Academic journal article Journal of Risk and Insurance

A New Premium Principle for Equity-Indexed Annuities

Academic journal article Journal of Risk and Insurance

A New Premium Principle for Equity-Indexed Annuities

Article excerpt

ABSTRACT

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.

INTRODUCTION

An equity-indexed annuity (EIA) is an insurance product with benefits linked to the performance of an equity market. It provides a limited participation in the performance of an equity index (e.g., S&P 500) while guaranteeing a minimum rate of return. Introduced by Keyport Life Insurance Co. in 1995, EIAs have been the most innovative annuity product over the last 10 years. They have become increasingly popular since their debut and the sales of EIAs have broken the $20 billion barrier ($23.1 billion) in 2004 and reached $27.3 billion in 2005. See the 2006 Annuity Fact Book from the National Association of Variable Annuities (NAVA).

There are two common approaches to deal with equity-linked products: the financial and the actuarial approaches. In the former, it is generally assumed that insurance companies can diversify the mortality risk. Working with this assumption and using the classical Black-Scholes-Merton (BSM) framework, Tiong (2000) and Lee (2002, 2003) use the Esscher transform method developed in Gerber and Shiu (1994) to obtain closed-form formulas for several equity-indexed annuities. Lin and Tan (2003) consider a more general model for equity-indexed annuities, in which the external equity index and the interest rate are general stochastic differential equations. In a discrete time setting, Gaillardetz and Lin (2006) propose loaded participation rates based on implied loaded mortality probabilities using standard life insurance information. The risk measures could also be used to evaluate equity-linked products. This method relies heavily on the choice of risk measures (see Artzner et al., 1997, 1999; Wirch and Hardy, 1999). The financial and actuarial approaches are compared by Boyle and Hardy (1997). Attempts are made by Jacques (2003) and Barbarin and Devolder (2005) to combine both approaches in the management of equity-linked products. Our goal is to integrate these two methodologies in order to protect the equity-linked issuers against the possible losses incurred by a fair valuation. The approach adopted in this article however differs from previous results in the literature; particularly, we obtain a loaded participation rate that is determined using the financial and actuarial approaches. Hence, the main contribution of this article is to propose a new premium principle that protects insurance companies against financial, mortality, and other risks.

The traditional insurance and annuity pricing methods calculate the net premium of a product as the expected present value of its benefits with respect to a mortality law. In order to protect insurance companies against risks, usually a loading based on certain premium principles is added to the net premium (see Bowers et al., 1997). The traditional actuarial pricing is difficult to extend directly to the valuation of equity-linked products since these products are embedded with various types of financial guarantees. In this article, we propose a premium principle for equity-indexed annuities that protects the issuer against the mortality and financial risks. In order to derive the premium principle, the fair participation rate based on a fair valuation of the equity-linked contract is obtained using the financial approach. …

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