Guaranteed Minimum Death Benefits Research Articles

Valuing equity-linked guaranteed minimum death benefits with European-style Asian payoffs under a regime switching jump-diffusion model

This paper presents a novel and efficient approach to pricing equity-linked guaranteed minimum death benefits (GMDB) with European-style geometric Asian and arithmetic Asian payoffs. Our method assumes that the underlying asset process follows a regime-switching Lévy model, which captures the key features of market dynamics in the continuous transition of the economy. To derive the approximate value of GMDB products, we employ the complex Fourier series (CFS) expansion method. Our error analysis demonstrates that this approach exhibits an exponential convergence rate. In our numerical experiments, we compare the CFS approach to other Fourier transform methods and Monte Carlo simulation. The results show that our method outperforms the other approaches in terms of both efficiency and accuracy. This paper contributes to the literature on pricing equity-linked GMDB products by proposing a novel and efficient approach based on regime-switching Lévy models and complex Fourier series expansion. The implications of our results may have significant practical implications for the insurance industry and financial markets.

Economic Representative Scenarios for Variable Annuity Dynamic Hedging of GMMB and GMDB

Variable annuities introduce significant risk for the insurers. To control this risk, insurers generally use dynamic hedging models. In practice, calculations for dynamic hedging models for variable annuities are computationally intensive since they require many nested stochastic scenario projections with outer and inner loops. In this article, we study the relationship between the hedging errors and the descriptive statistics of economic scenarios. Then, using the relationships between the hedging errors and the descriptive statistics of the economics scenarios, we present two novel scenario selection algorithms to determine economic representative scenarios (ERS) to run for dynamic risk hedging models. We do this for guaranteed minimum maturity benefit (GMMB) and guaranteed minimum death benefit (GMDB). We also benchmark these new algorithms against a control variate algorithm and Monte Carlo simulations. One of the two proposed novel scenario selection algorithms, the adapted Clara algorithm gives a lower mean absolute deviation of the conditional tail expectation (CTE) than the control variate method and Monte Carlo simulations for the same number of scenarios. Also, 500 representative scenarios selected by the adapted Clara algorithm generate a mean absolute deviation of the CTE95 of the hedging errors approximately equivalent to 1000 (or more in some cases) Monte Carlo simulations for 5 years GMMB and GMDB.

Valuation of Equity-Linked Death Benefits on Two Lives with Dependence

The purpose of this paper is to investigate equity-linked death benefits for joint alive and last survivor individuals. Utilizing Farlie–Gumbel–Morgenstern (FGM) type dependency modeling framework, we first analyze the joint distribution of the couple (joint alive and last survival density) when marginal distributions follow mixed exponentials and weighted exponentials distributions. Then, we derive the price of the guaranteed minimum death benefit (GMDB) product. In addition, we provide closed analytical expressions of the price of some financial contingent claim contracts (classical and exotic options). Furthermore, we present some numerical results to support our theoretical results. We show in our numerical example that it is important to model the dependency between two lives (couple) since the price changes as the copula parameter changes.

An economic scenario generator for embedded derivatives in South Africa

It is well known that interest rate risk is a dominating factor when pricing long-dated contingent claims. The Heston stochastic volatility model fails to capture this risk as the model assumes a constant interest rate throughout the life of the claim. To overcome this, the risk-free interest rate can be modelled by a Hull-White short rate process and can be combined with the Heston stochastic volatility model to form the so-called Heston-Hull-White model. The Heston-Hull-White model allows for correlation between the equity and interest rate processes, a component that is important when pricing long-dated contingent claims. In this paper, we apply the Heston-Hull-White model to price Guaranteed Minimum Maturity Benefits (GMMBs) and Guaranteed Minimum Death Benefits (GMDBs) offered in the life insurance industry in South Africa. We propose a further extension by including stochastic mortality rates based on either a continuous-time Cox-Ingersoll-Ross short rate process or a discrete-time AR(1)-ARCH(1) model. Our findings suggest that stochastic interest rates are the dominating factor when reserving for GMMB and GMDB products. Furthermore, a delta-hedging strategy can help reduce the variability of embedded derivative liabilities.

Analytic valuation of GMDB options with utility based asset allocation

A number of analytic solutions have been found for Variable Annuity Guaranteed Minimum Death Benefit (GMDB) option values under a variety of mortality laws. To date, the solutions are for Risk-Neutral valuation only. Where policyholder decisions are allowed, it is assumed that they act to maximize the risk-neutral value of the GMDB. We examine situations where the asset allocation decisions are made to maximize expected utility rather than option value. We find analytic solutions for both return of premium and ratchet options at small values of bequest motive for a number of mortality laws.

Pricing some life-contingent lookback options under regime-switching Lévy models

In this paper, we study the valuation problem of life-contingent lookback options embedded in variable annuity with guaranteed minimum death benefit (GMDB). Specifically, the underlying asset price process is assumed to be an exponential regime-switching Lévy process, which is observed periodically. The Fourier cosine series expansion method is applied to compute exponential moments of the discretely monitored maximum and minimum of the regime-switching Lévy process. Furthermore, some explicit pricing formulas for the life-contingent lookback options embedded in GMDB products are derived. Finally, numerical experiments confirm the accuracy and efficiency of our method.

Randomization and the Valuation of Guaranteed Minimum Death Benefits

Randomization and the Valuation of Guaranteed Minimum Death Benefits

Valuing Equity-Linked Guaranteed Minimum Death Benefits with European-Style Asian Payoffs Under the Regime-Switching Diffusion Process with Jump

Valuing Equity-Linked Guaranteed Minimum Death Benefits with European-Style Asian Payoffs Under the Regime-Switching Diffusion Process with Jump

Equity-linked Guaranteed Minimum Death Benefits with dollar cost averaging

In this paper, we analyze a form of equity-linked Guaranteed Minimum Death Benefit (GMDB), whose payoff depends on a dollar cost averaging (DCA) style periodic investment in the risky index, with rider premiums paid at regular intervals. This rider is a very natural insurance vehicle for equity-linked variable annuities, and the DCA feature has a tendency to reduce the uncertainty associated with the final payoff, beyond the minimum benefit guaranteed to the beneficiary upon death of the insured. This makes the insurer risk easier to manage as the contract ages. From the policyholder's perspective, the protection is cheaper than a standard GMDB whose payoff depends solely on the risky index value upon death, but still offers the upside potential from an investment made at regular intervals. We derive closed-form valuation formulas under the fairly broad class of exponential Lévy models for the risky index, which includes Black-Scholes as a special case. Closed-form valuation is provided by a fundamental link between this contract and a series of Asian options, for which valuation is well established. We provide several valuation strategies, and demonstrate the soundness of the framework.

Pricing equity-linked death benefits by complex Fourier series expansion in a regime-switching jump diffusion model

In this paper, we consider the valuation problem of equity-linked annuity with guaranteed minimum death benefit (GMDB) by complex Fourier series method under regime-switching jump diffusion models. We show that the price formulas can be expressed by some discounted density functions associated with the residual lifetime random variable. Fourier transforms for discounted density functions are derived, and complex Fourier series expansion method is applied to recover the discounted density functions. Some explicit formulas for computing the GMDB price are given. Finally, we give some numerical results to show effectiveness of our method.

Valuing Guaranteed Minimum Death Benefits by Cosine Series Expansion

Recently, the valuation of variable annuity products has become a hot topic in actuarial science. In this paper, we use the Fourier cosine series expansion (COS) method to value the guaranteed minimum death benefit (GMDB) products. We first express the value of GMDB by the discounted density function approach, then we use the COS method to approximate the valuation Equations. When the distribution of the time-until-death random variable is approximated by a combination of exponential distributions and the price of the fund is modeled by an exponential Lévy process, explicit equations for the cosine coefficients are given. Some numerical experiments are also made to illustrate the efficiency of our method.

PENENTUAN HARGA PREMI ASURANSI UNIT LINK DENGAN GARANSI MINIMUM

The goal of this research is to determine the pricing of unit-linked insurance after attaching the minimum guarantees, which are guaranteed minimum maturity benefit (GMMB) and guaranteed minimum death benefit (GMDB) using the Black-Scholes-Merton Method. Before the new price is determined, the previous steps are find the value of and for GMMB or and for GMDB. The result of pricing on the cases in this research, resulted if the new price included GMMB with the interest rate 6% and management expenses 0% and 2% are changed from Rp 21.000.000,00 become Rp 21.003.000,00 and Rp 21.031.000,00. On the other hand, the new price for interest rates 14% and 20% with both management expenses are constant. Furthermore, the new price included GMDB with management expense 0% and 2% also interest rates 6%, 14%, and 20% in succession are changed from Rp 21.000.000,00 become Rp 25.132.000,00; Rp 21.031.000,00; Rp 21.002.000,00; Rp 44.521.000,00; Rp 44.520.000,00 and Rp 44.520.000,00.

Valuing equity-linked death benefits in general exponential Lévy models

In this paper, a projection method combined with Fast Fourier Transform (FFT) is applied to value equity-linked death benefit products. Specifically, we focus on valuation of the products with Guaranteed Minimum Death Benefit (GMDB). The benefit amount is linked to the performance of the underlying asset, which is modeled by an exponential Lévy process. To implement the proposed method, B-spline functions as well as some auxiliary functions are introduced for series expansion, while FFT is adopted to calculate the projection coefficients. Valuation formulae are obtained for a class of payoff functions, and error analysis is also presented. Various numerical valuation results computed by B-spline functions with different orders are presented to demonstrate the method’s efficiency and accuracy.

Valuing guaranteed equity-linked contracts by Laguerre series expansion

Recently, more and more actuarial researchers have paid attention to valuing life insurance products. In this paper, we study the valuation of equity-linked annuity contracts with guaranteed minimum death benefits (GMDB). The discounted density function approach is used to value this class of life insurance products. We develop the density function of the time-until-death variable on the Laguerre basis and value the GMDB contracts with various equity-linked death benefit functions, including basic options and lookback options. Compared to the Laplace inversion transform method in the literature, our method exhibits remarkable accuracy and efficiency. We also implement life table data to fit the density function of the time-until-death variable by Laguerre functions and provide numerical results for some death benefit functions.

PERHITUNGAN NILAI GARANSI MINIMUM MANFAAT KEMATIAN PADA ASURANSI UNIT-LINK

The guarantee minimum on unit-linked insurance applies only to the most extreme situations of very bad rate of return on the fund’s policyholders. One of the investment guarantees commonly used in unit-link is guaranteed minimum death benefit (GMDB). The final value under unit-linked insurance contracts can be expressed in terms of options that can be calculated using the Black-Scholes-Merton method. The purpose of this study is to determine the effect of age to the guarantee minimum value calculated using the Black-Scholes-Merton method. The calculation of GMDB value based on case simulation in this study resulted that the increasing age of the insured the greater the minimum guarantee value to be obtained.

FAST COMPUTATION OF RISK MEASURES FOR VARIABLE ANNUITIES WITH ADDITIONAL EARNINGS BY CONDITIONAL MOMENT MATCHING

AbstractWe propose an approximation scheme for the computation of the risk measures of guaranteed minimum maturity benefits (GMMBs) and guaranteed minimum death benefits (GMDBs), based on the evaluation of single integrals under conditional moment matching. This procedure is computationally efficient in comparison with standard analytical methods while retaining a high degree of accuracy, and it allows one to deal with the case of additional earnings and the computation of related sensitivities.

Indifference fee rate for variable annuities

ABSTRACTIn this paper, we work on indifference valuation of variable annuities and give a computation method for indifference fees. We focus on the guaranteed minimum death benefits (GMDB) and the guaranteed minimum living benefits (GMLB) and allow the policyholder to make withdrawals. We assume that the fees are continuously paid and that the fee rate is fixed at the beginning of the contract. Following indifference pricing theory, we define indifference fee rate for the insurer as a solution of an equation involving two stochastic control problems. Relating these problems to backward stochastic differential equations (BSDEs) with jumps, we provide a verification theorem and give the optimal strategies associated to our control problems. From these, we derive a computation method to get indifference fee rates. We conclude our work with numerical illustrations of indifference fees sensibilities with respect to parameters.

Valuing inflation-linked death benefits under a stochastic volatility framework

In this paper we construct a framework to price the inflation-linked derivatives with the stochastic inflation rate, the stochastic interest rate, and stochastic risky assets with stochastic volatility. Because of the popularity of the guaranteed minimum death benefit (GMDB) in insurance market, we mainly study two types of GMDBs: the inflation guarantee and the combination guarantee. We consider the guaranteed minimum death benefit as an European option with a random maturity date, the closed-form pricing formulas for the GMDBs are derived by Fourier-based method. Moreover, we give an elaborate sensitivity analysis to explain economical behaviors of our models. The numerical results show that the death benefit of inflation guarantee is slightly overpriced in constant volatility of stock situation.

Valuing variable annuity guarantees on multiple assets

Guarantees embedded variable annuity contracts exhibit option-like payoff features and the pricing of such instruments naturally leads to risk neutral valuation techniques. This paper considers the pricing of two types of guarantees; namely, the Guaranteed Minimum Maturity Benefit and the Guaranteed Minimum Death Benefit riders written on several underlying assets whose dynamics are given by affine stochastic processes. Within the standard affine framework for the underlying mortality risk, stochastic volatility and correlation risk, we develop the key ingredients to perform the pricing of such guarantees. The model implies that the corresponding characteristic function for the state variables admits a closed form expression. We illustrate the methodology for two possible payoffs for the guarantees leading to prices that can be obtained through numerical integration. Using typical values for the parameters, an implementation of the model is provided and underlines the significant impact of the assets’ correlation structure on the guarantee prices.

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.