Equity-indexed Annuities Research Articles

Valuation of Equity-indexed Annuities with Stochastic Interest Rate and Jump Diffusion

This article considers the pricing of equity-indexed annuity (EIA). By employing the change of measure technique, we derive the closed-form solutions for the prices of both point-to-point and annual reset equity-indexed annuities. We also provide numerical results to illustrate the method and computational efficiency of our simulation scheme and the effects of various model parameters on the participation rate.

Pricing Equity-Indexed Annuities with Surrender Options in a Stochastic Interest Rates Environment

The aim of this paper is to value EIA contracts with surrender option at a fair price in a stochastic interest rate environment. A non Gaussian model is proposed where a new process called Kou Regime Switching process is introduced to give a more realistic modelling for financial prices. A general methodology is proposed and applied to mix GMDB and GMMB contract. Quasi-closed form solutions are given when the surrender clause is not present and this solution is incorporated into a LSMC algorithm to solve the pricing problem in the surrender case.

Efficient Valuation of Equity-Indexed Annuities Under LLvy Processes Using Fourier-Cosine Series

Equity-Indexed Annuities (EIAs) are deferred annuities which accumulate value over time according to crediting formulas and realized equity index returns. We propose an efficient algorithm to value two popular crediting formulas found in EIAs - Annual Point-to-Point (APP) and Monthly Point-to-Point (MPP) - under general Levy-process based index returns. APP contracts observe returns of referenced indexes annually and credit EIA accounts, subject to minimum and maximum returns. MPP contracts incorporate both local/monthly caps and global/annual floors on index credits. MPP contracts have payoffs of a cliquet option. Our algorithm, based on the COS method (Fang and Oosterlee, 2008), expands the present value of an EIA contract using Fourier-cosine series, expresses the value of the EIA contract as a series of terms involving simple characteristic function evaluations. We present several examples with different Levy processes, including the Black-Scholes model and the CGMY model. These examples illustrate the efficiency of our algorithm as well as its versatility in computing annuity market sensitivities, which could facilitate the hedging and pricing of annuity contracts.

Pricing of Equity Indexed Annuity under Fractional Brownian Motion Model

Fractional Brownian motion with Hurst exponentH∈(1/2,1)is a good candidate for modeling financial time series with long-range dependence and self-similarity. The main purpose of this paper is to address the valuation of equity indexed annuity (EIA) designs under the market driven by fractional Brownian motion. As a result, this paper presents an explicit pricing expression for point-to-point EIA design and bounds for the pricing of high-water-marked EIA design. Some numerical examples are given to illustrate the impact of the parameters involved in the pricing problems.

A dimension-reduction algorithm for the valuation of surrender options in EIA contracts with stochastic interest rates

This paper proposes a fast algorithm for the fair valuation of a ratchet-type equity-indexed annuity (EIA) endowment contract with surrender options under Vasicek stochastic interest rate models. Traditionally, the valuation for the indexed equity and interest rate of an American-type surrender option is performed under two-dimensional tree models, which is time-consuming for computation. This paper first applies the Black–Scholes method for ratchet-type options to reduce the two-dimensional tree structure to single one. Next, to overcome the path dependent problem inherent in the ratchet option, we also propose a recursive formula to implement the backward computation. By using the proposed algorithm, we are able to perform numerical analysis to verify that surrender options are more valuable with the increase of interest rates. High interest rate volatility enhances both the bonus and surrender option values entitled to the policyholder. A numerical experiment also shows that increasing interest rates may decrease the bonus option value but increase the surrender option value. These results can provide suggestions for insurance companies regarding the issue of EIA policies.

Pricing Equity Index Annuities with Surrender Options in Four Models

Equity-indexed annuities (EIAs) have generated a great deal of interest since Keyport Life first launched “Key Index” in February 1995. They are considered to be the most innovative products to appear on the market in years. EIAs are, essentially equity-linked deferred annuities which provide the policyholder with a guaranteed accumulation rate on their premium, and also at maturity, benefits from an additional return based on the performance of an equity mutual fund or a family of mutual funds or a stock index, typically of the Standard and Poor’s (S&P500) index, so the customer can profit from the growth of the stock market. Product designs of EIAs can vary, depending on the companies that sell them. In this article, we focus on the pricing of one of the product designs in the market which gives the possibility to surrender their policy before maturity. Such options can be valued using Monte Carlo simulation method proposed for pricing American options but, here, we use the least squares Monte Carlo suggested by Longstaff and Schwartz added to the control variate tool in order to construct efficient estimators. We analyze these equity-linked life insurance contracts under four different models. The frameworks differ from the way we model the price of the fund associated with the contract. The first setting is the usual Black and Scholes model, the second is the environment of jump diffusions, especially a Kou process, the third is the regime switching log normal model (RSLN) developed by Hamilton and the fourth is a mixed of a regime switching and a jump diffusion. The surrender option is priced using the Longstaff and Schwartz methodology.

Equity-linked products: evaluation of the dynamic hedging errors under stochastic mortality

In this paper, we analyze the effect of unpredicted changes in mortality rates on the risk incurred by the sale of equity-indexed annuities (EIAs). Jumps in mortality indices due to catastrophes (for example, the 1918 flu pandemic) may occur in the future, causing important financial losses to insurers selling products offering death benefits. Thus, we analyze the distribution of hedging errors extracted from the dynamic hedging strategy underlying the fair valuation. To model mortality jumps stochastically, we use a regime-switching model introduced by Milidonis et al. (North Am Actuarial J 15(2):266–289, 2011). We then employ Esscher transforms to obtain closed-form expressions for the price of a term-end EIA at any time between the inception of the contract and its maturity date. The hedging strategy is derived from that valuation, and hedging errors are extracted from this strategy since it is not self-financing. A detailed numerical analysis is performed for a term-end EIA.

Valuation of equity-indexed annuities with regime-switching jump diffusion risk and stochastic mortality risk

This paper extends the model and analysis of Lin, Tan and Yang (2009). We assume that the financial market follows a regime-switching jump-diffusion model and the mortality satisfies Levy process. We price the point to point and annual reset EIAs by Esscher transform method under Merton’s assumption and obtain the closed form pricing formulas. Under two cases: with mortality risk and without mortality risk, the effects of the model parameters on the EIAs pricing are illustrated through numerical experiments.

Pricing Ratchet Equity-Indexed Annuities with Early Surrender Risk in a CIR++ Model

In this article we propose a lattice algorithm for pricing simple Ratchet equity-indexed annuities (EIAs) with early surrender risk and global minimum contract value when the asset value depends on the CIR++ stochastic interest rates. In addition we present an asymptotic expansion technique that permits us to obtain a first-order approximation formula for the price of simple Ratchet EIAs without early surrender risk and without a global minimum contract value. Numerical comparisons show the reliability of the proposed methods.

Optimal surrender strategies for equity-indexed annuity investors with partial information

In this paper we consider an equity-indexed annuity (EIA) investor who wants to determine when he should surrender the EIA in order to maximize his logarithmic utility of the wealth at surrender time. We model the dynamics of the index using a geometric Brownian motion with regime switching. To be more realistic, we consider a finite time horizon and assume that the Markov chain is unobservable. This leads to the optimal stopping problem with partial information. We give a representation of the value function and an integral equation satisfied by the boundary. In the Bayesian case which is a special case of our model, we obtain analytical results for the value function and the boundary.

Pricing Equity Index Annuities with Surrender Options in Four Models

Since Keyport Life first launched ”Key Index” in February 1995, Equity-indexed annuities are considered to be the most innovative products to appear on the market in years. EIAs are, essentially equity-linked deferred annuities which provide the policyholder with a guaranteed accumulation rate on their premium, and also at maturity, benefits from an additional return based on the performance of an equity mutual fund or a family of mutual funds or a stock index, typically of the Standard and Poor’s (S&P 500) index, so the customer can profit from the growth of the stock market. Products designs of EIAs can vary , depending on the companies that sell them.In this paper, we focus on the pricing of one of the product design in the market which gives the possibility to surrender their policy before maturity. Such options can be valued using Monte Carlo simulation method proposed for pricing American options but, here, we use the least squares Monte-Carlo suggested by Longstaff and Schwartz added to the control variate tool in order to construct efficient estimators. We analyze these equity-linked life insurance contracts under four different models. The frameworks differ from the way we model the price of the fund associated with the contract. The first setting is the usual Black and Scholes model, the second is the environment of jump diffusions, especially a Kou process, the third is the regime switching log normal model developed by Hamilton and the fourth is a mixed of a regime switching and a jump diffusion. The surrender option is priced using the Longstaff and Schwartz methodology.

Valuation of Ratchet Equity-Indexed Annuities

Ratchet EIAs are the most popular equity-indexed annuities (EIAs) because returns are credited periodically with a guaranteed minimum and the account value never decreases once the return is credited. Pricing ratchet EIAs, however, is challenging. This paper derives pricing formulas that cover more contract features of ratchet EIAs than any in the literature. We obtain closed-form solutions in the Black-Scholes framework for both compound and simple versions of annual-reset ratchet products that may have a return cap and employ two types of geometric return averaging. Our numerical results demonstrate the impacts of individual contract features on contract value and the interactions among these features.

米国の株価指数連動年金に対する証券規制を巡る論議の動向

成長顕著な株価指数連動型年金に対して証券規制導入提案を行ったSECに対して，当該年金主力の生保業界が強い反対姿勢を示し，規則撤回を求めて連邦控訴裁判所に提訴した。裁判所は，規則の内容は妥当であるが，規則導入に必要な効率性等の分析が不十分であるとしてSECに対して再検討を指示した。SECは規則再提出意欲を示したが，おりからの金融危機後における金融規制改革法案審議の過程でSEC規制を排除する条項が組み込まれた同法案が成立することとなった。これにより，証券規制は阻止できたが，州の監督責任は一段と重くなり，その真価が問われている。

A New Premium Principle for Equity-Indexed Annuities

In this article, we introduce a premium principle for equity-indexed annuities (EIAs). Traditional actuarial loadings that protect insurance companies against risks cannot be extended to the valuation of EIAs since these products are embedded with various financial guarantees. We proposed a loaded premium that protects the issuers against the financial and mortality risks. We first obtain the fair premium based on a fair value of the equity-linked contract using arbitrage-free theory. Assuming a specific risk level for hedging errors, we obtain a new participation rate based on a security loading. A detailed numerical analysis is performed for a point-to-point EIA.

Efficient Pricing of Ratchet Equity-Indexed Annuities in a Variance-Gamma Economy

In this paper we propose a new method for approximating the price of arithmetic Asian options in a Variance-Gamma (VG) economy, which is then applied to the problem of pricing equityindexed annuity contracts. The proposed procedure is an extension to the case of a VG-based model of the moment-matching method developed by Turnbull and Wakeman and Levy for the pricing of this class of path-dependent options in the traditional Black-Scholes setting. The accuracy of the approximation is analyzed against RQMC estimates for the case of ratchet equityindexed annuities with index averaging.

Valuation of equity-indexed annuity under stochastic mortality and interest rate

An equity-indexed annuity (EIA) contract offers a proportional participation in the return on a specified equity index, in addition to a guaranteed return on the single premium. In this paper, we discuss the valuation of equity-indexed annuities under stochastic mortality and interest rate which are assumed to be dependent on each other. Employing the method of change of measure, we present the pricing formulas in closed form for the most common product designs: the point-to-point and the annual reset. Finally, we conduct several numerical experiments, in which we analyze the relationship between some parameters and the pricing of EIAs.

Pricing Asian Options and Equity-Indexed Annuities with Regime Switching by the Trinomial Tree Method

Equity-indexed annuities (EIAs) provide investors with a minimum rate of return and at the same time the opportunity of gaining a profit that is linked to the performance of an equity index. These properties make EIAs a popular product in the market. For modeling the equity index process and calculating the price of EIAs, as the maturity of EIAs usually is long, it is more reasonable to assume that the interest rate and the volatility of the equity index are stochastic processes. One simple way is to apply the regime-switching model, which allows these parameters depending on the market situation. However, the valuation of derivatives in such models is challenging, especially for the strong path-dependent options such as Asian options. A trinomial tree model is introduced to provide an efficient way to solve this problem. The valuation of Asian options is studied and extended to Asian-option-related EIAs.

“Pricing Asian Options and Equity-Indexed Annuities with Regime Switching by the Trinomial Tree Method”, Fei Lung Yuen and Hailiang Yang, April, 2010

(2010). “Pricing Asian Options and Equity-Indexed Annuities with Regime Switching by the Trinomial Tree Method”, Fei Lung Yuen and Hailiang Yang, April, 2010. North American Actuarial Journal: Vol. 14, No. 2, pp. 272-277.

Optimal surrender strategies for equity-indexed annuity investors

An equity-indexed annuity (EIA) is a hybrid between a variable and a fixed annuity that allows the investor to participate in the stock market, and earn at least a minimum interest rate. The investor sacrifices some of the upside potential for the downside protection of the minimum guarantee. Because EIAs allow investors to participate in equity growth without the downside risk, their popularity has grown rapidly. An optimistic EIA owner might consider surrendering an EIA contract, paying a surrender charge, and investing the proceeds directly in the index to earn the full (versus reduced) index growth, while using a risk-free account for downside protection. Because of the popularity of these products, it is important for individuals and insurers to understand the optimal policyholder behavior. We consider an EIA investor who seeks the surrender strategy and post-surrender asset allocation strategy that maximizes the expected discounted utility of bequest. We formulate a variational inequality and a Hamilton–Jacobi–Bellman equation that govern the optimal surrender strategy and post-surrender asset allocation strategy, respectively. We examine the optimal strategies and how they are affected by the product features, model parameters, and mortality assumptions. We observe that in many cases, the “no-surrender” region is an interval ( w l , w u ) ; i.e., that there are two free boundaries. In these cases, the investor surrenders the EIA contract if the fund value becomes too high or too low. In other cases, there is only one free boundary; the lower (or upper) surrender threshold vanishes. In these cases, the investor holds the EIA, regardless of how low (or high) the fund value goes. For a special case, we prove a succinct and intuitive condition on the model parameters that dictates whether one or two free boundaries exist.

Efficient Pricing of Ratchet Equity Indexed Annuities in a VG Economy

In this paper we propose a new method for approximating the price of arithmetic Asian options in a VG economy which is then applied to the problem of pricing Equity Indexed Annuity contracts. The proposed procedure is an extension to the case of a VG-based model of the moment-matching method developed by Turnbull and Wakeman (1991) and Levy (1992) for the pricing of this class of path-dependent options in the traditional Black-Scholes setting. The accuracy of the approximation is analysed against RQMC estimates for the case of ratchet Equity Indexed Annuities with index averaging.