“On the Merger of Two Companies,” Hans Gerber and Elias S. W. Shiu,July 2006

Abstract

Bankruptcy risk falls to pension plan participants if a plan sponsor fails when a defined benefit (DB) pension plan is underfunded. This article examines the incidence of that risk and how it changes when public policy provides a guarantee fund. Although government-based guarantee funds are in a unique position to provide pension protection, primarily because of the extent to which the risk of sponsor default is systematic in nature, a looming question is the extent to which such guarantees are exposed to moral hazard. The article focuses on that question using data from four Canadian provinces, including one (Ontario) that operates a guarantee fund for pensions. The findings show that plan assets per DB-plan participant increase with the earnings of workers and decrease with higher unemployment, and that level of assets also is moderated by the influence of taxes, with higher plan assets observed when and where tax rates are higher. Plans in Ontario had on average $20,035 less in asset value per participant, and Ontario plans covered by the guarantee fund had an average of $16,497 less per participant than other Canadian DB plans not backed by a guarantee fund. A separate model finds the presence of a guarantee fund to be one of a very small number of variables significant in explaining variability in the plans’ funded ratios. These empirical results are consistent with the existence of moral hazard.     

Mortality-Indexed Annuities Managing Longevity Risk Via Product Design

Abstract

Longevity risk has become a major challenge for governments, individuals, and annuity providers in most countries. In its aggregate form, the systematic risk of changes to general mortality patterns, it has the potential for causing large cumulative losses for insurers. Since obvious risk management tools, such as (re)insurance or hedging, are less suited for managing an annuity provider’s exposure to this risk, we propose a type of life annuity with benefits contingent on actual mortality experience.

Similar adaptations to conventional product design exist with investment-linked annuities, and a role model for long-term contracts contingent on actual cost experience can be found in German private health insurance. By effectively sharing systematic longevity risk with policyholders, insurers may avoid cumulative losses.

Policyholders also gain in comparison with a comparable conventional annuity product: Using a Monte Carlo simulation, we identify a significant upside potential for policyholders while downside risk is limited.     

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.     

New Swedish mortality table for annuity insurance

Abstract

The processive lowering of the general death rate during a considerable time past, which in spite of a temporary check such as the Spanish Flu epidemic of the years 1918–1920 is yet plainly evident, is a phenomenon of great significance for life insurance in general, and especially for life annuity insurance. Attention has been devoted to this trend by insurance men, and attempts have been made in several quarters to design mortality tables that could be regarded as affording ample security for the latter type of insurance. (Reference may in this connection be made to articles in ?Skandinavisk Aktuarietidskrift?, XII, p. 239 and XV, p. 45.)     

The impact of consumer inattention on insurer pricing in the Medicare Part D program

The Medicare Part D program relies on consumer choice to provide insurers with incentives to offer low‐priced, high‐quality pharmaceutical insurance plans. We demonstrate that consumers switch plans infrequently and search imperfectly. We estimate a model of consumer plan choice with inattentive consumers and show that high observed premiums are consistent with insurers profiting from consumer inertia. We estimate the reduction in steady state plan premiums if all consumers were attentive. An average consumer could save $1050 over three years; government savings in the same period could amount to $1.3 billion or 1% of the cost of subsidizing the relevant enrollees.     

The Poisson Log-Bilinear Lee-Carter Model

Abstract

Life insurance companies deal with two fundamental types of risks when issuing annuity contracts: financial risk and demographic risk. Recent work on the latter has focused on modeling the trend in mortality as a stochastic process. A popular method for modeling death rates is the Lee-Carter model. This methodology has become widely used, and various extensions and modifications have been proposed to obtain a broader interpretation and to capture the main features of the dynamics of mortality rates. In order to improve the measurement of uncertainty in survival probability estimates, in particular for older ages, the paper proposes an extension based on simulation procedures and on the bootstrap methodology. It aims to obtain more reliable and accurate mortality projections, based on the idea of obtaining an acceptable accuracy of the estimate by means of variance reducing techniques. In this way the forecasting procedure becomes more efficient. The longevity question constitutes a critical element in the solvency appraisal of pension annuities. The demographic models used for the cash flow distributions in a portfolio impact on the mathematical reserve and surplus calculations and affect the risk management choices for a pension plan. The paper extends the investigation of the impact of survival uncertainty for life annuity portfolios and for a guaranteed annuity option in the case where interest rates are stochastic. In a framework in which insurance companies need to use internal models for risk management purposes and for determining their solvency capital requirement, the authors consider the surplus value, calculated as the ratio between the market value of the projected assets to that of the liabilities, as a meaningful measure of the company’s financial position, expressing the degree to which the liabilities are covered by the assets.     

Intergenerational Transfers and Insurance Policy Design

Abstract

Group health insurance policies offering an identical benefit package to every member of the group result in lower expected health benefits for younger cohorts than older cohorts. The dispersion in insurance benefits across age groups differs among insurance policies. Simulation results presented in this paper demonstrate that a shift from comprehensive health insurance to high-deductible health insurance decreases the share of expected benefits going to younger cohorts. An estimated 81.5% of the 23-to-32-year-old cohort is expected to receive less than $500 in health benefits during a year for one prototypical high-deductible health plan. Low expected benefits for younger relatively healthy cohorts could increase the number of younger individuals who eschew health coverage. Age-rated premiums are probably the most straightforward way to stimulate demand for high-deductible health plans among younger healthier individuals.     

A Computationally Efficient Algorithm for Estimating the Distribution of Future Annuity Values Under Interest-Rate and Longevity Risks

Abstract

This paper proposes a computationally efficient algorithm for quantifying the impact of interestrate risk and longevity risk on the distribution of annuity values in the distant future. The algorithm simulates the state variables out to the end of the horizon period and then uses a Taylor series approximation to compute approximate annuity values at the end of that period, thereby avoiding a computationally expensive “simulation-within-simulation” problem. Illustrative results suggest that annuity values are likely to rise considerably but are also quite uncertain. These findings have some unpleasant implications both for defined contribution pension plans and for defined benefit plan sponsors considering using annuities to hedge their exposure to these risks at some point in the future.     

Determining the Optimum Guarantee Period for a One-Life Retirement Annuity

Abstract

We consider a risk averse retiree from a defined contribution plan who decides to purchase a onelife annuity with a guarantee period. Given the retiree has a bequest motive, we focus on the problem of determining the optimum length of the guarantee period. Assuming the retiree’s bequest function is proportional to his or her utility function, we determine necessary and/or sufficient conditions under which the retiree would choose an annuity with (i) no guarantee period, (ii) the maximum guarantee period, or (iii) an intermediate guarantee period.     

The Implied Longevity Yield: A Note on Developing an Index for Life Annuities

I develop an index for tracking the dynamic behavior of life (pension) annuity payouts over time, based on the concept of self‐annuitization. Our implied longevity yield (ILY) value is defined equal to the internal rate of return (IRR) over a fixed deferral period that an individual would have to earn on their investable wealth if they decided to self‐annuitize using a systematic withdrawal plan. A larger ILY number indicates a greater relative benefit from immediate annuitization. I use age 65—with a 10‐year period certain—compared against the same annuity at age 75 as the standard benchmark for the index, and calibrate to a comprehensive time series of weekly (Canadian) life annuity quotes from 2000 through 2004. I find that during this period the ILY varied from 5.45 percent to 6.90 percent for males and from 5.00 percent to 6.42 percent for females and was highly correlated with a duration‐weighted average yield of 10‐year and long‐term Government of Canada bonds. I believe our ILY metric can help promote and explain the benefits of acquiring lifetime payout annuities by translating the abstract‐sounding longevity insurance into more concrete and measurable financial rates of return.     

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.     

Some reflections on different disability models

Abstract

As a retired actuary of life and health insurance the author has wished to summarize his experience and reflections about different systems of disability insurance. He wants to show how the premium technique of the “classic” Hamza process presented in 1900 as a model of the invalidity insurance technique without any element of recovery can be described for an active person by three alternative formulas of expected value of a disability indemnity. His of interest to observe that the second of these formulas can be applied to the technique of sickness insurance based on the probability of being sick. This technique is also used in British and Norwegian long term disability insurance. The third formula leads after modification to the Swedish sickness annuity technique and, further, to the technique of basic continuance tables used in U.S.A. In describing the classic process the author has used the discontinuous approach, but otherwise the continuous approach with integrals instead of sums has been preferred.

Methods of calculating the premium for “waiver of premium” have in this paper only been touched upon as a result of a modification of the sickness annuity technique. It has been found necessary to refrain from the opportunity to illustrate the interesting development of the invalidity technique according to Schoenbaum and later to Simonsen who introduced a series of transition probabilities between different states of activity and disablement.     

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.     

Florida's Pension Election: From DB to DC and Back

During the year 2002, the State of Florida's 600,000 public employees were given the choice of converting their traditional defined benefit (DB) pension plan into an individual‐account defined contribution (DC) plan with full control over asset allocation and investment decisions. To mitigate some of the risk and uncertainty in the decision, the State granted each employee electing the DC plan an additional option to switch back (i.e., change their mind once) at any point prior to retirement. This option has been labeled the 2nd election by the State and the cost of reentry is fixed at the accumulated benefit obligation of their pension entitlement, which is the present value of the life annuity. Our article presents some original analytic insights relating to the optimal time and financial value of this unique 2nd election. Although our model is deterministic in nature, we believe that it provides a number of intuitive insights that are quite robust. Our results can be contrasted with Lachance, Mitchell, and Smetters (2003) . We estimate that the increase in retirement wealth that arises from having the 2nd election is equivalent to at most 30 percent in future value, and only when utilized optimally. Furthermore, for most State employees above the age of 45, the 2nd election has little economic value because the DB plan dominates the DC plan from day one. Of course, it remains to be seen what percent of Florida's 600,000 employees will elect to behave rationally with their newfound pension autonomy.     

Multivariate Analysis of Pension Plan Mortality Data

Abstract

This paper uses the logistic regression model to examine private pension plan data for 1989–95 collected by the Retirement Plans Experience Committee of the Society of Actuaries. When only one explanatory variable, such as annuity class size, is used in modeling mortality rates, the model provides a reasonable fit to the data. Multiple explanatory variables give less satisfactory results.     

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.     

The Shift to Defined Contribution Pension Plans

Abstract

There has been a strong shift away from defined benefit (DB) pension plans toward defined contribution (DC) pension plans in the United States over the last 20 years. A variety of reasons for this shift have been proposed. In another paper in this issue, Krzysztof Ostaszewski presents a new hypothesis to explain the shift to DC plans in the United States. He argues that the decline in importance of DB plans is due to a shift in the way relative returns to macroeconomic factors of production, that is, capital and labor, are being rewarded in the national economy.

This paper attempts to test the Ostaszewski hypothesis using Canadian data. In Canada there has been only a slight decrease in DB plan coverage. It is shown that the Ostaszewski theory does not fit the Canadian experience well. Instead, it is argued that pension regulation and tax legislation play a crucial role in pension design and reform. It is also argued that the difference in pension regulation and taxation in Canada versus the United States has directly influenced plan sponsors in considering their pension objectives, costs, and risks. Differences in the proportion of the workforce that is unionized may also be important. The paper concludes that pension regulation and taxation are more important variables than are macroeconomic reward systems in the use of DB versus DC pension plans.     

Market Discipline in the Individual Annuity Market

Theoretical expectations related to market discipline generally suggest a positive relationship between firm financial strength and price. We examine market discipline in the individual annuity market by measuring annuity contract yields during the accumulation phase and find that, among other results, firm financial strength is positively related to yield (i.e., negatively related to price). We argue that this apparent anomaly can be viewed as a form of market discipline itself, for at least four related reasons, the foremost reason being that in order to compete in the asset accumulation market, an insurer has an incentive to provide a track record of historically strong credited interest rates within the annuity. In addition, the credited interest rates within an annuity are only revealed ex post over time, thus diminishing consumer ability to impose traditional market discipline relating firm financial strength and price, and also enabling financially weaker insurers to impose higher ex post prices in the form of lower realized annuity yields.     

Self-Annuitization and Ruin in Retirement

Abstract

At retirement, most individuals face a choice between voluntary annuitization and discretionary management of assets with systematic withdrawals for consumption purposes. Annuitization–buying a life annuity from an insurance company–assures a lifelong consumption stream that cannot be outlived, but it is at the expense of a complete loss of liquidity. On the other hand, discretionary management and consumption from assets–self-annuitization–preserves flexibility but with the distinct risk that a constant standard of living will not be maintainable.

In this paper we compute the lifetime and eventual probability of ruin (PoR) for an individual who wishes to consume a fixed periodic amount–a self-constructed annuity–from an initial endowment invested in a portfolio earning a stochastic (lognormal) rate of return. The lifetime PoR is the probability that net wealth will hit zero prior to a stochastic date of death. The eventual PoR is the probability that net wealth will ever hit zero for an infinitely lived individual.

We demonstrate that the probability of ruin can be represented as the probability that the stochastic present value (SPV) of consumption is greater than the initial investable wealth. The lifetime and eventual probabilities of ruin are then obtained by evaluating one minus the cumulative density function of the SPV at the initial wealth level. In that eventual case, we offer a precise analytical solution because the SPV is known to be a reciprocal gamma distribution. For the lifetime case, using the Gompertz law of mortality, we provide two approximations. Both involve “moment matching” techniques that are motivated by results in Arithmetic Asian option pricing theory. We verify the accuracy of these approximations using Monte Carlo simulations. Finally, a numerical case study is provided using Canadian mortality and capital market parameters. It appears that the lifetime probability of ruin–for a consumption rate that is equal to the life annuity payout–is at its lowest with a well-diversified portfolio.     

Hedging Longevity Risk When Interest Rates are Uncertain

Abstract

This paper proposes an asset liability management strategy to hedge the aggregate risk of annuity providers under the assumption that both the interest rate and mortality rate are stochastic. We assume that annuity providers can invest in longevity bonds, long-term coupon bonds, and shortterm zero-coupon bonds to immunize themselves from the risks of the annuity for the equity holders subject to a required profit. We demonstrate that the optimal allocation strategy can lead to the lowest risk under different yield curves and mortality rate assumptions. The longevity bond can also be regarded as an effective hedging vehicle that significantly reduces the aggregate risk of the annuity providers.