Equity-indexed Annuities Research Articles

Efficient simulation and valuation of equity-indexed annuities under a two-factor G2++ model

Equity-indexed annuities (EIAs) with investment guarantees are pension products sensitive to changes in the interest rate environment. A flexible and common choice for modelling this risk factor is a Hull–White model in its G2++ variant. We investigate the valuation of EIAs in this model setting and extend the literature by introducing a more efficient framework for Monte-Carlo simulation. In addition, we build on previous work by adapting an approach based on scenario matrices to a two-factor G2++ model. This method does not rely on simulations or on Fourier transformations. In numerical studies, we demonstrate its fast convergence and the limitations of techniques relying on the independence of annual returns and the central limit theorem.

Analyzing the interest rate risk of equity-indexed annuities via scenario matrices

The financial return of equity-indexed annuities depends on an underlying fund or investment portfolio complemented by an investment guarantee. We discuss a so-called cliquet-style or ratchet-type guarantee granting a minimum annual return. Its path-dependent payoff complicates valuation and risk management, especially if interest rates are modelled stochastically. We develop a novel scenario-matrix (SM) method. In the example of a Vasicek-Black-Scholes model, we derive closed-form expressions for the value and moment-generating function of the final payoff in terms of the scenario matrix. This allows efficient evaluation of values and various risk measures, avoiding Monte-Carlo simulation or numerical Fourier inversion. In numerical tests, this procedure proves to converge quickly and outperforms the existing approaches in the literature in terms of computation time and accuracy.

A further examination of equity indexed annuities

Equity indexed annuities (EIAs) are deferred annuities that credit interest according to a formula tied to the performance of an underlying equity index. This research expands previous research, particularly that of Reichenstein (2009, 2011), by examining the distribution of returns that could have been created on a rolling monthly basis since 1928 for 11 through 15-year investment horizons.Second, we examine investment alternatives that include the options imbedded in EIAs. Third, rather than assuming constant cap rates we allow cap rates to vary with interest rates. We find that for long time horizons the opportunity costs of investing in EIAs is high.

Valuing rebate options and equity-linked products

In this article, we propose rebate options with multi-step barriers, which are an extension of rebate options with a constant barrier. Despite the applicability and marketability of rebate options, there have been only a few research on obtaining analytical formulas. Accordingly, in this paper, we derive closed-form pricing formulas for these options under the Black–Scholes framework. The rebate options with multi-step barriers allow a flexible barrier structure, and thus we propose complex equity-linked products embedded with rebate options with barriers and derive pricing formulas for them. We conduct numerical studies on the pricing of rebate options with multi-step barriers, equity-linked securities, and equity-indexed annuities. The numerical studies validate the prices obtained from the pricing formulas.

Pricing ratchet equity index annuity with mortality risk by complex Fourier series method

In this paper, we consider the pricing of the ratchet equity-indexed annuity with the incorporation of mortality risk by an efficient and accurate method under the general time-changed Lévy processes. The pricing of the contract is a two-dimensional problem: we apply the complex Fourier series method to the log return dimension of the asset and high-order quadrature approximation to the other dimensions. Depending on the subordinating dynamic of the volatility, we use the time-changed Lévy process to model stochastic volatility via the random time change in the Lévy process, which compensates for limitations in the common Lévy process. By combining the recursive algorithm and complex Fourier series method, we obtain the approximate price of the contract. Error analysis proves that the total error can achieve an exponential convergence rate, and numerical examples demonstrate the accuracy and efficiency of the complex Fourier series expansion method.

Pricing Equity-Indexed Annuities under a Stochastic Dividend Model

In this paper, we examine the valuations of equity-indexed annuities (EIAs) when their reference stocks distribute stochastic dividends. Due to the fact that stocks typically pay dividends at discrete times after the payment dates are announced, pricing EIAs with dividends is deemed to be practically significant. We directly model the discrete dividend payments using the jump diffusion process with regime switching, and then determine the dynamics of the stock price. The equivalent martingale measure of fair valuation in incomplete markets is determined by employing the Esscher transform. Finally, the pricing formulas of several of the most common EIAs in the market under the stochastic dividend model are obtained. Our model incorporates and extends the present literature on EIAs with accurate and effective valuation methods.

Dynamic hedging in incomplete markets using risk measures

Abstract In this paper, we consider the pricing of financial derivatives that involve dynamic hedging strategies and payments within the planning horizon. Equity-indexed annuities (EIAs), guaranteed investment certificates (GICs) and American and barrier options are typical examples of these products. Our exploration involves the use and comparison of alternative models that use risk measures. Although the hedging is done for each observation of the input stochastic process, the appropriate mix of risk measures and state dynamic equations helps the issuer to appropriately tailor the overall risk exercise. These different models are solved by a unified backward stochastic dynamic programming framework that we imbed with parametric techniques to shorten the running time and manage the curse of dimensionality in dynamic programming. To demonstrate the flexibility of this framework we present numerical examples featuring GICs and point-to-point EIAs. Finally, by using sampling techniques, optimal hedging strategies and specific indicators of the hedging performance, we make recommendations on how to fine tune the risk measure parameters.

Valuation of equity-indexed annuities under correlated jump–diffusion processes

This article studies the problem of valuation of equity-indexed annuities (EIA) when the stock index follows a Hawkes jump–diffusion model to account for the jump risk and clustering of index price jumps. Further, the interest rate is assumed to be driven by Vasicek type model, correlated to the dynamics of the stock index. In the proposed framework, an analytical expression is obtained for the price of annual reset and point-to-point designs of EIAs by employing the measure change technique. The effects of the model parameters governing the jump risk and the clustering of jumps on the EIAs pricing are illustrated through numerical experiments.

Worst-Case Valuation of Equity-Linked Products Using Risk-Minimizing Strategies

The impact of model risk when hedging equity-linked products and other investment guarantees is significant. We propose a model to determine the worst-case value of an equity-linked product through partial hedging. Risk control strategies based on conditional Value at Risk measures are used. The model integrates both mortality and financial risk associated with these products to find the worst-case value. We adopt robust optimization techniques to compute an optimal hedging strategy. To demonstrate versatility of the framework, numerical examples of point-to-point equity-indexed annuities are presented in multinomial lattice dynamics. We compare robustness of the model to super-replicating and quadratic hedging strategies by computing their capital requirements.

Pricing EIA with cliquet-style guarantees under time-changed Lévy models by frame duality projection

This paper presents an efficient and accurate method for pricing equity-indexed annuities (EIAs) with cliquet-style payoff structures under time-changed Lévy processes. We apply a square-root process to depict the activity rate process in stochastic time change, which enables us to use some Fourier transform methods to price EIAs. As the complexity of payoff structures, we make use of frame duality projection associated with the quadrature rule and the spectral filtering to realize our algorithm. Numerical experiments are reported to demonstrate the accuracy and efficiency of our approach.

Valuation and analysis on complex equity indexed annuities

Equity-indexed annuities (EIAs) are popular products that eliminate the downside risk while still providing upside potential. Among the three major categories of EIAs, ratchet EIAs are the most popular. Ratchet EIAs with quanto features emerge due to differences in asset returns across countries. The literature covers the pricing of the EIAs that are not quantos, and this paper fills the hole. In deriving pricing formulas, we add an exchange rate model as well as a foreign risk-free rate model to the pricing framework of Black and Scholes. Our formulas cover quanto ratchet EIAs for both compound and simple versions that may have a return cap and employ two types of geometric return averaging. The numerical analyses illustrate how contract features and market parameters affect contract values. The results also highlight the significance of quantos in contract pricing.

Valuing equity-indexed annuities with icicled barrier options

Inspired by the recent popularity of autocallable structured products, this paper intends to enhance equity-indexed annuities (EIAs) by introducing a new class of barrier options, termed icicled barrier options. The new class of options has a vertical (icicled) barrier along with the horizontal one of the ordinary barrier options, which may act as an additional knock-in or knock-out trigger. To improve the crediting method of EIAs, we propose a new EIA design, termed autocallable EIA, with payoff structure similar to the autocallable products except for the minimum guarantee, and further investigate the possibility of embedding various icicled barrier options into the plain point-to-point or the ratchet EIAs. Explicit pricing formulas for the proposed EIAs and the icicled barrier options are obtained under the Black–Scholes model. To the purpose, we derive the joint distribution of the logarithmic returns at the icicled time and the maturity, and their running maximum. As an application of the well-known reflection principle, the derivation itself is an interesting probability problem and the joint distribution plays a key role in the subsequent pricing stage. Our option pricing result can be easily transferred to EIAs or other equity-linked products. The pricing formulas for the EIAs and the options are illustrated through numerical examples.

Pricing and hedging equity-indexed annuities via local risk-minimization

ABSTRACTLin et al. (2009) employed the Esscher transform method to price equity-indexed annuities (EIAs) when the dynamic of the market value of a reference asset was driven by a generalized geometric Brownian motion model with regime-switching. Some rare events (release of an unexpected economic figure, major political changes or even a natural disaster in a major economy) can lead to brusque variations in asset prices, and hence we sometimes need to consider jump models. This paper extends the model and analysis in Lin et al. (2009). Specifically, we assume that the financial market has a regime-switching jump-diffusion model, under which we price the point-to-point, the Asian-end, the high water mark and the annual reset EIAs by exploiting the local risk-minimization approach. The effects of the model parameters on the EIAs pricing are illustrated through numerical experiments. Meanwhile, we present the locally risk-minimizing hedging strategies for EIAs.

Partial Hedging for Equity-Linked Products Using Risk-Minimizing Strategies

In this article, we consider the pricing and hedging of equity-indexed annuities (EIAs) using local risk-minimizing strategies as well as evaluating the capital requirement for these products. Since these products involve mortality as well as financial risks, we integrate mortality risk and propose partial hedging strategies that protect the insurer based on risk measures. The framework we present makes use of sequential local risk-minimizing strategies to take into account all intermediate requirements. To demonstrate the flexibility of this framework we present numerical examples featuring point-to-point EIAs with a two-state regime-switching equity model.

Equity-linked life insurance based on traditional products: the case of Select Products

Select Products are a German version of equity-indexed annuities common in the US and Canada, which are constructed by using a traditional life insurance contract and suitably leveraging the annual surplus distribution. In this paper, we aim to provide holistic insight into Select Products by means of a comparison with American equity-indexed annuities and a detailed description of design, hedging and accounting issues. We put a special emphasis on the unique features of traditional life insurance (particularly the collective savings process) which have been widely neglected in previous literature. In addition, we present a model framework for the most prominent type of Select Products, compare the product design when offered by a bank or an insurer, and compare its risk-return profile with traditional policies. Our analysis confirms that the current attractiveness of such products arises from the unique features of traditional life insurance by pooling financial risks as well as the utilization of the balance sheet in the current low interest rate environment. We discuss these aspects in detail and further address benefits as well as detriments of these products depending on the market conditions.

Equity-linked annuity pricing with cliquet-style guarantees in regime-switching and stochastic volatility models with jumps

In this paper, we develop a novel and efficient transform-based method to price equity-linked annuities (ELAs), including equity-indexed annuities (EIAs) and cliquet-style payoff structures popular in the insurance market under a general class of stochastic volatility models with jumps. We utilize frame duality and density projection combined with a continuous-time Markov chain (CTMC) weak approximation scheme and spectral filtering. Contracts considered include EIAs with return guarantees of a cliquet style. Models considered include exponential Lévy processes, regime-switching Lévy processes, and stochastic volatility models with a general jump size distribution including Heston, Scott’s, Hull–White, Schöbel–Zhu, and the 3/2 models. We also consider some recently proposed stochastic volatility models in the literature such as the α-Hypergeometric model, and the 4/2 model. Our framework encompasses and extends the current literature on EIAs with highly efficient and accurate valuation methods. Numerical experiments confirm our findings.

Impact of volatility clustering on equity indexed annuities

This study analyses the impact of volatility clustering in stock markets on the evaluation and risk management of equity indexed annuities (EIA). To introduce clustering in equity returns, the reference index is modelled by a diffusion combined with a bivariate self-excited jump process. We infer a semi-closed form or parametric expression of the moment generating functions in this framework for the equity return and the intensities of jumps. An econometric procedure is proposed to fit the model to a time series. Next, we develop a method, based on a normal inverse Gaussian approximation of the index return, to evaluate options embedded in simple variable annuities. To conclude, we compare prices, one-year value at risks, and tail value at risks of simple EIAs, computed with different models.

Pricing Equity-indexed Annuities When Discrete Dividends Follow a Markov-Modulated Jump Diffusion Model

The Equity-Indexed annuities (EIAs) provide investors not only a minimum rate of return but also an opportunity to gain a potential profit that is linked to the performance of an equity index, typically SP 500. These properties make EIAs become very popular in the market and receive considerable attention in the actuarial literature. In this article, we investigate the Equity-Indexed annuities valuation when the discrete dividend payments follow a Markov-modulated jump diffusion model. Using the generalized Itô formula, we obtain the explicit solution for dividend payments of the model. From Dividend Discount theory, which implies that the stock price should be equal to the net present value of all its future dividend payments, the stock price process is then deduced. Within this framework, closed-form solution to the point-to-point EIA pricing problem is derived via the characteristic function of the occupation times.

변액연금과 지수연동형 연금의 포트폴리오 리스크 관리

변액연금과 지수연동형 연금의 포트폴리오 리스크 관리

Risk-minimizing hedging strategy for an equity-indexed annuity under a regime switching model

The equity-indexed annuity (EIA) contract offers a proportional participation in the performance of a specified equity index, in addition to a guaranteed return on the single premium. How to manage the risk of the EIA is an important issue. This paper considers the hedging of the EIA. We assume that the parameters of the financial model depend on a continuous-time finite-state Markov chain and the Markov chain is observed, that is the Markov regime switching model. The state of the Markov chain can be interpreted as the state of an economy. Under the regime switching model, we obtain the risk-minimizing hedging strategy for the EIA.