mq.edu.au Received: 2 June 2020; Accepted: 24 June 2020; Published: 1 July 2020 Abstract: An essential input of annuity pricing is the future retiree mortality. From observed age-speciﬁc mortality data, modeling and forecasting can take place in two routes. On the one hand, we can ﬁrst truncate the available data to retiree ages and then produce mortality forecasts based on a partial age-range model. On the other hand, with all available data, we can ﬁrst apply a full age-range model to produce forecasts and then truncate the mortality forecasts to retiree ages. We investigate the difference in modeling the logarithmic transformation of the central mortality rates between a partial age-range and a full age-range model, using data from mainly developed countries in the Human Mortality Database (2020). By evaluating and comparing the short-term point and interval forecast accuracies, we recommend the ﬁrst strategy by truncating all available data to retiree ages and then produce mortality forecasts. However, when considering the long-term forecasts, it is unclear which strategy is better since it is more difﬁcult to ﬁnd a model and parameters that are optimal. This is a disadvantage of using methods based on time-series extrapolation for long-term forecasting. Instead, an expectation approach, in which experts set a future target, could be considered, noting that this method has also had limited success in the past. Keywords: age-period-cohort; Lee–Carter model with Poisson error; Lee–Carter model with Gaussian error; Plat model 1. Introduction Improving human survival probability contributes greatly to an aging population. To guarantee one individual’s financial income in retirement, a policyholder may purchase a fixed-term or lifetime annuity. A fixed-term or lifetime annuity is a contract offered by insurers guaranteeing regular payments in exchange for an initial premium. Since an annuity depends on survival probabilities and interest rates, pension funds and insurance companies are more likely to face a risk of longevity. Longevity risk is a potential systematic risk attached to the increasing life expectancy of policyholders, which can eventually result in a higher payout ratio than expected (Crawford et al. 2008). The concerns about longevity risk have led to a surge of interest in modeling and forecasting age-specific mortality rates. Many models for forecasting age-specific mortality indicators have been proposed in demographic literature (see Booth and Tickle 2008, for reviews). Of these, Lee and Carter (1992) implemented a principal component method to model the logarithm of age-specific mortality rates (m ) and extracted a single x,t time-varying index representing the trend in the level of mortality, from which the forecasts are obtained by a random walk with drift. Since then, the Lee–Carter (LC) method has been extended and modified (see Booth and Tickle 2008; Pitacco et al. 2009; Shang and Haberman 2018; Shang et al. 2011, for reviews). The LC method has been applied to many countries, including Belgium (Brouhns et al. 2002), Austria Risks 2020, 8, 69; doi:10.3390/risks8030069 www.mdpi.com/journal/risks Risks 2020, 8, 69 2 of 11 (Carter and Prkawetz 2001), England and Wales (Renshaw and Haberman 2003), and Spain (Debón et al. 2008; Guillen and Vidiella-i-Anguera 2005). Many statistical models focus on time-series extrapolation of past trends exhibited in age-speciﬁc mortality rates (see, e.g., Booth and Tickle 2008). We consider two modeling strategies: On the one hand, we can ﬁrst truncate all available data to retiree ages and then produce mortality forecasts (see, e.g., Cairns et al. 2006 2009). On the other hand, we can ﬁrst use the available data to produce forecasts and then truncate the mortality forecasts to retiree ages (see, e.g., Shang and Haberman 2017). In this paper, our contribution is to investigate the difference in modeling the logarithmic transformation of the central mortality rates between a partial age-range and a full age-range model, using data from mainly developed countries in the Human Mortality Database (2020). By evaluating and comparing the short-term point and interval forecast accuracies, we recommend the ﬁrst strategy by truncating all available data to retiree ages and then produce mortality forecasts. However, when we consider the long-term forecasts, it is unclear which strategy is better since it is more difﬁcult to ﬁnd a model and parameters that are optimal. This is a disadvantage of using methods based on time-series extrapolation for long-term forecasting. Instead, an expectation approach, in which experts set a future target, could be considered, noting that this method has also had limited success in the past (Booth and Tickle 2008). Our recommendations could be useful to actuaries for choosing a better modeling strategy and more accurately pricing a range of annuity products. The article is organized as follows: In Section 2, we describe the mortality data sets of 19 mainly developed countries. We revisit ﬁve time-series extrapolation models for forecasting age-speciﬁc mortality rates, which have been shown in the literature to work well across the full age range for some data sets (for more details, consult Shang 2012; Shang and Haberman 2018). Using these models as a testbed, we compare point and interval forecast accuracies between the two modeling strategies and provide our recommendations in Section 3. Conclusions are presented in Section 4. 2. Data Sets The data sets used in this study were taken from the Human Mortality Database (2020). For each sex in a given calendar year, the mortality rates obtained by the ratio between “number of deaths” and “exposure to risk”are arranged in a matrix for age and calendar year. Nineteen countries, mainly developed countries, were selected, and thus 38 sub-populations of age- and sex-specific mortality rates were obtained for all analyses. The 19 countries selected all have reliable data series commencing at/before 1950. Due to possible structural breaks (i.e., two world wars), we truncate all data series from 1950 onwards. The omission of Germany is because the Human Mortality Database (2020) for a reunited Germany only dates back to 1990. The selected countries and their abbreviations are shown in Table 1, along with their last year of available data (recorded in April 2019). To avoid fluctuations at older ages, we consider ages from 0 to 99 in a single year of age and the last age group is from 100 onwards. Should we consider all ages from 0 to 110+, and we may encounter the missing-value issue and observe mortality rates outside the range of [0, 1] for some years. Table 1. The 19 countries with the initial year of 1950 and their ending year listed below. Country Abbreviation Last Year Country Abbreviation Last Year Australia AUS 2014 Norway NOR 2014 Belgium BEL 2015 Portugal PRT 2015 Canada CAN 2011 Spain SPA 2016 Denmark DEN 2016 Sweden SWE 2016 Finland FIN 2015 Switzerland SWI 2016 France FRA 2016 Scotland SCO 2016 Italy ITA 2014 England & Wales EW 2016 Japan JPN 2016 Ireland IRE 2014 Netherland NET 2016 United States of America USA 2016 New Zealand NZ 2013 Risks 2020, 8, 69 3 of 11 3. Results 3.1. Forecast Evaluation We present 19 countries that begin in 1950 and end in the last year listed in Table 1. We keep the last 30 observations for forecasting evaluation, while the remaining observations are treated as initial ﬁtting observations, from which we produce the one-step-ahead to 30-step-ahead forecasts. Via an expanding window approach (see also Zivot and Wang 2006), we re-estimate the parameter in the time-series forecasting models by increasing the ﬁtted observations by one year and produce the one-step-ahead to 29-step-ahead forecasts. We iterate this process by increasing the sample size by one year until the end of the data period. The process produces 30 one-step-ahead forecasts, 29 two-step-ahead forecasts, . . . , one 30-step-ahead forecast. We compare these forecasts with the holdout samples to determine the out-of-sample forecast accuracy. 3.2. Forecast Error Criteria To evaluate the point forecast accuracy, we consider the mean absolute percentage error (MAPE) and root mean squared percentage error (RMSPE). The MAPE and RMSPE criteria measure how close the forecasts compare with the actual values of the variable being forecast, regardless of the error sign. The MAPE and RMSPE criteria can be expressed as: (31h) p m mb 1 x,j x,j MAPE = 100, x = 1, . . . , p, h å å p (31 h) m x,j j=1 x=1 (31h) p m mb 1 x,j x,j RMSPE = 100 h å å p (31 h) m x,j j=1 x=1 where m represents the actual holdout sample for age x in the forecasting year j, p denotes the total x,j number of ages, and mb represents the forecasts for the holdout sample. x,j To evaluate the pointwise interval forecast accuracy, we consider the interval score criterion of Gneiting and Raftery (2007). We consider the common case of the symmetric 100(1 a)% prediction intervals, with lower and upper bounds that were predictive quantiles at a/2 and 1 a/2, denoted by lb ub mb and mb . As deﬁned by Gneiting and Raftery (2007), a scoring rule for evaluating the pointwise x,j x,j interval forecast accuracy at time point j is n o lb ub ub lb lb lb b b b b b b S m , m ; m = m m + m m 1 m < m x,j x,j x,j x,j x,j x,j x,j x,j x,j n o ub ub m mb 1 m > mb , x,j x,j x,j x,j where 1fg denotes the binary indicator function. The optimal interval score is achieved when m x,j lb ub b b lies between m and m , with the distance between the upper and lower bounds being minimal. x,j x,j To obtain summary statistics of the interval score, we take the mean interval score across different ages and forecasting years. The mean interval score can be expressed as: (31h) lb ub S = S mb , mb ; m . a,h å å a x,j x,j x,j p (31 h) j=1 x=1 3.3. Comparison of Point and Interval Forecast Errors Modeling and forecasting mortality can be taken place in two routes. On the one hand, we can ﬁrst truncate the available data to certain ages, such as from 60 to 99 in a single year of age and 100+ as the last age, and then produce mortality forecasts for these retiree ages. On the other hand, we can ﬁrst Risks 2020, 8, 69 4 of 11 use the available data, i.e., age-speciﬁc mortality from 0 to 99 in a single year of age and 100+ as the last age, to produce forecasts for these 101 ages and then truncate the mortality forecasts to certain ages, such as 60 to 99 in a single year of age and 100+ as the last age. We study five time-series extrapolation models for forecasting age-specific mortality, which have been shown in the literature to work well across the full age range for some data sets. Please note that the Cairns-Blake-Dowd suite of models are not included in this paper, because they are designed just for ages 55 and over. The models that we have considered are subjective and far from extensive, but they suffice to serve as a testbed for comparing the forecast accuracy. These five models are: the Lee–Carter model with Poisson errors (see, e.g., Brouhns et al. 2002; Renshaw and Haberman 2003 2006), the Lee–Carter model with Gaussian errors (see, e.g., Booth et al. 2002; Koissi et al. 2006; Renshaw and Haberman 2003), age-period-cohort model (see, e.g., Renshaw and Haberman 2006) and the Plat model (Plat 2009). For the short-term forecast horizon (i.e., the one-step-ahead forecast horizon), we compute the mean of the MAPEs and RMSPEs to evaluate the point forecast accuracy. From Table 2, there is an advantage of directly modeling and forecasting the truncated series for the female mortality. For modeling the male mortality, the advantage of directly modeling and forecasting the truncated series intensiﬁes. By comparing the mean errors of the 19 countries, Table 2 shows that the most accurate forecasting method is the Plat model for providing best estimates of the female and male mortality forecasts. The Lee–Carter model with Poisson errors produces smaller MAPEs and RMSPEs than the Lee–Carter model with Gaussian errors. With the Lee–Carter model with Gaussian errors, it is advantageous to consider two components rather than only the ﬁrst component. Table 2. For the one-step-ahead forecast horizon h = 1, we compute the MAPE and RMSPE for each country and each model. The most accurate model for each country is highlighted in bold. For each model, we consider modeling the data with either a partial age range (termed as Partial) or a full age range (termed as Full). LC (Poisson) LC (Gaussian) LC (Gaussian) APC Plat Error Country Full Partial Full Partial Full Partial Full Partial Full Partial MAPE Female AUS 4.45 4.38 6.01 4.97 4.69 4.42 7.32 6.38 4.83 4.66 BEL 4.92 4.90 6.08 5.01 5.84 5.04 7.85 6.24 4.44 4.74 CAN 3.29 3.22 3.94 3.57 3.74 3.03 4.98 4.29 3.20 2.73 DEN 7.05 7.15 8.25 6.90 7.74 6.36 7.58 7.15 5.31 4.79 FIN 6.43 6.13 11.56 6.80 8.26 6.80 9.31 7.94 5.63 5.73 FRA 4.28 4.47 4.97 4.47 3.96 3.80 7.67 6.16 4.22 3.65 ITA 3.25 3.15 8.36 3.71 3.12 3.17 6.49 5.56 3.68 3.43 JPN 5.99 6.02 23.08 7.03 3.84 3.23 6.00 5.02 4.04 3.31 NET 3.62 3.61 4.58 3.67 4.39 3.65 7.53 6.29 4.19 3.78 NZ 7.70 7.75 8.65 7.90 8.33 7.85 10.02 9.38 7.75 7.74 NOR 5.25 5.28 6.92 5.54 6.41 5.41 8.18 7.58 5.66 5.56 PRT 6.06 5.42 11.68 5.84 7.82 5.78 8.75 7.30 6.13 6.58 SPA 5.15 5.04 12.99 5.73 6.03 4.19 7.92 6.24 4.80 5.21 SWE 4.03 4.03 4.31 4.09 4.55 4.11 8.19 7.00 4.91 4.31 SWI 4.69 4.72 5.99 4.65 5.89 4.84 8.94 7.39 5.28 5.03 SCO 6.79 6.83 8.78 6.55 7.62 6.49 7.21 6.97 5.48 5.15 EW 4.79 4.72 6.27 4.49 5.20 3.51 6.32 5.80 3.92 3.15 IRE 8.45 7.33 15.01 8.27 9.87 8.01 9.52 9.52 9.02 9.19 USA 3.33 3.22 3.84 3.20 3.81 2.37 4.50 4.28 3.43 2.13 Mean 5.24 5.12 8.49 5.39 5.85 4.85 7.59 6.66 5.05 4.78 Risks 2020, 8, 69 5 of 11 Table 2. Cont. LC (Poisson) LC (Gaussian) LC (Gaussian) APC Plat Error Country Full Partial Full Partial Full Partial Full Partial Full Partial Male AUS 5.39 4.84 7.85 5.39 5.74 5.58 7.71 5.98 5.15 5.27 BEL 7.86 6.11 10.53 9.74 8.43 7.43 6.29 5.44 5.31 4.86 CAN 5.54 4.28 7.58 5.05 5.85 4.76 5.19 4.25 4.12 3.31 DEN 8.19 6.67 10.29 10.51 7.77 8.33 7.77 6.53 5.78 5.52 FIN 8.12 7.49 11.84 8.57 10.02 8.43 8.11 7.10 6.61 6.59 FRA 3.71 3.17 4.42 3.96 4.11 3.66 7.54 6.02 4.16 3.32 ITA 5.85 4.21 11.29 5.96 4.63 3.73 5.16 4.96 3.74 2.87 JPN 3.95 3.76 9.58 4.31 3.78 4.13 4.92 4.61 3.59 3.27 NET 8.02 5.58 12.65 7.75 7.23 5.37 5.83 4.79 5.24 4.03 NZ 9.21 8.17 12.63 10.86 9.87 10.90 9.78 8.79 8.12 8.38 NOR 8.66 6.37 13.61 6.84 7.21 6.85 7.42 6.59 6.07 6.02 PRT 6.12 5.57 8.70 7.26 8.46 6.30 7.44 6.90 5.85 5.88 SPA 4.39 3.88 8.48 5.44 7.10 4.38 6.52 5.80 4.11 4.29 SWE 5.05 4.51 9.82 5.38 6.01 5.28 8.07 6.02 4.53 4.08 SWI 5.62 5.37 6.51 5.76 6.86 5.99 7.71 6.55 5.27 5.30 SCO 7.06 6.20 10.63 8.56 8.42 7.51 7.51 6.71 6.29 5.80 EW 4.78 4.04 8.52 5.15 5.99 4.28 6.73 4.93 3.49 3.43 IRE 13.24 9.34 17.28 12.43 12.59 10.55 10.07 9.84 9.86 9.71 USA 4.65 3.75 5.90 4.33 5.62 3.31 5.79 4.26 3.71 2.42 Mean 6.60 5.44 9.90 7.01 7.14 6.14 7.14 6.11 5.26 4.97 RMSPE Female AUS 5.84 5.77 7.62 6.49 6.11 5.82 8.86 7.87 6.20 6.19 BEL 6.40 6.43 7.55 6.39 7.37 6.47 9.46 7.64 5.74 6.30 CAN 4.17 4.11 4.96 4.59 4.74 3.98 6.06 5.31 3.96 3.50 DEN 9.36 9.70 10.05 9.23 9.55 8.41 9.32 8.83 6.67 6.10 FIN 8.51 8.32 14.54 8.85 10.88 9.14 11.69 10.12 7.46 8.05 FRA 5.57 5.87 6.22 5.51 5.03 4.85 9.15 7.53 5.33 4.94 ITA 4.14 4.01 9.80 4.62 4.05 4.19 7.90 6.92 4.75 4.59 JPN 7.13 7.54 25.03 8.26 4.78 4.07 7.32 6.30 5.00 4.51 NET 4.62 4.60 5.74 4.73 5.50 4.68 8.86 7.55 5.23 4.88 NZ 10.41 10.44 11.48 10.55 11.16 10.67 12.75 12.13 10.29 10.49 NOR 6.85 6.92 8.63 7.20 8.13 7.13 10.15 9.43 7.23 7.35 PRT 7.85 7.03 13.55 7.47 9.85 7.51 10.93 9.16 8.28 8.71 SPA 6.53 6.51 14.70 7.18 7.85 5.26 9.59 7.60 6.22 6.86 SWE 5.27 5.26 5.54 5.31 5.95 5.38 9.75 8.48 6.13 5.56 SWI 5.99 6.03 7.55 5.91 7.48 6.12 10.63 8.98 6.63 6.54 SCO 8.96 8.99 11.57 8.58 9.86 8.56 9.07 8.89 7.11 6.77 EW 6.33 6.25 8.29 5.94 6.74 4.42 7.69 7.15 4.86 3.96 IRE 11.10 9.63 18.75 10.82 12.88 10.58 12.22 12.43 11.76 12.26 USA 4.19 4.13 4.79 4.13 4.73 3.27 5.91 5.65 4.29 2.77 Mean 6.80 6.71 10.34 6.94 7.51 6.34 9.33 8.31 6.48 6.33 Male AUS 7.21 6.78 9.72 7.29 7.42 7.24 9.67 7.83 7.12 7.48 BEL 9.98 8.47 13.14 13.50 10.48 10.02 8.86 7.83 7.39 7.19 CAN 6.80 5.48 9.35 6.22 7.37 6.02 6.77 5.67 5.19 4.42 DEN 11.05 9.23 13.50 14.33 10.49 11.59 10.20 8.75 7.97 7.86 FIN 13.16 13.07 15.27 12.49 15.45 13.39 11.84 10.88 10.56 11.04 FRA 4.72 4.11 5.60 5.01 5.12 4.75 9.21 7.68 5.41 4.45 ITA 7.17 5.35 13.72 7.28 6.31 4.84 6.83 6.50 4.88 4.04 JPN 4.91 4.78 10.98 5.45 4.77 5.17 6.20 5.93 4.45 4.38 NET 9.68 7.05 15.47 9.82 8.96 6.80 7.61 6.36 5.60 5.54 NZ 13.36 12.08 16.86 15.04 13.59 14.72 12.94 11.90 11.72 12.27 Risks 2020, 8, 69 6 of 11 Table 2. Cont. LC (Poisson) LC (Gaussian) LC (Gaussian) APC Plat Error Country Full Partial Full Partial Full Partial Full Partial Full Partial NOR 11.32 8.98 17.02 9.45 9.82 9.73 9.89 8.98 8.42 8.62 PRT 8.11 7.59 10.55 9.59 10.56 8.60 9.72 9.07 8.06 8.46 SPA 5.70 5.31 9.97 7.01 9.17 5.73 8.23 7.71 5.47 5.64 SWE 6.56 6.04 11.71 6.99 7.75 7.04 9.72 7.66 5.89 5.60 SWI 7.56 7.30 8.50 7.77 9.24 8.19 10.02 8.60 7.16 7.34 SCO 9.21 8.43 12.98 10.73 11.10 10.28 10.04 9.05 8.50 7.97 EW 5.93 5.16 10.38 6.34 7.42 5.41 8.39 6.37 4.50 4.59 IRE 19.05 14.78 23.17 17.45 18.24 15.44 15.49 14.90 15.19 15.23 USA 5.59 4.82 6.95 5.30 6.66 4.08 7.87 6.14 4.78 3.26 Mean 8.79 7.62 12.36 9.32 9.47 8.37 9.45 8.31 7.28 7.13 For the short-term forecast horizon (i.e., the one-step-ahead forecast horizon), we compute the S to evaluate the interval forecast accuracy. From Table 3, there is an advantage of directly a=0.2 modeling and forecasting the truncated series for both female and male mortality. By comparing the mean errors of the 19 countries, Table 3 shows that the most accurate forecasting method is the Plat model for providing the interval forecasts of both female and male mortality rates. Table 3. For the one-step-ahead forecast horizon h = 1, we compute the mean interval score (100) for each country and each model. Country LC (Poisson) LC (Gaussian) LC (Gaussian) APC Plat Full Partial Full Partial Full Partial Full Partial Full Partial Female AUS 3.51 3.44 2.70 2.68 2.59 2.78 6.60 5.26 2.70 2.53 BEL 3.58 3.28 3.34 3.11 3.28 3.19 6.09 4.68 2.61 2.64 CAN 2.91 2.61 2.07 1.89 1.85 1.85 5.17 4.12 1.58 1.28 DEN 3.81 3.69 4.62 4.27 4.68 4.54 8.46 7.47 3.41 2.60 FIN 5.02 4.93 6.17 5.37 5.70 6.19 8.73 7.84 3.78 3.42 FRA 2.07 1.85 1.99 1.93 1.86 1.87 4.90 3.84 2.02 1.75 ITA 2.33 1.89 3.21 2.04 1.99 2.24 4.38 3.95 2.10 2.14 JPN 3.22 2.13 7.24 2.74 1.82 1.70 2.96 3.02 1.56 1.40 NET 3.20 3.08 2.76 2.43 2.71 2.52 7.58 5.99 2.49 2.22 NZ 5.87 5.75 4.86 4.68 4.90 5.63 9.18 8.19 5.01 4.63 NOR 4.36 4.29 3.38 3.20 3.48 3.23 6.92 6.19 3.23 2.94 PRT 5.34 4.33 5.54 4.21 4.58 4.20 5.57 4.69 4.24 3.92 SPA 4.00 3.32 3.85 2.59 2.91 2.11 4.24 3.39 2.62 2.69 SWE 3.00 2.93 2.83 2.76 2.94 3.18 8.19 6.44 2.46 2.14 SWI 3.09 3.02 4.30 4.14 4.34 4.57 7.12 5.11 2.62 2.64 SCO 4.79 4.62 4.01 3.80 4.12 4.45 7.67 7.51 4.12 3.40 EW 2.52 2.35 1.96 1.84 1.86 1.79 6.49 5.85 2.14 1.80 IRE 6.57 5.74 6.54 4.93 5.32 5.28 6.54 6.42 5.72 5.55 USA 3.18 2.92 2.03 1.78 1.91 1.27 5.89 5.51 2.67 1.19 Mean 3.81 3.48 3.86 3.18 3.31 3.29 6.46 5.55 3.00 2.68 Male AUS 5.90 5.53 4.75 4.44 4.35 4.57 11.27 7.63 4.60 4.57 BEL 8.79 7.71 7.73 8.24 6.87 7.81 9.87 8.85 5.99 5.49 CAN 5.31 4.78 3.86 3.39 3.62 3.31 8.68 6.42 3.55 2.65 DEN 8.23 7.04 7.48 8.33 7.08 7.97 11.80 9.96 6.37 5.49 FIN 10.36 9.95 9.13 8.63 9.65 9.15 13.67 12.30 5.20 7.39 FRA 3.04 2.66 2.91 2.85 2.99 3.05 10.62 8.20 3.32 2.73 Risks 2020, 8, 69 7 of 11 Table 3. Cont. Country LC (Poisson) LC (Gaussian) LC (Gaussian) APC Plat Full Partial Full Partial Full Partial Full Partial Full Partial ITA 4.75 3.45 4.16 3.44 3.05 3.33 6.68 6.35 3.15 3.04 JPN 3.32 2.52 4.97 3.62 3.31 3.85 4.72 4.76 2.38 2.22 NET 7.46 6.26 5.85 5.12 5.25 4.35 8.30 6.69 5.20 3.79 NZ 10.51 9.45 9.53 9.11 8.47 9.39 14.37 12.24 8.62 7.90 NOR 8.36 7.57 6.95 5.82 5.92 6.16 10.21 9.11 6.39 5.88 PRT 7.45 6.40 7.32 6.97 7.56 7.54 9.24 9.16 5.87 5.76 SPA 5.35 4.32 4.54 3.80 4.40 3.45 7.62 7.21 3.48 3.37 SWE 5.36 5.05 5.21 4.44 4.83 4.62 12.01 9.02 4.11 3.65 SWI 6.23 6.00 6.76 6.63 7.01 7.00 11.77 8.96 5.15 4.68 SCO 8.06 7.35 7.96 7.37 8.05 8.14 11.02 10.03 6.73 5.33 EW 4.29 3.71 4.03 3.26 3.63 3.19 9.37 6.86 2.81 2.79 IRE 15.37 12.27 11.78 10.10 10.43 10.13 13.84 13.32 11.78 11.12 USA 5.26 4.75 3.25 2.98 3.13 1.91 11.19 8.33 4.12 1.85 Mean 7.02 6.15 6.22 5.71 5.77 5.73 10.33 8.71 5.20 4.72 For the long-term forecast horizon (i.e., the 30-step-ahead forecast horizon), we also compute the mean of the MAPEs and RMSPEs to evaluate the point forecast accuracy. From Table 4, it is unclear from the comparison of the point forecast errors if there is an advantage of modeling and forecasting the truncated series for the female mortality. In contrast, there is an advantage of modeling and forecasting the male mortality and forecasting the whole series and then truncating the mortality forecasts. By comparing the mean errors of the 19 countries, Table 4 shows that the most accurate forecasting method is the Lee–Carter model with Poisson errors for providing best estimates of the female mortality forecasts. The most accurate forecasting method is the APC model for providing the best estimates of the male mortality forecasts. The Lee–Carter model with Poisson errors produces smaller MAPEs and RMSPEs than the Lee–Carter model with Gaussian errors. Table 4. For the 30-step-ahead forecast horizon h = 30, we compute the MAPE and RMSPE for each country and each model. The most accurate model for each country is highlighted in bold. For each model, we consider modeling the data with either a partial age range (termed as Partial) or a full age range (termed as Full). LC (Poisson) LC (Gaussian) LC (Gaussian) APC Plat Error Country Full Partial Full Partial Full Partial Full Partial Full Partial MAPE Female AUS 18.78 20.05 19.09 20.46 18.62 21.58 15.20 26.11 20.53 26.99 BEL 15.19 15.50 17.02 15.99 20.22 16.61 15.77 21.65 26.27 29.78 CAN 13.21 12.72 12.56 9.93 9.07 10.68 17.96 16.29 22.76 10.32 DEN 11.91 12.36 11.44 10.71 14.64 11.41 23.59 22.03 34.15 19.92 FIN 16.35 18.28 16.79 18.37 21.59 20.92 23.38 27.27 26.10 29.23 FRA 16.81 17.35 16.74 17.30 20.00 18.23 13.80 18.51 19.02 30.02 ITA 18.55 21.17 26.00 24.22 24.16 22.90 15.06 32.54 23.41 48.43 JPN 22.83 23.89 40.71 27.40 24.65 23.03 52.48 19.82 52.96 30.09 NET 15.04 14.42 20.25 14.41 11.70 12.82 14.91 16.98 31.19 15.17 NZ 34.61 35.91 38.23 40.11 38.56 36.21 28.84 41.59 35.94 44.18 NOR 10.17 9.52 12.32 11.39 10.44 10.44 14.22 19.68 35.73 12.53 PRT 41.89 34.61 53.01 40.15 52.41 45.16 17.40 46.54 47.46 54.60 SPA 25.09 24.33 52.66 27.03 27.00 28.85 35.96 27.82 35.92 40.34 SWE 14.54 13.73 15.85 12.50 13.50 12.55 16.02 16.26 24.55 12.05 SWI 11.27 10.93 10.99 10.83 12.25 12.30 17.56 19.76 37.25 18.94 SCO 16.91 18.29 17.55 19.72 19.75 20.56 10.77 25.86 31.53 20.71 EW 17.24 19.41 55.25 19.25 17.46 20.11 9.57 34.12 25.03 34.45 IRE 38.93 38.09 68.20 43.59 45.58 41.77 19.47 44.09 38.19 41.49 USA 8.90 7.51 11.13 5.98 7.75 5.55 16.65 13.24 12.56 7.83 Mean 19.38 19.37 27.15 20.49 21.54 20.62 19.93 25.80 30.56 27.74 Risks 2020, 8, 69 8 of 11 Table 4. Cont. LC (Poisson) LC (Gaussian) LC (Gaussian) APC Plat Error Country Full Partial Full Partial Full Partial Full Partial Full Partial Male AUS 50.13 52.47 56.68 55.41 49.33 55.95 38.21 52.54 53.81 72.66 BEL 58.34 54.88 65.69 92.47 58.83 85.97 31.63 46.42 87.24 46.11 CAN 45.46 42.26 47.85 44.46 47.80 44.78 33.70 43.26 79.94 49.49 DEN 63.45 63.01 63.36 67.67 66.87 67.44 66.01 57.71 73.57 56.30 FIN 43.39 48.11 44.34 96.58 47.78 51.77 20.85 31.29 75.96 52.26 FRA 32.81 33.83 35.81 34.77 37.00 35.36 39.09 35.86 60.49 39.45 ITA 64.41 62.35 70.38 67.22 53.63 62.03 32.11 61.05 81.64 60.60 JPN 12.30 12.68 20.98 12.79 12.91 12.82 39.53 20.94 32.12 15.61 NET 79.68 76.14 78.67 79.29 65.19 71.08 45.62 51.89 79.88 47.61 NZ 99.32 82.86 112.42 117.61 104.32 114.28 63.10 76.39 92.05 94.38 NOR 87.50 73.97 88.62 71.43 65.64 67.35 45.09 66.85 70.00 60.19 PRT 42.79 38.11 49.01 83.79 58.24 49.59 18.01 35.29 32.26 47.62 SPA 25.43 23.13 58.45 29.85 27.15 30.84 15.21 25.02 26.29 28.47 SWE 51.89 47.57 55.01 57.89 53.05 51.91 45.46 43.28 76.65 42.27 SWI 36.00 37.46 41.73 42.27 39.01 45.48 30.64 46.42 50.82 55.93 SCO 52.66 55.63 55.86 92.86 61.36 86.41 38.51 57.02 65.63 54.61 EW 52.44 55.96 52.81 60.03 50.06 55.52 35.42 48.16 47.26 54.41 IRE 110.20 91.02 112.71 111.84 111.82 104.52 43.10 97.26 63.07 93.34 USA 24.07 24.99 26.07 26.81 26.12 28.13 21.56 23.20 70.21 31.07 Mean 54.33 51.39 59.81 65.53 54.53 59.01 36.99 48.41 64.15 52.76 RMSPE Female AUS 21.74 23.11 22.17 23.51 21.82 24.60 19.09 28.34 24.41 30.76 BEL 18.25 18.92 20.82 19.44 24.70 20.38 18.67 24.89 31.54 35.64 CAN 15.37 14.95 14.53 12.18 11.31 13.01 22.76 21.38 28.37 14.13 DEN 14.08 14.44 13.33 13.46 16.62 13.50 25.58 25.11 40.84 23.08 FIN 19.82 22.37 20.14 22.67 26.18 25.80 27.37 30.78 33.02 35.66 FRA 19.01 19.60 19.27 20.17 23.47 21.18 17.97 21.60 24.55 34.67 ITA 21.18 23.45 28.25 26.36 26.44 25.17 18.28 36.15 28.19 57.57 JPN 25.73 26.91 47.72 30.06 27.27 25.71 53.75 27.37 55.50 33.61 NET 16.33 15.77 21.13 15.67 13.93 14.43 20.15 21.32 37.43 17.97 NZ 44.37 45.84 46.55 49.71 46.68 45.70 33.10 48.79 42.29 50.98 NOR 12.65 12.45 14.53 13.90 12.80 13.13 17.94 22.61 41.08 14.97 PRT 48.09 40.40 61.17 46.28 60.66 52.06 20.96 54.09 52.80 62.86 SPA 30.73 29.51 59.17 33.96 33.89 36.16 38.24 30.74 42.10 45.40 SWE 16.23 15.50 17.54 14.61 17.18 14.73 22.05 22.21 31.04 13.67 SWI 13.97 13.52 13.60 13.42 15.17 17.27 23.51 24.38 42.62 21.26 SCO 22.00 24.51 20.56 25.95 22.19 26.98 14.05 29.29 35.82 24.81 EW 21.84 24.08 61.19 23.89 21.62 24.44 11.78 38.79 29.59 39.16 IRE 44.06 43.59 73.27 48.92 50.75 47.72 24.33 57.14 42.55 47.34 USA 10.10 8.70 12.13 7.02 9.11 6.77 18.45 16.32 15.29 8.77 Mean 22.92 23.03 30.90 24.27 25.36 24.67 23.58 30.60 35.74 32.23 Male AUS 58.82 61.48 66.43 64.80 57.73 66.12 45.92 64.35 64.19 82.67 BEL 72.35 63.85 78.04 103.40 71.52 95.86 37.45 54.36 90.56 53.55 CAN 57.57 52.42 61.40 55.54 60.60 55.81 40.51 52.40 84.49 57.41 DEN 81.11 80.89 78.71 75.38 82.53 82.03 77.00 66.75 91.52 68.67 FIN 51.71 57.52 52.22 111.49 55.40 61.78 23.39 36.30 78.51 60.90 FRA 38.15 38.84 41.53 40.38 40.58 40.91 43.68 40.64 68.48 44.68 ITA 80.70 77.32 87.79 84.15 67.36 78.15 36.01 71.62 85.65 75.05 JPN 15.46 16.48 25.77 15.25 16.66 15.91 40.62 26.82 37.06 18.33 NET 102.58 95.23 100.24 102.76 81.69 89.74 53.64 60.81 87.97 64.18 NZ 112.83 98.95 128.69 136.49 117.87 132.75 74.82 91.95 102.96 107.61 NOR 108.62 85.81 110.96 82.25 80.62 79.43 50.60 77.04 79.85 76.25 PRT 49.93 44.03 56.63 95.17 64.89 55.57 20.40 41.52 38.67 55.07 SPA 32.82 30.04 63.13 33.14 32.15 34.60 19.17 27.51 30.23 33.06 SWE 62.19 56.65 66.37 68.98 62.41 62.39 52.86 51.01 84.63 50.20 SWI 43.71 45.27 48.84 50.17 45.93 53.22 36.91 55.66 61.52 60.67 SCO 64.25 66.37 68.29 105.02 72.18 97.90 44.12 67.61 71.81 66.18 EW 62.45 65.98 63.72 71.74 58.80 65.56 42.07 58.11 56.89 63.73 IRE 122.60 104.66 125.27 128.85 124.03 118.34 49.25 116.02 80.35 105.59 USA 28.44 29.33 30.26 31.41 30.41 32.05 25.85 26.09 72.70 34.40 Mean 65.59 61.64 71.28 76.65 64.39 69.37 42.86 57.19 72.00 62.01 Risks 2020, 8, 69 9 of 11 For the long-term forecast horizon (i.e., the 30-step-ahead forecast horizon), we also compute the S to evaluate the interval forecast accuracy. From Table 5, there is a slight advantage of directly a=0.2 modeling and forecasting the truncated series for the female mortality. For modeling the male mortality, there is an advantage of modeling and forecasting the whole series and then truncating the mortality forecasts. By comparing the mean errors of the 19 countries, Table 5 shows that the most accurate forecasting method is the Lee–Carter model with Poisson errors for providing the interval forecasts of the female mortality rates. The most accurate forecasting method is the APC model for providing the interval forecasts of the male mortality rates. The Lee–Carter model with Poisson errors produces smaller MAPEs than the Lee–Carter model with Gaussian errors. Table 5. For the 30-step-ahead forecast horizon h = 30, we compute the mean interval score (100) for each country and each model. LC (Poisson) LC (Gaussian) LC (Gaussian) APC Plat Country Full Partial Full Partial Full Partial Full Partial Full Partial Female AUS 8.25 7.20 9.30 8.94 7.75 8.80 13.42 10.46 12.31 9.29 BEL 5.99 4.99 6.85 6.18 7.83 10.10 8.24 11.80 19.53 12.75 CAN 12.64 10.76 7.54 8.09 5.11 7.09 23.15 21.93 23.71 4.47 DEN 4.81 5.23 5.37 5.07 6.35 18.66 9.72 8.85 28.91 8.85 FIN 12.80 9.57 9.23 9.29 12.38 162.23 25.85 20.57 27.49 16.37 FRA 5.39 4.51 5.49 7.33 5.45 5.69 11.37 11.82 14.18 9.99 ITA 8.81 4.54 12.62 10.21 6.16 8.46 6.16 7.09 7.25 13.39 JPN 6.73 4.30 12.47 10.43 5.40 6.06 27.24 20.42 39.63 7.77 NET 9.94 9.09 10.61 9.66 6.96 8.11 22.06 23.59 42.29 8.94 NZ 8.75 8.81 9.61 10.39 16.15 82.21 7.78 8.99 9.13 13.52 NOR 6.81 7.82 5.39 6.41 5.21 7.68 14.47 13.55 44.89 7.80 PRT 20.63 10.59 26.58 13.21 23.64 18.53 8.51 8.49 53.85 22.47 SPA 16.70 13.35 18.08 16.12 12.87 9.29 15.95 12.68 32.88 8.74 SWE 9.66 8.88 9.47 7.85 6.16 13.71 25.97 26.32 35.74 6.55 SWI 3.67 3.39 5.30 5.08 6.58 42.74 22.61 18.84 31.94 14.23 SCO 8.36 6.08 9.65 7.65 10.37 30.24 7.13 6.64 26.02 11.29 EW 3.84 4.04 11.42 4.45 4.81 6.20 5.09 7.44 13.97 8.55 IRE 17.25 12.39 20.57 16.64 18.89 21.43 8.06 9.81 15.26 20.33 USA 4.11 2.50 3.11 1.80 2.52 2.70 15.66 13.93 4.47 4.05 Mean 9.22 7.27 10.46 8.67 8.98 24.73 14.65 13.85 25.45 11.02 Male AUS 15.97 12.78 18.91 15.69 13.62 23.85 10.15 10.18 16.51 21.01 BEL 22.28 18.18 29.43 47.28 197.68 274.40 9.25 12.30 39.35 17.42 CAN 12.99 11.43 12.53 11.80 12.94 13.63 13.50 13.63 35.64 14.97 DEN 19.74 19.28 21.72 30.66 441.92 463.81 22.09 16.40 23.17 15.61 FIN 16.45 16.98 15.54 33.36 26.11 43.94 8.54 8.80 34.82 23.53 FRA 10.58 8.20 12.85 12.74 14.71 14.93 10.69 9.38 16.70 14.70 ITA 17.34 16.24 19.64 17.47 12.56 17.47 7.89 15.92 27.76 17.58 JPN 3.58 3.69 8.70 4.73 7.27 23.37 30.41 25.32 38.02 10.07 NET 23.74 21.63 24.04 20.78 18.91 23.43 14.96 10.42 28.78 15.71 NZ 45.00 26.13 54.39 41.50 52.59 95.29 18.23 16.61 28.10 42.92 NOR 26.45 25.79 25.86 27.40 16.89 23.49 14.77 13.63 22.20 16.68 PRT 19.87 13.68 24.01 37.26 30.92 118.32 8.50 12.32 12.26 22.79 SPA 27.44 14.80 35.05 19.88 15.81 14.26 8.92 10.80 11.09 10.42 SWE 17.71 13.46 18.53 17.60 19.87 20.01 14.96 11.34 25.19 10.65 SWI 9.37 9.05 14.48 13.32 28.57 28.63 9.04 11.19 14.00 22.49 SCO 20.85 19.01 21.61 43.03 84.51 96.68 10.22 12.77 25.07 18.27 EW 15.24 14.20 16.68 21.06 17.35 17.39 7.48 11.17 14.08 17.50 IRE 70.26 37.26 73.40 48.79 69.77 109.01 12.71 25.21 28.22 36.51 USA 10.95 9.65 10.60 9.26 10.47 10.34 12.59 11.09 42.64 11.69 Mean 21.36 16.39 24.10 24.93 57.50 75.38 12.89 13.60 25.45 18.97 4. Conclusions We consider forecasting retiree mortality using two modeling strategies. On the one hand, we can ﬁrst truncate the available data to retiree ages and then produce mortality forecasts. On the other Risks 2020, 8, 69 10 of 11 hand, we can ﬁrst use the available data to produce forecasts and then truncate the mortality forecasts to retiree ages. Using the empirical data from Human Mortality Database (2020), we investigate the short-term and long-term point and interval forecast accuracies. Between the two model strategies, we recommend the ﬁrst strategy by truncating all available data to retiree ages and then produce short-term mortality forecasts. Our recommendations could be useful to actuaries for choosing a better modeling strategy and more accurately pricing a range of annuity products. For the long-term mortality forecasts, we recommend the ﬁrst strategy for modeling female mortality but the second strategy for modeling male mortality. It is difﬁcult to recommend a strategy when the model and its parameters may not be optimal for the long-term forecasts. This is a disadvantage of using methods based on time-series extrapolation for long-term forecasting. Instead, an expectation approach, in which experts set a future target, could be considered, noting that this method has also had limited success in the past (Booth and Tickle 2008). There are several ways in which the present study can be further extended, and we brieﬂy mention two: (1) We could consider some machine learning methods to model mortality forecasts (see, e.g., Perla et al. 2020; Richman and Wüthrich 2020). (2) The results depend on the age range considered as well as the selected countries. The R code for reproducing the results can be provided upon request from the corresponding author. Author Contributions: Investigation, H.L.S.; Methodology, H.L.S.; Software, H.L.S.; Supervision, S.H. These authors contributed equally to this work. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Acknowledgments: The authors are grateful to the comments and suggestions received from three reviewers, and the scientiﬁc committee of the Living to 100 symposium in 2020. Conﬂicts of Interest: The authors declare no conﬂict of interest. References Booth, Heather, and Leonie Tickle. 2008. Mortality modelling and forecasting: A review of methods. Annals of Actuarial Science 3: 3–43. [CrossRef] Booth, Heather, John Maindonald, and Len Smith. 2002. Applying Lee-Carter under conditions of variable mortality decline. Population Studies 56: 325–36. [CrossRef] [PubMed] Brouhns, Natacha, Michel Denuit, and Jeroen K. Vermunt. 2002. A Poisson log-bilinear regression approach to the construction of projected life-tables. Insurance: Mathematics and Economics 31: 373–93. Cairns, Andrew J. G., David Blake, and Kevin Dowd. 2006. 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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

Risks – Multidisciplinary Digital Publishing Institute

**Published: ** Jul 1, 2020

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