Guaranteed Minimum Death Benefits Research Articles

On the Valuation of GMDB Options Using a Combination of Exponentials

This paper explores the applicability of a combination of exponentials to the valuation of guaranteed minimum death benefits (GMDBs) embedded in variable annuities. As is well known, the class of combinations of exponentials forms a weakly dense subset among the distributions of all nonnegative random variables. Moreover, explicit pricing formulas for various GMDBs are available under a simplified exponential mortality model. Motivated by these two facts, we first approximate the future lifetime distributions by using a combination of exponentials and then approximate GMDB option prices. Comparing with the numerical integration under the Makeham law of mortality, our numerical result shows that the approximation can be quite useful.

Valuation of variable annuities with guaranteed minimum withdrawal and death benefits via stochastic control optimization

In this paper we present a numerical valuation of variable annuities with combined Guaranteed Minimum Withdrawal Benefit (GMWB) and Guaranteed Minimum Death Benefit (GMDB) under optimal policyholder behavior solved as an optimal stochastic control problem. This product simultaneously deals with financial risk, mortality risk and human behavior. We assume that market is complete in financial risk and mortality risk is completely diversified by selling enough policies and thus the annuity price can be expressed as appropriate expectation. The computing engine employed to solve the optimal stochastic control problem is based on a robust and efficient Gauss–Hermite quadrature method with cubic spline. We present results for three different types of death benefit and show that, under the optimal policyholder behavior, adding the premium for the death benefit on top of the GMWB can be problematic for contracts with long maturities if the continuous fee structure is kept, which is ordinarily assumed for a GMWB contract. In fact for some long maturities it can be shown that the fee cannot be charged as any proportion of the account value — there is no solution to match the initial premium with the fair annuity price. On the other hand, the extra fee due to adding the death benefit can be charged upfront or in periodic installment of fixed amount, and it is cheaper than buying a separate life insurance.

Optimal allocation and consumption with guaranteed minimum death benefits, external income and term life insurance

Because human capital is often the largest asset an investor possesses when he is young, protecting human capital from potential risks should be considered as a part of overall investment advice. The risk of the loss of the policyholder’s human capital–the mortality risk–to the household can be partially hedged by a term life insurance policy. Guaranteed Minimum Death Benefits (GMDB) in Variable Annuities (VA) can also help policyholders hedge the risk of the loss of human capital. Therefore, GMDB options and term life insurance can be considered as substitute goods. However, they are not perfect substitute as GMDB and term life have their own properties: Term life insurance has no correlation with equity markets, and it is purely regarded as a protection for human capital; the variable annuity products follow the performance of equity markets, and the GMDB is a protection against downside risks on equity markets as well as human capital. We find that fairly priced GMDB options fail to add value to a VA contract if a term life policy is available.

Valuation of Variable Annuities with Guaranteed Minimum Withdrawal and Death Benefits via Stochastic Control Optimization

In this paper we present a numerical valuation of variable annuities with combined Guaranteed Minimum Withdrawal Benefit (GMWB) and Guaranteed Minimum Death Benefit (GMDB) under optimal policyholder behavior solved as an optimal stochastic control problem. This product simultaneously deals with financial risk, mortality risk and human behavior. We assume that market is complete in financial risk and mortality risk is completely diversified by selling enough policies and thus the annuity price can be expressed as appropriate expectation. The computing engine employed to solve the optimal stochastic control problem is based on a robust and efficient Gauss-Hermite quadrature method with cubic spline. We present results for three different types of death benefit and show that, under the optimal policyholder behavior, adding the premium for the death benefit on top of the GMWB can be problematic for contracts with long maturities if the continuous fee structure is kept, which is ordinarily assumed for a GMWB contract. In fact for some long maturities it can be shown that the fee cannot be charged as any proportion of the account value -- there is no solution to match the initial premium with the fair annuity price. On the other hand, the extra fee due to adding the death benefit can be charged upfront or in periodic instalment of fixed amount, and it is cheaper than buying a separate life insurance.

Quantile Hedging for Guaranteed Minimum Death Benefits with Regime Switching

The efficient hedging minimizes the average of the shortfall risk weighted by a loss function, where the hedging efficiency refers to the effectiveness of a hedge to accomplish the desired goal of risk management. Quantile hedging refers to the percentage of the hedge that can cover the contingent claim, which plays a key role for contingent claims in incomplete markets when perfect hedging is not possible. As observed in [8], the concept of quantile hedging can be considered as a dynamic version of the familiar value at risk concept (VaR). Treating regime switching diffusion models, this article focuses on guaranteed minimum death benefits (GMDBs), which are present in many variableannuity contracts, and act as a form of portfolio insurance. The GMDBs cannot be perfectly hedged due to the mortality component and incompleteness resulting from the regime switching, and as a result, quantile hedges are developed. Numerical examples are also presented to illustrate our results.

Optimal consumption and allocation in variable annuities with Guaranteed Minimum Death Benefits

We determine the optimal allocation of funds between the fixed and variable subaccounts in a variable annuity with a GMDB (Guaranteed Minimum Death Benefit) clause featuring partial withdrawals by using a utility-based approach. The Merton method is applied by assuming that individuals allocate funds optimally in order to maximize the expected utility of lifetime consumption. It also reflects bequest motives by including the recipient’s utility in terms of the policyholder’s guaranteed death benefits. We derive the optimal transfer choice by the insured, and furthermore price the GMDB through maximizing the discounted expected utility of the policyholders and beneficiaries by investing dynamically in the fixed account and variable fund and withdrawing optimally.

Valuing equity-linked death benefits and other contingent options: A discounted density approach

Motivated by the Guaranteed Minimum Death Benefits in various deferred annuities, we investigate the calculation of the expected discounted value of a payment at the time of death. The payment depends on the price of a stock at that time and possibly also on the history of the stock price. If the payment turns out to be the payoff of an option, we call the contract for the payment a (life) contingent option. Because each time-until-death distribution can be approximated by a combination of exponential distributions, the analysis is made for the case where the time until death is exponentially distributed, i.e., under the assumption of a constant force of mortality. The time-until-death random variable is assumed to be independent of the stock price process which is a geometric Brownian motion. Our key tool is a discounted joint density function. A substantial series of closed-form formulas is obtained, for the contingent call and put options, for lookback options, for barrier options, for dynamic fund protection, and for dynamic withdrawal benefits. In a section on several stocks, the method of Esscher transforms proves to be useful for finding among others an explicit result for valuing contingent Margrabe options or exchange options. For the case where the contracts have a finite expiry date, closed-form formulas are found for the contingent call and put options. From these, results for De Moivre’s law are obtained as limits. We also discuss equity-linked death benefit reserves and investment strategies for maintaining such reserves. The elasticity of the reserve with respect to the stock price plays an important role. Whereas in the most important applications the stopping time is the time of death, it could be different in other applications, for example, the time of the next catastrophe.

Fatalities among Civil Infrastructure Workers: Variable Annuity with Guaranteed Minimum Death Benefits Approach

AbstractMany studies are concerned with monitoring, evaluating, expanding, repairing, replacing, financing, or otherwise sustaining the civil infrastructure. Few studies address the ongoing and evolving financial problems associated with the workers responsible for developing civil infrastructure development. According to U.S. federal statistics, civil infrastructure activities are responsible for 22% of all work-related fatalities—the highest among all industries. In 2008, 429,000 nonfatal injuries and 1,005 fatal injuries were recorded on job sites. However, civil infrastructure workers still have some of the weakest coverage of health and retirement benefits. It is crucial to have a strong system that protects workers and their families in cases of work-related fatal and nonfatal injuries. This paper developed a financial model incorporating a variable annuity embedded with guaranteed minimum death benefits. The model is based on the principles of insurance pricing, option theory, Monte Carlo simulatio...

Numerical solutions of quantile hedging for guaranteed minimum death benefits under a regime-switching jump-diffusion formulation

This work develops numerical approximation methods for quantile hedging involving mortality components for contingent claims in incomplete markets, in which guaranteed minimum death benefits (GMDBs) could not be perfectly hedged. A regime-switching jump-diffusion model is used to delineate the dynamic system and the hedging function for GMDBs, where the switching is represented by a continuous-time Markov chain. Using Markov chain approximation techniques, a discrete-time controlled Markov chain with two component is constructed. Under simple conditions, the convergence of the approximation to the value function is established. Examples of quantile hedging model for guaranteed minimum death benefits under linear jumps and general jumps are also presented.

Impacts of Jumps and Stochastic Interest Rates on the Fair Costs of Guaranteed Minimum Death Benefit Contracts

The authors offer a new perspective to the field of guaranteed minimum death benefit contracts, especially for simple return premium and rising floor guarantees. A particular feature of these contracts is a guaranteed capital upon the insured's death. A complete methodology based on the generalized Fourier transform is proposed to investigate the impacts of jumps and stochastic interest rates. This paper thus extends Milevsky and Posner (2001). If jumps alone are considered, similar results are obtained, but, when stochastic interest rates are introduced, the fair costs of the guarantee feature are found to be substantially higher in this more general economy.

The Effect of Policyholder Transfer Behavior on the Value of Guaranteed Minimum Death Benefits

Variable annuity contracts frequently include both guaranteed minimum death benefit (GMDB) options and options to transfer funds between fixed and variable accounts. We model the difference between fixed and variable rates as the primary determinant of policyholder transfer behavior. We find that people tend to transfer their money into variable accounts at about 39% of the rate that would be required to maintain constant percentage rebalancing, but with the opposite sign. If these transfers are not taken into account, the GMDB options on the variable accounts will be overvalued and overhedged. Ignoring this effect can have a substantial impact on the size of the futures portfolio needed to hedge this risk and a nonnegligible impact on the earnings of the variable annuity portfolio.

Valuing the Guaranteed Minimum Death Benefit Clause with Partial Withdrawals

In this paper, we give a method for computing the fair insurance fee associated with the guaranteed minimum death benefit (GMDB) clause included in many variable annuity contracts. We allow for partial withdrawals, a common feature in most GMDB contracts, and determine how this affects the GMDB fair insurance charge. Our method models the GMDB pricing problem as an impulse control problem. The resulting quasi-variational inequality is solved numerically using a fully implicit penalty method. The numerical results are obtained under both constant volatility and regime-switching models. A complete analysis of the numerical procedure is included. We show that the discrete equations are stable, monotone and consistent and hence obtain convergence to the unique, continuous viscosity solution, assuming this exists. Our results show that the addition of the partial withdrawal feature significantly increases the fair insurance charge for GMDB contracts.

Quantile hedging for guaranteed minimum death benefits

Quantile hedging for contingent claims is an active topic of research in mathematical finance. It plays a role in incomplete markets when perfect hedging is not possible. Guaranteed minimum death benefits (GMDBs) are present in many variable annuity contracts, and act as a form of portfolio insurance. They cannot be perfectly hedged due to the mortality component, except in the limit as the number of contracts becomes infinitely large. In this article, we apply ideas from finance to derive quantile hedges for these products under various assumptions.

The Impacts of Guaranteed Minimum Death Benefit Options on the Profit Margins of Unit Linked Products

This paper shows a model of the profit testing and the process of assessing the profitability of the unit linked products. The impacts of the imbedded guaranteed minimum death benefit (GMDB) options on the profit margins of the unit linked policies are analysed.

Analytic Solution for Return of Premium and Rollup Guaranteed Minimum Death Benefit Options Under Some Simple Mortality Laws

Much attention has been focused recently on the issue of valuing guaranteed minimum death benefits embedded in annuity contracts. These benefits resemble a sequence of put options and their value should obey a differential equation similar to the Black-Scholes equation for simple put options. This paper derives a number of analytic solutions to this equation for a number of simple mortality laws.

Analytic Solution for Return of Premium and Rollup Guaranteed Minimum Death Benefit Options Under Some Simple Mortality Laws

Much attention has been focused recently on the issue of valuing guaranteed minimum death benefits embedded in annuity contracts. These benefits resemble a sequence of put options and their value should obey a differential equation similar to the Black-Scholes equation for simple put options. This paper derives a number of analytic solutions to this equation for a number of simple mortality laws.

A Universal Pricing Framework for Guaranteed Minimum Benefits in Variable Annuities

Variable Annuities with embedded guarantees are very popular in the US market. There exists a great variety of products with both, guaranteed minimum death benefits (GMDB) and guaranteed minimum living benefits (GMLB). Although several approaches for pricing some of the corresponding guarantees have been proposed in the academic literature, there is no general framework in which the existing variety of such guarantees can be priced consistently. The present paper fills this gap by introducing a model, which permits a consistent and extensive analysis of all types of guarantees currently offered within Variable Annuity contracts. Besides a valuation assuming that the policyholder follows a given strategy with respect to surrender and withdrawals, we are able to price the contract under optimal policyholder behavior. Using both, Monte-Carlo methods and a generalization of a finite mesh discretization approach, we find that some guarantees are overpriced, whereas others, e.g. guaranteed annuities within guaranteed minimum income benefits (GMIB), are offered significantly below their risk-neutral value.

A Universal Pricing Framework for Guaranteed Minimum Benefits in Variable Annuities

Variable Annuities with embedded guarantees are very popular in the US market. There exists a great variety of products with both, guaranteed minimum death benefits (GMDB) and guaranteed minimum living benefits (GMLB). Although several approaches for pricing some of the corresponding guarantees have been proposed in the academic literature, there is no general framework in which the existing variety of such guarantees can be priced consistently. The present paper fills this gap by introducing a model, which permits a consistent and extensive analysis of all types of guarantees currently offered within Variable Annuity contracts. Besides a valuation assuming that the policyholder follows a given strategy with respect to surrender and withdrawals, we are able to price the contract under optimal policyholder behavior. Using both, Monte-Carlo methods and a generalization of a finite mesh discretization approach, we find that some guarantees are overpriced, whereas others, e.g. guaranteed annuities within guaranteed minimum income benefits (GMIB), are offered significantly below their risk-neutral value.

Impacts of Jumps and Stochastic Interest Rates on the Fair Costs of Guaranteed Minimum Death Benefit Contracts

The authors offer a new perspective to the domain of guaranteed minimum death benefit contracts. These products have the particular feature to offer investors a guaranteed capital upon death. A complete methodology based on the generalized Fourier transform is proposed to investigate the impacts of jumps and stochastic interest rates. This paper thus extends Milevsky and Posner (2001). In contrast to their results, the fair costs of the guarantee feature are found to be substantially higher in this more general economy.

The Effect of the Real Option to Transfer on the Value of Guaranteed Minimum Death Benefits

AbstractVariable annuity contracts frequently have many options and option‐like features embedded in the contracts. Some are obvious, such as guaranteed minimum death benefits (GMDBs), while others are less obviously option‐like. In this article, we consider the effect of the real option to transfer funds between fixed and variable accounts. If a GMDB rider is considered in isolation, it is sometimes in the policyholder's interest to transfer to the fixed fund if the fixed fund earns less than the variable fund in a risk‐neutral world. On the other hand, the option to transfer will not be used if the entire annuity and rider are considered together.