2.1 Mortality improvements in aging Asian economies

Asia-Pacific is the fastest aging region in the world. By 2050, one in four people in the region will be 60 years old or older, whereas those 80 years or older will constitute about one-fifth of all older persons (United Nations, 2019). Economies that are most rapidly aging include Singapore, Hong Kong (SAR), South Korea, and Taiwan. Figure 1 compares the mortality rates of resident populations in these four economies from 1980 to 2017. Death rates are plotted for different age groups (ages 50–59, 60–69, 70–79, and 80+), and separately for males and females. The solid black lines in the plots reveal that Singapore has one of the highest rates of mortality improvements among its peers. We see that in the 1980s and early 1990s, Taiwan and Singapore have relatively comparable death rates. By the mid-1990s, however, Singapore's mortality levels have notably dropped below those of the former. Mortality decline in Singapore continued at a staggering pace over the next two decades. By around 2010, death rates of the resident Singapore population have declined to levels that are almost alike those of Hong Kong, which has the lowest mortality levels among the four economies since the mid-1980s.

Figure 1. Observed death rates of Hong Kong, South Korea, Singapore, and Taiwan from 1980 to 2017. Source: Authors' calculations derived from Human Mortality Database, Singapore Department of Statistics, and Census and Statistics Department of Hong Kong. Notes: For Singapore, annual mortality rates for 5-year abridged age groups are obtained from the Department of Statistics for 1980–2017 (SDOS, 2018). Death rates pertain only to the resident population (i.e., Singapore citizens and permanent residents). Mortality data for Hong Kong data are sourced from the Census and Statistics Department of Hong Kong (2017). Annual mortality data for South Korea and Taiwan are obtained from the Human Mortality Database (HMD, 2019).

In addition, mortality improvements in Singapore have been strong and persistent over the past four decades. Between 1980 and 2000, the observed annual rates of mortality decline were 3.8%, 2.9%, 2.1%, and 1.0%, respectively, for males ages 50–59, 60–69, 70–79, and 80+. Notably, the pace of mortality improvements has slowed somewhat for those aged 50–59; their death rates declined at about 2.6% per annum between 2001 and 2017. In contrast, the pace of mortality improvements has increased for those ages 60+ over time. Between 2001 and 2017, annual rates of decline were 3.4%, 2.5%, and 1.2%, respectively, for males ages 60–69, 70–79, and 80+. Similar patterns are observed for females. The higher rates of mortality improvement among older-old Singaporeans (as compared to previous decades) are worrisome from a longevity risk management standpoint. It implies that average life expectancy at older ages may continue to rise to new levels, and longevity risk would become more prominent going forward.Footnote ³

2.2 Stochastic mortality modeling

In the actuarial and demographic literature, several methods of stochastic mortality modeling have been proposed to allow researchers to capture the mortality trends of a population. One of the most widely-known approaches is the method proposed by Lee and Carter (Reference Lee and Carter1992), which demonstrated that the essential features of the mortality profile of a population over time can be described by an age-specific intercept plus a common trend for all age groups multiplied by an age-specific coefficient. Building on this study, various extensions have been proposed to improve the modeling procedure and forecasting performance (see, e.g., Li and Lee, Reference Li and Lee2005; Renshaw and Haberman, Reference Renshaw and Haberman2006; Cairns et al., Reference Cairns, Blake, Dowd, Coughlan, Epstein, Ong and Balevich2009; Li, Reference Li2013). The choice of a suitable model for forecasting survival probabilities for the resident population in Singapore is an important one. Given that the average life expectancy of Singaporeans has risen significantly in the past several decades, it is critical to adopt a statistical model that can accurately and robustly predict the longevity gains.

In this present analysis, we implement the augmented common factor Lee-Carter model based on Li and Lee (Reference Li and Lee2005). This is an extended version of the Lee and Carter (Reference Lee and Carter1992) mortality model, and has been widely applied in multi-population mortality forecasting. A key strength of the model is that it defines a common mortality index that represented the shared period effect for all populations (or subpopulations) considered, and additional factors to account for short-term deviations from the main trend for each population (or subpopulation). Li (Reference Li2013) applied the augmented common factor model to Australian population data and showed that it produces coherent mortality forecasts between males and females. This is because the time-series processes for the additional factors are chosen in such a way that the projected male-to-female ratio of death rates at each age converges to a constant over time.

Formally, let m(x, t, i) be the central death rate at age x in year t of sex i. The Li and Lee (Reference Li and Lee2005) model is specified as:

(1)$$\ln \,\;m( {x, \;t, \;i} ) = a( {x, \;i} ) + B( x ) \cdot K( t ) + b( {x, \;i} ) \cdot k( {t, \;i} ) + \varepsilon ( {x, \;t, \;i} ) , \;$$

where a(x, i) is the general mortality schedule over age, K(t) is the common mortality index over time with age-sensitivity B(x), and k(t, i) is the time-varying component of the additional factor with age-sensitivity b(x, i). Note that B(x) and K(t) are common parameters for both sexes being modeled, while the other parameters are specific to either sex. Following Li (Reference Li2013), we assume that the number of deaths follows the (over-dispersed) Poisson distribution and allow for more than one additional sex-specific factor.

2.3 Singapore's mortality improvements over time

Mortality data for the Singaporean resident population are sourced from the Singapore Department of Statistics (SDOS, 2018). Specifically, we use 1980–2017 abridged period life tables which specify sex-specific crude central death rates data in 5-year age groups, i.e., ages 0–4, 5–9, 10–14, and so on. The oldest observed age group is age 85 and over. Total exposure was 1.16 million (1.12 million) for males (females) in 1980 and this increased to 1.94 million and 2.02 million respectively in 2017. This dataset covers the entire resident Singaporean population, including citizens and permanent residents. Official mortality improvement tables are unavailable. To model and construct projected survival probabilities, we first fit the augmented common factor Lee-Carter model to the population data and then employ the estimated parameters from the fitted model to project sex-specific survival probabilities into the future using time-series processes.

Figure 2 shows the estimated parameters of the augmented common factor Lee-Carter model fitted to the 1980–2017 Singaporean mortality data. Importantly, we observe that the common mortality index K(t) exhibits a linearly decreasing trend over the last four decades. K(t) represents the shared period effect for both males and females in the resident population. The linear decline is not an inherent feature of the index, so that the fact that K(t) declines about linearly in Singapore's population for the past four decades is striking. In addition, the index declines at about the same pace during the first half of the period (1980–99) as it does during the second half (2000–17). Overall, this model fitting confirms our earlier observation from Figure 1 that Singapore's historical trends in mortality improvements have been steady and persistent, and also characterizes the nature of longevity risk for this population.

Figure 2. Parameter estimates of the augmented common factor Lee-Carter model fitted to 1980–2017 Singapore mortality data. Source: Authors' own. Notes: For age x in year t of sex i, a(x, i) is the general mortality schedule over age, K(t) is the common mortality index over time with age-sensitivity B(x), and k(t, i) is the time-varying component of the additional factor with age-sensitivity b(x, i). B(x) and K(t) are common parameters for both sexes, whereas the other parameters are specific to either sex; see text.

The linear decline in K(t) and its relatively constant variance are useful for mortality forecasting. We model K(t) by a random walk with drift in order to derive the projected common mortality index up to around 2080; see online Appendix A for further technical details. Results are shown in Figure 3 (left panel).Footnote ⁴ The plot illustrates the past values of K(t) for 1980–2017, along with the forecasts based on the time-series model and the associated 95% confidence interval. Not surprisingly, the variance of K(t) increases with the forecast horizon since there is greater uncertainty in predicting mortality improvements over the longer term. The fitted parameters and forecasts detailed here are subsequently employed to construct forecasts of death rates and survival probabilities required for annuity valuation. The relevant sets of cohort survival probabilities to extract for our analyses depend in part on the product design of the CPF annuities evaluated; an aspect we turn to next.

Figure 3. Projected common mortality index K _t from the Li and Lee (Reference Li and Lee2005) model (left) and its modified version (right). Source: Authors' own. Notes: The dotted lines on both sides of the mean projection values are the 95% prediction intervals.

3.1 Context and overview

The CPF is a fully-funded national DC scheme based on individual accounts. Established in 1955, it is one of the world's oldest DC schemes and has about 4 million members (as of June 2020). Coverage is almost universal since participation is compulsory for the majority of working Singaporeans and permanent residents, as well as self-employed persons who meet an annual income threshold. Monthly contribution rates range from 37% of wages (17% by employers and 20% by employees) for young working adults aged 35 and below, to 12.5% of wages for those aged 65 and over. At age 55, a plan participant's CPF savings – which may sit in differently purposed accounts – are consolidated and placed into a newly-created Retirement Account. As discussed earlier, the CPF Board requires participants to set aside a specific amount of monies (e.g., S$166,000 in 2017) before any excess accumulations in the Retirement Account can be withdrawn as a lump sum at age 55. The minimum retirement sum in the Retirement Account is preserved to age 65 before drawdowns are permitted. One of the most prominent reforms in the CPF system, to date, was switching the default decumulation option for this minimum retirement sum from phased withdrawal to mandatory annuitization.

The CPF longevity insurance scheme is known as the ‘CPF LIFE’. It was introduced on a pilot basis in September 2009. During this pilot phase, annuitization was entirely voluntary and annuitants could choose from four CPF annuities; see details in Fong et al. (Reference Fong, Mitchell and Koh2011).Footnote ⁵ In January 2013, the annuitization scheme was rolled out on a mandatory basis and the product menu was simplified to just two CPF annuities. In 2017, which is the year pertinent to our study, it remained that only two annuities were offered, namely the Standard plan (default) and the Basic plan. Both products, however, had level payout structures. In January 2018, a third plan called the ‘Escalating 2% plan’ was introduced to broaden the choice menu. It features an escalating payout structure to help protect consumers against inflation and was mooted by the CPF Advisory Panel (CPF, 2016).Footnote ⁶

Overall, the Singaporean approach to longevity risk pooling in its retirement system is unusual in some aspects. The CPF annuities are provided by the government, instead of by private insurance entities. In many countries with a history of annuitization in DC schemes, occupational pension annuities are offered by commercial providers. For example, the UK compulsory purchase market was serviced by retail life insurers offering a variety of annuity types (Cannon and Tonks, Reference Cannon and Tonks2010; Reference Cannon and Tonks2016).Footnote ⁷ Additionally, the choice menu is rather limited and it does not include joint-and-survivor annuities. The LIFE scheme is also designed to be non-profit in nature, as explicitly stated on the CPF website: ‘Being non-profit and administered by the CPF Board, the CPF LIFE scheme does not incur distribution costs from agents’ commissions. Costs can also be better spread out with CPF LIFE's large member base… CPF LIFE provides one of the highest payouts for every dollar committed compared to other private annuities… CPF LIFE is the only life annuity backed by the Singapore Government’ (CPF, 2019).

The CPF LIFE premium amounts and monthly payouts are determined by the CPF Board with advice from independent actuarial consultants. The benchmark premium amount is essentially the minimum retirement sum, which is $166,000 in 2017. Participants also have options to invest less than this (if they have insufficient savings) or more than this (if they have excess savings). Accordingly, the amount of monthly annuity payouts primarily depends on how much the participant paid in premium upfront, gender, and plan choice. For illustration, monthly payouts are $1,271 (male) and $1,184 (female) for the Standard plan given a premium of $166,000 in 2017 based on CPF's benefit schedule. Monthly payouts for all CPF annuities are nominal. CPF LIFE payouts may be adjusted over time to account for long-term changes in interest rates or life expectancy. According to the CPF Board, however, such adjustments (if any) are expected to be small and gradual (CPF, 2019). As a statutory board under the Ministry of Manpower, the CPF Board receives subvention from the government for its day-to-day operations including the administration of the LIFE scheme. There is, however, no indication on the CPF website or in any publicly-available materials that the benefits in the LIFE scheme are financed through subventions or state subsidies.

3.2 Key features of CPF annuities

In this subsection, we focus on four design features of the CPF annuities essential to our empirical analyses. The CPF annuities are single-premium, deferred, individual-life annuities with capital protection. First, entry age of the annuity contract. Annuitization age is set at age 55, being the age that CPF members are automatically included in the LIFE scheme if they have at least S$40,000 (US$27,580) in their Retirement Accounts. The average CPF savings across all members per aggregate statistics was approximately S$87,400 in 2016 (CPF, 2017); detailed breakdowns of balances by age groups were not available. Nonetheless, this suggests that the S$40,000 eligibility threshold enables the large majority of CPF members to annuitize, thus ensuring that the national annuity pool is sufficiently large under the annuitization mandate.Footnote ⁸

Second, annuity premiums. These are stipulated by the CPF Board and defined by the prevailing rules governing the LIFE scheme. In 2017, the stipulated premiums (or so-called Retirement Sums) are: S$83,000 (or Basic Retirement Sum), S$166,000 (Full Retirement Sum), and S$249,000 (Enhanced Retirement Sum).Footnote ⁹ We define these amounts, respectively, as low, medium, and high premium in subsequent analyses. Note that the LIFE scheme technically enforces only partial (rather than full) annuitization of retirement balances. If a plan participant has excess savings above the Full Retirement Sum, he or she may freely withdraw these monies instead of choosing to invest the Enhanced Retirement Sum.

Third, the CPF annuities are deferred life annuities. Annuitants are eligible for monthly annuity payouts only starting at age 65 (known as the LIFE payout eligibility age which is CPF-determined). Thus, there is a 10-year deferral (or waiting) period between ages 55 and 65. This deferral period inherent in the CPF products means that the provider, being the government, faces risks from predicting cohort mortality over an extended duration.

Finally, all the CPF annuities are capital protected in that any unused annuity premium (if any) will be paid out as a lump sum to the nominated beneficiaries or insured's estate if the insured dies. This unused annuity premium or bequest is defined as the single premium paid plus any interest earned, less the total amount of monthly payouts already made to date. Thus, the CPF member (and his or her estate) will always recover at least the amount that he or she paid for the annuity, in the form of payouts and/or bequest, regardless what age he or she lives to. Given that the three CPF plans vary in terms of monthly payout (as determined by CPF Board), it implies that bequest amounts will also vary across products since bequest is purely a function of the premium paid and monthly payouts. We show in the next section how each of these design features are factored into the MW valuation framework.

4.1 The MW model

The MWR is based on the calculation of the expected present discounted value (EPDV) of annuity payouts, relative to the premium or purchase price of the annuity (Mitchell et al., Reference Mitchell, Poterba, Warshawsky and Brown1999). For an immediate life annuity, the MWR is calculated according to the following formula:

(2)$$MW\;ratio = \displaystyle{{EPDV} \over {Premium}} = \displaystyle{1 \over {Premium}}\mathop \sum \limits_{t = 1}^T {}_t^{} p_{\rm a}\cdot {\rm \;}v^tA_t, \;$$

where a is the age at which the annuity is purchased (annuitization age), t is the number of months from annuitization age, ${}_t^{} p_a$ is the probability that an individual of age a survives after t months, $v_{}^t$ is the nominal discount factor at month t based on a term structure of interest rates, A _t is the annuity payout in month t, and T is the maximum age as per the survival tables. Annuitant survival tables are typically used in the context of voluntary annuity markets as those who elect to purchase a payout annuity tend to live longer than those who do not. In the context of compulsory annuity markets, population survival tables are used instead (Bütler and Ruesch, Reference Bütler and Ruesch2008; Cannon and Tonks, Reference Cannon and Tonks2010).

In their earlier study, Fong et al. (Reference Fong, Mitchell and Koh2011) modified the MW model to take into account the various design features of the CPF annuities, including the deferral period and capital protection, and showed that the MWR for these products can be represented as follows:

(3)$$\eqalign{& MWR = \displaystyle{1 \over {Premium}}\left[{\mathop \sum \limits_{t = D}^\infty {\rm \;}_tp_a\cdot {\rm \;}v^tA_t + \mathop \sum \limits_{t = 0}^\infty {\rm \;}_tp_a\cdot q_{a + t}\cdot v^{t + 1}\;B_{t + 1}} \right]. \cr & } $$

The additional terms in equation (3) are defined as follows. D is the deferred period in months (which is set at 120 months), $B_{t + 1{\rm \;}}^{}$ is the bequest/death benefit at time t + 1, and ${}_t^{} p_a\cdot q_{a + t}^{}$ is the probability of an annuitant age a surviving to t months and then dying between month t and month t  +  1. The first summation in the brackets captures the annuity benefits received over the lifetime of the insured if he or she lives to the point when payouts start, whereas the second summation reflects the lump-sum bequest paid out to the beneficiaries upon the insured's death.

Using 2009 data, Fong et al. (Reference Fong, Mitchell and Koh2011) found that the then newly-launched CPF annuities offered excellent value-for-money to the general population, with MWRs of 1.24–1.31 (1.26–1.34) for males (females) across four products. The interpretation of MWR exceeding unity is not straightforward. Even in well-functioning markets, MWR values equal or greater than unity are implausible in the long term because insurers have to factor in administrative loads, profit margins, and adverse selection (Poterba and Warshawsky, Reference Poterba, Warshawsky and Shoven2000). Consequently, in many countries, the expected present value of annuity payouts is only 80–90% of the premium cost (see, e.g., Mitchell et al., Reference Mitchell, Poterba, Warshawsky and Brown1999; Brown et al., Reference Brown, Mitchell, Poterba and Warshawsky2001; Cannon and Tonks, Reference Cannon and Tonks2004). Fong et al. (Reference Fong, Mitchell and Koh2011) thus posited that the high estimated annuity values in Singapore in 2009 were due to policy aimed at creating public buy-in for the new scheme, and it was uncertain whether the Singapore government could continue subsidizing payouts in the longer term. However, because the authors had used a deterministic projection of mortality, it is unclear what is the distribution of MW values for a given product across a range of population mortality scenarios. For instance, it could be possible that in low mortality improvement scenarios, the estimated MWRs lie below unity.

To provide a fuller picture, we adopt the Fong et al. (Reference Fong, Mitchell and Koh2011) method in valuing the CPF annuities but embed a stochastic mortality model within the MW valuation framework. This applied to 2017 data. In what follows, we describe the various data inputs required.

4.2 Annuity premiums and payouts

There were technically only two CPF annuity products offered in 2017, namely the Standard plan and the Basic plan. The third product, which is the Escalating 2% plan, was introduced in January 2018. Nonetheless, since data are available, we incorporate it in this present study to allow for richer analysis. The 2017 annuity payout and bequest schedules for all products are obtained from CPF (2018). Table 1 displays the monthly annuity payouts by sex, premium invested, and product type. As discussed earlier, the premium amounts for an age 55 CPF annuitant are: S$83,000 (low), $166,000 (medium), and $249,000 (high). For a $83,000 premium, for example, initial monthly annuity payouts at age 65 range from $500 to $646 for females and from $548 to $692 for males. Not unlike commercial annuities, monthly payouts for females are systematically lower because of their longer expected lifespans as compared to males.

Table 1. Monthly nominal payouts for CPF annuities by sex, premium, and product type (2017; payouts starting at age 65)

Source: Authors' own.

Notes: The three illustrative single-premium amounts correspond to the amounts stipulated in the CPF Retirement Sum scheme for an age 55 annuitant in 2017, namely S$83,000 (low), S$166,000 (medium), and S$249,000 (high). Members are assigned to the Standard plan by default if they do not make a choice. Monthly payouts are in nominal terms. Payouts for the Standard and Basic plans are level across time. Only the first-year payouts are shown for the Escalating plan, which offers 2% increase in payouts every year.

The default/Standard plan provides a higher level of monthly payout and leaves a lower bequest. The Basic plan provides a lower payout in exchange for a higher bequest. Figure 4 illustrates this trade-off in payouts and bequest across products. Solid lines depict the bequest amounts (primary vertical axis), whereas the dotted lines depict the monthly payouts (secondary vertical axis). For an illustrative S$83,000 premium, the graph shows that a male annuitant who survives to age 65 can start receiving level monthly payouts of $634 under the Basic plan or $692 under the Standard plan. Bequest amounts associated with the Standard plan are lower across all age values and diminish to zero by around age 80. In comparison, the Basic plan allows more to be left to beneficiaries and provides a positive death benefit even past age 90. The Escalating 2% plan is a back-loaded annuity: its starting payout is about 20.8% lower than that of the default/Standard plan but annuitants can look forward to a 2% fixed annual increase in payout thereafter.

Figure 4. Bequest amounts and monthly nominal payouts by plan type (male annuitant; illustrative single-premium of S$83,000 paid at age 55 in 2017). Source: Authors' computations from CPF (2018). Notes: The payout and bequest amounts are for a male annuitant who paid a lump-sum premium of S$83,000 at age 55 in 2017. We assume that the annuitant starts his payout at the earliest eligibility age of 65. Between ages 55 and 65, the premium set aside enjoys guaranteed returns of 4–5% per annum and thus the amount of bequest available at age 65 (approximately S$120,000) is larger than the initial premium invested (S$83,000). The solid lines show the bequest amounts (on the primary vertical axis), whereas the dotted lines show the monthly annuity payouts (on the secondary vertical axis).

4.3 Term structure of interest rates

The MW calculation requires a term structure of interest rates. Following Fong et al. (Reference Fong, Mitchell and Koh2011), we use the Singaporean Treasury bond rates to construct a riskless term structure of interest rates. A riskless term structure of interest rates is deemed more suitable than a corporate term structure of interest rates in our context since the CPF annuities are supplied by the government through the CPF Board. The Singapore government has a triple-A credit rating from international credit rating agencies. Also, the CPF annuities are capital protected.

Prices and yields of Singapore Government Securities issued 2017 with varying maturities are obtained from MAS (2018), including the 1-year Treasury bill and the 2-, 5-, 7-, 10-, 15-, 20-, and 30-year Treasury bonds. These are used to compute the riskless spot rates to proxy the yields on hypothetical zero-coupon bonds. We then linearly interpolate between intervals where spot rates are unavailable, for instance, between the 7- and 10-year spot rates, to derive the full term structure of interest rates. Our spot rates range from 1.40 to 3.71% per annum. Since the maximum duration available for the Singapore Government Securities is 30 years, we extrapolate the last spot rate for longer maturities. The long-term extrapolated rate of 3.71% used in our study is broadly consistent with the 3.44% reported in Fong et al. (Reference Fong, Mitchell and Koh2011).

4.4 Stochastic cohort survival probabilities

Our base stochastic mortality model is the augmented common factor Lee-Carter model based on Li and Lee (Reference Li and Lee2005) and Li (Reference Li2013) as outlined in Section 2. Population survival tables are used since the CPF annuitization scheme functions as a compulsory annuity market. The fitted parameters and forecasts from the base model are used to construct forecasts of death rates and survival probabilities for a male who is age 55 in 2017. Similar forecasts of death rates and future survival probabilities are derived for a female age 55 in 2017. Consistent with Fong et al. (Reference Fong, Mitchell and Koh2011), the terminal age for CPF annuitants is set to age 117. The set of cohort survival probabilities extracted from these mortality forecasts is called the ‘central projection’ and characterizes the future mortality experience of a typical age-55 annuitant in 2017. We also conduct Monte Carlo simulations to generate future mortality scenarios. This yields 5,000 sets of random sex-specific cohort survival probabilities, representing the uncertainty in predicting cohort mortality for male and female Singaporean annuitants over a long duration.Footnote ¹⁰

Figure 5 shows the survival fan chart for an age 55 female CPF annuitant in 2017 based on the 5,000 stochastic survival paths. Because the whole Singapore population is used for analysis, the modeled variability reflected in the fan chart is small. The chart also usefully demonstrates that uncertainty about death rates generally increases with the duration of the annuity contract, consistent with past studies (e.g., Blake et al., Reference Blake, Dowd and Cairns2008). In the first few years after annuity purchase, the probability of death is relatively low so there is little uncertainty. In later years and as the timespan lengthens, however, this uncertainty increases, especially between ages 75 and 90. This has implications from the supply-side perspective. In particular, the back-loaded Escalating annuity product presents a riskier liability for the provider than the other two level-payout annuities since a greater proportion of its present value will be paid over the period of greater uncertainty.

Figure 5. Fan chart of cohort survival probabilities for a female annuitant aged 55 in 2017. Source: Authors' own. Notes: This fan chart illustrates the uncertainties surrounding the projections of survival probabilities for a female CPF annuitant aged 55 in 2017. Sex-specific future cohort survival probabilities are generated from 5,000 simulated scenarios using the augmented common factor Lee-Carter model. The central heavy black line shows the median survival probabilities, whereas the two solid red lines on either side of the median show the 75th and 25th percentiles. The outer dotted lines show the 95th and 5th percentiles.

In what follows, we use the set of central mortality projections from the base mortality model to compute MW values for CPF annuitants. The 5,000 stochastic survival paths are also deployed to estimate the range of MWRs for a range of high and low population mortality improvements. Importantly, we then test the sensitivity of the MW results to the choice of mortality model. In robustness checks, we will separately implement the original Lee and Carter (Reference Lee and Carter1992) model and the product–ratio model proposed by Hyndman et al. (Reference Hyndman, Booth and Yasmeen2013) to derive alternative MW estimates and compare these against those from the base model.