Academic literature on the topic 'Variable annuities'

ABSTRACTThis paper provides a detailed overview of variable annuities. Consideration is given first to the definition of the term variable annuity. Common terminology used in the variable annuity market is introduced. The current state of the United Kingdom and other international markets is described. Then, by reference to a simplified product, an analysis of customer outcomes, pricing, reserving, risk management and hedging is carried out. The paper ends with a description of current U.K. pensions legislation and how it potentially constrains product development.

This paper explores the extent to which annuitants might be prepared to pay for protection against cohort-specific mortality risk, by comparing traditional indexed annuities with annuities whose payout rates are revised in response to differences between expected and actual mortality rates of the cohort in question. It finds that a man aged 65 with a coefficient of relative risk aversion of two would be prepared to pay 75p per £100 annuitised for protection against aggregate mortality risk while a man with risk aversion of twenty would be prepared to pay £5.75 per £100; studies put the actual cost at £2.70–£7 per £100, suggesting that unless annuitants are very risk averse it is likely that existing products tend to over-insure against cohort mortality risk.

AbstractMany retirees hope to continue earning capital market rewards on their saving while avoiding outliving their funds during retirement. We model a dynamic utility maximizing investor who seeks to benefit from holding both equity and longevity insurance. She is free to adjust her portfolio allocation of her financial wealth as well as of the annuity over time, and she can purchase variable payout annuities any time and incrementally. In this setting, we show that the retiree will not fully annuitize even without bequests; rather, she will combine variable annuities with withdrawals from her liquid financial wealth so as to match her desired consumption profile. Optimal stock exposures decrease over time, both within the variable annuity and the withdrawal plan. Welfare gains from this strategy can amount to 40% of financial wealth, depending on risk parameters and other resources; additionally, many retirees will do almost as well as the fully optimized outcome if they hold variable annuities invested 60/40 in stocks/bonds.

<span>This article presents an analysis of variable lifetime annuities and quantifies the advantages and disadvantages associated with this type of instrument. Given recent long-term rates of return and current low inflation rates, variable annuity contracts provide an effective means of compensating for inflation. An individual only needs to invest a small portion of retirement funds in variable annuities to protest the entire portfolio against the risk of long-term inflation without the risk of having ones entire retirement income based on variable annuities.</span>

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