Life Annuity Portfolio Research Articles

Solvency analysis of deferred annuities

AbstractWhile the solvency analysis of immediate life annuity portfolios has been extensively studied, the case of deferred annuities has received comparatively much less attention. We assess the importance and effect of stochastic mortality models and interest rates on the solvency analysis of a portfolio of deferred annuity contracts. Our analysis considers three steps: first, the benchmark case where mortality rates and interest rates are both deterministic; then, the case in which only mortality rates are stochastic is explored; finally, the full model where both mortality rates and interest rates are stochastic. The results demonstrate the model risk stemming from the uncertainty in the mortality models and its impact on the evaluation of solvency margins for life annuities. The role of the deferment period is thoroughly discussed and compared to the case of immediate annuities.

Special-Rate Life Annuities: Analysis of Portfolio Risk Profiles

Special-rate life annuities are life annuity products whose single premium is based on a mortality assumption driven (at least to some extent) by the health status of the applicant. The health status is ascertained via an appropriate underwriting step (which explains the alternative expression “underwritten life annuities”). Better annuity rates are then applied in presence of poor health conditions. The worse the health conditions, the smaller the modal age at death (as well as the expected lifetime), but the higher the variance of the lifetime distribution. The latter aspect is due to significant data scarcity as well as to the mix of possible pathologies leading to each specific rating class. A higher degree of (partially unobservable) heterogeneity inside each sub-portfolio of special-rate annuities follows, and this results in a higher variability of the total portfolio payout. The present research aims at analyzing the impact of extending the life annuity portfolio by selling special-rate life annuities. Numerical evaluations have been performed by adopting a deterministic approach as well as a stochastic one, according to diverse assumptions concerning both lifetime distributions and portfolio structure and size. Our achievements witness the possibility of extending the annuity business without taking huge amounts of risk. Hence, the risk management objective “enhancing the company market share” can be pursued without significant worsening of the annuity portfolio risk profile.

Optimal Strategies for Retirees in Australia With Realistic Risk Transfer

We investigate optimal investment and drawdown decisions for Australian retirees, allowing for a range of risk transferring options that could be implemented by superannuation funds. Retirees can allocate their wealth between a life annuity, deferred life annuity, and growth and defensive portfolios in an account-based pension. The preferred asset mix varies significantly with balance and home ownership, and preferences as modelled using both loss aversion and constant relative risk aversion utility. We find that strong loss aversion leads to hedging strategies to secure the target consumption, as reflected in high desire for annuities and subsequent asset allocation and drawdown decisions. Desire for annuities also emerges under risk aversion to both smooth consumption and limit its decline after exhaustion of the account-based pension. We also highlight interactions between the Age Pension and the decisions of Australian retirees. Our results have implications for the design of comprehensive income products for retirement in the Australian market.

Pricing longevity derivatives via Fourier transforms

Longevity-linked derivatives are one of the most important longevity risk management solutions for pension schemes and life annuity portfolios. In this paper, we decompose several longevity derivatives—such as geared longevity bonds and longevity-spread bonds—into portfolios involving longevity options. For instance, we show that the fair value of an index-based longevity swap can be broken down into a portfolio of long and short positions in European-style longevity caplets and floorlets, with an underlying asset equal to a population-based survivor index and strike price equal to the initial preset survivor schedule. We develop a Fourier transform approach for European-style longevity option pricing under continuous-time affine jump–diffusion models for both cohort mortality intensities and interest rates, accounting for both positive and negative jumps in mortality. The model calibration approach is described and illustrative empirical results on the valuation of longevity derivatives, using U.S. total population mortality data, are provided.

Heterogeneity in mortality: a survey with an actuarial focus

Heterogeneity in mortality is due to differences among the individuals, which are caused by various risk factors. Some risk factors are observable while others are unobservable. The set of observable risk factors clearly depends on the type of population addressed. The impact of observable risk factors on individual mortality, in particular when they also constitute “rating factors” in the calculation of premiums and other actuarial values, is usually expressed approximately, according to some pragmatic approach. For example, additive or multiplicative adjustments to the average age-specific mortality are frequently adopted. Heterogeneity due to unobservable risk factors can conversely be expressed by adopting the concept of individual “frailty”, which, however, can be interpreted and consequently modeled in several ways, according to the causes which are considered as originating the frailty itself: congenital characteristics, environmental features, lifestyle aspects, etc. In this paper, we provide a survey of scientific contributions on heterogeneity in mortality, focusing on modeling both observable and unobservable heterogeneity. We start with an overview of methodological contributions to heterogeneity and frailty modeling, coming from both the demographical and the actuarial context. We then shift to contributions analyzing the impact of frailty, in its various interpretations, on the results (cash flows, profits, etc.) of life insurance and life annuity portfolios and related risk profiles.

Managing Systematic Mortality Risk in Life Annuities: An Application of Longevity Derivatives

This paper assesses the hedge effectiveness of an index-based longevity swap and a longevity cap for a life annuity portfolio. Although longevity swaps are a natural instrument for hedging longevity risk, derivatives with non-linear pay-offs, such as longevity caps, provide more effective downside protection. A tractable stochastic mortality model with age dependent drift and volatility is developed and analytical formulae for prices of longevity derivatives are derived. The model is calibrated using Australian mortality data. The hedging of the life annuity portfolio is comprehensively assessed for a range of assumptions for the longevity risk premium, the term to maturity of the hedging instruments, as well as the size of the underlying annuity portfolio. The results compare the risk management benefits and costs of longevity derivatives with linear and nonlinear payoff structures.

Immunization and Hedging of Post Retirement Income Annuity Products

Designing post retirement benefits requires access to appropriate investment instruments to manage the interest rate and longevity risks. Post retirement benefits are increasingly taken as a form of income benefit, either as a pension or an annuity. Pension funds and life insurers offer annuities generating long term liabilities linked to longevity. Risk management of life annuity portfolios for interest rate risks is well developed but the incorporation of longevity risk has received limited attention. We develop an immunization approach and a delta-gamma based hedging approach to manage the risks of adverse portfolio surplus using stochastic models for mortality and interest rates. We compare and assess the immunization and hedge effectiveness of fixed-income coupon bonds, annuity bonds, as well as longevity bonds, using simulations of the portfolio surplus for an annuity portfolio and a range of risk measures including value-at-risk. We show how fixed-income annuity bonds can more effectively match cash flows and provide additional hedge effectiveness over coupon bonds. Longevity bonds, including deferred longevity bonds, reduce risk significantly compared to coupon and annuity bonds, reflecting the long duration of the typical life annuity and the exposure to longevity risk. Longevity bonds are shown to be effective in immunizing surplus over short and long horizons. Delta gamma hedging is generally only effective over short horizons. The results of the paper have implications for how providers of post retirement income benefit streams can manage risks in demanding conditions where innovation in investment markets can support new products and increase the product range.

Measuring Risk-Adjusted Performance and Product Attractiveness of a Life Annuity Portfolio

The paper proposes a new methodological approach for the product performance analysis into the actuarial context. Two indexes are proposed as restyled versions of the corresponding most popular ones: They have been adapted into the actuarial assessment preserving the plainness in the interpretation of the numerical results. The paper offers a practical implementation of the new approach in the case of a specific contract, containing itself innovative profiles: It concerns a life annuity in which the installments are scaled by a demographic index and contains an embedded option linked to the financial profit participating quota. It is a new life product linked at the same time to the financial and demographic volatility. The product project is studied in its profitability performance assuming stochastic hypotheses for the financial and demographic systematic risks. The indexes are implemented in a conditional quantile simulated framework and tables and graphs illustrate their trends as function of time. The results give an example of the usefulness of the proposed indexes in the phase of decisions about the product design feasibility. Moreover some suggestions concerning the consumer’s perception of the contract profitability are obtained by means of a utility-equivalent fixed annuity.

Life Annuities. Products, Guarantees, Basic Actuarial Models.

These Lecture Notes aim at introducing technical and financial aspects of the life annuity products, with a special emphasis on the actuarial valuation of life annuity benefits. The text has been planned assuming as target readers:- advanced undergraduate and graduate students in Economics, Business and Finance;- advanced undergraduate students in Mathematics and Statistics, possibly aiming at attending, after graduation, actuarial courses at a master level;- professionals and technicians operating in insurance and pension areas, whose job may regard investments, risk analysis, financial reporting, and so on, hence implying communication with actuarial professionals and managers.Given the assumed target, the use of complex mathematical tools has been avoided.We assume that the reader has attended courses providing basic notions of Financial Mathematics (interest rates, compound interest, present values, accumulations, annuities, etc.) and Probability (probability distributions, conditional probabilities, expected value, variance, etc). As mentioned, Mathematics has been kept at a rather low level. Indeed, all topics are presented in a “discrete” framework, thus not requiring analytical tools like differentials, integrals, etc.The Lecture Notes are organized as follows. Guarantees and options in life contingency products, and in life annuities in particular, are sketched in Chap. 1. In Chap. 2 the basic actuarial aspects are introduced with reference to conventional life annuity products, starting from expected present value definitions, then moving to premium and reserve calculations.A more general framework is proposed in Chap. 3, in order to introduce a wide range of life annuity products. The guarantee structure implied by various products is analyzed in Chap. 4, while Chap. 5 focusses on the time profile of the annuity benefits.Options and riders which can be added to life annuity products are dealt with in Chap. 6, while Chap. 7 focusses on annuity rates adopted to determine the annuity premiums. Cross-subsidy mechanisms working in life annuity portfolios are addressed in Chap. 8, where special attention is placed on “tontine” schemes.Strategies, which can be adopted to get the post-retirement income and, to some extent, can constitute alternatives to the immediate full annuitization of resources available at retirement, are described in Chap. 9.Long-term care insurance (LTCI) is briefly addressed in Chap. 10, with a special focus on life annuity benefits combined with LTCI benefits. Finally, Chap. 11 concludes with suggestions for further reading.

Redistribution of longevity risk: The effect of heterogeneous mortality beliefs

Existing literature regarding the natural hedge potential that arises from combining different longevity-linked liabilities typically does not address the question how changes in the liability mix can be obtained. We consider firms who aim to exploit the benefits of natural hedge potential by redistributing their risks, and characterize the risk redistributions that will arise when the parties bargain for a redistribution of risk that weakly benefits them all. We analyze the effects of heterogeneity in the beliefs regarding the probability distribution of future mortality rates on the properties of these risk redistributions, and provide a numerical illustration for a case where an insurer with a portfolio of term assurance contracts and a pension fund with a portfolio of life annuities redistribute their risks.

Frailty and Risk Classification for Life Annuity Portfolios

Life annuities are attractive mainly for healthy people. In order to expand their business, in recent years, some insurers have started offering higher annuity rates to those whose health conditions are critical. Life annuity portfolios are then supposed to become larger and more heterogeneous. With respect to the insurer’s risk profile, there is a trade-off between portfolio size and heterogeneity that we intend to investigate. In performing this, there is a second and possibly more important issue that we address. In actuarial practice, the different mortality levels of the several risk classes are obtained by applying adjustment coefficients to population mortality rates. Such a choice is not supported by a rigorous model. On the other hand, the heterogeneity of a population with respect to mortality can formally be described with a frailty model. We suggest adopting a frailty model for risk classification. We identify risk groups (or classes) within the population by assigning specific ranges of values to the frailty within each group. The different levels of mortality of the various groups are based on the conditional probability distributions of the frailty. Annuity rates for each class then can be easily justified, and a comprehensive investigation of insurer’s liabilities can be performed.

Frailty and Risk Classification for a Life Annuity Portfolio

Pools of voluntary life annuities typically report lower mortality rates than the general population, as when the product is quoted at standard rates, it is attractive mainly for healthy people. In order to expand their portfolio, in recent years some insurers have started offering higher annuity rates to those whose health conditions are critical. The portfolio can then become larger, but also more heterogeneous. While a larger size implies an improved pooling effect, a higher risk profile follows from the increased heterogeneity. We intend to investigate the result of this trade-off.In performing this analysis, there is a second, and possibly, more important issue that we address. In actuarial practice, mortality rates for life annuities are usually obtained from a population (projected) life table. The different mortality level arising from the (self-)selection process is represented by applying adjustment coefficients to the population mortality rates. According to common actuarial practice, such coefficients are chosen empirically, calibrated on the average ratio (possibly measured for age-groups) between the annuitants’ and the population mortality rates; conversely, a model formally justifying such a difference is not adopted.The heterogeneity of a population in respect of mortality can formally be described with a frailty model. The main findings of the model for a population as a whole are well-known. We suggest to adopt the model for risk classification. We identify risk groups (or classes) within the population by assigning specific ranges of values to the frailty within each group. Conditional probability distributions for the frailty are obtained for each risk class, which allow us to describe the different levels of mortality of the various groups. Different values for the annuity rate derive from the different assumptions about the frailty level of the specific risk class.Finally, the value of insurer’s liabilities is investigated, in terms of expected value and dispersion of the probability distribution of the present value of future benefits. Portfolios without and with risk classification are compared, so to measure the trade-off between portfolio size and heterogeneity.

A comparison of risk transfer strategies for a portfolio of life annuities based on RORAC

ABSTRACTThis paper aims to compare different reinsurance arrangements in order to reduce the longevity and financial risk originated by a life insurer while managing a portfolio of annuities policies. Linear and nonlinear reinsurance strategies as well as swap like agreements are evaluated via a discrete-time actuarial risk model. Specifically, longevity dynamics are represented by Lee–Carter type models, while interest rate is modeled by Cox–Ingersoll–Ross model. The reinsurance strategies effectiveness is evaluated according to the Return on Risk Adjusted Capital under a ruin probability constrain.

Assessing Longevity Risk in a Portfolio of Life Annuities

In this paper we focus on longevity risk when mortality laws are used for the patterns of death description in a population. The notion of longevity risk is strictly linked to parameter uncertainty that suggests the estimation output should be an interval instead of a single value for each parameter. To this aim we first create bootstrapped samples and then, for each cohort in a sample, estimate model parameters. The dynamics of each parameter is then modeled with an autoregressive process. Finally, we simulate future scenarios for yearly probabilities of death and use them in life annuity portfolio pricing.

Immunization and Hedging of Longevity Risk

Pension funds and life insurers offering annuities hold long term liabilities linked to longevity. Risk management of life annuity portfolios aims to immunize or hedge both interest rate and mortality risks. Standard fixed interest duration-convexity hedging must be adapted to allow for both interest rate and longevity risk. We develop an immunization approach along with a delta-gamma based approach allowing for both risks incorporating models for mortality and interest rate risk. The immunization and hedge effectiveness of fixed-income coupon bonds, annuity bonds, as well as longevity bonds, is compared and assessed using simulations of portfolio surplus outcomes for an annuity portfolio. Fixed-income annuity bonds can more effectively match cash flows and provide additional hedge effectiveness over coupon bonds. Longevity bonds, including deferred longevity bonds, reduce risk significantly compared to coupon and annuity bonds, reflecting the long duration of the typical life annuity and the exposure to longevity risk. Longevity bonds are shown to be effective in immunizing surplus over short and long horizons. Delta gamma hedging is shown to only be effective over short horizons.

Managing Systematic Mortality Risk in Life Annuities: An Application of Longevity Derivatives

Developing a liquid longevity market requires reliable and well-designed financial instruments. An index-based longevity swap and a cap are analyzed in this paper under a tractable stochastic mortality model. The model is calibrated using Australian mortality data and analytical formulas for prices of longevity derivatives are provided. Hedge effectiveness is examined under a hypothetical life annuity portfolio subject to longevity risk. The paper presents various hedging features exhibited by a longevity swap and a cap based on different assumptions underlying the market price of longevity risk, the term to maturity of hedging instruments, as well as the size of the underlying annuity portfolio. The results are demonstrated to have important implications for the optimal use of longevity hedging instruments with linear and nonlinear payoff structures.

Mortality regimes and longevity risk in a life annuity portfolio

This paper explores the presence of changes of trends or jumps in French mortality from 1947 to 2007, and assesses their implications on the longevity risk management of a life annuity portfolio. We accomplish this by extending the Poisson log-bilinear regression developed by Brouhns et al. (2002) with a regime-switching model. Estimation results show that French mortality is characterized by two distinct regimes. One refers to a strong uncertainty state, which corresponds to the longevity conditions observed during the decade following World War II. The second regime is related to the low volatility of longevity improvements observed during the last 30 years. We use these results to analyze the impact of mortality regimes on the longevity risk management of a life annuity portfolio. Simulation results suggest that the changes of trends in the mortality process have some implications for longevity risk management.

Forecasting mortality for small populations by mixing mortality data

In this paper we address the problem of projecting mortality when data are severely affected by random fluctuations, due in particular to a small sample size, or when data are scanty. Such situations may emerge when dealing with small populations, such as small countries (possibly previously part of a larger country), a specific geographic area of a (large) country, a life annuity portfolio or a pension fund, or when the investigation is restricted to the oldest ages. The critical issues arising from the volatility of data due to the small sample size (especially at the highest ages) may be made worse by missing records; this is the case, for example, of a small country previously part of a larger country, or a specific geographic area of a country, given that in some periods mortality data could have been collected just at an aggregate level.We suggest to ‘replicate’ the mortality of the small population by mixing appropriately the mortality data obtained from other populations. We design a two-step procedure. First, we obtain the average mortality of ‘neighboring’ populations. Three alternative approaches are tested for the assessment of the average mortality; conversely, the identification and the weight of the neighboring populations are obtained through (standard) optimization techniques. Then, following a sort of credibility approach, we mix the original mortality data of the small population with the average mortality of the neighboring populations.In principle, the approach described in the paper could be adopted for any population, whatever is its size, aiming at improving mortality projections through information collected from other groups. Through backtesting, we show that the procedure we suggest is convenient for small populations, but not necessarily for large populations, nor for populations not showing noticeable erratic effects in data. This finding can be explained as follows: while the replication of the original data implies the increase of the size of the sample, it also involves a smoothing of data, with a possible loss of specific information relating to the group referred to. In the case of small populations showing major erratic movements in mortality data, the advantages gained from the larger sample size overcome the disadvantages of the smoothing effect.

Forecasting Mortality for Small Populations by Mixing Mortality Data

In this paper we address the problem of projecting mortality when data are severely affected by random fluctuations, due in particular to a small sample size, or when data are scanty. Such situations may emerge when dealing with small populations, such as small countries (possibly previously part of a larger country), a specific geographic area of a (large) country, a life annuity portfolio or a pension fund, or when the investigation is restricted to the oldest ages. The critical issues arising from the volatility of data due to the small sample size (especially at the highest ages) may be made worse by missing records; this is the case, for example, of a small country previously part of a larger country, or a specific geographic area of a country, given that in some periods mortality data could have been collected just at an aggregate level. We suggest to 'replicate' the mortality of the small population by mixing appropriately the mortality data obtained from other populations. We design a two-step procedure. First, we obtain the average mortality of 'neighboring' populations. Three alternative approaches are tested for the assessment of the average mortality; conversely, the identification and the weight of the neighboring populations are obtained through (standard) optimization techniques. Then, following a sort of credibility approach, we mix the original mortality data of the small population with the average mortality of the neighboring populations. In principle, the approach described in the paper could be adopted for any population, whatever is its size, aiming at improving mortality projections through information collected from other groups. Through backtesting, we show that the procedure we suggest is convenient for small populations, but not necessarily for large populations, nor for populations not showing noticeable erratic effects in data. This finding can be explained as follows: while the replication of the original data implies the increase of the size of the sample, it also involves a smoothing of data, with a possible loss of specific information relating to the group referred to. In the case of small populations showing major erratic movements in mortality data, the advantages gained from the larger sample size overcome the disadvantages of the smoothing effect.

Managing longevity and disability risks in life annuities with long term care

The aim of the paper is twofold. Firstly, it develops a model for risk assessment in a portfolio of life annuities with long term care benefits. These products are usually represented by a Markovian Multi-State model and are affected by both longevity and disability risks. Here, a stochastic projection model is proposed in order to represent the future evolution of mortality and disability transition intensities. Data from the Italian National Institute of Social Security (INPS) and from Human Mortality Database (HMD) are used to estimate the model parameters. Secondly, it investigates the solvency in a portfolio of enhanced pensions. To this aim a risk model based on the portfolio risk reserve is proposed and different rules to calculate solvency capital requirements for life underwriting risk are examined. Such rules are then compared with the standard formula proposed by the Solvency II project.