Academic journal article Journal of Risk and Insurance

Pricing and Hedging Variable Annuities in a Levy Market: A Risk Management Perspective

Academic journal article Journal of Risk and Insurance

Pricing and Hedging Variable Annuities in a Levy Market: A Risk Management Perspective

Article excerpt

INTRODUCTION

Variable Annuities (VAs) are life insurance contracts linked to financial markets. Companies design VAs so that policyholders can benefit from favorable movements in the markets, yet remain protected when prices plummet. They are often grouped under the acronym GMxB. The G refers to guarantee, M to minimum, B to benefit, and x to a particular contract type: for example, M refers to maturity, D to death, A to accumulation, I to income, and W to withdrawal (see Hardy, 2003; Kalberer and Ravindran, 2009). These contracts have had great success in the United States, the United

Kingdom, Japan, and, to a lesser extent, in continental Europe. This success stems from the fact that VAs offer the opportunity to manage long-term savings and to potentially provide postretirement income. Specific tax advantages also provide incentives for investing in these products. VAs, together with similar contracts such as Equity Index Annuities (EIAs), represent a huge market. Taking into account the world's aging population, and the contribution that VAs can bring to the financing of postretirement income, it is likely that these markets will continue to expand over the coming years.

The pricing, hedging, and risk management of these products are the main areas of concern for insurers and represent a challenge for researchers. VAs involve different risks whose methods of interaction are unknown (Bacinello et al., 2011). Furthermore, these contracts contain nonstandard embedded options, for example, the Guarantee Minimum for Death Benefit (GMDB), in which the optional rider is an option whose expiration date is random (the policyholder's death). Milevsky and Posner (2001) name these options Titanic, give the fair fees for this type of contract, and compare their calculations with the fees actually charged by insurers.

A key assumption in the theoretical analysis of these contracts concerns the modeling of financial prices, and particularly the dynamics of the value of the referenced financial portfolio or market index. Many studies assume a Gaussian hypothesis for financial returns. However, the research now widely accepts that the distribution of these returns is not normal. Asymmetries and leptokurticities have to be taken into account. To cope with these stylized facts, the research has developed models with regime-switching schemes (Hardy, 2003), or with stochastic volatility (Heston, 1993). Furthermore, recent studies, such as Cont (2001) and Ait-Sahalia and Jacod (2009), show that jumps are present in financial prices. The embedded options in VAs are very sensitive to the tails of the underlying distribution, so jumps and/or stochastic volatility have to be taken into account. Pricing and hedging thus become more complex and must be performed in incomplete markets. This is not an easy task. The aim of this article is to suggest a general methodology for both pricing and hedging a subclass of VAs in a general Levy context so that insurers can evaluate fair mortality and expense fees, and can hedge the risk contained in the embedded option. This analysis is developed from an operational risk management point of view.

Pricing these contracts is a familiar theme in the literature. After the seminal article of Brennan and Schwartz (1976) using the principle of arbitrage in continuous time, an impressive flow of articles has been devoted to the pricing of life insurance contracts, in particular VAs and EIAs. Many risks have been considered-mortality risk, market risk, stochastic interest rate risk, default risk, and surrender risk-in the usual Black and Scholes economy, and more recently in a Levy environment (Kassberger, Kiesel, and Liebmann, 2008; Ballotta, 2005). The issue of hedging, extremely important for risk management purposes, has been far less frequently analyzed in the insurance literature than the issue of valuation. The theoretical analysis of hedging in incomplete markets is a well-established area of study in the world of mathematical finance. …

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