Optimal Annuitization Policies

Abstract

At, or about, the age of retirement, most individuals must decide what additional fraction of their marketable wealth, if any, should be annuitized. Annuitization means purchasing a nonrefundable life annuity from an insurance company, which then guarantees a lifelong consumption stream that cannot be outlived. The decision of whether or not to annuitize additional liquid assets is a difficult one, since it is clearly irreversible and can prove costly in hindsight. Obviously, for a large group of people, the bulk of financial wealth is forcefully annuitized, for example, company pensions and social security. For others, especially as it pertains to personal pension plans, such as 401(k), 403(b), and IRA plans as well as variable annuity contracts, there is much discretion in the matter.

The purpose of this paper is to focus on the question of when and if to annuitize. Specifically, my objective is to provide practical advice aimed at individual retirees and their advisors. My main conclusions are as follows:

? Annuitization of assets provides unique and valuable longevity insurance and should be actively encouraged at higher ages. Standard microeconomic utility-based arguments indicate that consumers would be willing to pay a substantial “loading” in order to gain access to a life annuity.

? The large adverse selection costs associated with life annuities, which range from 10% to 20%, might serve as a strong deterrent to full annuitization.

? Retirees with a (strong) bequest motive might be inclined to self-annuitize during the early stages of retirement. Indeed, it appears that most individuals—faced with expensive annuity products—can effectively “beat” the rate of return from a fixed immediate annuity until age 75?80. I call this strategy consume term and invest the difference.

? Variable immediate annuities (VIAs) combine equity market participation together with longevity insurance. This financial product is currently underutilized (and not available in certain jurisdictions) and can only grow in popularity.     

Auswirkungen von Enhanced Annuities auf den Bestand eines Versicherers

The value for money of a standard annuity is the higher, the longer the life expectancy of an insured, and therefore it is only acceptable for persons with an above average life expectancy. The discrepancy is intensified by tax regulations that favor lifelong annuity payments opposed to a lump sum. This discrimination of impaired insureds could be prevented if so-called enhanced annuities were offered, i.e. products where the annuity paid is the larger, the lower the person’s life expectancy. The article presents a quantitative comparison of the risk profile of insurance companies offering standard annuity contracts compared to enhanced annuities and an analysis of the impact of adverse selection on a standard insurer. By definition of individual mortality rates a heterogeneous insurance portfolio is specified. Besides we model the individual underwriting of enhanced annuities. A Monte Carlo Simulation provides results to compare the profit/loss situation of a portfolio of traditional annuity products and a portfolio of enhanced annuities with individual underwriting of different quality and to assess the impact of selection effects.     

Application of Relational Models in Mortality Immunization

The prediction of future mortality rates by any existing mortality models is hardly exact, which causes an exposure to mortality (longevity) risk for life insurers (annuity providers). Since a change in mortality rates has opposite impacts on the surpluses of life insurance and annuity, hedging strategies of mortality and longevity risks can be implemented by creating an insurance portfolio of both life insurance and annuity products. In this article, we apply relational models to capture the mortality movements by assuming that the realized mortality sequence is a proportional change and/or a constant shift of the expected one, and the size of the changes varies in the length of the sequences. Then we create a variety of non-size-free matching strategies to determine the weights of life insurance and annuity products in an insurance portfolio for mortality immunization, where the weights depend on the sizes of the proportional and/or constant changes. Comparing the hedging performances of four non-size-free matching strategies with corresponding size-free ones proposed by Lin and Tsai, we demonstrate with simulation illustrations that the non-size-free matching strategies can hedge against mortality and longevity risks more effectively than the size-free ones.     

To Hedge or Not to Hedge: Managing Demographic Risk in Life Insurance Companies

Demographic risk, i.e., the risk that life tables change in a nondeterministic way, is a serious threat to the financial stability of an insurance company having underwritten life insurance and annuity business. The inverse influence of changes in mortality laws on the market value of life insurance and annuity liabilities creates natural hedging opportunities. Within a realistically calibrated shareholder value (SHV) maximization framework, we analyze the implications of demographic risk on the optimal risk management mix (equity capital, asset allocation, and product policy) for a limited liability insurance company operating in a market with insolvency‐averse insurance buyers. Our results show that the utilization of natural hedging is optimal only if equity is scarce. Otherwise, hedging can even destroy SHV. A sensitivity analysis shows that a misspecification of demographic risk has severe consequences for both the insurer and the insured. This result highlights the importance of further research in the field of demographic risk.     

The Impact of Disability Insurance on a Portfolio of Life Insurances

The aim of this article is to study the impact of disability insurance on an insurer's risk situation for a portfolio that also consists of annuity and term life contracts. We provide a model framework using discrete time nonhomogeneous bivariate Markov renewal processes and in a simulation study focus on diversification benefits as well as potential natural hedging effects (risk-minimizing or risk-immunizing portfolio compositions) that may arise within the portfolio because of the different types of biometric risks. Our analyses emphasize that disability insurances are a less efficient tool to hedge shocks to mortality and that their high sensitivity toward shocks to disability risks cannot be easily counterbalanced by other life insurance products. However, the addition of disability insurance can still considerably lower the overall company risk.     

Ruin probabilities and optimal capital allocation for heterogeneous life annuity portfolios

A large part of the actuarial literature is devoted to the derivation of ruin probabilities in various non-life insurance risk models. On the contrary, very few papers deal with ruin probabilities for life insurance portfolios. The difficulties arise from the dependence and non-stationarity of the annual payments made by the insurance company. This paper shows that the ruin probability in case of life annuity portfolios can be computed from algorithms derived by De Pril (1989) and Dhaene & Vandebroek (1995). Approximations for ruin probabilities are discussed. The present article complements the works of Frostig et al. (2003) who considered whole life, endowment, and temporary assurances, and of Denuit & Frostig (2008) who considered homogeneous life annuities portfolios. Here, heterogeneous portfolios (with respect to age and/or face amounts) are studied. Particular attention is paid to the capital allocation problem. The total amount of reserve is shared among the risk classes in order to minimize the ruin probability. It is then fair to charge a higher margin to the risk classes requiring more capital.     

The Simple Analytics of a Pooled Annuity Fund

This article provides a formal analysis of payout adjustments from a longevity risk‐pooling fund, an arrangement we refer to as group self‐annuitization (GSA). The distinguishing risk diffusion characteristic of GSAs in the family of longevity insurance instruments is that the annuitants bear their systematic risk, but the pool shares idiosyncratic risk. This obviates the need for an insurance company, although such instruments could be sold through a corporate insurer. We begin by deriving the payout adjustment for a single entry group with a single annuity factor and constant expectations. We then show that under weak requirements a unique solution to payout paths exists when multiple cohorts combine into a single pool. This relies on the harmonic mean of the ratio of realized to expected survivorship rates across cohorts. The case of evolving expectations is also analyzed. In all cases, we demonstrate that the periodic‐benefit payment in a pooled annuity fund is determined based on the previous payment adjusted for any deviations in mortality and interest from expectations. GSA may have considerable appeal in countries which have adopted national defined contribution schemes and/or in which the life insurance industry is noncompetitive or poorly developed.     

Produktgarantien und Aktienkrise — Implikationen für die Kapitalanlagepolitik von Lebensversicherungsunternehmen

The present contribution analyzes the implications of two central factors influencing the asset allocation decision of (German) life insurance companies, the development of the equity market on one hand and the interest rate guarantees included in traditional life insurance products on the other. The adverse development of share prices in 2000–2002 implies the necessity to consider not only ?normal“ volatility but also worst case-developments for the purpose of risk control. Formally this is done by using the risk measures value-at-risk and conditional value-at-risk. The specific ?myopic“ nature of interest rate guarantees in German life insurance products, which are granted on a yearly basis implies — beyond the general control of the shortfall risk with respect to the guaranteed interest rates — the necessity to per-form the asset allocation on a yearly basis to be in conformity with the time horizon of the liabilities.On the basis of a quantitative approach corresponding model calculations are performed. Thereby not only a pure market valuation is considered but also institutional peculiarities (hidden reserves, accounting norms) of German life insurance companies. The possibility of a riskless one-year investment, either based on market values or on balance sheet values, is revealed to be crucial for giving interest rate guarantees on a yearly basis.     

寿险对消费内需的作用机制研究

采用预防性储蓄研究框架,通过建立纳入不确定因素的效用函数模型,分析了寿险产品中年金产品和非年金产品影响消费内需的机制及效果,并对2010年中国各地区及世界主要国家的寿险业发展和消费内需水平进行了横向比较。本文的研究表明,寿险业对于消费内需的作用机制在于:通过"风险汇聚"和"损失分担"的机制,消除居民面临的损失、收入、支出以及寿命的不确定性,使居民对未来形成稳定的预期,从而降低居民的预防性储蓄,增加现期消费。因此,寿险业对于消费内需具有重要的拉动作用。     

Markovian Approaches to Joint-Life Mortality

Abstract

Many insurance products provide benefits that are contingent on the combined survival status of two lives. To value such benefits accurately, we require a statistical model for the impact of the survivorship of one life on another. In this paper we first set up two models, one Markov and one semi-Markov, to model the dependence between the lifetimes of a husband and wife. From the models we can measure the extent of three types of dependence: (1) the instantaneous dependence due to a catastrophic event that affect both lives, (2) the short-term impact of spousal death, and (3) the long-term association between lifetimes. Then we apply the models to a set of jointlife and last-survivor annuity data from a large Canadian insurance company. Given the fitted models, we study the impact of dependence on annuity values and examine the potential inaccuracy in pricing if we assume lifetimes are independent. Finally, we compare our Markovian models with two copula models considered in previous research on modeling joint-life mortality.     

Two Paradigms for The Market Value of Liabilities

Abstract

Asset/liability management (ALM) theory and practices of insurers have matured and developed from early applications to guaranteed investment contracts (GICs) to all annuity and insurance products today. An important and logical next step of inquiry is the definition of, and calculation procedures for, the market value of an insurance liability. Because all ALM strategies have as their goal the management of some value of assets in relation to some value of liabilities, this inquiry will provide at last a canonical basis for ALM: the management of relative market values.

To set the stage for this exploration, the theory and application of pricing in a complete market are reviewed, as are the practical limitations of this theory in the real, and far from complete, financial markets. The notion of an ad hoc pricing model is developed, and examples are reviewed and critiqued. These models, though imperfect compared with pricing in a complete market, bridge the gap between pricing theory and practice.

The current state of the liabilities market is also discussed, and this market is seen to naturally split into a “long” and a “short” submarket. Of particular interest is the theoretical possibility of these markets becoming broad-based, deep and active, and the conclusions are relevant to the issue of long/short price equalization.

Two paradigms are then explored for defining and subsequently calculating an insurance liability market value. A “paradigm” is a generalized model or framework for accomplishing the task at hand. Each paradigm reflects observable market trading activity, however infrequent, and each is based on methods of valuation consistent with finance-theoretic approaches that are routinely used for the market valuation of assets.

In addition, each paradigm allows for a sequence of ad hoc valuation methodologies, which differ in the extent to which various risks are explicitly modeled versus judgmentally reflected in a risk spread. These paradigms are discussed and contrasted, and arguments made for the potential evolution of the respective values if a “liability” market began trading actively. Practical constraints on the realization of this evolution are also noted.

The last section of this paper discusses a host of considerations related to the application of option-pricing theory to insurance company liabilities.     

Long-Term Yield Rates for Actuarial Valuations

Consider the problem of valuing a life insurance or annuity on a person aged 20. The valuation formula requires that we know the prices of pure-discount bonds with maturities of up to 100 years. This article investigates the problem of estimating the yield rate for a pure-discount bond that matures in 100 years. It is shown how to estimate this yield rate with parametric and nonparametric models on the price of U.S. Treasury strips. Moreover, confidence intervals on these rates are constructed with bootstrap methods.     

公平视域下机关事业单位养老保险新政的功效分析

2015年1月国务院出台了机关事业单位养老保险新政,将机关事业单位工作人员与企业职工基本养老保险制度"并轨"。本文在围绕起点、过程、结果三个维度对新政出台前我国基本养老保险公平情况进行研究的基础上,分析新政带来的一定程度上促进基本养老保险公平发展、推动补充养老保险发展、促进人员合理流动的直接功效,并且进一步对职业年金的建立倒逼企业年金的发展,企业年金的发展倒逼基本养老保险公平发展的潜在功效进行了剖析。在此基础上,构建出底线公平养老保险体系的过渡性结构,随着政府财政支付社会统筹账户养老金能力的提高,这一结构最终应发展成为真正意义上底线公平养老保险体系的目标结构。     

Valuation of Equity-Linked Insurance and Annuity Products with Binomial Models

Abstract

In this paper we develop a valuation method for equity-linked insurance products. We assume that the premium information of term life insurances, pure endowment insurances, and endowment insurances at all maturities is obtainable within a company or from the insurance market. Using a method similar to that of Jarrow and Turnbull (1995), we derive three martingale probability measures associated with these basic insurance products. These measures are agedependent, include an adjustment for the mortality risk, and reproduce the premiums of the respective insurance products. We then extend the martingale measures to include the financial market information using copulas and use them to evaluate equity-linked insurance contracts and equity-indexed annuities in particular. This is different from the traditional approach under which diversification of mortality risk is assumed. A detailed numerical analysis is performed for various existing equity-indexed annuities in the North American market.     

Trading death: The implications of annuity replication for the annuity puzzle,arbitrage, speculation and portfolios

Annuities are perceived as being illiquid financial instruments, and this has limited their attractiveness to consumers and their inclusion in financial models. However, short positions in annuities can be replicated using life insurance and debt, permitting long positions in annuities to be offset, or short annuity positions to be created. The implications of this result for the annuity puzzle, arbitrage between the annuity and life insurance markets, and speculation on expected longevity are investigated. It is argued that annuity replication could help reduce the annuity puzzle, improve the price efficiency of annuity markets and promote the inclusion of annuities in household portfolios.     

Converting retirement benefit into a life care annuity with graded benefits*

This paper deals with life care annuities, i.e. bundled products comprising a life annuity and long-term care insurance. It aims to assess the cost of converting retirement benefit into a life care annuity with graded benefits using a pre-existing public pay-as-you-go pension scheme. With this objective in mind, we present an actuarial method based on array calculus for valuing this type of life care annuity. The health dynamics of the annuitant rely on a reversible illness-death multistate framework. The paper contains a numerical example in which mortality and disability assumptions are based on data from the USA and Australia, although this should be viewed simply as an illustration. In addition, in order to check the coherence of these data, we compute life expectancy for both healthy and dependent persons, and then for dependent persons in each of the states of dependence. The effect of ruling out the recovery assumption on the annuity’s cost is also assessed. The analysis provides valuable insights into how much it would cost to introduce these annuities and enables us to make some policy recommendations to help ensure that this combined pension scheme has a good actuarial design.     

Stochastic mortality under measure changes

We provide a self-contained analysis of a class of continuous-time stochastic mortality models that have gained popularity in the last few years. We describe some of their advantages and limitations, examining whether their features survive equivalent changes of measures. This is important when using the same model for both market-consistent valuation and risk management of life insurance liabilities. We provide a numerical example based on the calibration to the French annuity market of a risk-neutral version of the model proposed by Lee & Carter (1992).     

Immediate Annuity Pricing in the Presence of Unobserved Heterogeneity

Abstract

One of the acknowledged difficulties with pricing immediate annuities is that underwriting the annuitantis life is the exception rather than the rule. In the absence of underwriting, the price paid for a life-contingent annuity is the same for all sales at a given age. This exposes the market (insurance company and potential policyholder alike) to antiselection. The insurance company worries that only the healthiest people choose a life-contingent annuity and therefore adjust mortality accordingly. The potential policyholders worry that they are not being compensated for their relatively poor health and choose not to purchase what would otherwise be a very beneficial product.

This paper develops a model of underlying, unobserved health. Health is a state variable that follows a first-order Markov process. An individual reaches the state “death” either by accident from any health state or by progressively declining health state. Health state is one-dimensional, in the sense that health can either “improve” or “deteriorate” by moving farther from or close to the “death” state, respectively. The probability of death in a given year is a function of health state, not of age. Therefore, in this model a person is exactly as old as he or she feels.

I first demonstrate that a multistate, ageless Markov model can match the mortality patterns in the common annuity mortality tables. The model is extended to consider several types of mortality improvements: permanent through decreasing probability of deteriorating health, temporary through improved distribution of initial health state, and plateau through the effects of past health improvements.

I then construct an economic model of optimal policyholder behavior, assuming that the policyholder either knows his or her health state or has some limited information. the value of mortality risk transfer through purchasing a life-contingent annuity is estimated for each health state under various risk-aversion parameters. Given the economic model for optimal purchasing of annuities, the value of underwriting (limited information about policyholder health state) is demonstrated.     

Risk assessment of surplus appropriation schemes in participating life insurance

The aim of this paper is to analyze a life insurance company’s risk exposure with respect to different surplus appropriation schemes in participating life insurance. In this regard, three surplus appropriation schemes are considered, including the bonus system, the interest-bearing accumulation, and the system of shortening the contract term. We further examine an insurance company that offers all three schemes, i.e. each system is used for one third of the policyholders. Focus is laid on the effect of different asset portfolios and shocks to mortality on the insurer’s risk situation with respect to the policyholder’s age level at contract inception.     

Developing Mortality Improvement Formulas

Abstract

Longevity improvements have contributed to widespread underfunding of pension plans and losses in insured annuity portfolios. Insurers might reasonably expect some upside from the effect of lower mortality on their life business. Although mortality improvement scales, such as the Society of Actuaries Scale AA, are widely employed in pension and annuity valuation, the derivation of these scales appears heuristic, leading to problems in deriving meaningful measures of uncertainty. We explore the evidence on mortality trends for the Canadian life insurance companies, data, using stochastic models. We use the more credible population data to benchmark the insured lives data. Finally, we derive a practical, model-based formula for actuaries to incorporate mortality improvement and the associated uncertainty into their calculations.