Dissertations / Theses on the topic 'Variable annuities'

Thesis (Ph.D. in Applied Mathematics)--S.M.U., 2007.
Title from PDF title page (viewed Nov. 19, 2009). Source: Dissertation Abstracts International, Volume: 68-11, Section: B, page: 7372. Advisers: Zhangxin (John) Chen; Andrew H. Chen. Includes bibliographical references.

Dans cette thèse, nous étudions les variables annuities (VA) des produits avec garantie de prestations minimales (GMxB), un secteur à croissance rapide dans le domaine de l'assurance vie. Les produits GMxB ont attiré l'attention des praticiens et des universitaires à la fois en raison de leur longue échéance et des propriétés de conception complexes, et aussi à cause des comportements des assurés incertains, notamment en terme de taux d'échéance. Dans cette thèse, nous abordons le problème comme celui de l'évaluation d'une option de type Bermudienne pour l'assureur. Cette démarche d'évaluation correspond au prix qui permet aux assureurs de couvrir le risque quelle que soit la stratégie du titulaire. Nous avons également introduit de nouvelles idées de conception de produits basées sur cette approche garantissant une couverture totale quelle que soit les comportements d' exercices. Il est important de mentionner que jusqu'à présent, un taux d'échéance historique ou statistique est généralement admis pour la valorisation de ces garanties. Tant la théorie financière que les observations passées montrent que cette hypothèse peut conduire à une sous-estimation du risque associé à ces produits, les titulaires étant rationnels ou non. Sur le plan numérique, nous faisons appel à deux type de techniques différents: les méthodes de résolution d'EDP et la méthode de régression en grande dimension (HDR). Il est montré que la méthode PDE est précise à faible dimension (< 3), tandis que l'approche HDR est plus efficace quand il y a plus de trois variables d'état. Dans le modèle de Hull et White à taux d'intérêt stochastique nous montrons aussi comment un changement de numéraire peut être utilisé pour accélérer les algorithmes numériques de manière significative pour les politiques avec cliquet (lookback) propriétés. En outre, nous étendons également la traditionnelle solution semi-analytique pour les options américaines pour évaluer certains GMxB. Une méthode semi analytique est également introduite dans cette thèse pour estimer le prix des options américaines et leur prix d'exercice critique dans un modèle à volatilité stochastique (ex Heston modèle. En fait, cette méthode peut être étendue à d'autres processus de diffusion tant qu'il existe une méthode de tarification précise et rapide existent pour les produits européens correspondants
In this thesis we study the variable annuity (VA) products with guaranteed minimum benefits (GMxB), a fast growing business in the life insurance industry. The GMxB products attract the attention of practitioners and academics both because of its long maturity and complex design properties, and also because of uncertain policyholder behaviors, such as lapse rate. In this thesis, we address the pricing problem as the valuation of a Bermudan-style option for the insurer. This evaluation approach corresponds to the price that allows the insurers to hedge the risk whatever the lapse strategy of the holder is. We also introduce new product design ideas based on this evaluation approach to make sure insurers are fully protected form unexpected lapse waves in the future. It is worthy to mention that so far, a historical or statistical lapse rate has generally been assumed for pricing these guarantees. Both financial theory and past observations show that this assumption may lead to an underestimation of the risk associated to these products, the holders being rational or not. To evaluate the Bermudan-style liability, we apply two di_erent schemes: Partial Differential Equation (PDE) method and high-dimensional regression (HDR) method. It is shown that the PDE method is precise for low-dimensional problems (< 3), while the HDR is more efficient when there are more than three dimensions. In the Hull and White stochastic interest rate model, we also show how a change of numeraire technique can be used to accelerate the numerical algorithms significantly for policies with ratchet (lookback) properties. In addition, we also extend the traditional semi-analytical solution of American options to evaluate certain GMxB polices. A semi-analytical method is also introduced in this thesis to approximate both the American contingent claims and the critical exercise boundary of contingent claims in the stochastic volatility model (e. X. Heston model. In fact, this method can be extended to other diffusion processes as long as quick and accurate pricing methods exist for the corresponding European claims

Thesis (Ph. D.)--Ohio State University, 2004.
Title from first page of PDF file. Document formatted into pages; contains x, 86 p.; also includes graphics Includes bibliographical references (p. 84-86). Available online via OhioLINK's ETD Center

We determine the optimal allocation of funds between the fixed and variable sub-accounts in a variable annuity with a GMDB (Guaranteed Minimum Death Benefit) clause featuring partial withdrawals by using a utility-based approach. In section two, 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 the variable fund and withdrawing optimally. In section three, we add fixed and stochastic income to the model and find that both human capital and the GMDB will influence the insured's allocation and withdrawal decisions. Section four explores the GMDB effects if there is also a term life policy available in the market. Our work suggests that if term life insurance is available and is continuously adjustable, fairly priced GMDBs may not be useful investments and the existence of GMDBs does not affect term life policy demand significantly.

Policyholder exercise behavior presents an important risk factor for life insurance companies. Yet, most approaches presented in the academic literature – building on value maximizing strategies akin to the valuation of American options – do not square well with observed prices and exercise patterns. Following a recent strand of literature, in order to gain insights on what drives policyholder behavior, I first develop a life-cycle model for variable annuities (VA) with withdrawal guarantees. However, I explicitly allow for outside savings and investments, which considerably affects the results. Specifically, I find that withdrawal patterns after all are primarily motivated by value maximization – but with the important asterisk that the value maximization should be taken out from the policyholders’ perspective accounting for individual tax benefits. To this effect, I develop a risk-neutral valuation methodology that takes these different tax structures into consideration, and apply it to our example contract as well as a representative empirical VA. The results are in line with corresponding outcomes from the life cycle model, and I find that the withdrawal guarantee fee from the empirical product roughly accords with its marginal price to the insurer. I further consider the implications of policyholder behavior on product design. In particular – due to differential tax treatments and contrary to option pricing theory – the marginal value of such guarantees can become negative, even when the holder is a value maximizer. For instance, as I illustrate with both a simple two-period model and an empirical VA, a common death benefit guarantee may indeed yield a negative marginal value to the insurer.

2013/2014
The past twenty years have seen a massive proliferation in insurance-linked derivative products. The public, indeed, has become more aware of investment opportunities outside the insurance sector and is increasingly trying to seize all the benefits of equity investment in conjunction with mortality protection. The competition with alternative investment vehicles offered by the financial industry has generated substantial innovation in the design of life products and in the range of benefits provided. In particular, equity-linked policies have become ever more popular, exposing policyholders to financial markets and providing them with different ways to consolidate investment performance over time as well as protection against mortality-related risks. Interesting examples of such contracts are variable annuities (VAs). This kind of policies, first introduced in 1952 in the United States, experienced remarkable growth in Europe, especially during the last decade, characterized by “bearish” financial markets and relatively low interest rates. The success of these contracts is due to the presence of tax incentives, but mainly to the possibility of underwriting several rider benefits that provide protection of the policyholder’s savings for the period before and after retirement. In this thesis, we focus in particular on the Guaranteed Lifetime Withdrawal Benefit (GLWB) rider. This option meets medium to long-term investment needs, while providing adequate hedging against market volatility and longevity-related risks. Indeed, based on an initial capital investment, it guarantees the policyholder a stream of future payments, regardless of the performance of the underlying policy, for his/her whole life. In this work, we propose a valuation model for the policy using tractable financial and stochastic mortality processes in a continuous time framework. We have analyzed the policy considering two points of view, the policyholder’s and the insurer’s, and assuming a static approach, in which policyholders withdraw each year just the guaranteed amount. In particular, we have based ourselves on the model proposed in the paper “Systematic mortality risk: an analysis of guaranteed lifetime withdrawal benefits in variable annuities” by M. C. Fung, K. Ignatieva and M. Sherris (2014), with the aim of generalizing it later on. The valuation, indeed, has been performed in a Black and Scholes economy: the sub-account value has been assumed to follow a geometric Brownian motion, thus with a constant volatility, and the term structure of interest rates has been assumed to be constant. These hypotheses, however, do not reflect the situation of financial markets. In order to consider a more realistic model, we have sought to weaken these misconceptions. Specifically we have taken into account a CIR stochastic process for the term structure of interest rates and a Heston model for the volatility of the underlying account, analyzing their effect on the fair price of the contract. We have addressed these two hypotheses separately at first, and jointly afterwards. As part of our analysis, we have implemented the theoretical model using a Monte Carlo approach. To this end, we have created ad hoc codes based on the programming language MATLAB, exploiting its fast matrix-computation facilities. Sensitivity analyses have been conducted in order to investigate the relation between the fair price of the contract and important financial and demographic factors. Numerical results in the stochastic approach display greater fair fee rates compared to those obtained in the deterministic one. Therefore, a stochastic framework is necessary in order to avoid an underestimation of the policy. The work is organized as follows. Chapter 1. This chapter has an introductory purpose and aims at presenting the basic structures of annuities in general and of variable annuities in particular. We offer an historical review of the development of the VA contracts and describe the embedded guarantees. We examine the main life insurance markets in order to highlight the international developments of VAs and their growth potential. In the last part we retrace the main academic contributions on the topic. Chapter 2. Among the embedded guarantees, we focus in particular on the Guaranteed Lifetime Withdrawal Benefit (GLWB) rider. We analyze a valuation model for the policy basing ourselves on the one proposed by M. Sherris (2014). We introduce the two components of the model: the financial market, on the one hand, and the mortality intensity on the other. We first describe them separately, and subsequently we combine them into the insurance market model. In the second part of the chapter we describe the valuation formula considering the GLWB from two perspectives, the policyholder’s and the insurer’s. Chapter 3. Here we implement the theoretical model creating ad hoc codes with the programming language MATLAB. Our numerical experiments use a Monte Carlo approach: random variables have been simulated by MATLAB high level random number generator, whereas concerning the approximation of expected values, scenario- based averages have been evaluated by exploiting MATLAB fast matrix-computation facilities. Sensitivity analyses are conducted in order to investigate the relation between the fair fee rate and important financial and demographic factors. Chapter 4. The assumption of deterministic interest rates, which can be acceptable for short-term options, is not realistic for medium or long-term contracts such as life insurance products. GLWB contracts are investment vehicles with a long-term horizon and, as such, they are very sensitive to interest rate movements, which are uncertain by nature. A stochastic modeling of the term structure is thus appropriate. In this chapter, therefore, we propose a generalization of the deterministic model allowing interest rates to vary randomly. A Cox-Ingersoll-Ross model is introduced. Sensitivity analyses have been conducted. Chapter 5. Empirical studies of stock price returns show that volatility exhibits “random” characteristics. Consequently, the hypothesis of a constant volatility is rather “counterfactual”. In order to consider a more realistic model, we introduce the stochastic Heston process for the volatility. Sensitivity analyses have been con- ducted. Chapter 6. In this chapter we price the GLWB option considering a stochastic process for both the interest rate and the volatility. We present a numerical comparison with the deterministic model. Chapter 7. Conclusions are drawn. Appendix. This section presents a quick survey of the most fundamental concepts from stochastic calculus that are needed to proceed with the description of the GLWB’s valuation model.
Negli ultimi venti anni si `e assistito ad una massiccia proliferazione di prodotti de- rivati di tipo finanziario-assicurativo. Gli individui, infatti, sono diventati sempre piu` consapevoli delle opportunita` di investimento esistenti al di fuori del settore as- sicurativo e pertanto richiedono all’impresa di assicurazione non solo la protezione contro il rischio di mortalit`a/longevit`a, ma anche tutti i benefici di un investimento di capitali. Ed `e proprio per soddisfare le esigenze del mercato e per fronteggiare la concorrenza alimentata da altri competitors (banche, ecc.) che il mercato assi- curativo sta cambiando ed ha iniziato a sviluppare nuovi prodotti assicurativi ad elevato contenuto finanziario. Nell’ambito di questi prodotti, particolare interesse rivestono le cosiddette polizze variable annuities. Introdotte per la prima volta negli Stati Uniti nel 1952, esse hanno raggiunto ben presto un notevole sviluppo anche in Europa, soprattutto nell’ultimo decennio caratterizzato da mercati finanziari bearish e da tassi di interesse relativamente bassi. Il successo di questo tipo di contratti `e dovuto al favorevole trattamento fiscale di cui godono, ma soprattutto all’offerta di opzioni implicite che garantiscono una protezione dei risparmi degli investitori prima e dopo il pensionamento. In questo lavoro di tesi, ci siamo concentrati in particola- re sull’opzione Guaranteed Lifetime Withdrawal Benefit (GLWB). Essa permette di soddisfare esigenze di investimento di medio/lungo periodo e nello stesso tempo offre una discreta copertura al rischio dovuto alla volatilit`a dei mercati e al longevity risk. Infatti, a fronte di un capitale iniziale investito, garantisce all’assicurato un flusso di pagamenti futuri indipendente dalla performance della polizza sottostante per tutta la durata della sua vita. Piu` precisamente, in questo lavoro proponiamo un modello di valutazione per questo tipo di contratto, facendo ricorso a processi stocastici per descrivere la componente finanziaria e quella legata alla mortalità dell’assicurato. Analizziamo la polizza considerando sia il punto di vista del cliente che quello della compagnia di assicurazione. La nostra valutazione si è basata sul modello proposto da M. C. Fung, K. Ignatieva e M. Sherris nell’articolo “Systematic mortality risk: an analysis of guaranteed lifetime withdrawal benefits in variable annuities” (2014). Tuttavia le ipotesi alla base di questa analisi non trovano giustificazione nel mercato; in effetti, considerare un tasso di interesse ed una volatilità costanti sembra poco sensato. Proprio per proporre un modello più fedele al mercato, si è pensato di indebolire questi assunti, prendendo in considerazione un processo stocastico a sé stante per descrivere la dinamica del tasso di interesse e della volatilità. Dapprima abbiamo analizzato separatamente l’impatto dei due processi sul prezzo equo dell’opzione, per poi considerare anche il loro effetto congiunto. Come parte integrante del lavoro, abbiamo implementato il modello teorico proposto impiegando un approccio Monte Carlo. A questo scopo abbiamo creato codici ad hoc utilizzando il linguaggio di programmazione MATLAB, sfruttando al meglio tutte le sue potenzialità di calcolo matriciale. Sono state condotte analisi di sensitività per analizzare l’impatto sul prezzo equo dell’opzione di alcuni importanti parametri finanziari e demografici. I risultati numerici mostrano come effettivamente l’impiego di un approccio stocastico sia più capace di descrivere le fluttuazioni del mercato e quindi permetta di ottenere risultati più realistici. Il valore equo delle commissioni applicate dalla compagnia di assicurazione per l’attivazione della garanzia GLWB aumenta quando si passa da un approccio deterministico ad uno stocastico (soprattutto se quest’ultimo considera congiuntamente tassi di interesse e volatilità stocastici), rivelando come un adeguato modello stocastico sia necessario per evitare una sottovalutazione di tali polizze. Il lavoro è strutturato come segue: Capitolo 1. Questo capitolo ha un ruolo introduttivo e mira a fornire una descrizione delle caratteristiche principali delle polizze variable annuities. Si analizza l'evoluzione storica di tali polizze ed il loro sviluppo nei principali mercati internazionali. Segue una breve rassegna dei principali contributi accademici sulla valutazione di tali contratti e si spiegano le ragioni alla base di questo lavoro. Capitolo 2. Tra le varie garanzie implicite nei contratti variable annuity ci soffermiamo sull'opzione Guaranteed Lifetime Withdrawal Benefit. In questo capitolo analizziamo il modello di valutazione del contratto proposto da M. Sherris (2014); introduciamo le due componenti del modello (il mercato finanziario e l'intensità di mortalità) dapprima descrivendole separatamente, poi combinandole. Nella seconda parte del capitolo studiamo le formule per il calcolo del prezzo equo del contratto considerando due punti di vista, quello dell'assicurato e quello dell'assicuratore. Capitolo 3. In questo capitolo implementiamo il modello teorico creando codici ad hoc con il linguaggio di programmazione MATLAB. Le nostre valutazioni sono state realizzate utilizzando un approccio Monte Carlo. Diverse analisi di sensitività sono state condotte per analizzare l’impatto sul prezzo equo dell’opzione di alcuni importanti parametri finanziari e demografici. Capitolo 4. In questo capitolo si propone una generalizzazione del modello deterministico indebolendo l'ipotesi di struttura a termine dei tassi di interesse costante. Per descrivere la dinamica del tasso di interesse si introduce in particolare un processo Cox- Ingersoll- Ross. Capitolo 5. In questo capitolo si indebolisce l'ipotesi che considera costante la volatilità del fondo d'investimento prevedendo una dinamica descritta dal processo di Heston. Capitolo 6. Si descrive un modello che considera congiuntamente un processo stocastico per i tassi di interesse (CIR) e per la volatilità (Heston). Si conducono analisi di sensitività e si mostrano i risultati ottenuti. Capitolo 7. In questo capitolo traiamo le conclusioni del nostro lavoro. Appendice. Proponiamo una breve rassegna delle principali nozioni di calcolo stocastico necessarie per meglio comprendere la descrizione del modello di valutazione.
XXVII Ciclo
1986

La présente thèse s'intéresse à trois acteurs du financement des régimes de retraite : le législateur, l'assureur et l’individu. Dans un environnement en proie à un comportement déviant du marché financier et à des évolutions démographiques défavorables, le rôle de ces parties prenantes doit impérativement faire l’objet d’une réévaluation pour relever le défi de la pérennité du financement des retraites. L’étude de la règlementation et de la conception des régimes a été réalisée en intégrant des caractéristiques types du futur paysage des retraites, telles que le poids de plus en plus important du risque assumé par l’individu ou l’éventuelle participation d'investisseurs boursiers dans l’offre de contrats. Les conclusions de cette étude permettent de dégager des orientations en vue de la gestion du risque de longévité pour les individus, une évaluation de l’attrait de l’exposition au risque de longévité pour les investisseurs, des informations sur l’élaboration des contrats pour les assureurs ainsi que des propositions, pour les décideurs politiques, de mesures règlementaires favorisant la durabilité du paysage des retraites
This dissertation examines three stakeholders in pension finance: the individual, the policymaker, and the pension provider (e.g., an insurer or a pension fund). In a setting beset by unforseen financial market circumstances and demographic changes that disfavor financial security in retirement, a re-evaluation of these stakeholders’ role is necessary. We explore the regulation and design of retirement plans by incorporating features that characterize the future retirement landscape, such as the increasing burden of risk borne by the individual, and the potential involvement of market investors in the provision of retirement contracts. The implications of our findings encompass guidance for individuals in managing longevity risk, evaluation of the appeal of longevity risk exposure to investors, insights on contract design for the insurer, and proposals to the policymaker on regulatory measures that foster a sustainable retirement environment

A general methodology is described in which policyholder behaviour is decoupled from the pricing of a variable annuity based on the cost of hedging it, yielding two sequences of weakly coupled systems of partial differential equations (PDEs): the pricing and utility systems. The utility systems are used to generate policyholder withdrawal behaviour, which is in turn fed into the pricing systems as a means to determine the cost of hedging the contract. This approach allows us to incorporate the effects of utility-based pricing and factors such as taxation. As a case study, we consider the Guaranteed Lifelong Withdrawal and Death Benefits (GLWDB) contract. The pricing and utility systems for the GLWDB are derived under the assumption that the underlying asset follows a Markov regime-switching process. An implicit PDE method is used to solve both systems in tandem. We show that for a large class of utility functions, the two systems preserve homogeneity, allowing us to decrease the dimensionality of solutions. We also show that the associated control for the GLWDB is bang-bang, under which the work required to compute the optimal strategy is significantly reduced. We extend this result to provide the reader with sufficient conditions for a bang-bang control for a general variable annuity with a countable number of events (e.g. discontinuous withdrawals). Homogeneity and bang-bangness yield significant reductions in complexity and allow us to rapidly generate numerical solutions. Results are presented which demonstrate the sensitivity of the hedging expense to various parameters. The costly nature of the death benefit is documented. It is also shown that for a typical contract, the fee required to fund the cost of hedging calculated under the assumption that the policyholder withdraws at the contract rate is an appropriate approximation to the fee calculated assuming optimal consumption.

The Guaranteed Minimum Withdrawal Benefits (GMWBs) are optional riders provided by insurance companies in variable annuities. They guarantee the policyholders' ability to get the initial investment back by making periodic withdrawals regardless of the impact of poor market performance. With GMWBs attached, variable annuities become more attractive. This type of guarantee can be challenging to price and hedge. We employ two approaches to price GMWBs. Under the constant static withdrawal assumption, the first approach is to decompose the GMWB and the variable annuity into an arithmetic average strike Asian call option and an annuity certain. The second approach is to treat the GMWB alone as a put option whose maturity and payoff are random. Hedging helps insurers specify and manage the risks of writing GMWBs, as well as find their fair prices. We propose semi-static hedging strategies that offer several advantages over dynamic hedging. The idea is to construct a portfolio of European options that replicate the conditional expected GMWB liability in a short time period, and update the portfolio after the options expire. This strategy requires fewer portfolio adjustments, and outperforms the dynamic strategy when there are random jumps in the underlying price. We also extend the semi-static hedging strategies to the Heston stochastic volatility model.

A guaranteed minimum income benefit (GMIB) is a long-dated option that can be embedded in a deferred variable annuity. The GMIB is attractive because, for policyholders who plan to annuitize, it offers protection against poor market performance during the accumulation phase, and adverse interest rate experience at annuitization. The GMIB also provides an upside equity guarantee that resembles the benefit provided by a lookback option. We price the GMIB, and determine the fair fee rate that should be charged. Due to the long dated nature of the option, conventional hedging methods, such as delta hedging, will only be partially successful. Therefore, we are motivated to find alternative hedging methods which are practicable for long-dated options. First, we measure the effectiveness of static hedging strategies for the GMIB. Static hedging portfolios are constructed based on minimizing the Conditional Tail Expectation of the hedging loss distribution, or minimizing the mean squared hedging loss. Next, we measure the performance of semi-static hedging strategies for the GMIB. We present a practical method for testing semi-static strategies applied to long term options, which employs nested Monte Carlo simulations and standard optimization methods. The semi-static strategies involve periodically rebalancing the hedging portfolio at certain time intervals during the accumulation phase, such that, at the option maturity date, the hedging portfolio payoff is equal to or exceeds the option value, subject to an acceptable level of risk. While we focus on the GMIB as a case study, the methods we utilize are extendable to other types of long-dated options with similar features.

The objective of this thesis is to provide an analysis of the Variable Annuity (VA) products. In particular, we focus on the actuarial and financial valuation of some embedded guarantees in VAs, derive No-arbitrage pricing models and study the impact of mortality risk. We have decided to deal with this products in the light of significant international success obtained by VAs and we believe that perspectives of their development in Italy market and throughout Europe and Asia are favourable. One of the reasons of this success is the presence of guarantees which offer partial protection against the downside movements of the interest rates or the equity market, an attractive feature for the individual retirement security. The shift from defined benefit to self-directed defined contribution plans and the reform of the Social Retirement System in many countries, so that it includes personal accounts, have encouraged the proliferation of new kind of products. Owing to the long term horizon of their commitments, pension funds are exposed to important financial risk due to the volatility of interest rates and equity markets. At this regard, VAs were first introduced by insurance companies in the 1970s in the United States to compete with mutual funds. Over the years, many practical and academic contributions have been offered for describing the VAs and the guarantees embedded. Most of the earlier literature (e.g., Rentz Jr. (1972) and Green (1973)) is constituted by empirical works dealing with product comparisons rather than pricing and hedging issues. It was not until recently that some guarantees were discussed by practitioners ( e.g., J.P.Morgan (2004), Lehman Brothers (2005), Milliman (2007)); they highlight the growing opportunities to introduce VAs in new markets. Recently, the academic literature has shown a fervent interest to the topic too (cfr. Bauer et al. (2006), Chen et al. (2008), Coleman et al. (2006), Dai (2008), Holz (2006), Milevsky and Panyagometh (2001), Milevsky and Posner (2001), Milevsky M.A and Promislow S.D (2001), Milevsky and Salisbury(2002)., Milevsky and Salisbury (2006), Nielsen and Sandmann (2003)). Bauer et al. offer the first universal general framework in which any design of options and guarantees currently offered within Variable Annuities can be modeled. Besides the valuation of a contract assuming that the policyholder follows a given strategy with respect to surrender and withdrawals, they are able to price contracts with different embedded options. Milevsky und Posner (2001) price various types of guaranteed minimum death benefits. They present closed form solutions for this option in case of an exponential mortality law and numerical results for the more realistic Gompertz-Makeham law. They find that in general these guarantees are overpriced in the market. In Milevsky und Salisbury (2006), the authors price GMWB options. Besides a static approach, where deterministic withdrawal strategies are assumed, they calculate the value of the option in a dynamic approach. Here, the option is valuated under optimal policyholder behavior. They show that under realistic parameter assumptions optimally at least the annually guaranteed withdrawal amount should be withdrawn. Furthermore, they find that such options are usually underpriced in the market. This result is in contrast with the common belief that the guarantees embedded in variable annuity policies are overpriced (see Clements (2004)). This thesis aims at following this literature by proposing some theoretical and practical innovative works. Our original contributions lie in: 1) describe how the value of Guaranteed Minimum Death Benefit (GMDB) options evolves over time and in the presence of mortality changes and produce an application to Italian data; 2) study the insurance surplus over time for a portfolio of Variable Annuities with GMDB Options and offer a model that can be used for an evaluation of the adequacy of solvency; 3) develop a sensitivity analysis for the value of Guaranteed Lifelong Withdrawal Benefit (GLWB) options under the hypothesis of a static withdrawal strategy; 4) decompose a VA with a GLWB option into a life annuity plus a portfolio of Quanto Asian Put Options, with decreasing strikes and increasing expiration dates, and verify that this product is underpriced on US market.

Several of the more complex optimization problems in finance can be characterized as impulse control problems. Impulse control problems can be written as quasi-variational inequalities, which are then solved to determine the optimal control strategy. Since most quasi-variational inequalities do not have analytical solutions, numerical methods are generally used in the solution process. In this thesis, the impulse control problem framework is applied to value two complex long-term option-type contracts. Both pricing problems considered are cast as impulse control problems and solved using an implicit approach based on either the penalty method or the operator splitting scheme. The first contract chosen is an exotic employee stock option referred to as an infinite reload option. This contract provides the owner with an infinite number of reload opportunities. Each time a reload occurs, the owner pays the strike price using pre-owned company shares and, in return, receives one share for each option exercised and a portion of a new reload option. Numerical methods based on the classic Black-Scholes equation are developed while taking into account contract features such as vesting periods. In addition, the value of an infinite reload option to it's owner is obtained by using a utility maximization approach. The second long-term contract considered is a variable annuity with a guaranteed minimum death benefit (GMDB) clause. Numerical methods are developed to determine the cost of the GMDB clause while including features such as partial withdrawals. The pricing model is then used to determine the fair insurance charge which minimizes the cost of the contract to the issuer. Due to the long maturity of variable annuities, non-constant market parameters expressed through the use of regime-switching are included in the GMDB pricing model.

本研究主要探討附保證最低提 (Guaranteed Minimum Withdrawal Benefits, GMWB)的變額(Variable Annuity, VA) 在隨機模型下之定價。保證最低提是變額的一種附加約 (rider) 並在市場下跌的情況下為變額持有人提供保障。它保證持有人在合約期內的總提少於一個預先訂的額，而變額的投資表現。一般，這個保證額相等於變額的初始投資額。本研究的融模型假設投資標的基價格符合對維過程 (exponential Lévy process)，而隨機則符合由維過程驅動的瓦西克模型 (Vasiček model)。融模型中的個維過程的相依結構 (dependence structure) 會由維關結構 (Lévy Copula) 描述。這個方法的好處是可描述同型的相依結構。用一個配合維關結構而有效的蒙地卡模擬方法，我們研究在同相依結構及模型下保證最低提的價值變化。在固定的特別情況下，保證最低提的價值能夠透過卷積方法 (convolution method) 而得到半解析解 (semi-analytical solution) 。最後，我們將本研究中的學模型擴展以研究近期出現由保證最低提演化而成的一種保證產品。這個產品名稱為保證終身提 (Guaranteed Lifelong Withdrawal Benefit, GLWB)，而此產品的到期日則與持有人的壽命相關。
In this thesis, we study the problem of pricing the variable annuity(VA) with the Guaranteed Minimum Withdrawal Benefits (GMWB) under the stochastic interest rate framework. The GMWB is a rider that can be elected to supplement a VA. It provides downside protection to policyholders by guaranteeing the total withdrawals throughout the life of the contract to be not less than a pre-specied amount, usually the initial lump sum investment, regardless of the investment performance of the VA. In our nancial model, we employ an exponential L´evy model for the underlying fund process and a Vasiček type model driven by a L´evy process for the interest rate dynamic. The dependence structure between the two driving L´evy processes is modeledby the L´evy copula approach whichis exible to model a wide range of dependence structure. An effcient simulation algorithm on L´evy copula is then used to study the behavior of the value of the GMWB when the dependence structure of the two L´evy processes and model parameters Vry. When the interest rate is deterministic, the value of the GMWB can be solved semi-analytically by the convolution method. Finally, we extend our model to study a recent variation of GMWB called Guaranteed Life long Withdrawal Benefits (GLWB) in which the maturity of the GLWB depends on the life of the policyhodler.
Detailed summary in vernacular field only.
Chan, Wang Ngai.
Thesis (M.Phil.)--Chinese University of Hong Kong, 2012.
Includes bibliographical references (leaves 115-121).
Abstracts also in Chinese.
Abstract --- p.i
Acknowledgement --- p.iv
Chapter 1 --- Introduction --- p.1
Chapter 1.1 --- Variable Annuity & Guaranteed Minimum Withdrawal Benefit --- p.1
Chapter 1.2 --- Literature Review --- p.4
Chapter 1.3 --- Financial Model for GMWB --- p.7
Chapter 2 --- L´evy Copulas and the Simulation Algorithm --- p.12
Chapter 2.1 --- Definitions and Theorem --- p.15
Chapter 2.2 --- Examples of L´evy Copulas --- p.19
Chapter 2.2.1 --- Independence case --- p.19
Chapter 2.2.2 --- Complete Dependence --- p.20
Chapter 2.2.3 --- The Clayton L´evy Copula --- p.21
Chapter 2.3 --- Simulation algorithm for two-dimensional dependent L´evy process --- p.22
Chapter 3 --- Model Formulation for GMWB --- p.26
Chapter 3.1 --- Financial Model for GMWB --- p.27
Chapter 3.2 --- Underlying Fund of VA and the Interest Rate --- p.30
Chapter 3.3 --- A Special Case of Deterministic Interest Rate --- p.34
Chapter 4 --- Numerical Implementation --- p.38
Chapter 4.1 --- The Clayton L´evy Copula --- p.39
Chapter 4.2 --- The Underlying Fund and the Interest Rate Processes --- p.42
Chapter 4.3 --- Kendall’s Tau Coefficient --- p.47
Chapter 4.4 --- The GMWB Option Value --- p.49
Chapter 4.4.1 --- Control Variate for Simulation --- p.49
Chapter 4.4.2 --- Simulation Results --- p.51
Chapter 4.5 --- Deterministic Interest Rate --- p.52
Chapter 5 --- GMWB Pricing Behavior --- p.56
Chapter 5.1 --- L´evy model for the underlying fund --- p.57
Chapter 5.1.1 --- The Skewness --- p.57
Chapter 5.1.2 --- The Kurtosis --- p.65
Chapter 5.2 --- The Vasiček model driven by L´evy process --- p.73
Chapter 5.2.1 --- The Volatility Parameter ôV --- p.73
Chapter 5.2.2 --- The Mean Reverting Parameter aV --- p.77
Chapter 5.3 --- Dependence between the underlying fund and rate processes --- p.81
Chapter 5.3.1 --- The jump direction dependence parameter n{U+1D9C} --- p.83
Chapter 5.3.2 --- The jump magnitude dependence parameter θ{U+1D9C} --- p.90
Chapter 6 --- GMWB for Life --- p.96
Chapter 6.1 --- Model Formulation --- p.98
Chapter 6.1.1 --- Mortality model --- p.99
Chapter 6.1.2 --- Financial Model for GLWB --- p.101
Chapter 6.2 --- GLWB product from John Hancock --- p.103
Chapter 6.3 --- GLWB Pricing Behavior --- p.104
Chapter 6.3.1 --- The correlation effect --- p.106
Chapter 7 --- Conclusion --- p.108
A Proofs --- p.113
Chapter A.1 --- Proof of Equation 3.1 --- p.113
Chapter A.2 --- Proof of Equation 3.3 --- p.114
Bibliography --- p.115

Product risk is important to firms’ enterprise risk management. This dissertation focuses on product risk in the U.S. life insurance and health insurance industries. In particular, we add new dimensions to the measurement of product risk for these industries, and we explore how these industries manage product risk in a context of other enterprise risks. In this dissertation, we identify new product risks, propose new measures, and study the management of these risks. In the life insurance industry, we identify a new type of product risk, the guarantee risk, caused by variable annuities with guaranteed living benefits (VAGLB). We propose a value-at-risk type measure inspired by the risk-based capital C3 Phase II to quantify the guarantee risk. In the health insurance industry, where the degree of uncertainty varies for different types of health insurance policies, we develop four exposure-based risk measures to capture health insurers’ product risks. Then we study how life and health insurers manage product risks (and asset risks) by using capital in the context of other risks and appropriate controls. We add to the literature in the life insurance industry by examining the relationship between capital and risks when the guarantee risk is accounted for. In the health insurance industry, to our knowledge, no similar research on the relationship between capital and risks has been conducted. In view of the current topicality of health insurance, our research therefore adds a timely contribution to the understanding of health insurer risk management in an era of health care reform. Capital structure theories, transaction cost economics, and insurers’ risk-taking behaviors provide the theoretical foundation for our research. As to methodology, we implement standard capital structure models for the life and health insurance industries using data from the National Association of Insurance Commissioners (NAIC) annual filings of life/health insurers and health insurers. Simultaneous equations modeling is used to model life and health insurers’ enterprise risk management. And the estimation is conducted by the generalized estimation equations (GEE). We find that both U.S. life/health insurers and health insurers prudently build up capital as they experience more product risk and asset risk controlling for the other enterprise risks. We also find that life/health insurers may be using derivatives as a partial substitute for capital when managing new product risk caused by VAGLB, the guarantee risk.
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Le modèle GARCH à changement de régimes est le fondement de cette thèse. Ce modèle offre de riches dynamiques pour modéliser les données financières en combinant une structure GARCH avec des paramètres qui varient dans le temps. Cette flexibilité donne malheureusement lieu à un problème de path dependence, qui a empêché l'estimation du modèle par le maximum de vraisemblance depuis son introduction, il y a déjà près de 20 ans. La première moitié de cette thèse procure une solution à ce problème en développant deux méthodologies permettant de calculer l'estimateur du maximum de vraisemblance du modèle GARCH à changement de régimes. La première technique d'estimation proposée est basée sur l'algorithme Monte Carlo EM et sur l'échantillonnage préférentiel, tandis que la deuxième consiste en la généralisation des approximations du modèle introduites dans les deux dernières décennies, connues sous le nom de collapsing procedures. Cette généralisation permet d'établir un lien méthodologique entre ces approximations et le filtre particulaire. La découverte de cette relation est importante, car elle permet de justifier la validité de l'approche dite par collapsing pour estimer le modèle GARCH à changement de régimes. La deuxième moitié de cette thèse tire sa motivation de la crise financière de la fin des années 2000 pendant laquelle une mauvaise évaluation des risques au sein de plusieurs compagnies financières a entraîné de nombreux échecs institutionnels. À l'aide d'un large éventail de 78 modèles économétriques, dont plusieurs généralisations du modèle GARCH à changement de régimes, il est démontré que le risque de modèle joue un rôle très important dans l'évaluation et la gestion du risque d'investissement à long terme dans le cadre des fonds distincts. Bien que la littérature financière a dévoué beaucoup de recherche pour faire progresser les modèles économétriques dans le but d'améliorer la tarification et la couverture des produits financiers, les approches permettant de mesurer l'efficacité d'une stratégie de couverture dynamique ont peu évolué. Cette thèse offre une contribution méthodologique dans ce domaine en proposant un cadre statistique, basé sur la régression, permettant de mieux mesurer cette efficacité.
The Markov-switching GARCH model is the foundation of this thesis. This model offers rich dynamics to model financial data by allowing for a GARCH structure with time-varying parameters. This flexibility is unfortunately undermined by a path dependence problem which has prevented maximum likelihood estimation of this model since its introduction, almost 20 years ago. The first half of this thesis provides a solution to this problem by developing two original estimation approaches allowing us to calculate the maximum likelihood estimator of the Markov-switching GARCH model. The first method is based on both the Monte Carlo expectation-maximization algorithm and importance sampling, while the second consists of a generalization of previously proposed approximations of the model, known as collapsing procedures. This generalization establishes a novel relationship in the econometric literature between particle filtering and collapsing procedures. The discovery of this relationship is important because it provides the missing link needed to justify the validity of the collapsing approach for estimating the Markov-switching GARCH model. The second half of this thesis is motivated by the events of the financial crisis of the late 2000s during which numerous institutional failures occurred because risk exposures were inappropriately measured. Using 78 different econometric models, including many generalizations of the Markov-switching GARCH model, it is shown that model risk plays an important role in the measurement and management of long-term investment risk in the context of variable annuities. Although the finance literature has devoted a lot of research into the development of advanced models for improving pricing and hedging performance, the approaches for measuring dynamic hedging effectiveness have evolved little. This thesis offers a methodological contribution in this area by proposing a statistical framework, based on regression analysis, for measuring the effectiveness of dynamic hedges for long-term investment guarantees.