Actuarial and Financial Risks: Models, Statistical Inference, and Case StudiesView this Special Issue
Pricing Equity-Indexed Annuities under Stochastic Interest Rates Using Copulas
We develop a consistent evaluation approach for equity-linked insurance products under stochastic interest rates. This pricing approach requires that the premium information of standard insurance products is given exogenously. In order to evaluate equity-linked products, we derive three martingale probability measures that reproduce the information from standard insurance products, interest rates, and equity index. These risk adjusted martingale probability measures are determined using copula theory and evolve with the stochastic interest rate process. A detailed numerical analysis is performed for existing equity-indexed annuities in the North American market.
An equity-indexed annuity is an insurance product whose benefits are linked to the performance of an equity market. It provides 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, equity-indexed annuities have been the most innovative insurance product introduced in recent years. They have become increasingly popular since their debut and sales have broken the $20 billion barrier ($23.1 billion) in 2004, reaching $27.3 billion in 2005. Equity-indexed annuities have also reached a critical mass with a total asset of $93 billion in 2005 (2006 Annuity Fact Book (Tables 7-8) from the National Association for Variable Annuities (NAVA)). See the monograph by Hardy  for comprehensive discussions on these products.
The traditional actuarial pricing approach evaluates the premiums of standard life insurance products as the expected present value of its benefits with respect to a mortality law plus a security loading. Since equity-linked products are embedded with various types of financial guarantees, the actuarial approach is difficult to extend to these products and often produces premiums inconsistent with the insurance and financial markets. Many attempts have been made to provide consistent pricing approaches for equity-linked products using financial and economical approaches. For instance, Brennan and Schwartz  and Boyle and Schwartz  use option pricing techniques to evaluate life insurance products embedded with some financial guarantees. Bacinello and Ortu [4, 5] consider the case where the interest rate is stochastic. More recently, Møller  employs the risk-minimization method to evaluate equity-linked life insurances. Young and Zariphopoulou  evaluate these products using utility theory1. Particularly for equity-indexed annuities, Tiong  and Lee  obtain closed-form formulas for several annuities under the Black-Scholes-Merton framework. Moore  evaluates equity-indexed annuities based on utility theory. Lin and Tan  and Kijima and Wong  consider more general models for equity-indexed annuities, in which the external equity index and the interest rates are general stochastic differential equations.
The liabilities and premiums of standard insurance products are influenced by the insurer financial performance. Indeed, insurance companies adjust their premiums according to the realized return from their fixed income and other financial instruments as well as market pressure. Therefore, mortality security loadings underlying insurance pricing approach evolve with the financial market. With the current financial crisis, a flexible approach for equity-linked products that allows interdependency between risks should be used. Hence, we generalize the approach of Gaillardetz and Lin  to stochastic interest rates. Similarly to Wüthrich et al. , they introduce a market consistent valuation method for equity-linked products by combining probability measures using copulas. Indeed, the deterministic interest rate assumption may be adequate for short-term derivative products; however, it is undesirable to extrapolate for longer maturities as for the financial guarantees embedded in equity-linked products. Therefore, we use the approach of Gaillardetz  to model standard insurance products under stochastic interest rates. It supposes the conditional independence between the insurance and interest rate risks. Here, this approach is generalized to models that are based on copulas.
Similarly to Gaillardetz and Lin , we assume that the premium information of term life insurances, pure endowment insurances, and endowment insurances at all maturities is obtainable. We obtain martingale measures for each standard insurance product under stochastic interest rates. To this end, it is required to assume that the volatilities for standard insurance prices are given exogenously. Gaillardetz  provides additional structure to find an implicit volatilities for the standard insurance and annuity products. Then, the martingale probability measures for the insurance and interest rate risks are combined with the martingale measure from the equity index. These extend martingale measures are used to evaluate equity-linked insurance contracts and equity-indexed annuities in particular.
This paper is organized as follows. The next section presents financial models for the interest rates and equity index as well as insurance model. We then derive martingale measures for those standard insurance products under stochastic interest rates in Section 3. In Section 4, we derive the martingale measures for equity-linked products. Section 5 focuses on recursive pricing formulas for equity-linked contracts. Finally, we examine the implications of the proposed approaches on the EIAs by conducting a detailed numerical analysis in Section 6.
2. Underlying Financial Models
In this section, we present a multiperiod discrete model that describes the dynamic of a stock index and the interest rate. These lattice models have been intensively used to model stocks, stock indices, interest rates, and other financial securities due to their flexibility and tractability; see Panjer et al.  and Lin , for example. Moreover, as it often happens when working in a continuous framework, it becomes necessary to resort to simulation methods in order to obtain a solution to the problems we are considering. Moreover, the premiums obtained from discrete models converge rapidly to the premiums obtained with the corresponding continuous models when considering equity-indexed annuities.
2.1. Interest Rate Model
Similarly to Gaillardetz , it is assumed that the short-term ratefollows that ofBlack et al.  (BDT), which means that the short-term rate follows a lattice model that is recombining and Markovian. Particularly, the short-term rate can take exactly distinct values at year denoted by . Indeed, represents the short-term rate between time and that has made “” up moves. The short-term rate today, , is equal to , and in the case where , the short-term rate at time , can only take two values, either (decrease) or (increase). We consider the short-term rate process under the martingale measure and hence, the discounted value process is a martingale. represents the price at time of a default-free, zero-coupon bond paying one monetary unit at time and , the money market account, represents one monetary unit () accumulated at the short-term rate
Let be the probability under that the short-term rate increases at time given . That is for , which is set to be under the BDT model. Figure 1 describes the dynamic of the short-term rate process.
The BDT model also assumes that short-term rate process matches an array of yields volatilities (), which is assumed to be observable from the financial market. This vector is deterministic, specified at time 0, and each element is defined by for and Hence, is larger than thus, (2.3) may be rewritten as follows: Equation (2.4) holds for and leads to for . Equations (2.4) and (2.5) lead to
By matching the market prices and the model prices, we have where represents the expectation with respect to . Replacing , , in (2.7) using (2.5) leads to a system of two equations (2.6) and (2.7) with two unknowns and , which can be solved for all .
2.2. Index Model
Similar to Gaillardetz and Lin , we suppose that each year is divided into trading subperiods of equal length , which means that the set of trading dates for the index is . We also assume a lattice index model such that the index process , , has two possible outcomes and given for the time period , where is the initial level of the index. The index level at time represents the index level when the index value goes down and represents the index level when the index values goes up. Since the short-term rate is a yearly process, we assume that the values and are constant for each year. Hence, we may write and for . Because of the number of trading dates per year, the time- value of the money-market account is given by for Here, is the floor function.
A martingale measure for the financial model needs to be determined for the valuation of equity-linked products. This martingale measures should be such that , and , remain martingales. Note that the goal of this section is to derive the conditional distribution of the index process. Hence, the constraints imposed by the martingale discounted value process are to be discussed later. Let be the conditional probability that the index value goes up during the period given , that is, for and Supposing that the discounted value process is a martingale implies for and From (2.10) it is obvious that is constant over each year, that is, for . The no-arbitrage thus requires for . The previous conditions may not be respected for the BDT model when long maturity or high volatility are considered. In this case, the bounded trinomial model from Hull and White  would be more suitable.
Under this model, the ratio takes possible values denoted , , which are defined by Their corresponding conditional martingale probabilities are for .
The model assumes the usual frictionless market: no tax, no transaction costs, and so forth. Furthermore, for practical implementation purposes, one may also use current forward rates for .
Figure 2 presents the conditional index process tree under stochastic interest rates when for the time period .
For notational convenience, let
which represents the index's realization up to time with for , where .
2.3. Insurance Models
In this subsection, we introduce lattice models for the standard insurance products under stochastic interest rates. We will use the standard actuarial notation which can be found in Bowers et al. . Let be the future lifetime of insured of age and the curtate-future-lifetime the number of future complete years lived by the insured prior to death. For notational purposes, let represent the realization of the short-term rate process up to time with , , where .
For integers and (), let denote, respectively, the time- prices for the -year term life insurance , -year pure endowment insurance , and -year endowment insurance () given that the short-term rate followed the path .
The value process of -year term life insurance is defined by with . Note that represents the interest rate information known by the process, but does not stand as an indexing parameter.
Similarly, define to be the value process of the -year pure endowment insurance and it is given by with .
Finally, let denote the value process generated by the -year endowment insurance with
The processes , , represent the intrinsic values of the standard insurance products and are presented in Figure 3.
3. Martingale Measures for Insurance Models
In this section, we employ a method similar to the approach of Gaillardetz  to derive a martingale probability measure for each of the value processes introduced in the last section. Gaillardetz  derives these martingale measures under conditional independence assumptions. Here, we relax this assumption by using copulas to describe possible dependence structures between interest rates and insurance products. It is important to point out that these probabilities are age-dependent and include an adjustment for the mortality risk since we use the information from the insurance market.
The martingale measures are defined such that and are martingales. As mentioned in Section 2, we assume that the time-0 premiums , of the term life insurance, pure endowment insurance, and endowment insurance are given exogenously. The annual short-term rate process is governed by the BDT model with and volatilities are given exogenously for and The conditional martingale probability of each possible outcome is defined by for . These martingale probabilities are presented above each branch in Figure 3.
The main objective of this section is to determine ’s that will be used to evaluate equity-indexed annuities in later sections.
To ensure that the discounted value process is a martingale, we must have for , , and all . Note that the martingale mortality and survival probabilities are given, respectively, by
As in the short-term rate model, additional structure is needed to set the time- premiums. Similar to Black et al. , we suppose that the volatilities of insurance liabilities are defined at time by for , and all . Here, represents the conditional variance with respect to . We assume that the volatilities are deterministic but vary over time and are given exogenously. Gaillardetz  uses the natural logarithm function to ensure that each process remains strictly positive. Since is close to 0, it directly uses to ensure that the process remains strictly greater than 0. On the other hand, it uses for to ensure that the processes are strictly smaller than 1 since 's are closer to 1.
In order to identify the martingale probabilities , Gaillardetz  assumes the independence or the conditional independence between the interest rate process and the insurer's life. Here, the additional structure is provided by the choice of copulas. Indeed, the dependence structure between the interest rates and the premiums of insurance products is modeled using a copula. The main advantage of using copulas is that they separate a joint distribution function in two parts: the dependence structure and the marginal distribution functions. We use them because of their mathematical tractability and, based on the Sklar’s Theorem, they can express all multivariate distributions. A comprehensive introduction may be found in Joe  or Nelsen . Frees and Valdez , Wang , and Venter  have given an overview of copulas and their applications to actuarial science. Cherubini et al.  present the applications of copulas in finance.
There exists a wide range of copulas that may define a joint cumulative distribution function. The simplest one is the independent copula where and are marginal cumulative distribution functions. Extreme copulas are defined using the upper and lower Frechet-Hoeffding bounds, which are given by One of the most important families of copulas is the archimedean copulas. Among them, the Cook-Johnson (Clayton) copula is widely used in actuarial science because of its desirable properties and simplicity. The Cook-Johnson copula with parameter is given by The Gaussian () copula, which is often used in finance, is defined as where is the bivariate standard normal cumulative distribution function with correlation coefficient and is the inverse of the standard normal cumulative distribution function. Hence, the parameter in formulas (3.9) and (3.10) indicates the level of dependence between the insurance products and interest rates.
3.1. Term Life Insurance
Proposition 3.1. For given (), copulas , and volatilities ( and all ), the age-dependent, mortality risk-adjusted martingale probabilities are given by for , where the price at time is defined recursively using for and all .
With the martingale structure identified, the -year term life insurance premiums may be reproduced as the expected discounted payoff of the insurance
3.2. Pure Endowment Insurance
Proposition 3.2. For given (), copulas , and volatilities ( and all ), the age-dependent, mortality risk-adjusted martingale probabilities are given by (45) for , where the price at time is defined recursively using for and all .
With the martingale structure identified, the -year pure endowment insurance premiums may be reproduced as the expected discounted payoff of the insurance
3.3. Endowment Insurance
There is no general solution for the endowment insurance products since the -year endowment insurance price at time may not be expressed using only either mortality or survival probabilities. For the -year term-life insurance, the time-() price is determined based on the death martingale probabilities and the -year pure endowment price may be obtained using the survival probabilities at time . Therefore, once you combine both products to form an endowment insurance, there is no way to solve explicitly for the martingale probabilities. However, closed-from solutions may be derived for the independent, upper and lower copulas. Numerical methods need to be used for the Cook-Johnson and Gaussian copulas. Furthermore, the width of the participation rate bands for the unified approach is narrow under deterministic interest (see Gaillardetz and Lin ). For these reasons, we are focusing on the independent and Frechet-Hoeffding bounds.
Proposition 3.3. For given (), copulas , and volatilities ( and all ), the age-dependent, mortality risk-adjusted martingale probabilities are given by for , where the price at time is defined recursively using for and all .
Proof. The proof can be found in Gaillardetz .
Since we suppose that the time-0 insurance prices, the insurance volatilities, the zero-coupon bond prices, and the interest rate volatility are given exogenously, it is possible to extract the stochastic structure of each insurance products using Propositions 3.1, 3.2, and 3.3. There are constraints on the parameters because the martingale probabilities should be strictly positive. However, there is no closed-form solution for the stochastic interest models.
Theoretically, there exists a natural hedging between the insurance and annuity products. However, Gaillardetz and Lin  argue that it is reasonable to evaluate insurances and annuities separatelysince in practice due to certain regulatory and accounting constraints and issues such as moral hazard and anti-selection.
3.4. Determination of Insurance Volatility Structure
For implementation purposes, we now relax the assumption of exogenous insurance volatilities. In Subsections 3.1, 3.2, and 3.3, the volatilities of insurance liabilities defined by either (3.4) or (3.5) were supposed to be known. However, identifying these volatilities is extremely challenging due to the lack of empirical data and studies. Similar to Gaillardetz , we extract an implied volatility from the insurance market under certain assumptions.
There are three different sources that define the insurance volatilities: the interest rates, the insurance prices, and the martingale probabilities. The implied insurance volatilities is obtained assuming that the short-term rate has no impact on the martingale probabilities. Thus, we extract the insurance volatility such that the martingale probabilities in the case of an up move from the interest rate process are equal to the martingale probabilities in the case of a down move. Let (, , and all ) denote the implied volatilities defined by (3.4) for and (3.5) for under the following constraint: for . In other words, insurance companies that do not react to the interest rate change should have an insurance volatility close to . Gaillardetz [13, 27] explain that behavior of insurance companies facing the interest rate shifts could be understood through these volatilities. They also describe recursive formulas to obtain numerically the implied volatilities. In the following examples, equity-indexed annuity contracts are evaluated using the implied volatilities, which are obtained from (3.33).
4. Martingale Measures for Equity-Linked Products
Due to their unique designs, equity-linked products involve mortality and financial risks since these type of contracts provide both death and accumulation/survival benefits. Moreover, the level of these benefits are linked to the financial market performance and an equity index in particular. Hence, it is natural to assume that equity-linked products belong to a combined insurance and financial markets since they are simultaneously subject to the interest rate, equity, and mortality risks. Similar to Section 3, we evaluate these types of products by evaluating the death benefits and survival benefits separately. Under this approach, two martingale measures again need to be generated: one for death benefits and another for survival benefits. Furthermore, these martingale measures should be such that they reproduce the index values in Section 2 and the premiums of insurance products under stochastic interest rates in Section 3. In other words, the marginal probabilities derived in the previous sections should be preserved, and the martingale measures are such that , and and will remain martingales. Let denote the martingale probability under such that survives and or the martingale probability such that dies and between and given , , and as illustrated in Figure 4. The function is given explicitly by (2.12).
What remains is to determine the probabilities 's for all and . We introduce the dependency between the index process, the short-term rate, and the premiums of insurance products using copulas. Let , j = 1,2,3, denote this joint conditional cumulative distribution function over time and . That is As explained, the marginal cumulative distribution functions of the insurance products and the index are preserved under the extended measures, that is, which are determined using (3.18), (3.19), and (3.20) for , (3.24), (45), and (3.26) for , (3.29), (3.30), as well as (3.31) for , and (2.13) for the index. Let be the choice of copula, then the cumulative distribution function is defined by where represents the free parameter between and that indicate the level of dependence between the insurance product, interest rate, and the index processes. Here, the copula could be defined using either (3.6), (3.7), (3.8), (3.9), or (3.10). Note that in some cases, for example, the lower copula (3.8), the function would not be a cumulative distribution function. We also remark that 's are functions of , but for notational simplicity we suppress .
The martingale probabilities can be obtained from the cumulative distribution function and are given by for for for and for and , where and is obtained using (4.5). Similarly, for for for , for , and for .
Consider now an equity-linked product that pays