Estimation of Longevity Risk and Mortality Modelling

Dissertation presented as partial requirement for obtaining the master’s degree in Statistics and Information Management Risk, with a specialization in Analysis and Risk Management Estimation of Longevity Risk and Mortality Modelling TABI ROSY CHRISTY ATEMNKENG NOVA Information Management School Instituto Superior de Estatística e Gestão de Informação Universidade Nova de Lisboa Estimation of Longevity Risk and Mortality Modelling by TABI ROSY CHRISTY ATEMNKENG Dissertation presented as partial requirement for obtaining the master’s degree in Statistics and Information Management, with a specialization in Analysis and Risk Management. SUPERVISOR: Prof. Dr. Jorge Miguel Ventura Bravo November 2021 Acknowledgement First, I thank God for giving me knowledge and patience to carry out this research. My Family, Relative and Lecturers, your support is appreciated. A Special thanks to my Supervisor Prof. Dr. Jorge Miguel Ventura Bravo for your guidance and assistance through the entire project. Abstract Previous mortality models failed to account for improvements in human mortality rates thus in general, human life expectancy was underestimate. Declining mortality and increasing life expectancy (longevity) profoundly alter the population age distribution. This demographic transition has received considerable attention on pension and annuity providers. Concerns have been expressed about the implications of increased life expectancy for government spending on old-age support. The goal of this paper is to lay out a framework for measuring, understanding, and analyzing longevity risk, with a focus on defined pension plans. Lee-Carter proposed a widely used mortality forecasting model in 1992. The study looks at how well the Lee-Carter model performed for female and male populations in the selected country (France) from 1816 to 2018. The Singular Value Decomposition (SVD) method is used to estimate the parameters of the LC model. The mortality table then assesses future improvements in mortality and life expectancy, taking into account mortality assumptions, to see if pension funds and annuity providers are exposed to longevity risk. Mortality assumptions are predicted death rates based on a mortality table. The two types of mortality are mortality at birth and mortality in old age. Longevity risk must be effectively managed by pension and annuity providers. To mitigate this risk, pension providers must factor in future improvements in mortality and life expectancy, as mortality rates tend to decrease over time. The findings show that failing to account for future improvements in mortality results in an expected provision shortfall. Protection mechanisms and policy recommendations to manage longevity risk can help to mitigate the financial impact of an unexpected increase in longevity. Keywords: Lee-Carter (LC) model, Mortality modeling, Forecasting, Life expectancy, Singular value decomposition (SVD), INDEX Table of Contents Acknowledgement ............................................................................................................ 3 Abstract ............................................................................................................................. 4 LIST OF FIGURES .......................................................................................................... 7 LIST OF TABLES ........................................................................................................... 7 1. INTRODUCTION ..................................................................................................... 8 1.1. Background of the study .................................................................................... 8 1.2. Statement of the Problem ................................................................................. 11 1.3. Objective .......................................................................................................... 12 1.3.1. Specific Objective..................................................................................... 12 1.4. Justification of the Study ................................................................................. 13 2. LITERATURE REVIEW ........................................................................................ 14 2.1. Theoretical Background: Longevity Risk and Mortality Risk ......................... 14 2.1.1. Products With Longevity Risk Exposure ............................................................ 16 2.2. Decomposition of Mortality Risk .................................................................... 18 2.3. Management And Quantification of Longevity Risk....................................... 19 3. METHODOLOGY .................................................................................................. 23 3.1. Basic Mortality Functions ................................................................................ 23 3.2. Modelling Structure and Specification ............................................................ 26 3.3. Risk Model Classification ................................................................................ 28 3.4. Identification and Structure of the APC Stochastic Mortality Model ............. 29 3.5. Stochastics Mortality Models .......................................................................... 30 4. EMPIRICAL ANALYSIS ...................................................................................... 41 4.1. Uncertainty About Mortality and Life Expectancy ......................................... 41 4.1.1. The relationship between mortality and life expectancy: Life Table ....... 41 4.1.2. The uncertainty surrounding the improvement in mortality ..................... 42 4.1.3. Mortality and life expectancy forecasting methods .................................. 43 4.2. Measuring mortality and longevity improvement uncertainty ........................ 44 4.2.1. Lee Carter Model Measurement ............................................................... 44 4.2.2. Fitting and Estimating the Parameters of the Lee-Carter Model .............. 47 4.3. The impact of longevity risk on defined-benefit private pension plans .......... 49 4.3.1. How does longevity risk affect DB private pension plans?...................... 49 4.3.2. How private pension funds account for future improvements in mortality and/or life expectancy?............................................................................................ 50 5. CONCLUSIONS ..................................................................................................... 52 5.1. Policy issues ..................................................................................................... 52 5.2. Areas in which additional research is required ................................................ 53 6. REFERENCES ........................................................................................................ 55 APPENDIX .................................................................................................................... 58 TABLES AND FIGURES .............................................................................................. 58 LIST OF FIGURES Figure 1: Male Death Rate, France 1816 -2018. ............................................................ 44 Figure 2: Female Death Rate, France 1816-2018. .......................................................... 45 Figure 3: 𝑎𝑥 and 𝑏𝑥 for France population based on life tables (1989 to 2018) .......... 45 Figure 4: Pattern of age according to death rates for France Population ....................... 46 Figure 5: Pattern of Death rate based on Year (1918-2018)........................................... 46 Figure 6: Estimated parameter of 𝑎𝑥,𝑏𝑥𝑘𝑡 .................................................................... 47 Figure 7: Projected value of 𝑘𝑡 for 100 years................................................................. 48 Figure 8: Pattern of Past and Projected rates for people aged 65 ................................... 48 Figure 9: Life Expectancy at age 65 in 2050 .................................................................. 63 LIST OF TABLES Table 1: Mortality tables and improvement required by regulation and used in practice ........................................................................................................................................ 38 Table 2:Mortality Projection Scale AA compiled by the Society of Actuaries.............. 39 Table 3: Comparing Life Expectancy at selected age groups, France 1985-2018 ......... 43 Table 4: Life table, France 2018 Males .......................................................................... 58 Table 5: An increase in the annuity payments' net present value ................................... 58 Table 6: Age group specific central death rates female population, France 1998-2018 59 Table 7: Natural logarithm of death rates for female, France 1989-2018 ...................... 60 Table 8: 𝑎𝑥 and 𝑏𝑥 Estimate, France 1989 to 2018 ...................................................... 61 Table 9: Estimate of 𝑘𝑡, France 1989 to 2018 (Male and Female) ................................ 61 1. INTRODUCTION 1.1. Background of the study The continuous improvements in longevity bring new problems and challenges at different levels of political, social, economic, and regulatory. However, one of the most observed effects of this improvements in longevity is on pensions. In recent decades, most high-income countries have responded to continuous life expectancy increases, below replacement-level fertility, an upward trend in old-age dependency ratios, low productivity gains and economic growth, a rapidly shifting labor market and declining financial market returns with systemic (e.g., the switch towards a Non-Financial Defined Contribution (NDC) scheme in Sweden, Italy, Poland, Latvia and Norway; pension financialization, i.e., the expansion of private complementary occupational and personal pre-funded defined-contribution (DC) pensions) and/or gradual parametric reforms in national public pension schemes (e.g., updates in the early and normal retirement ages, modifications in the defined benefit (DB) pension formula) as part of their efforts to reduce or eliminate short-term and long-term imbalances between revenues and expenditures, alleviating the pressure on public finances, together with efforts to preserve minimum pension adequacy (OECD, 2019; Bravo & Herce, 2020). For national public pension schemes, a common denominator of most reforms has been to introduce automatic adjustment or stabilization mechanisms specifically designed to correct for the financial imbalance of the pension system, mechanically updating the scheme’s parameters to demographic and/or economic developments. A common denominator in most pension reforms adopted in developed countries has been to automatically link pension benefits to life expectancy developments observed at retirement ages. The link has been established and reinforced in multiple ways (Ayuso, Bravo & Holzmann, 2021b; Bravo & Ayuso, 2020, 2021): i) by indexing normal and early retirement ages to life expectancy (e.g., Denmark, The Netherlands, Portugal, UK); (ii) by linking entry pensions to sustainability factors (e.g., Finland, Portugal), (iii) by indexing the eligibility requirements to the contribution length (e.g., France); (iv) by conditioning the annual pension indexation (e.g., The Netherlands, Luxembourg); (v) by introducing longevity-linked risk-sharing life annuities in public and private pension schemes (Bravo & El Mekkaoui, 2018; Bravo, 2019, 2020, 2021a). In 2009, most companies in developing countries closed the defined benefit retirement plans (such as 401(K) plans in the United States) offer to their employees. The pension plans provided to the employer can either be defined contribution or defined benefit. These plans ensure employees receive a certain amount at retirement. In addition, defined benefit pension plans have been replaced by defined contribution plans. A defined-benefit program is a promise of lifetime retirement benefits and the most vital risk for retirement resulting from longevity. Longevity risk is the risk that insurance companies or pension funds faced when assumptions about life expectancies and mortality rates are inaccurate. Mortality rates and longevity trend risk are the main indicator considered when attempting to transfer longevity risk. The insurance sectors faced the risk arising from increased longevity i.e., the trend of longevity improvement will significantly change in the future. To face this long-term risk, more capital must be set asides. Hence it has become more important for life office (insurance companies, pension funds) to find efficient and suitable method to transfer part of longevity risk to capital or financial market. However, longevity risk cannot be transferred so easily since it is difficult to understand and manage due to its long-term nature, precisely projections for longevity are sensitive and the modeling of integrated interest rate risk remains challenging. Two main factors when transferring the longevity risk for a particular pension plan or insurer must be considered. The first is the current mortality levels, which can be observed but vary considerably between socio-economic and health categories. The second is the risk of longevity, which is the risk trajectory for the ageing population and is systemic. Systematic mortality trend risk can be offset directly by keeping exposure to increased mortality. One reason for ceding the risk is the uncertainty concerning the longevity risk, especially because of the systematic nature, in a pension plan or an insurance company. To manage the risk of longevity better, Individuals and life offices needs to fully understand longevity risk, or consider its implications, when they come to plan their retirement income. Three causes for this are uncertainty, underestimation, and complexity. To help us better understand these terms, (Yinglu Deng et al., 2012a) assert that “Longevity risk describes the risk that an individual or group will live longer life than expected thus their mortality rate will be lower than expected, while mortality risk describes the risk that an individual or group will live a shorter life than expected thus their mortality rate will be higher than expected.”, pp. 697,2012. Longevity risk for pensioners refers to “the possibility that they will live to such an advanced age that they will deplete their retirement savings and have to rely solely on Social Security and Medicare for their expenses”. Longevity risk in retirement planning can be defined as “the risk that members of some reference population might live longer on average than anticipated” (Stamp duty and land tax for non-resident owners of Australian property, n.d.). Longevity risk for individuals with DC pension savings can have significant implications when they retire. The risk of people outliving their retirement savings or the risk of people underspending their savings leads to lower pension incomes. For pension plans, longevity risk refers to the increase in retirement pension duties because of longer lifespans. For individuals, longevity risks mean a person's possibility of outliving on their pension assets. In the first place, the longevity risk is due to the fact that people are now living longer due to various factors like medical and health. This means that one can reasonably expect to add another from the false certainty of a single projection, and a step toward explicit recognition of the uncertainty surrounding the path of future improvements. 2.1.1. Products With Longevity Risk Exposure Longevity risk exists in any product in which the issuer is exposed to financial losses if policyholders live longer than expected. This is common when payments from the issuer are contingent on the policyholder's survival. Traditionally, these products have been issued by insurance companies and used to hedge against an individual outliving their assets. In recent years, the number and variety of products exposed to longevity risk has grown. This can happen even if transferring longevity risk is not the primary goal of the transaction. We examine some of the products on the market that are vulnerable to longevity risk. We also take into account the other risks that these products face, such as financial risk, pricing risk, and regulatory risk. Conversely, longevity risk is generally defined as the exposure of a company to lower-than-expected mortality(Owusu et al., 2016). i. Immediate annuities An immediate annuity is a product that usually provides payments for life in exchange for a lump sum. The frequency and payment amount may vary over the course of the contract. They can be designed to provide a fixed level payment, a stream of payments that increase at a predetermined rate, or a stream of payments that is linked to an underlying equity index. Immediate annuities can be purchased as either single life or joint-and-survivor policies. In the latter case, annuity payments continue as long as one of the two lives is alive, though the size of the annuity payment may decrease if the primary insured dies. Immediate annuities are also subject to pricing risk. Companies that set prices for their products that are inconsistent with best estimate assumptions face a greater risk that the actual experience will differ from what was expected. Because annuity rates are simple to understand and compare for insurers, pricing for longevity risk is competitive. ii. Enhanced and impaired life annuities Impaired or enhanced annuities provide higher annuity payments to people who can demonstrate that they are in poor health or are terminally ill. For the insurer, there is a greater risk of medical breakthroughs in a single condition extending an individual's life, which necessitates that enhanced products be priced at a higher margin than standard annuities. This also has implications for estimating future mortality improvements. The risks associated with enhanced and impaired life annuities are similar to those associated with standard immediate annuities. However, given the higher expected mortality rates assumed for these policies, the longevity risk may be exacerbated, as there is likely to be less data on the mortality experience of subgroups of the population. iii. Deferred annuities Traditional Deferred annuities are primarily used to accumulate tax-deferred savings, which can then be distributed as an immediate annuity or as a lump sum payment. Fixed, variable, and equity-indexed annuities are the three types of deferred annuities available in the United States. As a result, they are less vulnerable to the risks associated with aging. The addition of guarantees to product offerings has introduced longevity risk as the market has developed and become more competitive. When a deferred annuity is annuitized at maturity, it is subject to a number of risks that are not present when it is distributed in lump sum. These products have been in place for a long time, and it is difficult to protect the cash flows due to a scarcity of assets with the appropriate duration. As a result, deferred annuities are subject to reinvestment risk. iv. Advanced Life Delayed Annuities ALDAs (advanced-life delayed annuities) are a type of longevity insurance. ALDAs are inflation-linked annuities sold to people in their early twenties that begin paying out at the age of 80, 85, or 90. There is no cash value, and no mortality insurance benefits that can be repaid at any time. ALDAs are designed to mimic a defined benefit pension benefit at advanced ages for people who do not have access to this type of protection. Traditional deferred annuities may be better suited to protecting against catastrophic longevity (Owusu et al., 2016). v. Corporate pensions There are two types of corporate pension plans: defined benefit (DB) and defined contribution (DC). The employee receives a fixed income stream based on his or her salary, years of service, retirement age, and other factors under a DB plan. Typically, the benefit stream is set. Contributions are made into individual accounts by each employee under a DC plan, and the employer may make a matching contribution. When you retire, you can take a lump sum equal to the value of your current account. The lump sum can be used to supplement retirement income. vi. Structured settlements Structured settlements are payments made as the result of a general insurance liability involving human life (e.g., serious injury, medical negligence, or occupational injury). Payments are sometimes made in the form of a lump sum for the injured party's lost earnings and/or the cost of care if they are seriously injured. Annuities payable for life, on the other hand, have recently been used as a type of settlement. vii. Life settlements Purchasers of life settlements face longevity risk because lower mortality means they must pay insurance premiums for a longer period of time and receive the death benefit later than expected. Most buyers of this type of contract are not in the business of profiting from mortality. Life Settlements are a way for an investment bank or hedge fund to diversify risk while potentially achieving a high rate of return, as has historically been the case with these portfolios. 2.2. Decomposition of Mortality Risk Mortality risk is generally defined as a company's exposure to greater-than-expected mortality. The International Actuarial Association divides mortality and longevity risk into four categories: level, trend, volatility, and catastrophe. Risk can be classified into two types: systematic risk and specific risk. The term "systematic risk" refers to incorrect base assumptions (level and trend), whereas "specific risk" refers to volatility that surrounds the base assumptions (volatility and catastrophe). Specific risk is decreasing, but the systematic risk cannot be diversified as the number of lives covered increases. There are considerable and increasing costs of systematic risk for pension plans and insurers. Mortality risk is a vital risk factor for insurance companies and mortality risk is broken up into subcategories, systemic risk, unsystematic risk, and adverse remedies. The risk of mortality refers to the risk of a person living for a shorter life than expected and is, therefore, higher than expected. The interest of life insurers and pensioners in longevity risk to the design of a defined benefit plan has increased (Gatzert & Wesker, 2014). i. Unsystematic Mortality risk: The risk of individual deaths is a random variable with a certain probability (see Biffiss, Denuit, and Devolder, 2010). Thus, it may be diversified through natural hedges, or transfers through mortality to the capital market, Contingent bonds (MCBs). ii. Systematic Mortality risk: The risk of systematic mortality is the risk of sudden changes to underlying population mortality, for example as a result of common factors affecting deaths of the entire population that trigger life dependencies and cannot be diversified by broadening the portfolio (see Wills and Sherris, 2010). iii. Adverse Selection: This referred to the fact that, for various populations of assured persons, for example, life insurers and pensioners, the probability distribution differs in age level and trend (see Brouhns, Denuit and Vermunt, 2002a). In addition, adverse selection is a major source of risk when hedging longevity risk via MCB or other capital markets instruments, because of individual mortality heterogeneity and information asymmetries between the insurance company and Insured (see, e.g., Sweeting, 2007). iv. Basic Risk: This occurs when hedge population mortality does not coincide with the portfolio hedge mortality. This means that there is a base risk in longevity hedges in the differences in population mortality and mortality of the insured pensioners caused by adverse selection. In this analysis, we explicitly consider the fundamental risk in hedges and models every kind of mortality risk in order to analyze its impact on the risk situation of the life insurer. 2.3. Management And Quantification of Longevity Risk To ensure that insurers' exposure to longevity risk is effectively managed, actuaries must first be aware of the current methods for quantifying and managing this risk. Only then can they take an active part in identifying and building additional risk management techniques that are more effective in addressing longevity risks. Companies are required to maintain a certain percentage of their net risk or reserves to cover the risk that death is different from expected. As a result, most companies continue to quantify the risk of longevity with relatively fundamental methodologies. Because the risk of long life for insurers is increasing, major annuity authors and reinsurers look for ways to manage their costs effectively. To date, product design, contracting, natural hedging, and reinsurance are the conventional methods that direct authors use. Furthermore, companies have started to use their longevity risk exposure solutions to the financial markets. • Buy-ins, A pension scheme's liabilities such as pensioners' in-payment, are covered by buy-in. The policy pays an income equivalent to the members' benefits, removing the danger of insufficient assets to fund future commitments. • Bulk Annuity and reinsurance transactions to transfer rents between insurers and reinsurers. • These solutions are also insurance. • Longevity bonds which transfer a long-term risk to another party in the form of a security from a pension plan or annuity portfolio. These are solutions to the capital markets. • Longevity swaps to transfer longevity only to another party from a pension scheme or annuity portfolio. These can either be insurance solutions or solutions for the capital markets. • Mortality catastrophe and swapping, transferring from life insurer or reinsurer to other parties, the risk of devastating (catastrophe) increases in mortality due, e.g., to a pandemic or natural disaster. These are solutions to the capital markets. • Life securitizations that transfer risks related to a specific block of insurance undertakings, as a security, to capital markets. These are solutions to the capital markets. • US life settlements transactions transferring to investors small portfolios of U.S. life insurance policies. These are solutions to the capital markets. • Pensions buy-outs that transfer pension obligations and all associated risks and obligations to insurers (also known as pension plan terminals). These are the solutions for insurance. The hedging instrument is the third feature of risk transactions with a pure longevity. The longevity swap for survivors has previously been the most common structure. Mortality forward (q-forward) A forward mortality contract is often known as a forward, as the letter 'q' stands for actuarial mortality rate symbols. It is the simplest type of longevity (and mortality) risk transfer instrument (Coughlan et al. 2007b) and was the first type of capital markets that were used for longevity hedges. This was an agreement between UK Lucida and J.P. Morgan pension insurers and is described in the next section. The importance of qforwards is that they form fundamental blocks from which other life-related derivatives can be built. A q-forwards portfolio can be used, if appropriately designed, to replicate and safeguard a lifetime exposure or to protect a life insurance book or a pension liability. A q-forwards shall be defined as an agreement between two parties in which a sum proportional to the actual mortality rates performed for a given population (or subpopulation) is exchanged in exchange for the sum proportional to a fixed death rate agreed upon at the outset to be payable in the future (the maturity of the contract). If there is a fair price of the q-forward, there is no change in payment hands at the start of the trade, but at maturity one of the two counterparties makes a net payment (unless the fixed and actual mortality rates happen to be the same). The maturity payment is based on the net amount payable and is proportional to the difference between the fixed mortality rate (the forward rate transacted) and the reference rate realized. If in the reference year the rate is lower than the fixed rate (that is, a lower death rate), the settlement is positive, and the settlement payment is received by the pension plan to make up for the increase in its liability value. Where, on the other hand, the reference rate is higher than the fixed rate (ie. higher mortality), the repayment is negative, and the pension plan pays the hedge provider the settlement payment, which is offset by the decline in the value of the payment. The net liability value is therefore locked with regards to the mortality rates. The scheme is protected against unexpected mortality rate changes. Survivor forward (S-forward) A survivor forward, also known as a “S-forward,” is similar to a q-forward in concept but uses survival rates rather than mortality rates. It is an agreement between two parties to exchange an amount proportional to the actual, realized survival rate of a given population (or subpopulation) in exchange for an amount proportional to a fixed survival rate that has been mutually agreed upon at the contract's inception to be payable at the contract's maturity. As such, it entails exchanging a notional amount multiplied by a pre-agreedupon fixed survival rate for the same notional amount multiplied by the realized survival rate for a specified cohort over a specified time period (Coughlan et al., 2008b; Dawson et al., 2010). If the contract has a one-year maturity, a survivor forward is the inverse of a mortality forward. However, if the contract maturity exceeds a year, this simple relationship no longer exists because survival rates over longer time periods are nonlinear functions of annual mortality rates. Because it is a function of several mortality rates at different ages and times, a survivor forward is more complex than a q-forward. In some situations, it can nevertheless be a useful building block. Longevity swaps A longevity swap can be classified as either a capital markets derivative or an insurance contract. In either case, it is a financial instrument that involves exchanging actual pension payments for a series of pre-agreed-upon fixed payments (Dowd et al., 2006; Bravo & Nunes, 2021). Each payment is based on an amount weighted survival rate. In any longevity swap, the hedger of longevity risk (for example, a pension plan) receives the actual payments it must make to pensioners from the longevity swap provider and, in exchange, makes a series of fixed payments to the hedge provider. As a result, if retirees live longer than expected, the higher pension amounts that the pension plan must pay are offset by the higher payments received from the longevity swap provider. As a result, the swap offers the pension plan a long maturity, customized cash flow hedge of its longevity risk. The July 2008 Canada Life-J.P. Morgan transaction (Trading Risk 2008; Life & Pensions 2008). Variants on longevity swaps The transaction carried out by Aegon and Deutsche Bank in January 2012 is one variant of the standard longevity swap. This was an “out-of-the-money” longevity swap because it only transferred the longevity risk associated with a significant increase in life expectancy (or equivalently, a very large and sustained fall in mortality rates). Aegon, the hedger, receives no incremental payment for modest increases in life expectancy until a certain threshold, or "attachment point," is crossed. Aegon will then be paid for which the life span increases until a certain maximum level of protection is attained when life expectancy rises to a very extreme level. This swap is indeed a standard long-life swap, except that it has floating caps and floors. The swap in capital markets was based on indexes over 20 years and the index matched the national population data of the Netherlands. This swap also included, like the Aviva-RBS transaction, a swap payment at maturity to protect the longevity of any responsibility cash flow that exceeds the maturity date. Longevity bonds Since the start of this market, longevity bonds have been widely spoken to prevent the risks of longevity. A longevity bond (or a survivor bond as it was originally called) is a bond that pays coupons that proportionally correspond to the number of survivors still living on the coupon payment date in the population cohort specified. (Wolff, 2001; Blake et al., 2006a, 2006a; Dowd, 2003). The cash flows of a single longevity vanilla bond are the same as those of a longevity swap floated bearing. However, longevity bonds with different structures have recently been proposed. The cash flows of the bond are indexed to the mortality experienced in the United Kingdom by 65-year-old men. There is a 10year deferment period before the start of payment and a terminal switching payment at 105 years is made to cover the risk of a long life after 105 years. If more people survive at each age, then the bond pays more; if fewer people survive, then the bond pays less (similar to the floating leg of the RBS-Aviva longevity swap). 3. METHODOLOGY Life expectancy is the most common statistical indicator of the average remaining lifespan an individual is expected to live (Ayuso et al., 2021). 3.1. Basic Mortality Functions Let (𝑥) denote a life that survives to the age 𝑥. The life (𝑥) is called a life-age-𝑥. Let 𝐷𝑥𝑡 be a random variable, in a population who die at aged (𝑥) last birthday during a calendar year t. 𝑑𝑥𝑡 denote the observed number of persons who die between ages (𝑥) and (𝑥+𝑡) 𝑙𝑥 denote number of persons who attain age x according to the mortality table. 𝑞𝑥 denote the probability that (𝑥) will die within 1 year 𝑝𝑥 denote the probability that (𝑥) will live 1 year. 𝑑𝑥 denote 3.1.1. Initial Mortality Rate 𝑞𝑥 is called the mortality rate at age 𝑥, in actuarial terminology 𝑞𝑥 is the probability that (x) dies before age (𝑥 + 1). We can also subscribe a (𝑡) to get 𝑞𝑥 𝑡 which is the probability that (x) dies before age 𝑥 + 𝑡, 𝑞𝑥= 𝑑𝑥 𝑙𝑥 (1) 3.1.2. Probability of Survival The survival function of 𝑇𝑥 is denoted by 𝑝𝑥 𝑡. It is the probability that a life aged 𝑥 survives 𝑡 more years or is the probability that an age (𝑥) survives to at least age (𝑥 + 𝑡). In simplicity removing (𝑡), we get. 𝑝𝑥= 𝑙𝑥+1 𝑙𝑥 (2) 3.1.3. Central Death Rate The number of people who died during the year divided by the total number of people who were alive during the year. The Central death Rate (𝑚𝑥) denotes as the central death rate for the year of age (𝑥) to (𝑥+1). 𝑚𝑥= 𝑑𝑥 𝑙𝑥 (3) In the actuarial modeling literature, we use the following standard definitions (Dickson et al. (2013; 2009); Pitacco et al. (1998)). Let 𝑇𝑥 denote the remaining life expectancy of an individual of age 𝑥. The cumulative function of distribution and survival of 𝑇𝑥 is written as 𝜏 𝑞𝑥 = 𝑃(𝑇𝑥≤ 𝜏 ) and τ𝑝𝑥 = 𝑃(𝑇𝑥 > 𝜏 ) respectively. For an individual aged 𝑥, the force of mortality at age 𝑥 + 𝜏 is defined as 𝜇𝑥+𝜏 ∶= lim ℎ→01 ℎ𝑃(𝑇𝑥<𝜏+ℎ|𝑇𝑥>𝜏)=− 𝑑 𝑑𝜏ln𝜏𝜌𝑥 Let 𝑓𝑥 (𝑡) be the density function of 𝑇𝑥, then from (1) we have. 𝜏𝑞𝑥=∫ 𝑓𝑥(𝑠)𝑑𝑠 𝜏 0=∫ 𝑠𝜌𝑥 𝜏 0𝜇𝑥+𝑠 𝑑𝑠 The central death rate for 𝑥-year-old, where 𝑥 𝜖 ℕ, is defined as 𝑚𝑥:= 𝑞𝑥 ∫𝑠𝑝𝑥𝑑𝑠 1 0=∫𝑠𝑃𝑥𝜇𝑥+𝑠 𝑑𝑠 1 0∫𝑠𝑃𝑥𝑑𝑠 1 0 which is a weighted average of mortality force (𝑞𝑥∶= 𝑞𝑥 1). Taking account, the socalled constant force of mortality assumption, µ𝑥+𝑠 = µ𝑥 where 0 ≤𝑠 <1 and 𝑥 ∈ ℕ, from (2), we have 𝑚𝑥= µ𝑥. If a Poisson assumption is denoting of the actual number of deaths, then the maximum likelihood estimates of the force of mortality µ𝑥 is given by µ𝑥= 𝐷𝑥𝐸𝑥 ⁄= 𝑚𝑥 where 𝐷𝑥 denotes the recorder number of deaths at age 𝑥 last birthday and exposure to risk 𝐸𝑥 is the average number of individuals in the observation year who were 𝑥 years old on their last birthday. Notice that 𝐸𝑥 is based on a population estimate of people who were 𝑥 years old on their last birthday in the middle of the observation year. 𝐸𝑥𝑡 𝑐 represent the central exposed to risk at age 𝑥 in year 𝑡, and 𝐸𝑥 𝑜 denotes the initial exposed to risk for all arrays of 𝑥-age and 𝑡-year comprising ages (on the rows) 𝑥 = 𝑥1,𝑥2,𝑥3 ..., 𝑥𝑘, and calendar years (on the columns) 𝑡 = 𝑡1,𝑡2,𝑡3 ...,𝑡𝑛, 3.1.4. The force of mortality (𝝁𝒙,𝒕) 𝜇𝑥,𝑡 represents the hazard rate for mortality for an individual at exactly age x and dies at the exact t years. The force of mortality related to the death probability as 𝜇𝑥,𝑡 = lim 𝑑𝑥→0+𝑃𝑟[𝑇0≤𝑥+𝑑𝑥|𝑇0>𝑥] 𝑑𝑥 𝜇𝑥𝑑𝑥≈ lim 𝑑𝑥→0+𝑃𝑟[𝑇0≤𝑥+𝑑𝑥|𝑇0>𝑥] 𝜇𝑥=−𝑑 𝑑𝑥𝑆0(𝑥) 𝑆0(𝑥) (4) 𝐺𝑜𝑚𝑝𝑒𝑟𝑡𝑧: 𝜇𝑥=𝐵𝐶𝑥,0<𝐵<1,𝑐>0 3.1.5. Life expectancy (𝒆𝒙,𝒕) 𝑒𝑥,𝑡 means that an individual of the given age 𝑥 can expect to live with time 𝑡 an additional number of years on average. Life expectancy, which is equivalent to the total life span, is most common at birth. 𝑒𝑥=𝑇𝑥 𝐿𝑥 (5) But life expectancy for a given age in which the age plus life expectancy is equal to the total life expectancy. Consider the mortality model that represents the model which examines the structure of probability of death or central mortality rates across ages and or years. ln[𝑚𝑥(𝑡)−𝛼×]≈𝜌1𝑈𝑥,𝑖𝑉𝑖,𝑡 (14) Thus, estimates of 𝛽𝑥 and 𝑘𝑡 can be obtained: 𝛽𝑥=𝑈𝑥,𝑖 ∑𝑈𝑥,𝑖𝑥 , (15) 𝑘𝑡=𝜌1𝑉1,𝑡∑ 𝑈𝑥,1 𝑥 (16) The fitting effect of the singular value decomposition depends on the efficiency of extracting from the ln𝑚𝑥(𝑡)−𝛼𝑥 matrix. It is generally considered that the method can explain more than 90% of the sum of squares of deviations. After obtaining the estimated values of the parameters, it can be found that 𝛼𝑥 𝑎𝑛𝑑 𝛽𝑥 are fixed over time, and the variety of mortality over time is mainly reflected by (𝑘𝑡). The prediction value of future mortality can be obtained by extrapolating (𝑘𝑡). It is believed that (𝑘𝑡) is a random walk with drift or ARIMA process. According to the BIC information criterion, (𝑘𝑡). Should be the AR IMA (0,1,1) Model with drift term. Advantages of the Lee-Carter model It provides a good fit for the historic data. The 𝛼𝑥 aging function makes it possible for a model to be employed at all ages, even young ages, when the life table shape can be very complex, while the k t term represents the prevalent tendency in mortality evolution. It is simple to fit with relatively few parameters, particularly compared to other complicated models and both the original decomposition of the single value and Brouhns et al (2002) are well understood and easy to implement Poisson Likelihood fitting model. The project is easy. Because of the common linear trend of most datasets in 𝜅𝑡's, the random walk-in drift time series is used extensively for estimating the future central mortality rates. It is a simple concept to grasp. Both 𝛼𝑥 and 𝜅𝑡 are easily understood as the shape of mortality across ages and the level of mortality each year, which is useful when reporting results to a larger audience. Disadvantages of Lee-Carter Model It has only one-period term 𝜅𝑡, which indicates that the change in all the central mortality rates in each year of the projection is perfectly tied to the unrealistic problem and to the risk of liabilities and securities, based on the central mortality rate. 𝛽𝑥 does not have universal interpretation and can make unpredictable projections? The shape of a 𝛽𝑥 becomes important when the central mortality rate is projected because a model fitted into a long range of historical data will continue to show high rates of improvement at the younger age and, at higher age rates, which might be unlikely. There is no provision for “cohort” impacts based on a person's birth year. Renshaw and Haberman were among the first to propose models based on the Lee-Carter model but integrating cohort effects (2006). 3.5.2. Other Models II. The Cairns-Blake-Dowd model To address perceived problems with the Lee-Carter model and overcome problems with projected death rates in single age/period term models, Cairns et al. introduced one of the most popular competing models of the LC model, the Cairns-Blake-Dowd model (2006). The Cairns-Blake-Dowd model presumes that death probabilities can be modeled as 𝑙𝑜𝑔𝑖𝑡(𝑞𝑥,𝑡)= 𝜅𝑡 (1)+(𝑥−𝑥)𝜅𝑡 (2) (17) The logit of death probabilities is a linear age function, which is reasonable for high age (about 50 years old) but is not true for the younger age. It is assumed. The 𝜅𝑡 (1)parameter determines death levels over all years for a certain year in the Cairns-Blake-Dowd model. The 𝜅 𝑡 (2)parameter determines the 'aging rate' of each year, i.e., an increase in mortality between one age and the following age. Cairns et al. presented a predictor structure with two age-period terms (𝑁 = 2), agemodulating parameters 𝛽𝑥 (1) = 1 and 𝛽𝑥 (2)= 𝑥 – 𝑥,, no static age function, and no cohort effect (2006). The CBD model predictor is provided by: 𝜂𝑥𝑡 = 𝜅𝑡 (1) + (𝑥 – 𝑥)𝜅𝑡 (2) (18) Where: 𝑥 represent the average age. Advantage of the Cairns-Blake-Dowd model The Cairns-Blake-Dowd model is a commonly used mortality model, particularly among practitioners concerned with the riskiness of liabilities tied to high-risk death probability, such as annuities. In comparison to the Lee-Carter model, it provides for a more sophisticated correlation structure between distinct death probability. This is especially significant when assessing the possible riskiness of liabilities, such as for insurance solvency considerations. It is simple to put together. Because there are no age functions in the model, it can be fitted using least squares or likelihood maximization approaches to produce a satisfactory fit to the historical data when utilized over long periods of time. It provides smooth estimates for death probabilities for every given year. This is preferable if it is believed that the basic processes determining mortality should not change as people age. It is simple to project. For projecting the 𝜅𝑡 parameters across a number of countries, the bivariate random walk with drift has proven to be a reliable and robust model. In addition to aggregate measures of longevity such as period life expectancy, it provides stochastic forecasts with confidence ranges for individual 𝑞𝑥,𝑡's that are deemed to be realistic in contrast to previous evidence. Disadvantages of Cains. Blake-Dowd model Models based on the Cairns-Blake-Dowd model that include cohort effects have recently been presented, most notably in Cairns et al (2009) and Platts et al (2009). It does not fit data well across the board. The assumption of linearity in 𝑙𝑜𝑔𝑖𝑡(𝑞𝑥,𝑡) is no longer reasonable below the age of 50, and it may not be reasonable even at highly advanced ages (above 90). There have been attempts to accommodate this by introducing an age function 𝛼𝑥, similar to that found in the Lee-Carter model, for example in Platt (2009). III. The P-splines model Currie et al. (2004) proposed the P-splines model as a mechanism for reliably smoothing and predicting central mortality rates. It is founded on Eilers' and Marx's use of penalized B-splines (1996). A "spline" is a piecewise polynomial function defined across a range of values. A family of splines known as a basis of splines (also known as B-splines) is large enough to cover the complete range of an interest. The linear sum of the B-splines can then be used to smooth any discontinuous function over this range. The number of splines employed and where the knots are placed have a significant impact on the smoothing accomplished by this method. This P-spline was used by Currie et al (2004) on two-dimensional mortality data to smooth the crude estimates of central death rates over ages and years. They also predicted central mortality rates into the future by using missing values in the model for future years. The P-splines model implies that the force of mortality may be represented as a linear combination of smooth functions over time and space, i.e. log𝑚(𝑥,𝑡)=∑𝜃𝑖𝑗𝛽𝑖𝑗(𝑡,𝑥) 𝑖𝑗 (19) Where: 𝛽𝑖𝑗(𝑡,𝑥) is the predetermined foundation function with regularly spread knots, and the 𝛽𝑖𝑗 is the age and cohort parameters to be calculated. It is commonly recognized that the use of splines can result in over-fitted functions, resulting in unnecessarily lumpy fitted mortality surfaces. Advantages of P-splines model The P-splines method has become widely used for smoothing historical data, most notably by the Continuous Mortality Institute for producing deterministic mortality projections – for example, in CMI (2002) and CMI (2004). (2009b). It gives values that are smooth across age and time for central mortality rates and is thus excellent in removing the effect of random noise from the crude data. It's relatively unpleasant. The smoothing procedure reduces the total number of model parameters and reduces the effective number of free parameters further with the penalty function. It provides projections to allow for changes in the central mortality rates of various ages on the basis of the observations. Disadvantages of P-splines model Its explanation and implementation are complex. There is no intuitive meaning to parameters, and the fitting procedure used by Currie et al (2004) and Currieet al (2006) involves manipulating very large matrices that reduce the fitting speed and can cause computer memory allocation problems. The surfaces are fitted that can be considered too smooth. The P-spline method itself tries to reduce the impact of shocks on the data to alleviate potentially valid characteristics such as a one-off increase in the central death rate due to an epidemic. There are no stochastic projections available. Instead of allowing future rates to be generated by a stochastic process, the P-splines model fits a deterministic surface to the data and extends it into the future. Currie (2006) attempts to provide “confidence intervals” for future projections, but these are dependent on errors in estimating the underlying parameters rather than being truly stochastic. It does not consider “cohort” impacts. " If desired, the P-splines model can be changed from an age/period to an age/cohort model, as described by CMI (2006), although this removes the period effects, which are frequently dominating and cause problems because some cohorts have limited observations. IV. The CMI Model The Continuous Mortality Investigation (CMI) developed the CMI mortality projection model (2009). It is a model for mortality improvement rates rather than mortality rates themselves, as the previous models for mortality were. The mortality improvement rates are defined as 𝑟𝑥𝑡 = 1− 𝑞𝑥𝑡 𝑞𝑥,𝑡−1 (20) To derive the pattern of mortality improvements, the structure of mortality rates in a population is analyzed over age, time, and years. The age/period and cohort components discovered are then assumed to persist for several years before blending into a userspecified “long-term rate of improvement.” Advantage of CMI Model Based on a single and relatively simple input from the user, it can quickly generate a central projection of mortality rates. This is extremely beneficial for actuarial consultants who work primarily in deterministic environments (for instance, valuation of pension schemes or reserving for life assurance). In this context, it can also serve as a "common currency" for translating the pattern of improvements in mortality rates or life expectancy observed in another model (for example, the Lee-Carter model) into a roughly equivalent long-term rate of improvement. Disadvantage of CMI Model The CMI model's inability to generate stochastic projections of mortality rates means that it is unsuitable for measuring the risk inherent in any projection, except when comparing competing scenarios. It is also a very complex model when compared to the other models used, though this complexity is largely hidden from the intended end user and is only visible here because the methodology must be applied to different datasets. 3.6. Regulatory Framework of Mortality Assumption Mortality assumptions used in the valuation of pension and annuity liabilities are typically presented in the form of a table, with the probability of death over the next years, 𝑞𝑥, given for each individual age 𝑥. Usually, different assumptions are used for males and females, however certain districts' regulations necessitate the use of unisex rates. Tables of mortality can be one-dimensional, accounting just for differences in death by age, or two-dimensional, accounting for mortality evolution through time. Onedimensional tables, often known as static tables, have only one death rate for each age group. As many years of sufficient mortality experience are required, establishing assumptions for predicted mortality improvement needs substantially more data and is thus more difficult to set. As a result, mortality improvement assumptions are frequently based on general population mortality. After the mortality assumptions have been determined, they can be applied to the initial mortality level to establish a generational table giving the mortality assumption at any future point in time. They are commonly used in the following ways, where 2000 is the year in which the initial level of mortality was determine and 𝑟 is the annualized rate of mortality improvement for age 𝑥: 𝑞𝑥,2000+𝑡 =𝑞𝑥,2000((1−𝑟𝑥)𝑡 (21) In practice, 𝑟 may vary over time, but it usually just varies by age and gender. 3.6.1. Mortality Assumptions in Practice and Regulation The regulatory framework may demand the use of specialized mortality tables. These tables indicate minimal mortality assumptions and may or may not account for future improvements in mortality and life expectancy. However, when minimum tables are necessary, pension funds and annuity providers are often allowed to employ mortality tables that are more conservative than those required in order to account for and prepare for significant future improvements in mortality and life expectancy if deemed suitable. Where the legislative framework does not create specific mortality tables, pension funds and annuity providers may use their own tables, or the tables most commonly used by the industry. The extent to which mortality assumptions are regulated varies greatly between countries and is not always uniform between pension funds and annuity providers within the same country. Table 1 illustrates whether the regulation mandates minimum mortality assumptions or whether the regulation requires that future improvements in mortality be accounted for in the assessment of pension and annuity liabilities, while the specific assumptions to be used are not required. The analysis evaluates whether it is standard market practice to account for future mortality improvement in the pricing of liabilities, even if regulation does not demand it. In half of the countries, neither pension funds nor annuity providers are required to account for future mortality improvement. Despite the lack of a legislative obligation, the majority of countries do so in practice, with annuity providers doing so more frequently than pension funds. Table 1: Mortality tables and improvement required by regulation and used in practice Country Minimum table required by Regulations Mortality Improvement required by Regulations Mortality Improvements used in Practice Annuity Providers Pension Plans Annuity Providers Pension Plans Annuity Providers Pension Plans Brazil No Yes No No No No Canada No Yes Yes Yes Yes Yes Chile Yes Yes Yes Yes Yes Yes China Yes Yes No No No No France Yes Yes Yes Yes Yes Yes Germany Yes Yes/No Yes Yes Yes Yes Israel Yes Yes Yes Yes Yes Yes Japan No Yes No No Yes No Korea No No No No No No Mexico Yes No Yes No Yes No Netherland No No Yes Yes Yes Yes Peru Yes Yes No No Some Some Spain No No Yes Yes Yes Yes Switzerland No No No No Yes Some United Kingdom No No Yes Yes Yes Yes United States Yes Yes No Yes Yes Yes Source: OECD Notes: The statistical data for Israel are supplied by and under the responsibility of the relevant Israel authorities. The use of such data by the OECD is without prejudice to the status of the Golan Heights, Wast Jerusalem and Israeli settlements in the West Bank under the terms of international law. 1. For non/regulated Pensionskassen and insurance oriented Pensionsfonds. 2. For regulated Pensionskassen and non/insurance oriented Pensionsfonds. Despite the lack of a legal obligation to provision for mortality improvement, the majority of countries do so in practice, with annuity providers doing so more frequently than pension funds. In practice, thirteen of the sixteen nations' annuity providers use mortality improvement assumptions, whereas only eleven of the sixteen countries' pension funds do. 3.6.2. Standard Mortality Table The analysis is based on the situation in which conventional mortality tables are used by pension funds and annuity providers. Mortality rates for most plans will be based on standard tables created and published by the Society of Actuaries or governmental organization. Tables are often titled based on (1) the types and characteristics of data underlying the table and (2) because mortality rates generally change over time, the calendar year of experience that the mortality rates are assumed to represent. In most cases, detailed information about the data's source is included in the report that is published alongside the table. In addition, a breakdown of table rates for subgroups may be provided. 3.6.3. Mortality Improvement Current mortality tables, which have been specifically constructed for the retirement area, typically have no room for future mortality improvement. Most Society of Actuaries mortality tables used in the retirement area, however, include projection scales for use in estimating future mortality improvement. These scales are typically differentiated by age and gender. Table 2:Mortality Projection Scale AA compiled by the Society of Actuaries Age Male Female 60 .016 .005 61 .015 .005 62 .015 .005 63 .014 .005 64 .014 .005 65 .014 .005 66 .013 .005 67 .013 .005 68 .014 .005 69 .014 .005 70 .015 .005 Source: Mortality Projection Scale AA compiled by the Society of Actuaries For full scale see Table 7-3 in RP-2000 Mortality Table, https://www.soa.org/globalassets/assets/Files/Research/ExpStudy/rp00_mortalitytables.pdf These scales are used to reduce the likelihood of death in the following way: 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝑑𝑒𝑎𝑡ℎ 𝑤𝑖𝑡ℎ 𝑛 𝑦𝑒𝑎𝑟𝑠 𝑜𝑓 𝑚𝑜𝑟𝑡𝑎𝑙𝑖𝑡𝑦 𝑖𝑚𝑝𝑟𝑜𝑣𝑒𝑚𝑒𝑛𝑡 = (𝑚𝑜𝑟𝑡𝑎𝑙𝑖𝑡𝑦 𝑟𝑎𝑡𝑒 𝑎𝑡 𝑎𝑔𝑒 𝑥) (1 − 𝑝𝑟𝑜𝑗𝑒𝑐𝑡𝑖𝑜𝑛 𝑠𝑐𝑎𝑙𝑒 𝑣𝑎𝑙𝑢𝑒 𝑎𝑡 𝑎𝑔𝑒 𝑥)𝑛 3.6.4. Generational Mortality Improvement If it is assumed that the forces leading to mortality improvement will continue in the future, then mortality rates will vary by both age and the calendar year of attainment of age, because those attaining the age later will be exposed to the forces leading to mortality improvement for a longer time period. Thus, the probability of dying at 60 would be higher for a person turning 60 in 2012 than for a person becoming 60 in 2016. Another way to look at it is that various generations (those born in 1952 versus those born in 1956) will have different mortality rates at the age of 60. To account for this difference, projection scales for the number of years between the valuation year and the year the individual reaches a certain age can be used. This is known as the generational approach for projecting mortality improvement. For example, if a valuation is being performed as of January 1, 2014, using a mortality table with mortality rates representative of 2014, a present value factor at age x would use the following mortality rates, where the superscript represents the calendar year in which the individual attains a given age. 𝑞2014𝑥, 𝑞2015𝑥+1 =𝑞2014𝑥+1(1−𝑠𝑐𝑎𝑙𝑒𝑥+1), 𝑞2016𝑥+2 =𝑞2014𝑥+2(1−𝑠𝑐𝑎𝑙𝑒𝑥+2)2,....., 4. EMPIRICAL ANALYSIS This section examines how defined benefit (DB) pension plans would be affected by uncertainty about future mortality and life expectancy outcomes. In this regard, the first step is to assess the uncertainty surrounding future changes in mortality and life expectancy, also known as longevity risk. Second, it considers the impact of longevity risk on defined benefit (DB) pension plans provided by employers. The link between mortality and life expectancy, as well as how life tables are constructed from mortality data, is examined in order to assess the uncertainty surrounding future mortality and life expectancy outcomes. Finally, a focus on the most pressing issue confronting pension funds: forecasting the future path of mortality and life expectancy in order to determine their future liabilities. As a result, the section presents a stochastic approach to modeling mortality and life expectancy uncertainty. It provides the results of estimating the Lee-Carter model for the selected country in this regard. France data was chosen for this study for estimation and modeling. The data for the estimation came from the Society of Actuaries Annuity Mortality database and the human mortality database. The Author used R-programming for analysis of the data. 4.1. Uncertainty About Mortality and Life Expectancy 4.1.1. The relationship between mortality and life expectancy: Life Table For a given population, life tables provide a summary of mortality, survivorship, and life expectancy. They can contain data for each and every year of life (complete life tables) or by 5or 10-year intervals (abridged life tables). A life table can be created in its most basic form by combining a set of age-specific death rates. Age-specific death rates are calculated as the ratio of deaths in a given year to the population size. They're usually expressed in terms of people per 1,000. Mortality rates, on the other hand, are the chances that someone of a specific age will die during the time period under consideration (i.e., the probability of dying). The numerator is the number of individuals from this generation who die between age n and age n+1, and the denominator is the size of the generation who reach age n during the year in question. The annual death rate is different from the annual probability of dying by age because the latter is the proportion of people of that age who die during the year, whereas the probability of dying is the proportion of people of that age dying during the age interval. Therefore, life tables provide a link between mortality and life expectancy. As a result, life tables establish a connection between mortality and life expectancy. The mean number of years still to be lived by a person who has reached that exact age (i.e., age-specific life expectancies) if subjected to the current age-specific probabilities of dying for the rest of his or her life is the final result of a life table. Table 4 shows a life table for males in France 2018. The first column lists the Figure 7: Projected value of 𝒌𝒕 for 100 years Forecasting is the main aim behind the stochastic modeling. One of the noteworthy properties of the LC model is that, once it is fitted (i.e., once values of 𝑎𝑥, 𝑏 𝑥, and 𝑘 𝑡 are found), only the mortality index (𝑘𝑡) over time needs to be forecasted for future time points. Lee and Carter (1992) fitted autoregressive integrated moving average (ARIMA) (0,1,0) (i.e., random walk with drift) for modeling mortality index for French population. The figure below shows the future projected values of 𝑘𝑡𝑠 up to 60years. Figure 8: Pattern of Past and Projected rates for people aged 65 Source: Human Mortality Database (http://www.mortality.org/index.html). Notes: HMD France 5x1 (age by year), Author Calculations Finally, the entire rate pattern is simple to deduce. In this matrix, past and projected rates are both blinded. We present here a pattern of past and projected rates for people over the age of 65 based on different populations. Figure 8 clearly shows the expected improvement. This could be attributed to HIV/AIDS Pandemics, disease, and drugs. We observed that in next decades mortality is expected to decline for both female and male population in France. This is due to decreasing nature of 𝑘𝑡. We have forecasted values of age specific death rate, by using estimated parameters 𝑎𝑥, 𝑏𝑥 and forecasted values of mortality index 𝑘𝑡. 4.3. The impact of longevity risk on defined-benefit private pension plans The impact of longevity risk on employer-provided DB private pension schemes is examined in this section. The previous section demonstrated that forecasting mortality and life expectancy using a stochastic approach allows you to assign probabilities to a variety of possible projections and hence estimate the uncertainty surrounding future mortality and life expectancy outcomes. Private pension funds, on the other hand, are concerned about the impact of this uncertainty on their pension commitments. This section assesses the changes in the net present value of annuity payments as mortality and life expectancy evolves, as this is the principal impact of longevity risk on net pension obligations. These adjustments are assessed for members of pension funds of various ages, as well as pension funds with various age membership structures. 4.3.1. How does longevity risk affect DB private pension plans? Longevity risk has the greatest influence on the net pension liabilities of employerprovided DB private pension plans because of annuity payments. An annuity is a contract in which one person or organization agrees to pay a stream or series of payments to another person or organization (the annuitant) (annuity payments). Annuities are designed to give a constant stream of income to the annuitant over a period of time, which can begin immediately or at any time in the future. Capital gains and investment profits are usually tax-deferred. There are numerous types of annuities. They can be classified in a variety of ways, including: (1) by the underlying investment into fixed or variable; (2) by the primary purpose, i.e., accumulation or pay-out, into deferred or immediate; (3) by the nature of the pay-out commitment into fixed period, fixed amount, or lifetime; and (4) by the premium payment arrangement into single or flexible premium. In a fixed annuity, the insurance company or pension fund guarantees the principle as well as a minimum rate of interest, but in a variable annuity, the annuity payment is based on the underlying portfolio's investment performance. An immediate annuity is intended to pay a lump sum or a series of payments immediately after the annuity is purchased, whereas a deferred annuity pays the annuitant at a later date. Fixed period annuities pay an income for a set length of time (e.g., 10 years), whereas lifetime annuities pay income for the rest of the annuitant's life. A single premium annuity is one that is funded with a single payment, whereas a flexible premium annuity is one that is funded over a series of payments. Only deferred annuities are flexible. Because employer-provided DB private pensions promise their members a guaranteed future stream of payments at retirement for the rest of their lives, the research concentrates on the impact of longevity risk on fixed, deferred, lifetime, and flexible premium annuities throughout. Longevity risk would have a greater impact on annuities that are fixed, deferred, and for the annuitant's lifetime once retirement age is achieved. The impact of longevity risk on fixed period annuities, on the other hand, is less obvious. Furthermore, the extent of the impact of longevity risk on annuity payments would be determined not just by the type of annuity guarantees, but also by how pension funds account for improvements in mortality and life expectancy when calculating the net present value of annuity payments. 4.3.2. How private pension funds account for future improvements in mortality and/or life expectancy? Pension funds do not appear to account fully for projected increases in mortality and life expectancy. Recent study, particularly that of the Actuarial Profession and Cass Business School (2005), discovered that current practice differs significantly across the EU. Pension funds in certain countries account for predicted future improvements in mortality, whilst others use tables based on mortality recorded in the past, without accounting for the possibility that life expectancy will continue to rise (Belgium, Denmark, Norway, Sweden, and Switzerland). Of those countries incorporating an allowance for future improvements in mortality, Austria, France, Germany (for only 25 years and using 1996 as the base year), Ireland (improvements incorporated only until 2010), Italy, the Netherlands, Spain, and the United Kingdom use forecasts; while Canada, Finland, and the United States, despite of having mortality tables with built in mechanisms to take into account future changes in mortality, generally do not use them. Furthermore, there is no standardized or consistent mechanism for accounting for future increases in mortality and life expectancy. In this aspect, assessing longevity risk is challenging due to the lack of a consistent methodology, which makes mortality projections arbitrary and impossible to compare among pension funds, let alone countries. As a result, the impact of the longevity risk is amplified. The impact of future improvements in mortality and life expectancy (i.e., longevity risk) on employer-provided DB private pension plans is compounded by the fact that few actuaries and pension schemes account for future improvements in mortality and life expectancy, and those that do so only partially. Furthermore, even with adjustments for anticipated improvements in mortality, the base tables used for demographic assumptions are nearly ten years old, dating from the early to mid-1990s. Furthermore, the lack of standard methods to forecast mortality and life expectancy, and the fact that these methods are generally far from being fully stochastic complicate any comparative analysis and make the task of examining the impact of longevity risk on pension fund liabilities fuzzier. Furthermore, the lack of standard methods for forecasting mortality and life expectancy, as well as the fact that these methods are far from being totally stochastic, complicates any comparison study and makes the task of analyzing the impact of longevity risk on pension fund liabilities even more hazy. 5. CONCLUSIONS Life expectancy forecasts are necessary for estimating future healthcare and pension costs. The Lee-Carter (LC) model (1992), which forecasts age-specific death rates log bilinearly, is a commonly used model to anticipate mortality. The LC model is employed because parameter estimation is simple, and it provides a good fit over a wide range of ages. The data collection includes data on France's population mortality from 1816 to 2018. The parameters of the LC model are estimated using the Singular Value Decomposition (SVD) method. The mortality values are forecasted using the Auto Regressive Integrated Moving Average (ARIMA) time series model. We forecasted the time-index using a random walk with drift, which is typically found to be an appropriate mode (Callot et al. 2016). The overall pattern of mortality ( 𝑎𝑥) for both female and male populations revealed high infant mortality, an accidental hump around the age of 20, and a nearly exponential increase at older ages. The sensitivity of mortality ( 𝑏 𝑥) has revealed that mortality declines at a higher rate for females aged 25-34 years and males aged 15-24 years than for other ages. The Mortality index ( 𝑘 𝑡) has been declining. Female mortality improvement has outpaced male mortality as well as the series of the general indices clearly tend to decrease, although not monotonically over time. For the first half of the period, there is a significant increase in female mortality over men, which decreases significantly in the second half of the period. The sensitivity of mortality has shown that mortality declines at a rapid rate for people aged 20 to 25. Since World War I and World War II, the mortality index has shown a decreasing trend with two spikes. The predicted Lee Carter model fits France population data well over a wide age range but performs poorly below the age of four and after the age of 55. 5.1. Policy issues Longevity risk, defined as the uncertainty surrounding future developments in mortality and life expectancy, has a non-negligible impact on the liabilities of employer-provided pension plans because lifetime annuity payments are based on the length of time people are expected to live, according to the paper. The impact of this on the net present value of annuity payments for a "theoretical pension fund" was calculated in Table 5. It was discovered that the amount of this influence is determined by the pension fund membership's age structure. As a result, pension funds with a younger membership structure will be more affected by longevity risk since they will be exposed to uncertain changes in mortality and life expectancy for a longer period of time. Unfortunately, the impact of longevity risk is aggravated by the fact that few pension plans account for future changes in mortality and life expectancy, and those that do only account for partial improvements. To make matters worse, most pension funds rely on mortality tables that are almost a decade old. Furthermore, the lack of a consistent technique for calculating longevity risk makes determining the optimal way to account for gains in mortality and life expectancy difficult. Using a common methodology to predict death rates and life expectancy has an obvious advantage in this regard. This research argues for the use of a stochastic model in this case because it allows for the attachment of probabilities and consequently the assessment of the degree of uncertainty around future mortality and life expectancy outcomes. Unfortunately, many small and medium-sized pension funds may lack the financial and technical capabilities to create forecasts using a standardized technique. Government entities may be able to develop them if they have the necessary resources and technical expertise. However, assumptions about total populations rather than specific membership groups of private pension plans may not be useful. Governmental entities might create forecasts for the overall population as well as for various subgroups based on gender, age, wealth, and educational attainment. As a result, separate pension funds could use the subpopulation that most closely reflects their current membership composition. Using mortality tables that differentiate based on socioeconomic position and gender, on the other hand, has its own set of issues because it may give rise to discriminatory issues. Arguments in favor of distinguishing tables include the fact that adopting an average life expectancy index penalizes persons with greater life expectancy (e.g., women, well educated, and well-off people) while rewarding people with lower life expectancy (e.g., men, low educated and low-income people). Furthermore, private pension plans must hedge against their own longevity risk, i.e., the risk associated with their own membership structure, rather than an average longevity risk. Finally, in addition to incorporating mortality improvements through the adoption of a standard methodology and average or differentiated mortality tables, the impact of longevity risk on employer-provided DB plans can be mitigated in part by indexing pension benefits to life expectancy. Indexing benefits to life expectancy, on the other hand, moves some of the longevity risk back to individuals, reducing one of the main reasons people buy annuities. Differentiating between individual and aggregate or cohort longevity risk can be useful in this regard. Individual risk is unique to each person, but it can be easily mitigated by sharing risks. As a result, assuming it by pension funds would be more efficient, as they are best positioned to pool individual unique risks. On the other side, the aggregate or cohort risk is more difficult to address or mitigate. As a result, by indexing benefits to cohort longevity changes, this risk can be borne more easily by pension funds and people. 5.2. Areas in which additional research is required We made an effort to be thorough by identifying and reviewing literature on the subject of longevity risk. However, mortality risk is dynamic, and continual study is required to ensure that the industry is up to date on the current trends. In recent years, this has included increasingly extensive analysis of characteristics such as separating lives into cohorts, focusing on specific causes of death as drivers of mortality, and increasing the roughness of the risk variables used in mortality investigations. There is still opportunity for more complex study, which would only serve to better understanding of mortality and longevity risk profile. On the topic of stochastic mortality models, there is a lot of literature, primarily from academics. These are usually concerned with the shape of the models and how well they fit historical data. One area where there is far less information is the discussion about the practical application of such models. It would be useful to see some in-depth analysis from a company standpoint of the relative costs and benefits of implementing stochastic mortality analysis in various stages of the product cycle (pricing, reserving, managing capital, hedging longevity risk, and so on) and across different product categories (payout annuities, life settlements, etc.). This could be because insurance businesses specialize in this sector, thus all product advancements will most likely originate from within the industry. Insurers, on the other hand, are frequently required to satisfy a variety of stakeholders. Suggestions for new and unique product concepts could be fascinating to see 6. REFERENCES (PDF) Addressing Longevity Heterogeneity in Pension Scheme Design and Reform. (n.d.). Retrieved October 28, 2021, from https://www.researchgate.net/publication/311921892_Addressing_Longevity_Hete rogeneity_in_Pension_Scheme_Design_and_Reform Antolin, P. (2007). 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