Minimum Maturity Benefit Research Articles

Optimal Static Hedging of Variable Annuities with Volatility-Dependent Fees

Variable annuities (VAs) and other long-term equity-linked insurance products are typically difficult to hedge in the incomplete markets. A state-dependent fee tied with market volatility for VAs is designed to contribute the risk-sharing mechanism between policyholders and insurers. Different from prior research, we discuss several aspects on a fair valuation, fee-rate determination and hedging with volatility-dependent fees from the perspective of a VA hedger. A method of efficient hedging strategy as a benchmark compared to other strategies is developed in the stochastic volatility setting. We illustrate this method in guaranteed minimum maturity benefits (GMMBs), but it is also applicable to other equity-linked insurance contracts.

Valuation of guaranteed minimum maturity benefits under mean reversion and jump models with surrender risk

We propose an efficient valuation approach for guaranteed minimum benefits embedded in variable annuity (VA) contracts, where the price of underlying asset is modeled by a mean reversion process with jumps. We consider the case in which early surrender is permitted, and an intensity-based framework is used to capture surrender behavior and other asset price volatilities. To analyze the contract within this complex stochastic setting, we rely on the Fourier cosine series expansion method to approximate the expectation of the value function, which serves as an estimation of the value of VA benefits. We conduct numerical examples to demonstrate the efficiency and accuracy of the proposed method. Sensitivity analysis of involved parameters is given.

Guaranteed Minimum Maturity Benefits in a Self-Exciting Stochastic Mortality Model: Pricing, Estimation and Calibration

The guaranteed minimum maturity benefit (GMMB) is quite a popular feature embedded in several unit-linked policies offered by insurance companies. The value of this benefit depends on several processes assumed to describe both the mortality and the financial dynamics, typically represented by interest rates and by the fund associated to the unit-linked policy. A large literature is devoted to the valuation of GMMB for different mortality models, in particular when the mortality dynamics is described by affine models of diffusion type. In the present article we assume a self-exciting behavior for the mortality dynamics, described by a Hawkes-type process with exponential kernel. This allows us to keep both the Markov and affine features but introduces jumps with a stochastic intensity. These types of dynamics, exhibiting a jump clustering property, are quite convenient for describing mortality in some critical situations, like epidemics, where contagion phenomena make the probability of jump arrivals higher whenever a jump occurs. By assuming diffusion-type dynamics for both the fund and the interest rate and introducing all of the possible correlations among the diffusion processes necessary to obtain realistic dynamics, we take advantage of the affine features of the model proposed and compute in a semi-explicit form the value of a GMMB. Lastly, we calibrate our model and perform an empirical analysis to determine the effects of excess mortality on GMMB prices as well as a death benefit portfolio.

Valuation of variable annuities with guaranteed minimum maturity benefits and periodic fees

This paper focuses on the valuation of variable annuities with a guaranteed minimum maturity benefit under a regime-switching Lévy model. The model allows policyholders to surrender their annuities and receive a surrender benefit at predetermined tenor times before maturity. Additionally, we consider a state-dependent periodic fee structure where fees are deducted from the policyholder's account if it exceeds a certain level at discrete time points. Incorporating this fee structure, the Fourier cosine series expansion method based on characteristic functions is employed to determine the values and optimal surrender strategies for variable annuity contracts. Finally, we provide a comprehensive set of numerical examples to demonstrate and assess the effectiveness of our approach thoroughly.

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.

Pricing a Guaranteed Minimum Maturity Benefit in Uncertain Markets

Pricing a Guaranteed Minimum Maturity Benefit in Uncertain Markets

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.

Efficient valuation of guaranteed minimum maturity benefits in regime switching jump diffusion models with surrender risk

We present an efficient valuation approach for guaranteed minimum maturity benefits (GMMBs) embedded in variable annuity (VA) contracts in a regime-switching jump diffusion model. We allow early surrender of the VA contract and impose surrender charges, which are important in practice to discourage early termination/lapse of the contract. We consider both continuously-monitored and discretely-monitored surrender behaviors before maturity, and utilize an intensity-based framework. Based on the continuous-time Markov chain (CTMC) approximation combined with the Fourier cosine series expansion method, we find that the valuation problem can be solved under a regime-switching jump diffusion framework. Both error analysis and numerical experiments demonstrate the accuracy and efficiency of the proposed method.

Valuing variable annuities with path-dependent surrender guarantees under regime-switching Lévy models

Variable annuities with complex surrender features are nowadays increasingly popular for managing longevity risks. The study of their accurate pricing and sensitivity analysis is one of the main actuarial research topics. This paper studies the valuation problem of variable annuity contracts with guaranteed minimum maturity benefits on a set of predetermined discrete tenor dates under regime-switching Lévy models. Extending from existing vanilla payoffs, we consider the guaranteed minimum maturity benefits with lookback and geometric average features. We customise the dynamic programming principle to solve the corresponding optimal stopping problem, relying on some semi-analytical valuation formulae resulting from an acute Fourier cosine series expansion. Finally, numerical illustrations are provided to show the accuracy and efficiency of the proposed method. We also demonstrate the use of our proposed method in a range of sensitivity analysis exercises, which shed light on the pricing and risk management of complex variable annuity products.

Valuation of guaranteed minimum maturity benefits under generalised regime-switching models using the Fourier Cosine method

This paper presents a flexible valuation approach for variable annuity (VA) contracts embedded with guaranteed minimum maturity benefit (GMMB) riders written on an underlying fund that evolves according to a general regime-switching framework. Unlike the classical regime-switching models which only allow model parameters to change upon regime switches, our framework allows, more importantly, model structures to vary. With mild assumptions on the characteristic function of the log-stock price, our model settings enable the study of fundamental features of the market dynamics, such as stochastic volatility and jumps, on the underlying fund value of GMMB in a unified framework. This novel idea is illustrated by a three-regime model whose environments can be characterised by either the geometric Brownian motion process, the Heston (1993) stochastic volatility process, or double exponential process. Three versions of the GMMB riders are considered; a fixed or roll-up guarantee, a geometric average guarantee and a ratchet guarantee. With the Fourier Cosine (COS) method which utilises characteristic functions, explicit valuation expressions for various contracts are derived, and numerical illustrations are performed to analyse the efficiency of the approach in terms of computational speed and accuracy. The paper makes a unique contribution by presenting regime-dependent bounds and an algorithm for determining the optimal grid points required for the COS method to achieve a specific level of accuracy. Numerical experiments for the valuation framework reveal that i) as the likelihood of regime shifts increases, the price difference of VA contracts with different initial regimes diminishes, which is consistent with financial intuition; ii) the generalised regime-switching models are more suitable for VA pricing when the time-to-maturity spans over several decades, during which structural changes, although infrequent, are inevitable.

Portfolio Optimization With a Guaranteed Minimum Maturity Benefit and Risk-Adjusted Fees

Portfolio Optimization With a Guaranteed Minimum Maturity Benefit and Risk-Adjusted Fees

Valuation of a Guaranteed Minimum Maturity Benefit in a Self-exciting Stochastic Mortality Model

Valuation of a Guaranteed Minimum Maturity Benefit in a Self-exciting Stochastic Mortality Model

Valuing guaranteed minimum accumulation benefits by a change of numéraire approach

Three correlated risk factors, namely, financial, mortality and lapse risks, are modelled in an integrated way to support the valuation of guaranteed minimum accumulation benefits (GMAB). The dynamics of such risk factors under the risk-adjusted measure needed for valuation are derived. A sequence of reference probability measure changes is employed to obtain a compact and implementable price representations. We demonstrate how a change of numéraire approach facilitates the valuation of a GMAB rider in a variable annuity. A numerical illustration is presented to confirm the computational efficiency and accuracy of our proposed valuation methodology. In particular, our proposed approach on average takes only 0.07% of the computing time for the Monte-Carlo (MC) simulation technique to complete the valuation calculation. Furthermore, the standard errors of our approach's results are lower than those obtained from the MC-based computations. A sensitivity analysis is also performed with respect to various parameters. When there are no renewal options in a GMAB contract, we get the special case of a guaranteed minimum maturity benefit for which a closed-form valuation solution is established.

Optimal Surrender Policy of Guaranteed Minimum Maturity Benefits in Variable Annuities with Regime-Switching Volatility

This study investigates valuation of guaranteed minimum maturity benefits (GMMB) in variable annuity contract in the case where the guarantees can be surrendered at any time prior to the maturity. In the event of the option being exercised early, early surrender charges will be applied. We model the underlying mutual fund dynamics under regime-switching volatility. The valuation problem can be reduced to an American option pricing problem, which is essentially an optimal stopping problem. Then, we obtain the pricing partial differential equation by a standard Markovian argument. A detailed discussion shows that the solution of the problem involves an optimal surrender boundary. The properties of the optimal surrender boundary are given. The regime-switching Volterra-type integral equation of the optimal surrender boundary is derived by probabilistic methods. Furthermore, a sensitivity analysis is performed for the optimal surrender decision. In the end, we adopt the trinomial tree method to determine the optimal strategy.

Analysis of VIX-linked fee incentives in variable annuities via continuous-time Markov chain approximation

We consider the pricing of variable annuities (VAs) with general fee structures under a class of stochastic volatility models which includes the Heston, Hull-White, Scott, α-Hypergeometric, 3/2, and 4/2 models. In particular, we analyze the impact of different VIX-linked fee structures on the optimal surrender strategy of a VA contract with guaranteed minimum maturity benefit (GMMB). Under the assumption that the VA contract can be surrendered before maturity, the pricing of a VA contract corresponds to an optimal stopping problem with an unbounded, time-dependent, and discontinuous payoff function. We develop efficient algorithms for the pricing of VA contracts using a two-layer continuous-time Markov chain approximation for the fund value process. When the contract is kept until maturity and under a general fee structure, we show that the value of the contract can be approximated by a closed-form matrix expression. We also provide a quick and simple way to determine the value of early surrenders via a recursive algorithm and give an easy procedure to approximate the optimal surrender surface. We show numerically that the optimal surrender strategy is more robust to changes in the volatility of the account value when the fee is linked to the VIX index.

The Valuation of a Guaranteed Minimum Maturity Benefit under a Regime-Switching Framework

Global insurance markets have become more sophisticated in recent times in response to the evolving needs of populations that tend to live longer. Policy holders desire the benefits of longevity/mortality protection while taking advantage of investment growth opportunities in equity markets. As a result, insurers incorporate payment guarantees in new insurance products, known as equity-linked contracts, whose values are dependent on prices of risky assets. A guaranteed minimum maturity benefit (GMMB) is now common in many equity-linked contracts. We develop an integrated pricing framework for a GMMB focusing on segregated fund contracts. More specifically, we construct hidden Markov models (HMMs) for a stock index, interest rate, and mortality rate. The dependence between these risk factors is characterized explicitly. We assume that the stock index follows a Markov-modulated geometric Brownian motion and the interest and mortality rates have Markov-modulated affine dynamics. A series of measure changes is employed to obtain a semi-closed-form solution for the GMMB price. A Fourier transform method is applied to numerically approximate the prices more efficiently. Recursive HMM filtering is used in our model calibration. Numerical investigations in our article demonstrate the accuracy of GMMB prices and an extensive analysis is included to systematically examine how risk factors affect the value of a GMMB.

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.

Portfolio Optimization With a Guaranteed Minimum Maturity Benefit and Risk-Adjusted Fees

We study a portfolio optimization problem involving the rational policyholder of a variable annuity with a guaranteed minimum maturity benefit. This financial guarantee is fi nanced via a fee withdrawn directly from the investment account, which impacts the net investment return. We propose a new fee structure that adjusts to the investment mix. A fair pricing constraint is de fined in terms of the risk-neutral value of the fi nal contract payout. We seek the investment strategy that maximizes the policyholder's expected utility of terminal wealth after the application of a fi nancial guarantee and subject to the fair pricing constraint. We solve the associated constrained stochastic control problem using a martingale approach and analyze the impact of the fee structure on the optimal investment strategies and payoff. Numerical results show that it is possible to nd an optimal portfolio for a wide range of fees, while keeping the contract fairly priced.

Optimal surrender strategies and valuations of path-dependent guarantees in variable annuities

In this paper, we consider variable annuity contracts with guaranteed minimum maturity benefits in which the embedded guarantees can be surrendered any time before the maturity date has been reached. We derive the optimal surrender strategies and the values of three different types of guarantees embedded in VA contracts: (1) constant guarantees, (2) constant guarantees with an up-and-out barrier, and (3) path-dependent guarantees with a lookback feature. The valuation of the embedded guarantees with a surrender option can be formulated as a free boundary problem, and we can obtain a corresponding non-homogeneous partial differential equation (PDE). We apply the Mellin transform to the non-homogeneous PDE in order to obtain an integral equation that produces the value of the embedded guarantee as well as the optimal surrender boundary in a variable annuity contract. We present the properties of the fair fee rate and the optimal surrender strategy with numerical examples.

Optimal surrender of guaranteed minimum maturity benefits under stochastic volatility and interest rates

In this paper we analyse how the policyholders’surrender behaviour is influenced by changes in various sources of risk impacting a variable annuity (VA) contract embedded with a guaranteed minimum maturity benefit rider that can be surrendered anytime prior to maturity. We model the underlying mutual fund dynamics by combining a Heston (1993) stochastic volatility model together with a Hull and White (1990) stochastic interest rate process. The model is able to capture the smile/skew often observed on equity option markets (Grzelak and Oosterlee 2011) as well as the influence of the interest rates on the early surrender decisions as noted from our analysis. The annuity provider charges management fees which are proportional to the level of the mutual fund as a way of funding the VA contract. To determine the optimal surrender decisions, we present the problem as a 4-dimensional free-boundary partial differential equation (PDE) which is then solved efficiently by the method of lines (MOL) approach. The MOL algorithm facilitates simultaneous computation of the prices, fair management fees, optimal surrender boundaries and hedge ratios of the variable annuity contract as part of the solution at no additional computational cost. A comprehensive analysis on the impact of various risk factors in influencing the policyholder’s surrender behaviour is carried out, highlighting the significance of both stochastic volatility and interest rate parameters in influencing the policyholder’s surrender behaviour. With the aid of the hedge ratios obtained from the MOL, we construct an effective dynamic hedging strategy to mitigate the provider’s risk and compare different hedging performances when the policyholders’ surrender behaviour is either optimal or sub-optimal.