An Optimal Product Mix for Hedging Longevity Risk in Life Insurance Companies: The Immunization Theory Approach

This article investigates the natural hedging strategy to deal with longevity risks for life insurance companies. We propose an immunization model that incorporates a stochastic mortality dynamic to calculate the optimal life insurance–annuity product mix ratio to hedge against longevity risks. We model the dynamic of the changes in future mortality using the well‐known Lee–Carter model and discuss the model risk issue by comparing the results between the Lee–Carter and Cairns–Blake–Dowd models. On the basis of the mortality experience and insurance products in the United States, we demonstrate that the proposed model can lead to an optimal product mix and effectively reduce longevity risks for life insurance companies.     

Self‐Funding of Health‐Care Plans by Small Firms: Risks and Reforms

Small employers that offer health insurance have usually offered fully insured products through traditional health plans. Recently, the Patient Protection and Affordable Care Act (ACA) has created new requirements for fully insured products that will entice more small firms to fund their own health‐care benefits. However, self‐funding poses significant risks to these small firms, their employees, and state exchanges. To mitigate some of these risks within current political realities, we recommend advance disclosures—to small firms of material changes in their stop‐loss policies, and to their employees that premium subsidies are available only on ACA exchanges. We also suggest strengthening Small Business Health Options Program exchanges by broadening the availability of subsidies and building partnerships with brokers. Finally, we recommend an expanded role for brokers and third‐party administrators in helping small firms improve their choice of health‐care insurance.     

Can the Life Insurance Market Provide Evidence for a Bequest Motive?

Using U.K. microeconomic data, we analyze the empirical determinants of participation in the life insurance market. We find that term insurance demand is positively correlated with measures of bequest motives like being married, having children, and/or subjective measures of strong bequest motives. We then show that a life‐cycle model of life insurance demand, saving, and portfolio choice can rationalize quantitatively the data in the presence of a bequest motive. These findings provide evidence supporting the presence of a bequest motive.     

Regret in health insurance post‐purchase behavior

Unlike other forms of insurance, individuals with health insurance generally expect to make claims through the policy period. Selecting an appropriate level of cost‐sharing is difficult and individuals may, ex‐post, regret the choice of a less‐than‐suitable coverage amount. Using a national health insurance survey of private market consumers from 2013 to 2017, we evaluate the potential for post‐purchase regret in the health plan purchasing decision. We employ an ordered logistic model and find that consumers whose plan choices were likely financially dominated by a foregone alternative are significantly more likely to express regret through reporting significantly lower likelihood of renewal, even when controlling for confounding considerations including affordability, self‐assessed risk, and satisfaction with the plan.     

Dynamic Portfolio Choice with Stochastic Wage and Life Insurance

We study optimal insurance, consumption, and portfolio choice in a framework where a family purchases life insurance to protect the loss of the wage earner's human capital. Explicit solutions are obtained by employing constant absolute risk aversion utility functions. We show that the optimal life insurance purchase is not a monotonic function of the correlation between the wage and the financial market. Meanwhile, the life insurance decision is explicitly affected by the family's risk preferences in general. The model also predicts that a family uses life insurance and investment portfolio choice to hedge stochastic wage risk.     

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.     

The shock of COVID-19 and the shock of rapid vaccine development on the value of senior life settlement contracts

In this paper, we develop a model that can capture how COVID-19 and the subsequent rapid vaccine development against COVID-19 impacts the value of pools of senior life settlements. The pandemic unexpectedly boosted the mortality rates of senior citizens who had prior diagnoses of certain health conditions. Our model accounts for the existence and concentration of these COVID-19 comorbidities in portfolios of senior life settlements. It is the concentration of assets linked to the mortality rates of a group who is at elevated risk to COVID-19 and who is also the primary beneficiaries of the COVID-19 vaccine that we examine. We illustrate how the shock of the pandemic increases the value of senior life settlements and how the accelerated development and distribution of COVID-19 vaccines moderated this increase. Our model is general enough to simulate the impact on other financial contracts that are linked to individual mortality rates. These would include life insurance contracts, annuities, and health insurance policies.     

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.     

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.     

The Demand for Health Insurance in the Group Setting: Can You Always Get What You Want?

To what extent do health benefits obtained in the employment‐based setting reflect individual preferences? We examine this question by comparing the relationship between person‐level characteristics and the plans they obtain in a group setting to the relationship observed in the individual insurance market, using data from the 1996‐1997 and 1998‐1999 Community Tracking Study's Household Surveys. We also examine the effect of unions on group choice. Our structural models of the demand for insurance indicate that plans obtained in the group setting often reflect underlying individual preferences for insurance, but we consistently observe significantly different effects of ethnicity and unionization.     

The Impact of the Secondary Market on Life Insurers' Surrender Profits

Life insurers often claim that the life settlement industry reduces their surrender profits and leads to an adverse shift in their portfolio of insured risks; that is, high risks remain in the portfolio instead of surrendering. In this article, we aim to quantify the effect of altered surrender behavior––subject to the health status of an insured––in a portfolio of life insurance contracts on the surrender profits of primary insurers. Our model includes mortality heterogeneity by applying a stochastic frailty factor to a mortality table. We additionally analyze the impact of the premium payment method by comparing results for annual and single premium payments.     

Questionnaire based mortality assessment: an exploratory analysis of NHANES III

The need for speed and efficiency in life insurance underwriting has never been greater. With increasing acquisition costs, the need for timeliness to complete the sales process, and the need for accurate mortality prediction a great deal of resources have been devoted to the development of non-traditional underwriting processes and criteria in hopes of streamlining the process. In this paper, we describe an exploratory analysis of mortality linked data from the US NHANES III study. Our goal for this analysis was to identify mortality predictors that would be good candidates for further investigation in insurance applicants, especially in the context of simplified issue products which focus on younger age applicants. We identified aerobic exercise, dental health, alcohol beverage type and quantity, and dietary components and patterns as factors that may hold promise as mortality predictors in persons age 17-65. Though the limitations of this analysis preclude the immediate use of these factors as underwriting criteria, it does appear that several of these hold promise, and should be tested in an insurance application context. Focus groups, market testing, or even insertion as non-actionable questions in a subset of applications may be ways to collect further data on them.     

Explaining Low Annuity Demand: An Optimal Portfolio Application to Japan

Using an optimizing financial planning model in the tradition of Merton and Richard we explore how individuals should determine their life insurance and annuity choices, given uncertainty about investment returns and mortality. Both consumption and bequests appear as arguments in the individual's preference function. The model explicitly recognizes the existence of social security in retirement, and of loadings on insurance premiums, due to administration costs in the life insurance and annuities markets. The model sheds light on the reasons for the thinness of voluntary life annuity markets worldwide. The relative importance of pre‐existing annuitization through social security, the role of bequests, and premium loadings are quantitatively assessed within a single optimizing framework. Results are presented for a model specification calibrated to Japan.     

Use of a Markov Model to Estimate Long-Term Insured Lives’ Mortality Risk Associated With BRCA1 and BRCA2 Gene Mutations

Abstract

There is uncertainty regarding the degree of insurance risk associated with BRCA1/2, the gene mutations associated with breast cancer. Most reports to date have been based on high-risk populations selected from families with multiple and/or early-onset cancers; more favorable data have been reported in studies without this selection bias.

This paper discusses use of a Markov model to estimate mortality risk associated with BRCA1/2 gene mutations in female life insurance applicants. The goal is to derive a range of risk estimates based on different assumptions of breast and ovarian cancer incidence. A particular strength of the model is that transition probabilities after cancer diagnosis vary with age and cancer stage, as do excess hazard rates.

Data calculated by the model indicate that no single mortality curve characterizes risk for all life insurance applicants with a BRCA1/2 mutation. Rather, mortality risk depends on breast and ovarian cancer incidence rates and subsequent mortality rates, and on the method used to deal with competing breast and ovarian cancer incidence and mortality rates. Further refinement of risk estimates will depend on better incidence data and on resolution of complex statistical problems, such as informative censoring.

Widespread use of genetic information by insurance consumers could have important economic implications. For companies that sell individually underwritten products, profitability might decrease. Consumers might find higher prices and reduced availability, with a corresponding decrease in quantity of insurance purchased. Insurance and consumer ramifications would vary by cover, with living-benefit products, such as critical-illness insurance, most adversely affected. Societal choices are limited. Given assumptions in the cited scenario, it is likely premiums would rise and quantity of insurance purchased would decrease even with no change in existing social policy; attempted legal or regulatory remedies would further accentuate price increases and reductions in quantity purchased.     

Reference mortality K2004 of personal life insurance policies in Finland

This article presents the reference mortality model K2004 approved by the Actuarial Society of Finland and the technique that was implemented in developing it. Initially, I will present the historical development of individual mortality rates in Finland. Then, the requirements posed for a modern mortality modelling will be presented. Reference mortality model K2004 is based on total population mortality rates, which were adjusted to correspond with that portion of the population that has a life insurance policy. First, the model presents a margin of the observed life insurance mortality rate in the total population with a Lee-Carter method together with a forecast, where the downward trend in mortality rates is expected to continue at the rate illustrated since the 1960s. Then, the mortality rate has been adjusted into life insurance mortality per age so that it corresponds to the differences observed between total population and the portion of population that has a life insurance during 1991–2001. Finally, a cohort and gender-specific functional margin will be presented to obtained data.     

中国寿险市场存在逆向选择吗?——来自CHARLS数据的经验证据

我国寿险市场是否存在逆向选择问题,在理论和实证两个方面缺乏细致的讨论。本文利用CHARLS数据和正相关理论检验了我国定期寿险和终身寿险市场中的逆向选择问题。我们选取了死亡率这一远期指标和健康状况这一近期指标来衡量消费者的死亡风险,从广延边际和集约边际两个方面利用正相关理论进行了深入的研究。实证结论表明,以死亡率和健康状况衡量的死亡风险与寿险消费负相关或者不相关,即我国寿险市场并不存在逆向选择问题。然后,我们讨论了模型的内生性问题,并根据年龄变量检验了结论的稳健性,实证结果表明我们的结论是稳健的。最后,本文利用双变量Probit模型设计了一个机制,并利用该机制验证了利他动机是我国寿险市场不存在逆向选择的原因之一。     

Securitization of Mortality Risks in Life Annuities

The purpose of this article is to study mortality‐based securities, such as mortality bonds and swaps, and to price the proposed mortality securities. We focus on individual annuity data, although some of the modeling techniques could be applied to other lines of annuity or life insurance.     

Guaranteed renewable life insurance under demand uncertainty

Guaranteed renewability (GR) is a prominent feature in many health and life insurance markets. We develop a model that includes unpredictable (and unobservable) fluctuations in demand for life insurance as well as changes in risk type (observable) over individuals' lifetimes. The presence of demand type heterogeneity leads to the possibility that optimal GR contracts may have a renewal price that is either above or below the actuarially fair price of the lowest risk type in the population. Individuals whose type turns out to be high risk but low demand renew more of their GR insurance than is efficient due to the attractive renewal price. This results in imperfect insurance against reclassification risk. Although a first‐best efficient contract is not possible in the presence of demand type heterogeneity, the presence of GR contracts nonetheless improves welfare relative to an environment with only spot markets.     

中国寿险市场存在逆向选择吗?——来自CHARLS数据的经验证据

我国寿险市场是否存在逆向选择问题,在理论和实证两个方面缺乏细致的讨论。本文利用CHARLS数据和正相关理论检验了我国定期寿险和终身寿险市场中的逆向选择问题。我们选取了死亡率这一远期指标和健康状况这一近期指标来衡量消费者的死亡风险,从广延边际和集约边际两个方面利用正相关理论进行了深入的研究。实证结论表明,以死亡率和健康状况衡量的死亡风险与寿险消费负相关或者不相关,即我国寿险市场并不存在逆向选择问题。然后,我们讨论了模型的内生性问题,并根据年龄变量检验了结论的稳健性,实证结果表明我们的结论是稳健的。最后,本文利用双变量Probit模型设计了一个机制,并利用该机制验证了利他动机是我国寿险市场不存在逆向选择的原因之一。     

Markov Aging Process and Phase-Type Law of Mortality

Abstract

In this article, we propose a finite-state Markov process with one absorbing state to model human mortality. A health index called physiological age is introduced and modeled by the Markov process. Under this model the time of death follows a phase-type distribution. The model possesses many desirable analytical properties useful for mortality analysis. Closed-form expressions are available for many quantities of interest including the conditional survival probabilities of the time of death and the actuarial present values of the whole life insurance and annuity. The heterogeneity or frailty effect of a cohort can be expressed explicitly. The model is also able to explain some stylized facts of observed mortality data. We fit the model to some Swedish population cohort data and life tables compiled by the U.S. Social Security Administration. The fitting results are very satisfactory.