Optimal investment-consumption-insurance with random parameters

This paper discusses an optimal investment, consumption, and life insurance purchase problem for a wage earner in a complete market with Brownian information. Specifically, we assume that the parameters governing the market model and the wage earner, including the interest rate, appreciation rate, volatility, force of mortality, premium-insurance ratio, income and discount rate, are all random processes adapted to the Brownian motion filtration. Our modeling framework is very general, which allows these random parameters to be unbounded, non-Markovian functionals of the underlying Brownian motion. Suppose that the wage earner’s preference is described by a power utility. The wage earner’s problem is then to choose an optimal investment-consumption-insurance strategy so as to maximize the expected, discounted utilities from intertemporal consumption, legacy and terminal wealth over an uncertain lifetime horizon. We use a novel approach, which combines the Hamilton–Jacobi–Bellman equation and backward stochastic differential equation (BSDE) to solve this problem. In general, we give explicit expressions for the optimal investment-consumption-insurance strategy and the value function in terms of the solutions to two BSDEs. To illustrate our results, we provide closed-form solutions to the problem with stochastic income, stochastic mortality, and stochastic appreciation rate, respectively.     

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.     

Optimal Gradual Annuitization: Quantifying the Costs of Switching to Annuities

We compute the optimal dynamic annuitization and asset allocation policy for a retiree with Epstein–Zin preferences, uncertain investment horizon, potential bequest motives, and pre‐existing pension income. In our setting the retiree can decide each year how much he consumes and how much he invests in stocks, bonds, and life annuities, while the prior literature mostly considered restricted so‐called deterministic or stochastic switching strategies. We show that postponing the annuity purchase is no longer optimal in the gradual annuitization (GA) case since investors are able to attain the optimal mix between liquid assets (stocks and bonds) and illiquid life annuities each year. In order to assess potential utility losses, we benchmark various restricted annuitization strategies against the unrestricted GA strategy.     

Discussion of the Papers By Robert L. Brown and Thomas Trauth

Abstract

We find the minimum probability of lifetime ruin of an investor who can invest in a market with a risky and a riskless asset and who can purchase a deferred life annuity. Although we let the admissible set of strategies of annuity purchasing process be the set of increasing adapted processes, we find that the individual will not buy a deferred life annuity unless she can cover all her consumption via the annuity and have enough wealth left over to sustain her until the end of the deferral period.     

个人保险、消费和储蓄决策

本文通过失业保险,研究了家庭保险、消费和储蓄的决策问题。在连续时间情形下,本文将随机问题转化为确定性问题来考察,运用最优控制理论,考察了家庭的决策过程。分析了购买保险对于个人消费和储蓄决策的影响,以及如何通过保险稳定其财富和提高终生效用,同时探讨了人均消费和人均资本存量的动态变化过程,得到了关于经济增长路径的结论。     

Portfolio Choice and Life Insurance: The CRRA Case

We solve a portfolio choice problem that includes life insurance and labor income under constant relative risk aversion (CRRA) preferences. We focus on the correlation between the dynamics of human capital and financial capital and model the utility of the family as opposed to separating consumption and bequest. We simplify the underlying Hamilton–Jacobi–Bellman equation using a similarity reduction technique that leads to an efficient numerical solution. Households for whom shocks to human capital are negatively correlated with shocks to financial capital should own more life insurance with greater equity/stock exposure. Life insurance hedges human capital and is insensitive to the family's risk aversion, consistent with practitioner guidance.     

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.     

Asset Allocation Via The Conditional First Exit Time or How To Avoid Outliving Your Money

The risk of outliving your money (or shortfall) with low risk, low return investments is very often more serious than the risk of losing money on high risk investments, until quite late in life. A stochastic process model incorporating mortality tables for men and women of retirement age, random rates of return and fixed initial wealth and desired level of consumption provides the analytical tool. A simulation using Canadian mortality tables and rates of return shows that almost all retirees should invest some of their wealth in equity, and for many the optimal allocation is 70–100% equity. The risk of shortfall is surprisingly high for a reasonable range of values of the variables, especially for an allocation of 100% in treasury bills. Women face much greater risk of shortfall than men. The analytical model also permits calculation of the distribution of the bequest and hence allows an individual to trade off changes in shortfall risk against changes in the expected bequest to the heirs.     

Optimal life insurance with no-borrowing constraints: duality approach and example

We solve an optimal portfolio choice problem under a no-borrowing assumption. A duality approach is applied to study a family’s optimal consumption, optimal portfolio choice, and optimal life insurance purchase when the family receives labor income that may be terminated due to the wage earner’s premature death or retirement. We establish the existence of an optimal solution to the optimization problem theoretically by the duality approach and we provide an explicitly solved example with numerical illustration. Our results illustrate that the no-borrowing constraints do not always impact the family’s optimal decisions on consumption, portfolio choice, and life insurance. When the constraints are binding, there must exist a wealth depletion time (WDT) prior to the retirement date, and the constraints indeed reduce the optimal consumption and the life insurance purchase at the beginning of time. However, the optimal consumption under the constraints will become larger than that without the constraints at some time later than the WDT.     

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.     

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.     

On systematic mortality risk and risk-minimization with survivor swaps

A new market for so-called mortality derivatives is now appearing with survivor swaps (also called mortality swaps), longevity bonds and other specialized solutions. The development of these new financial instruments is triggered by the increased focus on the systematic mortality risk inherent in life insurance contracts, and their main focus is thus to allow the life insurance companies to hedge their systematic mortality risk. At the same time, this new class of financial contract is interesting from an investor's point of view, since it increases the possibility for an investor to diversify the investment portfolio. The systematic mortality risk stems from the uncertainty related to the future development of the mortality intensities. Mathematically, this uncertainty is described by modeling the underlying mortality intensities via stochastic processes. We consider two different portfolios of insured lives, where the underlying mortality intensities are correlated, and study the combined financial and mortality risk inherent in a portfolio of general life insurance contracts. In order to hedge this risk, we allow for investments in survivor swaps and derive risk-minimizing strategies in markets where such contracts are available. The strategies are evaluated numerically.     

Wealth, Income, and Optimal Insurance

This article considers the decision to purchase insurance against possible losses of a property or wealth. The decision involves a standard economic trade‐off between the benefit of protection against loss and the cost of insurance premium. The premium is paid out of the income and decreases the consumption of other goods and services, rather than out of wealth and decreases the property or wealth. The demand for insurance depends mainly on the income and preferences. As a result, unlike in the standard model, a fair premium is neither necessary nor sufficient for the optimality of full coverage insurance. Rather, the individuals with higher incomes purchase full coverage insurance even at unfair prices of insurance while the individuals with lower income purchase partial coverage insurance at a fair price.     

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.     

The Annuity Puzzle and an Outline of Its Solution

In his seminal 1965 paper, Yaari showed that, assuming actuarially fair annuity prices, uncertain lifetimes, and no bequest motives, utility-maximizing retirees should annuitize all of their wealth on retirement. Nevertheless, the markets for individual immediate life annuities in the United States, the United Kingdom, and several other developed countries have been small relative to other financial investment outlets competing for retirement savings. Researchers have found this situation puzzling, hence the so-called “annuity puzzle.” There are many possible explanations for the annuity puzzle, including “rational” explanations such as adverse selection, bequest motives, and incomplete markets; and “behaviorial” explanations, such as mental accounting, cumulative prospect theory, and mortality salience. We review the literature on the various plausible explanations given for the existence of the annuity puzzle, suggest ways of stimulating the demand for annuities, and suggest a few of the ingredients needed for further development of hybrid annuity products that may provide a solution to the puzzle.     

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.     

One-Period Model of Individual Consumption, Life Insurance, and Investment Decisions

This article studies individuals' optimal decisions on consumption, life insurance, and stock purchases in a one‐period framework. With exponential utility functions, individuals' life insurance and stock purchases are independent of each other; life insurance purchases are affected only by individuals' future income, bequest intensity, risk attitude, survival probability, and the insurance risk premium; stock purchases are affected only by individuals' risk attitude, the risk‐free rate of return, the stock return, and stock volatility. With power utility functions, life insurance and stock purchases are positively related with each other and are affected by all the factors.     

The impact of a partial borrowing limit on financial decisions

We consider a consumption, investment, life insurance, and retirement decision problem in which an economic agent is allowed to borrow against only a part of future income. The closed-form solution is attained by applying a dual approach that directly imposes the conditions for the borrowing limit on a dual value function. We provide analytic comparative statics for optimal strategies with rigorous proofs. It is confirmed that a more stringent borrowing limit leads to less consumption and less life insurance purchase. However, even with a tighter borrowing limit, an agent with weak incentive to retire can invest more when the wealth level is high enough. We also show that a more stringent borrowing limit can delay or hasten the optimal retirement timing depending on the agent's current wealth level.     

Structured settlement annuities, part 1: overview and the underwriting process

Structured settlement underwriting is the underwriting of medically impaired lives for the purchase of an annuity to fund the settlement. Other than risk assessment, structured settlement (SS) underwriting has little in common with traditional life insurance underwriting. Most noteworthy of these differences is the relative lack of actuarial data on which to base decisions about mortality and the necessity for prospective thinking about risk assessment. The purpose of this paper is to provide a foundation for understanding the structured settlement business and to contrast the underwriting of structured settlements with that of traditional life insurance. This is the first part of a two-part article on SS annuities. Part 2 deals with the mortality experience in SS annuitants and the life-table methodology used to calculate life expectancy for annuitants at increased mortality risk.     

Mortality Risk and the Value of a Statistical Life: The Dead-Anyway Effect Revis(it)ed

In the expected-utility theory of the monetary value of a statistical life, a well-known result found by Pratt and Zeckhauser [1996] asserts that an individuals’ willingness to pay (WTP) for a marginal reduction in mortality risk increases with the initial level of risk. Their reasoning is based on the so-called “dead-anyway effect” which states that marginal utility of a dollar in the state of death is smaller than in the state of survival. However, this explanation is based on the absence of markets for contingent claims, i.e. annuities and life insurance. This paper reexamines the relationship between WTP and the level of risk under more general circumstances and establishes two main results: first, when insurance markets are perfect, for a risk-averse individual without a bequest motive, marginal WTP for survival does increase with the level of risk but this occurs for a different reason, namely an income effect. Secondly, when the individual has a bequest motive and is endowed with a sufficient amount of wealth from human capital, the effect of initial risk on WTP for survival is reversed: the higher initial risk the lower the value of a statistical life. In the imperfect-markets case we interpret this result as a “constrained-bequest effect”.JEL Classification No.: D8, H43, I18