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.     

The Value of Deposit Insurance in the Presence of Interest Rate and Credit Risk

In this paper we employ the theory of the term structure of interest rates and the pricing of interest contingent contracts to determine the fair value of insurance for depository institutions. The balance sheet of a bank is taken to consist of long and short positions in various fixed income securities. Deposit insurance for the bank is a put option on the value of the assets. The value of deposits, assets, the implied exercise price of the put and the value of the put are all determined simultaneously as part of the same valuation solution. The approach is developed initially for a single‐state term structure. It is extended to incorporate credit risk on bank assets.
The most important policy implication is that for a bank whose assets are longer term than its liabilities and whose borrowers are not excessively leveraged the properly calculated, risk‐adjusted deposit insurance premia are increasing functions of the level of interest rates. Sensitivity analyses also treat such factors as the bank's deposit to asset ratio, duration gap, interest volatility, the volatility of assets backing the bank loans, and the bank's borrowers' debt to equity ratio.     

The pricing of equity-linked life insurance policies with an asset value guarantee

This paper considers the equilibrium pricing of equity-linked life insurance policies with an asset value guarantee; such policies provide for benefits which depend upon the performance of a reference portfolio subject to a minimum guaranteed benefit. The benefit is decomposed into a sure amount and an immediately exercisable call option on the reference portfolio. A numerical procedure for determining the value of the call option is presented and the risk minimizing investment strategy to be followed by the issuer of the policy is derived.     

Vergleich von ökonomischen und biometrischen Risiken in der Personenversicherung

In life insurance both the time and the amount of future payments between insurer and policyholder may be stochastic; biometrical as well as financial risks are transferred to the insurer. We present an approach that allows to decompose the randomness of the discounted value of future benefits and premiums to a sum whose addends correspond to the uncertainty of the policy development, the interest rates, the probabilities of death, the probabilities of disablement, etc. Upon modeling the actuarial assumptions stochastically, we quantify these risk factors for typical life insurance contracts and compare them with each other. Contrary to a common folklore, the examples show that the systematic biometrical risks are in many cases not marginal compared to the interest rate risk.     

Valuation of an early exercise defined benefit underpin hybrid pension

In this paper, we consider three types of embedded options in pension benefit design. The first is the Florida second election (FSE) option, which has been offered to public employees in the state of Florida since 2002. The state runs both defined contribution (DC) and defined benefit (DB) pension plans. Employees who initially join the DC plan have the option to convert to the (DB) plan at a time of their choosing. The cost of the switch is assessed in terms of the ABO (Accrued Benefit Obligation), which is the expected present value of the accrued DB pension at the time of the switch. If the ABO is greater than the DC account, the employee is required to fund the difference. The second is the DB Underpin option, also known as a ‘floor offset’ or a ‘Greater-of-benefit’ plan, under which the employee participates in a DC plan, but with a guaranteed minimum benefit based on a traditional DB formula. The third option can be considered a variation on each of the first two. We remove the requirement from the FSE option for employees to fund any shortfall at the switching date. The resulting plan is similar to the DB underpin, but with the possibility of early exercise. We adopt an arbitrage-free pricing methodology to value each option. We analyse and value the optimal switching strategy for the employee by constructing an exercise frontier, and we illustrate numerically the difference between the FSE, DB Underpin and Early Exercise DB Underpin options.     

Uniqueness of the fair premium for equity-linked life insurance contracts

An equity-linked life insurance contract combines an endowment life insurance and an investment strategy with a minimum guarantee. The benefit of this contract is determined by the guaranteed amount plus a bonus equal to a call on the portfolio. This bonus is similar to an Asian option. This article analyzes the relationship between the periodic insurance premium and its proportional share invested into the portfolio. For a general model of the financial risks we show the existence and uniqueness of an insurance premium. Furthermore the premium is strictly increasing and convex as a function of the share invested.     

Effects of Noise on Optimal Exercise Decisions: The Case of Risky Debt Secured by Renewable Lease Income

This paper considers the valuation and default exercise policy of risky coupon debt that is secured by a lease-encumbered noisy real asset. For parameter values used in our analysis, asset value noise is shown to reduce the value of waiting to default. Moreover, the borrower is shown to delay default exercise until the noisy signal of asset value is far into-the-money. This latter finding provides an information-based explanation for the apparent under-exercise of the mortgage default option that has been observed in the literature. An implication of this finding is that, if the claimholder recognizes that noise exists, but the empiricist—who is trying to compare observed exercise policy with that predicted by a noiseless model of asset prices—does not, a “sub-optimal” exercise policy may be inferred when in fact the policy is rational given the information available. This explanation is consistent with evidence from mortgage default studies as to why the observed default exercise boundary is lower than that predicted by standard theoretical option-based models.     

Insurance contracts portfolios with heterogenous parametric life distributions

In this paper we consider two portfolios: one of m endowment insurance contracts and one of m whole life insurance contracts. We introduce the majorization order, Schur functions, and parametric families of distribution functions. We assume that the owners of the portfolios are exposed to different members of a known parametric family of distributions and study the effect of this stochastic heterogeneity on the premiums and death benefits of the insurance contracts. We show that the premiums paid in both contracts are Schur concave and that the death benefit awarded in the whole life contract is Schur convex. We provide upper and lower bounds for the premiums and for the death benefit, and compute the bounds for four parametric families of distribution functions used frequently in the Actuarial Sciences.     

Principle of equivalent utility and universal variable life insurance pricing

In this paper we study the pricing problem for a class of universal variable life (UVL) insurance products, using the idea of principle of equivalent utility. As the main features of UVL products we allow the (death) benefit to depend on certain indices or assets that are not necessarily tradable (e.g., pension plans), and we also consider the “multiple decrement” cases in which various status of the insured are allowed and the benefit varies in accordance with the status. Following the general theory of indifference pricing, we formulate the pricing problem as stochastic control problems, and derive the corresponding HJB equations for the value functions. In the case of exponential utilities, we show that the prices can be expressed explicitly in terms of the global, bounded solutions of a class of semilinear parabolic PDEs with exponential growth. In the case of general insurance models where multiple decrements and random time benefit payments are all allowed, we show that the price should be determined by the solutions to a system of HJB equations, each component corresponds to the value function of an optimization problem with the particular status of the insurer.     

Creating Customer Value in Participating Life Insurance

The value of a life insurance contract may differ depending on whether it is looked at from the customer's point of view or that of the insurance company. We assume that the insurer is able to replicate the life insurance contract's cash flows via assets traded on the capital market and can hence apply risk‐neutral valuation techniques. The policyholder, on the other hand, will take risk preferences and diversification opportunities into account when placing a value on that same contract. Customer value is represented by policyholder willingness to pay and depends on the contract parameters, that is, the guaranteed interest rate and the annual and terminal surplus participation rate. The aim of this article is to analyze and compare these two perspectives. In particular, we identify contract parameter combinations that—while keeping the contract value fixed for the insurer—maximize customer value. In addition, we derive explicit expressions for a selection of specific cases. Our results suggest that a customer segmentation in this sense, that is, based on the different ways customers evaluate life insurance contracts and embedded investment guarantees while ensuring fair values, is worthwhile for insurance companies as doing so can result in substantial increases in policyholder willingness to pay.     

Lifetime asset allocation with idiosyncratic and systematic mortality risks

This paper considers a lifetime asset allocation problem with both idiosyncratic and systematic mortality risks. The novelty of the paper is to integrate stochastic mortality, stochastic interest rate and stochastic income into a unified framework. An investor, who is a wage earner receiving a stochastic income, can invest in a financial market, consume part of his wealth and purchase life insurance or annuity so as to maximize the expected utility from consumption, terminal wealth and bequest. The problem is solved via the dynamic programming principle and the Hamilton–Jacobi–Bellman equation. Analytical solutions to the problem are derived, and numerical examples are provided to illustrate our results. It is shown that idiosyncratic mortality risk has significant impacts on the investor’s investment, consumption, life insurance/annuity purchase and bequest decisions regardless of the length of the decision-making horizon. The systematic mortality risk is largely alleviated by trading the longevity bond. However, its impacts on consumption, purchase of life insurance/annuity and bequest as well as the value function are still pronounced, when the decision-making horizon is sufficiently long.     

Leverage, volatility and executive stock options

This paper studies how an optimal wage contract can be implemented using stock options, and derives the properties of the optimal contract with stock options. Specifically, we show how the exercise price and the size of the option grant should change in response to changes in exogenous parameters. First, for a fixed exercise price of executive stock options, the size of the option grant decreases in the riskiness of a desired investment policy, decreases in the volatility of return from the risky project, and increases in leverage. Second, for a fixed size of the option grant, the optimal exercise price of managerial stock options increases in the riskiness of a desired investment policy, increases in the volatility of return from the risky project, and decreases in leverage. Several empirical predictions are drawn from these conclusions regarding the pay-performance sensitivity of management compensation.     

Maturity and exercise price of executive stock options

Using a simple three-period model in which a manager can gather information before making an investment decision, this paper studies optimal contracts with various stock options. In particular, we show how the exercise price of executive stock options is related to a base salary, the size of the option grant, leverage, and the riskiness of a desired investment policy. The optimal exercise price increases in the size of grant and the base salary and decreases in leverage and the riskiness of a desired investment policy. Other things equal, the optimal exercise price of European options with a longer maturity should increase more for an increase in the base salary and the size of grant and decrease more for an increase in leverage than the one with a shorter maturity. The optimal exercise price of American options is determined by the optimal exercise prices of European options with different maturities. Given the fixed exercise price, the size of the option grant does not decrease in the face value of debt.     

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.     

Anorexia nervosa: a 63-year population-based survival study

As eating disorders attract increasing publicity, more affected individuals will seek medical attention. Many will have needs for life insurance. Due to selection bias, most of the literature on anorexia nervosa (AN) presents an unfavorable prognosis. Therefore, the impairment is considered an adverse life insurance risk. This review is from an unselected, community population. The demographics of the study population and its expected mortality are similar to a population purchasing life insurance products. Comparative experience over 63 years of follow-up reveals mortality ratios and excess death rates similar to those expected for the population. High-risk comorbid diagnoses of depression and alcoholism are discussed.     

The early exercise of American puts

This paper shows that American puts on dividend paying stocks are most likely to be exercised either just after an ex-dividend date or just prior to expiration. At any other time the option to exercise an American put early may have less value. Thus, put writers and converters can predict when protection against premature exercise will be most valuable. The probability of early exercise is shown to be sensitive to managerial policy regarding the suspension of dividend payments, transaction costs, and interest rates. However, dividend payments are demonstrated to be the primary deterrent to early exercise.     

Valuation of Participating Life Insurance Liabilities

Surrender and paid-up states are incorporated in the valuation of guaranteed benefits and payments of a level premium paying life insurance policy.

We present different valuation methods and examine to what extent they avoid capitalizing and releasing future loadings which are associated with the payment of future premiums.

We demonstrate how to avoid capital being required in the future to cover valuation strains. The paid-up benefit valuation method is being extended so that it does not require the premium basis to be on the safe-side of the valuation basis. We obtain a unification and integration of the level premium and paid-up valuation principles.     

Unisex-Tafeln in der betrieblichen Altersversorgung

Due to the so-called “Test-Achats”-decision of the European Court the insurance business is stipulated to use unisex-calculations from December 21th, 2012 onwards. According to prevailing view this also impacts occupational pension schemes. The present article deals with an exact as possible implementation of the unisex-approach in the mortality table RT05G of Klaus Heubeck as this mortality table is relevant for the pension schemes in Germany. For pension schemes—in contrast to the private insurance industry—one has to deal with the difficulties that on the one hand the different kinds of benefit like old age pension, disability benefits and widow’s pension are evaluated at the same time, and on the other hand the mortality and disability rates contain no (or little) safety margins. Therefore one has to establish an exact as possible approach for all biometric probabilities simultaneously. Apart from the natural demixing due to mortality a closer look has to be taken at the probabilities concerning the surviving dependants—probabilty of being married at death, age of the surviving dependants and mortality of surving dependants.     

Assessing the Risk Potential of Premium Payment Options in Participating Life Insurance Contracts

Most life insurance contracts embed the right to stop premium payments during the term of the contract (paid-up option). Thereby, the contract is not terminated but continues with reduced benefits and often provides the right to resume premium payments later, thus increasing the previously reduced benefits (resumption option). In our analysis, we start with a basic contract with two standard options, namely, an interest rate guarantee and annual surplus participation. Next, in addition to the features of the basic contract, a paid-up and resumption option is included in the framework. The valuation process is not based on assumptions about a particular policyholders' exercise strategy but instead assesses the risk potential from the insurer's viewpoint by providing an upper bound for any possible exercise behavior. This approach provides important information to the insurer about the potential hazard of offering the paid-up and resumption option. Further, the approach allows an analysis of the impact of guaranteed interest rate, annual surplus participation, and investment volatility on the values of the premium payment options.     

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.