Assessment of longevity risk: credibility approach

Abstract

To correctly measure the effect of mortality rates on the stability of insurance and pension provider's financial risk, longevity risk should be considered. This paper aims to investigate the future mortality and longevity risk with different age structures for different countries. Lee–Carter mortality model is used on the historical census data to forecast future mortality rates. Turkey, Germany, and Japan are chosen concerning their expected life and population distributions. Then, the longevity risk on a hypothetical portfolio is assessed based on static and dynamic mortality table approaches. To determine the impact of longevity risk, which is retrieved using a stochastic mortality model, a pension insurance product is taken into account. The net single premium for an annuity is quantified under the proposed set up for the selected countries. Additionally, the credibility approach is proposed to establish a reliable estimate for the annuity net single premium.

1. Introduction

Prediction of future life expectancy is an essential issue for life insurance, pension fund, and related areas, as especially underestimation of the liabilities may cause catastrophic financial losses. An expected remaining lifetime based on the probability of survival constitutes the vital input in pricing life insurance and pension products. The risk of people living longer than expected is called longevity risk, and it influences the risk and premium of life and pension products. The key driver of longevity risk is mortality rates.

For some Organization for Economic Cooperation and Development (OECD) countries changes in the life expectancy at age x ( $e_x$) can be observed from Figure 1 [20], which expose the increase in aging between 1960 (bottom line) and 2020 (top line) for every decade, for a person whose age is 0 and 65. Life expectancy over the last five decades improved significantly, and globalization of life expectancy plays an important role. Rectangularization of the survival function is due to the demographic transition from high to low mortality trends. Reduction in mortality rates and increasing life expectancy stems from more nutritious foods, clean water supply, better housing conditions, access to primary and medical care, vaccinations, etc. Over the years, causes of death are changed from more infectious diseases to degenerative and chronic diseases. Besides those, socio-economic class significantly affects the mortality trends. When examining mortality at different periods, it can be seen in Figure 1 that every country has a different mortality trend and life expectancy. It can be noticed that the average increase in $e_0$ in the last 57 years is around 12.7 years, whereas $e_{65}$ yields an average increase of 8.7 years. Additionally, the country-specific history also has influence on the lifetimes, such as Korean war in late 1950s, economic boom in Turkey after 1980s. Under declining mortality at old ages, graduation and revision of life tables are vital, and studies show that a year increase in life expectancy at age 65 results in at least a 3% increase in the present value of UK pension liabilities [5].

Figure 1.

Expected lifetime ( $e_x$) at age 0 (top) and 65 (bottom), for selected OECD Countries.

To identify if the state of the economy has relation to longevity risk, we consider a developing and two developed countries. Turkey, as a developing country, has an age structure that is different and remarkable in the young age population compared to most of the developed countries, such as Germany, USA, and Canada, whose economic welfare is significantly better than Turkey.

Turkey has a young population compared to Germany and Japan, median ages are 31.4, 47.4, and 47.7, respectively [28]. However, projections indicate that the fertility rate for Turkey in 1980 has a decreasing trend from 5.39 to 2.12 in 2010 and is expected gradually decrease to 1.75 in 2070 [19]. Moreover, the annual support ratio which is the major indicator in sustainability of a pension system is expected to decline rapidly to the level of countries like Germany and Japan, as can be seen in Figure 2. The historical (1960–2020) and projected (2021–2100) old age support ratios for the selected OECD countries show a rapid decrease for Germany and Japan, and a relatively slower rate of decline for Turkey. Due to less frequent census studies (in 5 year intervals), there are not many Turkey specific mortality and longevity studies in the literature.

Figure 2.

Historical and projected old age support ratios for 1950–2100.

Germany and Japan have a higher life expectancy than Turkey, and a rapid decrease in Turkish mortality rates point to a future high life expectancy. Turkey appears to have an over 61% increase in life expectancy at birth, whereas Germany and Japan have 17% and 24% increases. Similarly, it can be seen in Figure 1 that Germany and Japan have a life expectancy at age 65 with a rise of 9.1 and 12.8 years, respectively, and life expectancy at old ages rises almost every year and for every country.

With the unexpected decrease in death rates, variability in the age of death declined, and deaths are concentrated to the older years of life, therefore, causing deficits in pension funds, which are built upon the underestimated rates. Having a young population with decreasing fertility rates is not promising for pension fund valuations because the pension scheme structure leads to a growing deficit in pension funds. Despite the radical revisions on pension schemes, due to the change in mortality rates and its depending indicators, Turkey is expected to prone to longevity risk. Therefore, the determination of demographic projections along with recent reforms requires precaution on the impact of longevity risk on mortality projections in Turkey.

In literature, an extensive amount of studies exist on the topic of mortality forecasting. For instance, Booth and Tickle [7] classified the mortality forecasting into three approaches: Expectation, explanation, and extrapolation. Among extrapolative methodologies, Lee–Carter [17] produces successful results and is implemented mortality studies. The performance of the Lee–Carter model for mortality forecasting evaluated using actual and hypothetical forecasts is done by Lee and Miller [18]. Their findings on several data sets over long periods suggested the predictions be reliable. Haberman and Renshaw [12] examine the Lee–Carter method in the frame of age-period-cohort effects on life expectancy and annuity values. The choice of AR-ARCH model in forecasting the mortality using Lee–Carter studied by Giacometti et al. [11] leading to the improvement in forecast ability. Many developments are made based on the Lee–Carter model. Brouhns et al. [8] modify the model by using Poisson distribution, and Cossette et al. [10] offer a binomial process to Lee–Carter and Koissi and Shapiro [16] proposes fuzzy model.

In recent years recognizable studies on the methodology to quantify the longevity risk have been done. In order to assess the longevity risk, three methodologies have been classified: discounted present value of liabilities [21], funding ratio volatility [23], and probability of ruin [22]. Stevens et al. [26] present the longevity risk in terms of the discounted present value of liabilities, funding ratio volatility, and the probability of ruin. Olivieri and Pittacco [24] analyzes the financial impact of longevity risk on life annuities and long term care products, solvency, and reinsurance aspects.

The possible effect of longevity on pension funds has also been investigated quite often in the recent past. Antolin [2] examines the financial results of longevity on private pension plans. Hardy and Panjer [14] used Bühn-Straub credibility to find an estimate of the mortality loss ratio for a company, relative to a standard table. Bisetti and Favero [6] shows that pension systems in which the retirement age is not indexed to expected life have a very large exposure to longevity risk by using Lee–Carter model and cohort approach.

In order to reduce the exposure to longevity risk, Baione et al. [3] suggest reinsurance strategies based on risk criteria. Tontine annuities have recently been used as a form of longevity risk management in aging populations by Baker and Siegelman [4] and Weinert and Gründl [29]. A risk-sharing scheme is proposed by Hanbali et al. [13] between the insurer and insured via a dynamic equivalence principle.

As a consequence, in the light of past mortality trends and its financial impact, to protect the balance of a pension fund, it is vital predicting the future mortality trend that represents the real population. The main purpose of this study is to assess the longevity risk based on different methods and determine its influence in pension valuation. Focusing on Turkey, as a young aged population and developing country, we aim to depict the impact of mortality changes on pension schemes in the characteristics of developing and developed systems as well as young and elderly populations. For this reason, Japan and Germany are selected to be a comparative scale for detecting the longevity risk in the Turkish pension system. Additionally, the implementation of credibility approach to determine the risk premium under longevity risk for the selected countries is performed.

The remainder of this paper is structured as follows: After a brief examination of Turkish, Germany, and Japan Pension Schemes is given in Section 2, stochastic mortality and annuity premiums are introduced in Section 3. Longevity risk and its components for Turkey, Germany, and Japan, projected mortality tables based on the Lee–Carter model are developed in Section 4. In Section 5, the credibility approach is introduced to find a reliable estimate of the annuity premium with a different perspective. The impact of longevity risk, specifically, on life annuities under different scenarios for certain retirement ages, are evaluated and compared using a hypothetical portfolio approach in Section 6. Finally, Section 7 closes with comments and conclusions.

2. Pension systems

Pension plans are arranged into two basic schemes: defined contribution (DC), defined benefit (DB) plans, and Hybrid plans, which are composed of DC and DB schemes. In defined contribution plans, each participant pays a fixed ratio or fixed amount of their monthly earnings to the fund. That amount is invested in assets during the accrual period. When the participant earns the right to be a retiree, the accumulated amount can be withdrawn by an annuity or in another way. The individual controls the fund both in the working period (active period) and in the retirement period (passive period), so the participant takes the entire investment risk. In defined benefit plans, the benefit that the participant receives depends on the length of service or a specific ratio of compensation or a specified amount at retirement. The plan sponsor contributes to the plan to fund the defined benefit at retirement. Table 1 summarizes basic pension system characteristics of the countries [20]. These are the scheme type, retirement age (early age; eligible age for retirement with pension penalty, and normal age; eligible age for retirement with full pension benefits), ratio of pension wealth (PW) to annual individual earnings (AIE).

Table 1.

Country pension system characteristics of Turkey, Germany, and Japan.

	Scheme	Early age	Normal age	PW/AIE ${}^{\mathrm{a}}$
Turkey	DB	–	Men 60	Men 13.4
			Women 58	Women 15
Germany	Points	–	65	Men 8.3
				Women 9
Japan	Basic/DB	60	65	Men 6.6
				Women 7.9

${}^{\mathrm{a}}$PW/AIE: pension wealth/annual individual earnings.

The Turkish pension system is based on an earnings-related public pension scheme with average annual worker earnings of 10,438 USD (OECD average is 36,622 USD) [20]. The current pension eligibility age is 60 years for men and 58 for women with the condition that at least 7200 days of contribution. The pension eligibility age will gradually rise to 65 for both men and women until 2044. Pension benefits are paid quarterly. Pension income is not taxed and is not subject to social security contributions.

Turkey experiences an increasing deficit in the last decades because of social security expenses [25]. In light of this information, together with increasing life expectancy, decreasing fertility rate, and the support ratio, the funds are expected to be insufficient to cover the liabilities in pension systems.

The social security system in Turkey offers universal coverage to workers employed by public and private sectors since its establishment in 1945. It is based on a defined benefit model and financed on a PAYG basis. The private pension system is a voluntary, defined contribution system intended to be a complementary scheme to the mandatory social security scheme. As of October 2001, Individual Pension System (BES) offers a complimentary plan to the existing social security system. A decree-law by 1 January 2013 is put in action to encourage participation in individual pension savings in Turkey. The reform has removed tax incentives and proposed government contribution as 25% of annual contribution with a limit of 25% of the yearly minimum wage. The primary purpose of the new system is, in the short term, to decrease the savings gap and, in the long run, to cover the liabilities of the public pension system. The main reforms include a gradual increase in the retirement age, which is set to 58 for women and 60 for men till 2035 and will be raised gradually afterward to age 65 for both genders. Here, we should state that total investments as a percentage of GDP are very low for Turkey, compared to Germany and Japan.

The public pension system in Japan has two tiers: a basic flat-rate scheme and an earnings-related plan. Average annual worker earnings is 43,692 USD [20]. Pension eligibility age is 65 years, with a minimum of 10 years of contribution. A full basic pension requires 40 years of contribution, and benefits are adjusted proportionally for shorter or longer periods. There are no special rules for the taxation of pension income. Contributions to health insurance are levied on the pension scheme.

Germany's public pension system is statutory and is an earnings-related PAYG system. Average annual worker earnings is 50,307 USD [20]. The old-age pension is payable from age 65 with a minimum of 5 years of contribution. In 2016, 72% of the pension was taxable. From 2020 to 2040, the taxable part of the pension will increase by one percentage point per year. Pensioners pay social security contributions in pension income to the health care system, which is 8.4% in 2016.

3. Stochastic mortality and life annuities

There are many methodologies on the theoretical background of quantifying the mortality rates, based on the historical changes. Extrapolation is the most commonly used technique. It assumes that future mortality trends will follow past trends; hence, it uses techniques from time series analysis to capture stochasticity. Indeed, historical population data shows that mortality has a stable trend throughout the years. In one-factor models, parameterization functions which are known as laws of mortality, such as De Moivre, Gompertz, Makeham, and Heligman-Pollard are utilized.

As a well-known and commonly referred two-factor mortality forecasting model Lee–Carter [17] suggests a simple log-bilinear model in the variables x (age) and t (calendar year) for forecasting the logarithm of $m_{x,t}$ (force of mortality) and is defined as

$$\ln (m_{x,t})={\alpha }_x+{\beta }_x{k}_t+{\epsilon }_{x,t},$$ (1)

where ${\alpha }_x$ is the average age-specific pattern of mortality, ${\beta }_x$ describes the pattern of deviations from the age profiles as time index $k_t$ varies and the error term ${\epsilon }_{x,t}$ expresses the deviation of the model from observed log-central death rates and it is assumed to be normally distributed with zero mean and constant variance. The ${\alpha }_x$ and ${\beta }_x$ are time-independent variables, $k_t$ is time-dependent index of the level of mortality. For a unique solution of Equation (1), Lee–Carter assumes the following constraints:

$$\sum _x{\beta }_x=1\mathrm{and}\sum _t{k}_t=0.$$ (2)

Lee–Carter estimates the parameters of the model by a series of steps. First, we start with the estimation of ${\alpha }_x$

$${\alpha }_x=\frac{1}{T}\sum _t\ln (m_{x,t}).$$ (3)

Then, by Singular Value Decomposition (SVD) which provides a convenient way for breaking a matrix into simpler meaningful partitions, a least square solution to $[\ln (m_{x,t})-{\alpha }_x]$, the parameters ${\beta }_x$ and $k_t$ are estimated. However, the estimated parameter, ${\hat{k}}_t$, does not guarantee to capture the pattern between the observed and fitted number of deaths since the SVD parameter is designed for minimizing the error of the logarithm of the death rate, not the actual death rate. Therefore, the $k_t$ parameter is re-estimated to ensure equity. Then, the second stage estimation proceeds as follows:

$$D_t=\sum _x[\exp ({\hat{a}}_x+{\hat{\beta }}_x{\tilde{k}}_t)N_{x,t}],$$ (4)

where $N_{x,t}$ is the population of age x in year t, $D_t$ is the total number of deaths in year t and ${\tilde{k}}_t$ is the re-estimated parameter. After that the second stage estimation in order to predict the future mortality rate the ${\tilde{k}}_t$ are extrapolated by using Box–Jenkins method

$${\tilde{k}}_t=c+k_{t-1}+{\delta }_t$$ (5)

which is modeled as Autoregressive Integrated Moving Average Model $(p,d,q)$ (mostly ARIMA(1,1,0)) to project $k_t$ parameter. Here, c denotes the drift term and ${\delta }_t$ denotes the error term that follows a normal distribution with zero mean and constant variance, ${\sigma }^2$.

A standard financial instrument in both the private and public sectors is a life annuity, which consists of a series of payments that are made while the annuitant (of initial age x) lives. Hence, the uncertainty on the expected value of payments is originated from the remaining lifetime of the annuitant. To ensure the future payments insured needs to make contributions (net premium) to the fund. The amount of the payment stream is pre-determined, and the expected present value of payments yields the value of the life annuity. Pension plans periodically evaluate their liabilities, gains, and reserves according to the value of the life annuity, which depends on the mortality table and the interest rate.

Let K be the remaining curtate lifetime of the annuitant and let Y denote the present value of the regular one unit payment stream with a discount factor ν shown as

$$Y={\ddot{a}}_{K+1}=\frac{1-{\nu }^{K+1}}{d},K=0,1,\dots ,$$ (6)

where $d=\mathrm{i}\nu$. The net single premium (expected present value, $\mathbb{E}(Y)$) of a life annuity, ${\ddot{a}}_x$, becomes

$${\ddot{a}}_x=\sum _{k=0}^{\mathrm{\infty }}{\ddot{a}}_{k+1}{}_k{p}_x{q}_{x+k}.$$ (7)

The variance of Y identifies the size of the risk which is expressed as

$$\mathrm{Var}(Y)=\mathrm{Var}\left(\frac{1-Z}{d}\right)=\frac{\mathrm{Var}(Z)}{d^2}.$$ (8)

It can be seen that the variance is increased with respect to the discount factor of the future present value of the remaining lifetime for years $K=0,1,\dots$. Here, ν is assumed to be constant, but it can be taken as stochastic.

4. Quantification of longevity risk

An expected lifetime based on a fixed calendar year, $e_x$, named here as a ‘static’ life expectancy, does not include the cohort effect of time. Considering the concept of life annuities, the net single premium depends on the expected remaining lifetime of the insured. However, the net single premium is evaluated based on the time when the annuity contract starts. If the contract is evaluated using the life table that belongs to the current year, the value of the contract would be underestimated since the mortality improvement is only based on age. This valuation ignores mortality development over the years. For this reason, a ‘dynamic’ life expectancy approach is defined. The effect of changing mortality rates depending on the time aspect can be determined by including the cohort probabilities of living.

Survival probability of a person aged x for τ years, is given by ${}_{\tau }{p}_{x,t}$,

$${}_{\tau }{p}_{x,t}=p_{x,t}\times p_{x+1,t+1}\times \cdots \times p_{x+\tau -1,t+\tau -1},$$ (9)

where x denotes the age of the participant and t is the base year. Based on this, dynamic expected lifetime, $e_{x,t}$ can be evaluated as

$$e_{x,t}=\sum _{\tau =1}^{\mathrm{\infty }}{}_{\tau }{p}_{x,t}.$$ (10)

The performance of a pension portfolio and the risk taken by the insurer is measured by the variance of the annuity, which is a significant risk indicator for reserve valuation. Assuming that the future lifetime of each one of N annuitant in a portfolio, denoted by $Y_i$, $i=1,\dots ,N$ , are independent and identically distributed, the aggregate liability, Υ, expected value, $\mathbb{E}(\mathrm{{\rm Y}})$, and its variance, $\mathrm{Var}(\mathrm{{\rm Y}})$, with given order are as follows:

$$\begin{array}{rl}\mathrm{{\rm Y}}& =\sum _{i=1}^N{Y}_i,\end{array}$$ (11)

$$\begin{array}{rl}\mathbb{E}(\mathrm{{\rm Y}})& =N\cdot \mathbb{E}(Y),\end{array}$$ (12)

$$\begin{array}{rl}\mathrm{Var}(\mathrm{{\rm Y}})& =N\cdot \mathrm{Var}(Y).\end{array}$$ (13)

$\mathrm{Var}(\mathrm{{\rm Y}})$ in Equation (13) is hereby named as the variance in ‘deterministic approach’. However, the uncertainty of future mortality needs to be adapted to the variance of the portfolio. For this purpose, we consider the distributional assumption, whose variance is called as ‘stochastic approach’. Let $F_t=\{q_{x,t+\tau }|\tau \ge 1\}$ be the distribution function of the projected probabilities of death. Then, the expected value and variance in the ‘stochastic approach’ become

$$\begin{array}{rl}\mathbb{E}(\mathrm{{\rm Y}})& =\mathbb{E}[\mathbb{E}(\mathrm{{\rm Y}}|F_t)]\\ & =N\cdot \mathbb{E}[\mathbb{E}(Y|F_t)]\\ & =N\cdot \mathbb{E}(Y),\end{array}$$ (14)

$$\begin{array}{rl}\mathrm{Var}(\mathrm{{\rm Y}})& =\mathbb{E}[\mathrm{Var}(\mathrm{{\rm Y}}|F_t)]+\mathrm{Var}[\mathbb{E}(\mathrm{{\rm Y}}|F_t)]\\ & =N\cdot \mathbb{E}[\mathrm{Var}(Y|F_t)]+N^2\cdot \mathrm{Var}[\mathbb{E}(Y|F_t)],\end{array}$$ (15)

respectively. The first term on the right-hand side of variance in Equation (15), as in the deterministic approach, measures the random fluctuations around the expected values. The second term demonstrates the systematic deviations in projected values, which is corresponding to the longevity risk [21]. For the fair pricing of the portfolio, pooling is implemented to the model which results in

$$\begin{array}{rl}\mathrm{Var}\left(\frac{\mathrm{{\rm Y}}}{N}\right)& =\mathrm{Var}\left(\frac{1}{N}\sum _{i=1}^N{Y}_i\right)\\ & =\mathbb{E}\left[\mathrm{Var}\left(\frac{1}{N}\sum _{i=1}^N{Y}_i|F_t\right)\right]+\mathrm{Var}\left[\mathbb{E}\left(\frac{1}{N}\sum _{i=1}^N{Y}_i|F_t\right)\right]\\ & =\frac{\mathbb{E}[\mathrm{Var}(Y|F_t)]}{N}+\mathrm{Var}[\mathbb{E}(Y|F_t)].\end{array}$$ (16)

One can see that the first term on the right-hand side of Equation (16) is sensitive to the portfolio size N. As the number of policies in the portfolio increases, the risk resulting from random fluctuations around the expected value will diminish due to the pooling effect. On the contrary, the second term quantifying the longevity risk is numb to N. On the other hand, the longevity risk can also be captured by the coefficient of variation $c=\sqrt{\mathrm{Var}(\mathrm{{\rm Y}})}/\mathbb{E}(\mathrm{{\rm Y}})$, which enables us to define the partition of variance with respect to its components given as

$$c={(\frac{1}{N}\frac{\mathbb{E}[\mathrm{Var}(Y|F_t)]}{\mathbb{E}(Y{)}^2}+\frac{\mathrm{Var}[\mathbb{E}(Y|F_t)]}{\mathbb{E}(Y{)}^2})}^{1/2}.$$ (17)

Here, the longevity risk on the second term of Equation (17) remains persistent even in the case of the increasing size of the portfolio.

5. Credibility analysis

Credibility theory is widely used in actuarial science for pricing insurance contracts. It provides a means for the use of group-specific and portfolio specific experience together. While historically, the credibility method is mostly used in nonlife actuarial data, in 2013 scope of Actuarial Standard of Practice No. 25 is expanded to include life, health, pension, and property/casualty practice areas [1].

We propose the credibility analysis to find an estimate for the life annuity premium by a weighted average of the deterministic mortality rates and the dynamic approach. Bühlmann credibility is adopted to recognize the interaction of two sources of variability in the data. These are (i) the diversifiable risk and (ii) the longevity risk. Therefore, Bühlmann credibility parameter, denoted by k, which is the expected value of the process variance divided by the variance of hypothetical mean becomes

$$k=\frac{\mathbb{E}(\mathrm{Var}(Y\mid F_t))}{\mathrm{Var}(\mathbb{E}(Y\mid F_t))}.$$ (18)

Correspondingly, Bühlmann credibility factor, Z, is defined as

$$Z=\frac{m}{m+k},Z\in [0,1],$$ (19)

where m is the number of years of data collected to calculate the future mortality rates and estimates the stochastic annuity premium value. As m increases, the value of Z converges to 1, and the higher values of Z provide more confidence to the projected mortality data. We define the risk premium, ${\mu }_x$, as

$${\mu }_x=Z{\ddot{a}}_{x,t}+(1-Z){\ddot{a}}_x,$$ (20)

where ${\ddot{a}}_x$ denotes the static annuity premium and ${\ddot{a}}_{x,t}$ the stochastic annuity premium value. The ${\mu }_x$ defined here contributes to the influence of both deterministic and stochastic risk premiums in corporation with the k, which has longevity component as proposed in this paper.

6. Implementation

Turkish mortality studies are scarce because of insufficient history on mortality and population census data. After the implementation of Address Based Population Registration System (ABPRS) in 2007, the population dynamics are available but not long enough to construct a reliable mortality table. In this study, regression levels from the Construction of Turkish Life and Annuity Tables Project [15] are used as the primary input between the years 1931 and 2015. The project assumes that Turkish Life Tables and the Cole-Demeny West Model Life Tables [9] are alike. The data used in the project is derived initially from the Turkish Population Census Data. Using these regression levels, we regenerate Turkish mortality life tables over the years 1931 to 2015. The data for Germany between 1990 and 2017 and Japan between 1947 and 2016 were obtained from the Human Mortality Database [27]. Even though the time periods for each country do not coincide, the projections done based on these data sets are expected to reflect country-specific behavior. For this reason, due to the availability of the data, we remain on the maximum available time periods to set up the models.

We use mortality data for each country to illustrate the proposed approach in understanding the influence of population characteristics on risk premium. Based on these data sets, Lee–Carter parameters are estimated, which enables us to forecast mortality rates for the years between 2020 and 2075. The expected lifetimes are calculated based on the Turkish pre- and post-reform retirement age limits (45, 58, and future targeted retirement age 65). Figure 3 illustrates the forecasted life expectancy values for Turkey at birth and different retirement ages. According to Lee–Carter forecast life expectancy at birth continues to increase and reaches 75.03 years for males and 82.86 years for females, respectively, in 2075. Moreover, we see that life expectancy for females inclines faster than males. It is noteworthy to see that the pattern for females deviates dramatically for males as life expectancy increases.

Figure 3.

Expected lifetime forecasts for age 0, 45, 58, and 65 for Turkey, based on Lee–Carter forecast.

Considering the demographic properties of Turkey, such as decreasing mortality, increasing life expectancy, and lowering fertility rates, two countries (Japan and Germany) whose mortality characteristics are different from Turkey, are studied. To emphasize the effect of changing mortality rates on the remaining lifetime, static and dynamic approaches are used to compute the life expectancy on forecasted mortality tables from 2020 to 2075. Table 2 illustrates the life expectancy at different retirement ages. The difference between dynamic and static approaches for both genders is found to be significant. It shows that the increase from static to dynamic life expectancy in Germany data is higher for males of all ages. On the contrary, the increase is higher for female life expectancy at all ages for the Turkish case. Meanwhile, for Japan, the increase is higher for males and at birth, but for the other ages, it is higher for females. It should also be noted that the base year is taken as 2020.

Table 2.

Life expectancy with static ( $e_x$) and dynamic ( $e_{x,t}$) approaches ( $t_0=2020$).

		Female		Male
	Age	$e_x$	$e_{x,t}$	$e_x$	$e_{x,t}$
Turkey	0	76.5861	81.6651	68.0238	72.0361
	45	33.4361	35.9256	28.3884	29.6252
	58	21.4831	22.8414	17.4984	18.0506
	65	15.59646	16.4359	12.5338	12.8435
Germany	0	83.1723	88.4863	78.8876	87.1671
	45	39.0066	42.0445	35.0943	39.3679
	58	26.9741	28.8453	23.6344	26.0856
	65	20.8677	22.1449	18.0173	19.6022
Japan	0	85.2240	93.8547	78.6202	88.0138
	45	41.1392	46.7913	35.2304	40.1628
	58	28.8888	32.4884	23.6038	26.4058
	65	22.5417	25.0679	17.8942	19.7111

To determine the longevity risk, different mortality reduction scenarios on mortality projections are proposed as: Deterministic (Case I) and stochastic (Case II). The Case I considers the mortality table with base year 2020 and calculates annuity premium and variance components according to the base year. Case II takes into account 95% confidence interval bounds. These intervals are constructed using Monte Carlo (MC) simulations, generating the random error terms that are the result of repeated estimation of the Lee–Carter model. The upper limit of the confidence interval named as maximum scenario represents the scenario of maximum reduction in mortality, and the lower bound named as minimum scenario represents the scenario of minimum reduction in mortality. Based on the projected Turkish, Germany, and Japan mortality tables, three version of mortality reduction; low (min), medium (med), and high (max) risk scenarios are constructed using 95% confidence interval [21].

The present value of the life annuity for a hypothetical portfolio for different retirement ages with a constant interest rate of 5% is calculated concerning these scenarios. In Table 3, the expected life annuity, its variance, and the coefficient of variation are summarized for the Case I and a single annuitant. The expected value of life annuity is higher for females for all countries, and as expected, it decreases as the retirement age increase. Annuity variance is higher for males than females at all ages, and it increases with the retirement age. The coefficient of variation, c, is also higher for males and increases with age. It should be noted that c illustrates the influence of longevity risk with respect to systematic risk. Hence, the higher-risk group of the deterministic (static) Case I, for all countries, is the portfolio consisting of 65-year-old males.

Table 3.

Life annuity indicators based on deterministic approach, t = 2020.

Country	Age	Scenario	Female	Male
Turkey	45	$\mathbb{E}(Y\mid F_t)$	16.5331	15.2189
		$\mathrm{Var}(Y\mid F_t)$	7.3559	11.7619
		c	0.1640	0.2253
	58	$\mathbb{E}(Y\mid F_t)$	13.3127	11.6897
		$\mathrm{Var}(Y\mid F_t)$	12.2327	15.8879
		c	0.2627	0.3410
	65	$\mathbb{E}(Y\mid F_t)$	10.9900	9.4698
		$\mathrm{Var}(Y\mid F_t)$	14.2007	15.5352
		c	0.3429	0.4162
Germany	45	$\mathbb{E}(Y\mid F_t)$	17.4954	16.6881
		$\mathrm{Var}(Y\mid F_t)$	6.5555	9.4814
		c	0.1463	0.1845
	58	$\mathbb{E}(Y\mid F_t)$	15.0077	13.9034
		$\mathrm{Var}(Y\mid F_t)$	10.5039	13.9656
		c	0.2160	0.2688
	65	$\mathbb{E}(Y\mid F_t)$	13.1590	11.9738
		$\mathrm{Var}(Y\mid F_t)$	11.7751	14.6606
		c	0.2608	0.3198
Japan	45	$\mathbb{E}(Y\mid F_t)$	17.8556	16.7445
		$\mathrm{Var}(Y\mid F_t)$	5.3788	8.8749
		c	0.1299	0.1779
	58	$\mathbb{E}(Y\mid F_t)$	15.5369	13.9025
		$\mathrm{Var}(Y\mid F_t)$	8.9282	13.5554
		c	0.1923	0.2648
	65	$\mathbb{E}(Y\mid F_t)$	13.7305	11.9072
		$\mathrm{Var}(Y\mid F_t)$	10.9840	14.8321
		c	0.2414	0.3234

For Case II, stochastic annuity premium its variance components and c are calculated as can be seen in Table 4, starting from the base year of 2020 and increasing from thereon to highest possible living age. Comparison between Case I and II shows that the stochastic approach has higher expectation and variance in general due to changing (decreasing) mortality rates in time.

Table 4.

Life annuity indicators based on stochastic approach with $r_{\min }=0.1$, $r_{\mathrm{med}}=0.8$, $r_{\max }=0.1$, and base year is taken as $t_0=2020$.

			Female			Male
Country	Age	Scenario	Min	Med	Max	Min	Med	Max
Turkey	45	$\mathbb{E}(Y\mid F_t)$	14.8480	16.6051	17.5615	13.8630	15.3848	16.5682
		$\mathrm{Var}(Y\mid F_t)$	4.8318	8.6927	16.1386	8.2540	13.1376	19.2324
		c	0.1480	0.1776	0.2288	0.2072	0.2356	0.2647
	58	$\mathbb{E}(Y\mid F_t)$	11.5012	13.4689	14.7609	10.5183	11.9324	13.2732
		$\mathrm{Var}(Y\mid F_t)$	10.6261	14.4019	19.1597	15.5007	18.4072	20.4923
		c	0.2834	0.2818	0.2965	0.3743	0.3596	0.3411
	65	$\mathbb{E}(Y\mid F_t)$	9.3380	11.1959	12.5693	8.4950	9.7388	11.0168
		$\mathrm{Var}(Y\mid F_t)$	14.5771	16.8774	18.5854	17.7916	18.6604	18.8428
		c	0.4089	0.3669	0.3430	0.4965	0.4436	0.3940
Germany	45	$\mathbb{E}(Y\mid F_t)$	17.1126	17.6523	18.0903	16.0837	16.8908	17.7042
		$\mathrm{Var}(Y\mid F_t)$	5.7079	7.1442	8.7881	7.2487	10.0034	12.4310
		c	0.1396	0.1514	0.1639	0.1674	0.1873	0.1991
	58	$\mathbb{E}(Y\mid F_t)$	14.4277	15.2500	15.9286	13.0112	14.1530	15.3445
		$\mathrm{Var}(Y\mid F_t)$	10.1540	12.0250	13.7636	12.7414	15.9884	18.0591
		c	0.2209	0.2274	0.2329	0.2743	0.2825	0.2769
	65	$\mathbb{E}(Y\mid F_t)$	12.4386	13.4650	14.3073	10.9873	12.2917	13.6632
		$\mathrm{Var}(Y\mid F_t)$	11.7100	13.9540	15.9711	14.2596	17.3533	18.9982
		c	0.2751	0.2774	0.2793	0.3437	0.3389	0.3190
Japan	45	$\mathbb{E}(Y\mid F_t)$	15.4661	18.0595	19.0339	13.9590	17.0163	18.5301
		$\mathrm{Var}(Y\mid F_t)$	2.1714	5.6122	13.9501	3.8767	9.1669	17.3283
		c	0.0953	0.1312	0.1962	0.1411	0.1779	0.2246
	58	$\mathbb{E}(Y\mid F_t)$	12.1943	15.8788	17.4029	10.2923	14.3205	16.5711
		$\mathrm{Var}(Y\mid F_t)$	4.5085	9.5616	17.8112	7.6968	14.4656	19.4466
		c	0.1741	0.1947	0.2425	0.2696	0.2656	0.2661
	65	$\mathbb{E}(Y\mid F_t)$	9.9659	14.1707	16.0385	8.1033	12.4055	15.0109
		$\mathrm{Var}(Y\mid F_t)$	6.5053	11.9804	18.1915	10.1847	16.3545	19.0175
		c	0.2559	0.2443	0.2659	0.3938	0.3260	0.2905

The following step in the application is to introduce randomness to the algorithm. We use previously defined three mortality projections (min, med, max) and assign weights to each scenario at which the sum is one. Each of the weights is interpreted as a possibility of the corresponding mortality reduction projection. The weights are chosen as $r_{\min }=0.1$, $r_{\mathrm{med}}=0.8$, $r_{\max }=0.1$ to the scenarios low, medium and high, respectively. Also as stated in the literature, results do not change significantly when other values of weights are used [21]. The coefficient of variation of the present value of future liabilities is taken to be the risk measurement on longevity. Change on c for the male population is higher when r and portfolio size (N) increases. Table 5 shows sensitivity of portfolio size on the variance and the coefficient of variation. As the number of policies (N) increases, the variance increases faster than the number of annuitants. This increase stems from the $\mathrm{Var}(Y\mid F_t)$ (longevity risk), which is increasing at $N^k$. The coefficient of variation has an inverse ratio with N, and it increases with retirement age. Germany and Japan's male populations have a higher risk than their female population when combined in a portfolio for all N. On the contrary, for Turkey, there is no stable pattern. For retirement age 45, longevity risk is lower for females than males, where, as for age 58, female longevity risk is higher for the portfolio with greater than ${10}^3$ number of annuitants. Similarly, for age 65, female longevity risk is higher for the portfolio with greater than ${10}^2$ number of annuitants. From which it is concluded that there is less room for the development of male mortality rates for Turkey than Germany and Japan. As a consequence, the riskiest portfolio for Germany and Japan is the one with 65-year-old male annuitants, where for Turkey, it is the one with 65-year-old female annuitants.

Table 5.

Sensitivity to N for the annuity portfolios under stochastic approach.

				N
Gender	Country	Age	Indicator	1	${10}^2$	${10}^3$	${10}^4$
Female	Turkey	45	$\mathrm{Var}(Y\mid F_t)$	9.445	4843.164	402856.003	39470996.100
			c	0.186	0.042	0.038	0.038
		58	$\mathrm{Var}(Y\mid F_t)$	15.050	6945.326	564031.477	55098136.548
			c	0.290	0.062	0.056	0.055
		65	$\mathrm{Var}(Y\mid F_t)$	17.359	6996.495	548285.817	53314944.953
			c	0.374	0.075	0.066	0.066
	Germany	45	$\mathrm{Var}(Y\mid F_t)$	7.213	1198.652	55380.500	4893203.405
			c	0.152	0.020	0.013	0.013
		58	$\mathrm{Var}(Y\mid F_t)$	12.125	2335.695	125463.827	11465326.251
			c	0.229	0.032	0.023	0.022
		65	$\mathrm{Var}(Y\mid F_t)$	14.107	3152.631	189880.975	17734276.304
			c	0.279	0.042	0.032	0.031
	Japan	45	$\mathrm{Var}(Y\mid F_t)$	6.843	8023.479	747430.722	74193900.835
			c	0.146	0.050	0.048	0.048
		58	$\mathrm{Var}(Y\mid F_t)$	11.424	16419.890	1553057.719	154416459.353
			c	0.216	0.082	0.080	0.079
		65	$\mathrm{Var}(Y\mid F_t)$	14.116	21828.539	2074367.612	206351898.372
			c	0.270	0.106	0.103	0.103
Male	Turkey	45	$\mathrm{Var}(Y\mid F_t)$	13.629	5030.757	383747.226	37181437.866
			c	0.241	0.046	0.040	0.040
		58	$\mathrm{Var}(Y\mid F_t)$	18.705	5629.248	397998.995	38150641.805
			c	0.363	0.063	0.053	0.052
		65	$\mathrm{Var}(Y\mid F_t)$	18.910	5039.500	336623.929	31989132.051
			c	0.446	0.073	0.060	0.058
	Germany	45	$\mathrm{Var}(Y\mid F_t)$	10.102	2310.096	141273.347	13229972.118
			c	0.188	0.029	0.022	0.022
		58	$\mathrm{Var}(Y\mid F_t)$	16.143	4310.176	288180.687	27389700.034
			c	0.284	0.046	0.038	0.037
		65	$\mathrm{Var}(Y\mid F_t)$	17.567	5302.746	375398.898	35991132.926
			c	0.341	0.059	0.050	0.049
	Japan	45	$\mathrm{Var}(Y\mid F_t)$	10.594	12345.671	1149480.939	114097232.686
			c	0.193	0.066	0.064	0.063
		58	$\mathrm{Var}(Y\mid F_t)$	16.384	22404.620	2111880.318	209902214.728
			c	0.286	0.106	0.103	0.102
		65	$\mathrm{Var}(Y\mid F_t)$	18.505	26609.385	2516904.441	250250103.273
			c	0.352	0.133	0.130	0.129

As noted before, with the pooling, the second component of the variance ( $\mathrm{Var}[\mathbb{E}(Y|F_t)]$) becomes independent of the number of annuitants and is persistent through periods, as given in Equation (16). The first component ( $\mathbb{E}[\mathrm{Var}(Y|F_t)]$), however, is reduced with the number of annuitants in the pool, and as a consequence, the total variance of the annuity descends. Figure 4 illustrates that the pooling argument reduces the variance. It can be concluded that even though the longevity risk can be decreased with increasing the size of the portfolio, it remains after pooling.

Figure 4.

The sensitivity of pooled variances to portfolio size. (a) Turkey, (b) Germany, (c) Japan.

The weights given at each level are altered to see the impact of change on the variance and coefficient of variation. Other possible values for r's do not make significant changes to the results, which is consistent with Olivieri [21]. Figure 5 illustrates the effect of the assigned probability of the central scenario and its sensitivity to the risk level of the portfolio. As the weight of the medium scenario decreases, the risk increases. For a portfolio that has a small number of annuitants, the difference in the weight assigned to the central scenario (med) does not make a significant change since the first component of the variance dominates the total variance. However, when the size of the portfolio increases, the first term of the variance gets very small, and hence the longevity risk becomes more significant.

Figure 5.

The sensitivity of coefficient of variation to portfolio size at median scenario at age 58. (a) Turkey, (b) Germany, (c) Japan.

Finally, in the light of these findings, the credibility premium of the annuity is calculated for all three countries based on variance components, and the number of years at which historical mortality rates being available for each country. The highest m value (number of years of data collected) belongs to Turkey with 85 years of past mortality data, generating more confidence to the stochastic approach. Whereas Japan has 70 years and Germany has the lowest number with 28 years of past mortality data. Hence the smallest credibility factor belongs to Germany, and Japan has the highest Z of all countries, with 65 years male group. Table 6 provides the results of the credibility parameter, credibility factor, and credibility premium computations for the three possible retirement ages. The premium for younger ages is higher than the older ages as expected. Turkey has larger premiums compared to Germany and Japan. Females yield higher credibility premiums than males for all selected countries.

Table 6.

Credibility premium based on stochastic approach with weights, for three scenarios.

Country	Age	Scenario	Female	Male
Turkey	45	k	22.9839	35.7871
		Z	0.7872	0.7037
		${\mu }_x$	16.5314	15.2580
	58	k	26.3863	48.2653
		Z	0.7631	0.6378
		${\mu }_x$	13.3337	11.7750
	65	k	31.6448	58.4588
		Z	0.7287	0.5925
		${\mu }_x$	11.0327	9.5808
Germany	45	k	148.6028	75.9367
		Z	0.1585	0.2694
		${\mu }_x$	17.6189	16.8367
	58	k	105.8750	58.2820
		Z	0.2092	0.3245
		${\mu }_x$	15.1880	14.0754
	65	k	79.1780	48.0426
		Z	0.2612	0.3682
		${\mu }_x$	13.3715	12.1789
Japan	45	k	8.2310	8.2928
		Z	0.8948	0.8941
		${\mu }_x$	17.8600	16.7569
	58	k	6.4032	6.8111
		Z	0.9162	0.9113
		${\mu }_x$	15.5475	13.9238
	65	k	5.8449	6.3992
		Z	0.9229	0.9162
		${\mu }_x$	13.7464	11.9347

7. Conclusion

This study aims to justify the existence of longevity risk for a population with a significant young population and illustrates its effect on the pension funds. Turkey, as a good example of the young population and emerging market, Germany, and Japan, which are known as long life expectancy and economically robust, are studied through the stochastic mortality model, Lee–Carter. The life expectancy is reviewed based on three different retirement ages concerning the Turkish pension system. A portfolio of N annuitants are taken into account to evaluate the impact of longevity risk under deterministic and stochastic approaches. Deterministic and stochastic methods show that there is a significant difference in life expectancy when the cohort effect is taken into account. This difference can and will affect the robustness of pension funds; therefore, it should be taken into account by the pension funds. A comparison between the three countries provided the longevity structure information. Even though having a lack of historical data on Turkish mortality, this study illustrates that opposite to other countries, female longevity risk is higher than males in Turkey and remains persistent over the years. Also, due to long historical data, credibility analysis assigns more weight to the estimated stochastic annuity premium. This shows that the proposed longevity risk and its impact on annuity net single premium yield reliable results.

Therefore based on the findings, we infer for the Turkish public pension system, which has a growing liability and causing a more significant deficit in the government budget, longevity risk has to be taken into account based on recent reforms. The outcomes and further extensions of this study can be utilized to modify the existing pension systems and individual pension insurance in Turkey. Additionally, it will help to develop custom-tailored life insurance products. In international platforms, it guides to design longevity bond products.

Disclosure statement

No potential conflict of interest was reported by the author(s).