Valuation of Equity Linked Securities with Guaranteed Return

Equity-linked securities with a guaranteed return become very popular in financial markets ether as investment instruments or life insurance policies. The contract pays off a guaranteed amount plus a payment linked to the performance of a basket of equities averaged over a certain period. This paper presents a new model for valuing equity-linked securities. Our study shows that the security price can be replicated by the sum of the guaranteed amount plus the price of an Asian style option on the basket. Analytical formulas are derived for the security price and corresponding hedge ratios. The model appears to be accurate over a wide range of underlying security parameters according to numerical studies. Finally, we use our model to value a segregated fund with a guarantee at maturity.Comment: 20 pages, 4 figure

Guaranteed Equity-Linked Security Analytics

Equity-linked securities with a guaranteed return become popular in a volatile market environment. This paper presents a new model for valuing guaranteed equity-linked notes. We consider a security whose value depends on the performance of a basket of equities averaged over certain points in time, but that is floored by a guaranteed amount. We show that the security’s price is given by the sum of the guaranteed amount plus the price of an Asian style option on the basket above. The model provides analytical formulas for the security’s price as well as for corresponding hedge ratios; these respective formulas appear to be accurate over a wide range of underlying security parameter values, based on numerical testing against a Monte Carlo benchmark, Finally, we apply our method to value a type of segregated fund with a maturity guarantee

The valuation of GMWB variable annuities under alternative fund distributions and policyholder behaviours

In this paper, we present a dynamic programming algorithm for pricing variable annuities with Guaranteed Minimum Withdrawal Benefits (GMWB) under a general Lévy processes framework. The GMWB gives the policyholder the right to make periodical withdrawals from her policy account even when the value of this account is exhausted. Typically, the total amount guaranteed for withdrawals coincides with her initial investment, providing then a protection against downside market risk. At each withdrawal date, the policyholder has to decide whether, and how much, to withdraw, or to surrender the contract. We show how different policyholder’s withdrawal behaviours can be modelled. We perform a sensitivity analysis comparing the numerical results obtained for different contractual and market parameters, policyholder behaviours and different types of Lévy processes

Essays on modeling, hedging and pricing of insurance and financial products

Cette thèse est composée de trois articles abordant différentes problématiques en relation avec la modélisation, la couverture et la tarification des risques d’assurance et financiers. “A general class of distortion operators for pricing contingent claims with applications to CAT bonds” est un projet présentant une méthode générale pour dériver des opérateurs de distorsion compatibles avec la valorisation sans arbitrage. Ce travail offre également une nouvelle classe simple d’opérateurs de distorsion afin d’expliquer les primes observées dans le marché des obligations catastrophes. “Local hedging of variable annuities in the presence of basis risk” est un travail dans lequel une méthode de couverture des rentes variables en présence de risque de base est développée. La méthode de couverture proposée bénéficie d’une exposition plus élevée au risque de marché et d’une diversification temporelle du risque pour obtenir un rendement excédentaire et faciliter l’accumulation de capital. “Option pricing under regime-switching models : Novel approaches removing path-dependence” est un projet dans lequel diverses mesures neutres au risque sont construites pour les modèles à changement de régime de manière à générer des processus de prix d’option qui ne présentent pas de dépendance au chemin, en plus de satisfaire d’autres propriétés jugées intuitives et souhaitables.This thesis is composed of three papers addressing different issues in relation to the modeling, hedging and pricing of insurance and financial risks. “A general class of distortion operators for pricing contingent claims with applications to CAT bonds” is a project presenting a general method for deriving probability distortion operators consistent with arbitrage-free pricing. This work also offers a simple novel class of distortions operators for explaining catastrophe (CAT) bond spreads. “Local hedging of variable annuities in the presence of basis risk” is a work in which a method to hedge variable annuities in the presence of basis risk is developed. The proposed hedging scheme benefits from a higher exposure to equity risk and from time diversification of risk to earn excess return and facilitate the accumulation of capital. “Option pricing under regime-switching models: Novel approaches removing path-dependence” is a project in which various risk-neutral measures for hidden regime-switching models are constructed in such a way that they generate option price processes which do not exhibit path-dependence in addition to satisfy other properties deemed intuitive and desirable

A Dynamic Programming Algorithm for the Valuation of Guaranteed Minimum Withdrawal Benefits in Variable Annuities

In this paper we present a dynamic programming algorithm for pricing variable annuities with Guaranteed Minimum Withdrawal Benefits (GMWB) under a general Lévy processes framework. The GMWB gives the policyholder the right to make periodical withdrawals from her policy account even when the value of this account is exhausted. Typically, the total amount guaranteed for withdrawals coincides with her initial investment, providing then a protection against downside market risk. At each withdrawal date, the policyholder has to decide whether, and how much, to withdraw, or to surrender the contract. We show how different levels of rationality in the policyholder’s withdrawal behaviour can be modelled. We perform a sensitivity analysis comparing the numerical results obtained for different contractual and market parameters, policyholder behaviours, and different types of Lévy processes

Pricing variable annuity guarantees in South Africa under a Variance-Gamma model

The purpose of this study is to investigate the pricing of variable annuity embedded derivatives using a suitably refined model for the underlying assets, in this case the Johannesburg Securities Exchange FTSE/JSE All Share Index (ALSI). This is a practical issue that life insurers face worldwide in the management of embedded derivatives. We consider the Variance-Gamma (VG) framework to model the underlying data series. The VG process is useful in option pricing given its ability to model higher moments, skewness and kurtosis and to capture observed market dynamics. The framework is able to address the inadequacies of some deterministic pricing approaches used by life insurers, given the increasing complexity of the option-like products sold.The third author is funded by NRF CSUR project no. 90313.http://www.actuarialsociety.org.za/Professionalresources/SAActuarialJournal.aspxam201

Quantitative Risk Management under the Interplay of Insurance and Financial Risks

En esta tesis, abordamos algunos problemas relacionados con la interacción de los riesgos de seguros y financieros. Primero, consideramos una compañía de seguros o financiera con la intención de asignar el capital de riesgo retenido para su cartera de inversión general entre sus constituyentes. Brevemente, suponemos que la compañía calcula el capital de riesgo a través de la medida de riesgo Haezendonck - Goovaerts, y establecemos la regla de asignación de capital única consistente con un enfoque RORAR (retorno sobre capital ajustado al riesgo). Además, presentamos algunas asintóticas y proponemos un estimador consistente para la regla de asignación de capital. Finalmente, realizamos algunos estudios numéricos. Luego, resolvemos el problema de valorar algunos derivados vinculados a la mortalidad empleando el enfoque de precios de indiferencia de utilidad. De manera sucinta, suponemos que el riesgo de mortalidad emana de una cartera de asegurados de vida, cuyas vidas restantes se modelan como tiempos aleatorios condicionalmente independientes. Al adaptar algunos resultados de la teoría del riesgo de crédito, calculamos una expresión explícita para el precio de indiferencia de la utilidad cuando el derivado es una combinación lineal de dotaciones puras. Al considerar una reclamación contingente más general, utilizamos técnicas de ecuaciones diferenciales estocásticas hacia atrás (BSDE) para caracterizar el precio de indiferencia en términos de una solución a un BSDE no lineal con un generador no Lipschitz. Finalmente, consideramos a un individuo con el objetivo de elegir de manera óptima sus estrategias de inversión, consumo y compra de seguros de vida en un mercado financiero completo. Al suponer que el criterio de optimización es la maximización de la utilidad esperada del individuo, la cual depende del estado de la economía, resolvemos el problema de elección óptima en una configuración general, que incluye varias funciones de utilidad empleadas en la literatura.In this thesis, we tackle some problems concerning the interplay of insurance and financial risks. First, we consider an insurance or financial company intending to allocate the risk capital withheld for its overall investment portfolio among its constituents. Shortly, we assume that the company computes the risk capital through the Haezendonck--Goovaerts risk measure, and we establish the unique capital allocation rule consistent with a RORAR (return on risk-adjusted capital) approach. Besides, we present some asymptotics and propose a consistent estimator for the capital allocation rule. Finally, we conduct some numerical studies. Then, we solve the problem of valuing some mortality-linked derivatives by employing the utility indifference pricing approach. Succinctly, we suppose that the mortality risk emanates from a portfolio of life insurance policyholders, whose remaining lifetimes are modeled as conditionally independent random times. By adapting some results from credit risk theory, we compute an explicit expression for the utility indifference price when the derivative is a linear combination of pure endowments. By considering a more general contingent claim, we use techniques of backward stochastic differential equations (BSDE) to characterize the indifference price in terms of a solution to a non-linear BSDE with a non-Lipschitz generator. Finally, we consider an individual aiming to optimally choose its investment, consumption, and life insurance purchase strategies in a complete financial market. By assuming that the optimality criterion is the maximization of the individual's expected state-dependent utility, we solve the optimal choice problem in a general setup, which includes several utility functions employed in the literature.Doctor en CienciaDoctorad

Risk management of variable annuity portfolios using machine learning techniques

Variable annuities (VAs) are increasingly becoming popular insurance products in many developed countries which provide guaranteed forms of income depending on the performance of the equity market. Insurance companies often hold large VA portfolios and the associate valuation of such portfolios is a very time-consuming task. There have been several studies focusing on inventing techniques aimed at reducing the computational time including the selection of representative VA contracts and the use of a metamodel to estimate the values of all contracts in the portfolio. In this thesis, LASSO regression is used to select a set of representative scenarios after the representative contracts are chosen, which in turn allows for the set of representative contracts to expand without significant increase in computational load. The proposed approach leads to a remarkable improvement in the computational efficiency and accuracy of the metamodel. Stochastic reserving and calculation of capital requirement require VA providers to calculate risk measures such as Value at Risk and Conditional Tail Expectation. An emulation framework is proposed to calculate these risk measures by building a neural network to model the net liability of a VA contract at some given scenario. The surrogate model is faster at estimating net liability than the exact calculation. Efficiency is improved thanks to faster computing of net liability for any contract at any scenario in the Monte Carlo simulation. This approach can also be used to select scenarios where the estimated portfolio liabilities are in the top quantile. The true liabilities of the portfolio at these top-quantile scenarios can be computed which can then be used to compute the risk measures. This results in a reduction in computational time because the Monte Carlo method is performed on only a fraction of the original scenarios. As an equity-linked insurance products, VA is exposed to significant market risks due to the underlying assets in the mutual funds that its contributions are invested in. To hedge against these market risks, insurers need to construct a hedging portfolio consisting of the underlying assets whose hedge positions can be determined by the Greeks of the portfolio such as the partial dollar Deltas. For a large portfolio, the calculation of the Greeks using Monte Carlo simulation is very slow, so a metamodeling approach can be used to estimate the Greeks. Assuming that the mutual funds of the VA insurers is a mixture of major market indices, there is likely a dependence between the partial dollar Deltas of the portfolio on the market indices. This dependent relationship can be incorporated into the model using multi-output regression approaches and the resulting improvement in the effectiveness of the metamodel or the lack thereof will be studied in the thesis

Innovations in Quantitative Risk Management

Quantitative Finance; Game Theory, Economics, Social and Behav. Sciences; Finance/Investment/Banking; Actuarial Science