VALUATION OF CLIQUET-STYLE GUARANTEES WITH DEATH BENEFITS

. In this paper, we consider the problem of valuing an equity-linked insurance product with a cliquet-style payoff. The premium is invested in a reference asset whose dynamic is modeled by a geometric Brownian motion. The policy delivers a payment to the beneficiary at either a fixed maturity or the time upon the insured’s death, whichever comes first. The residual lifetime of a policyholder is described by a random variable, assumed to be independent of the asset price process, and its distribution is approximated by a linear sum of exponential distributions. Under such characterization, closed- form valuation formulae are derived for the contract considered. Moreover, a discrete-time setting is briefly discussed. Finally, numerical examples are provided to illustrate our proposed approach.


1.
Introduction. Equity-linked annuity (ELA) is a hybrid between a life insurance (or annuity) policy and a pure investment product. ELAs allow the policyholder to invest the premium in the equity market and give the policyholder an opportunity to participate in the upside potential while being protected against the downside risk. These contracts are so flexible that they provide both life insurance and guaranteed minimum accumulation benefits. A buyer pays the premium at the initiation of an ELA contract, then it involves an accumulation phase and a payout phase, at which the insurer promises to pay the policyholder either a fixed annuity or a lump sum in the future. The benefit return is credited to the account and benefit payout amount is directly linked to the performance of a reference asset, say a stock or an equity index. One of the most commonly used indices is the S&P 500 Index. In addition, ELAs are tax-deferred, which makes both policyholders and insurers prefer them to other long-term investments with lower yields.
The equity-index annuity (EIA) structure was first introduced in Keyport Life Insurance in 1995 (see [26] for details) and has been considered to be a successful innovation in the market. Usually, some guarantees are embedded in EIAs to meet different demands from customers. A participation rate is offered which specifies a proportion of the return that can be credited to the policyholder's account value. Common indexing methods include point-to-point design, cliquet-style design, and high-water-mark design. The cliquet-style (also ratchet, or ratchet-type 1 ) EIA seems to be the most popular type since the earning is credited annually, based on the greater of the return on the linked equity with a participation rate during the year, and a guaranteed minimum return. The annual reset feature protects the earnings from future market declines, thus policyholder's account value will never decrease once the return is credited.
ELAs may include various guarantees at the maturity of the contract and/or on the death of the insured. The aim of this paper is to value a cliquet-style contract that also pays death benefits if the insured dies before the contract maturity. Immediately following the insured's death, the provision assures a payment, which is the greater of the then-current account value and the accrued return amount. If the insured survives the maturity, the contract pays the accumulated benefits in a lump sum to the recipient.
In this paper, we assume such a contract is linked to a stock (or stock index). At any time t ≥ 0, the price of a stock follows a geometric Brownian motion where we set S(0) = 1, and {X(t), t ≥ 0} is a drifted Brownian motion: in which µ, σ ∈ R + and W (t) is a Wiener process.
The payoff of such products possibly depends on the historical prices of the stock. In the following, we denote the running maximum of X(t) by For an x-aged insured, we denote the remaining lifetime by a random variable T x (T x > 0). Further, we suppose that the market is frictionless, and no transaction fees are required. In addition, it is reasonable to assume that mortality risk and the market are independent, that is, T x and S(t) are independent. We shall make this assumption throughout this paper.
There is an extensive literature on the valuation of equity-linked insurance contracts. Pioneering work on this topic includes [1] and [3], in which the pricing and hedging of investment guarantees in life insurance are studied under the Black-Scholes model. [26] studies the valuation of equity-indexed annuities (EIAs) with major designs and explicit valuation formulas are obtained via the renowned Esscher transform. Many also pay attention to ratchet-type products. [17] proposes a lattice method for valuing ratchet EIAs. [21] discuss the pricing of ratchet EIAs under a stochastic rate model. [24] exploit the valuation of EIAs under stochastic mortality and interest rate (two factors are assumed to be independent). Recently, more complicated settings are adopted to study the EIA valuation problem. To capture how the status of the market affects the value of an equity-linked life insurance contract, both [18] and [6] render the valuation problem under a regime-switching framework while the latter involves stochastic volatility models with jumps. [5] consider ratchet EIAs with the quanto feature.
There are ways to specify the mortality model. Some directly use life table data, see [21] and [18], therein. Others apply analytical laws of mortality, see [27] and [28] where the constant force of mortality and De Moivre's mortality law are adopted, respectively. In [29], the Makeham's law of mortality model is fitted from a life table. Noticing the fact that any distribution on positive half line can be approximated by a linear combination of exponential distributions (see [7,8]), [13] and [14] propose to use linear combination of exponential distributions to approximate the distribution of T x . Some examples of fitting life table data by linear combination of exponential distributions can be found in [25] and [11]. In [23], the distribution of T x is approximated by the piecewise constant force of mortality.
We are interested in valuing a cliquet-style contract with payout time defined by T x ∧ n, the minimum of the time upon insured's death and a fixed maturity. Assuming n is an integer, the contract matures in n years. The valuation of the contract can be mathematically formulated as where G(·) denotes the cliquet-style benefit function and its specific forms are elaborated in Section 2. In addition, δ denotes the constant force of interest throughout the paper. This paper is organized as follows. Payoff formulations for the cliquet-style equity-linked contract considered are introduced in Section 2. As we use a linear combination of exponential distributions to approximate the distribution of T x , we find that it suffices to study (1) by replacing T x with an independent exponential random variable. Next, we discuss two types of products which credit returns differently and derive closed-form valuation formulae in Section 3 and Section 4, respectively. A discrete-time setting and a modified valuation problem under such case is discussed in Section 5. Finally, numerical examples are displayed in Section 6 and concluding remarks are given in Section 7.
2. Cliquet-style design. Within every year, the greater of a minimum guaranteed return and partial return that the chosen asset attains will be credited to the investment account. Once the return is credited, then it is locked in and will not be affected by future downturns of the linked asset. Since we assume the initial value of the linked equity is 1 unit, the total earning of a cliquet-style EIA contract maturing in n years is a multiplication of returns obtained from each period, denoted by where g > 0 is a fixed minimum guaranteed return rate and assumed to stay the same for every year. The participation rate is denoted by α, a constant specifying the portion of asset's return that will be credited to the investment account. Usually, α ≤ 1 in practice. In this case, we take G(S(n)) = R X n . The return rate increment between two consecutive years is denoted by X j , where j = 1, 2, ...n, and it follows that indicating that {X j } j=1,2,...,n forms a sequence of i.i.d. normal random variables Another type of cliquet-style product is also of interest. We denote the peak value of return rate increment that the linked underlying attains during year j − 1 to j by then the total return accrued to n years is denoted by It follows from (4) that G(S(n)) = R M n in this case. The return earned is based on the highest return rate attained during a year over the value at the beginning of that year. To distinguish how the return rate increment is credited, we use letter X and M as superscripts in (2) and (4), respectively. Remark 1. The cliquet-style design considered in this paper corresponds to the compound or multiplicative type of ratchet EIAs, where returns are compounded yearly. Another common type is the simple ratchet EIA, where the returns are added from every year together to calculate the final payout. Contributions on simple EIAs are referred to, for example, [17], [6], [18].
Since the distribution of T x can be approximated by a linear combination of exponential distributions, then when calculating (1), the valuation problem can be simplified. To see this, as in (1), we let G(·) denote the cliquet-style payoff and decompose (1) into two cases, where 1 (·) denotes the indicator function throughout the paper.
As T x is a non-negative random variable, we use f Tx to denote its probability density function. Due to the fact that linear combinations of exponential functions are weakly dense in the positive real line, then where for each k, τ k ∼ Exp(λ k ) and p k 's are constants satisfying k p k = 1, in which a negative value of p k is allowed. Using the independence between T x and S(t), we have Therefore V (δ, T x ∧ n) can be also expressed as which means that we can replace T x with an exponential random variable τ ∼Exp(λ) (assumed to be independent of S(t)), and it is now sufficient to study Combining (2) and (4), we have and It is obvious that (9) is always greater than (8) and the policyholder benefits more from the latter valuation scheme.

3.
Valuing V X (δ, τ ∧ n). In this section, we calculate (8). Similar to (5), we write (8) as a sum of two terms, with [τ ] taking the integer part of τ throughout this paper. When the insured's death occurs before or upon maturity n, the total earning is a multiplication of returns up to [τ ] years, i.e., R X [τ ] , and max e g(τ −[τ ]) , e α(X(τ )−X([τ ])) , the return obtained from last fractional period. The guaranteed return rate in the last noncomplete year is proportionally calculated. Otherwise when the insured survives the maturity, a benefit R X n will be paid.
Remark 2. The payment is due immediately following the insured's death which resembles the guaranteed minimum death benefits (GMDBs) provision in a variable annuity, a different equity-linked insurance product. Readers may find latest research on GMDBs in [30] and [20], among others.
We will discuss V X n (δ, n) and V X τ (δ, τ ) one by one. Since we assume τ follows an exponential distribution with rate parameter λ and it is independent of Equation (11) implies that it suffices to calculate V X (δ, n), since V X (δ + λ, n) can be obtained by replacing the first variable δ by δ + λ.
Based on our assumptions, all X j 's are i.i.d. and follow N (µ, σ 2 ). Throughout this paper, the cumulative distribution function of a standard normal variable is denoted by N (·). After some calculation, we obtain Letting δ = 0 in (12) immediately gives the result: Remark 3. Formula (12) indeed corresponds to the value of a compound ratchet EIA maturing in n years. The compound ratchet form is also mentioned in [17] and [4] in contrast to the simple ratchet form. Compared to simple ratchet EIA, explicit valuation formulae for compound ratchet EIA are easier to obtain if proper assumptions are presumed. In addition, formula (12) is equivalent to formula (5.5) in [26]. The author mainly pays attention to obtain nice closed-form pricing formulas for common EIA designs when the mortality risk is not considered.
which is regarded as a sum of consecutive payments incurring every year. The present value of each payment is where the second equation is true since R X i−1 is independent of both τ and the return rate increment attained from the remaining non-complete year. In addition, E R X i−1 can be obtained from (13).
To calculate E e αX(s) 1 (X(s)>ms) , we need the following lemma and provide its proof in Appendix A.
Using Lemma 3.1 and the notation ϑ = µα + 1 2 σ 2 α 2 , we obtain The integrals in (17) display a specific form that the integrand incorporates an exponential function and a normal c.d.f. To calculate such integrals, the following lemma is proposed and its proof is given in Appendix B.
Lemma 3.2. For real numbers w (w > 0) and u, we have the following result.
The valuation formula for V X (δ, τ ∧n) is summarized in the following proposition.
Proposition 1. By formula (13), Lemma 3.1 and Lemma 3.2, a closed-form expression for A i is given by Thereby, we obtain where V X (δ + λ, n) is obtained by replacing δ with δ + λ in (12).

4.
Valuing V M (δ, τ ∧n). Many equity-linked insurance products have the so-called high-water-mark feature, the payoff of this kind product is similar to a lookback option's payoff. Lookback options have been studied by many researchers, see, for example, [9,10,19]. For related research on EIAs products, readers are referred to [26,22,12]. Different from the last section, we now consider an alternative design where the formulations are given in (3) and (4). Within a year, the earning is based on the highest return rate attained over the value in the beginning. In other words, the earning is calculated via the high-water-mark method every year. Suppose s > 0, X(s) is a drifted Brownian motion, then the joint distribution of it and its maximum process M (s) can be found in [16] (pp. [7][8][9][10][11][12][13][14]: By letting x ↗ y in above joint distribution and note that {M (s) ≤ y} ⊆ {X(s) ≤ y}, we obtain where D = 1 2 σ 2 . It immediately follows from (21) that the density function of M (s) is We recall the formulation of highest value of return rate increment during a year in (3), and it implies that for j = 1, 2, · · · , n, Note that {M j } j=1,2,··· ,n forms a sequence of i.i.d. variables and M j d = M (1). Similar to the previous section, we now rewrite (9) and obtain Since τ is independent of the underlying process and M j 's are i.i.d., we have To calculate E max e g , e αM (1) , the following lemma is useful and its proof is presented in Appendix C.
In particular, if c = 0, then (25) is the moment generating function of M (s), that is, By (21) and (25), we have and we immediately obtain Next, we proceed to solve V M τ (τ ) in (23). Similarly, we decompose V M τ (τ ) into a serial sum as follows, where for i = 1, 2, ..., n, the value of each term is defined by where the second equation is true due to that R M i−1 is independent with both τ and the highest return rate increment from last fractional period. By discovering In the following proposition, we provide a closed-form expression for V M (δ, τ ∧n).

5.
Valuation problem under a discrete-time setting. In previous sections, we assume a continuous-time setting for valuation. However, many actuaries tend to use a discrete-time framework since insurance products are monitored periodically in practice. In line with this reason, some scholars assume that the payment occurs at the end of the year of death, see Section 6 in [21], Section 3.3.2 in [18], among others. We render the valuation problem in a discrete-time setting. To this end, we introduce an integer-valued random variable K x , denoting the curtate-futurelifetime for an insured at age x, that is, the time until the beginning of the year of the insured's death. Except that the death benefit is paid at the end of the year that death incurs, that is, K x + 1 and the payout amount is based on the accumulated benefits up to time K x , other assumptions introduced in earlier sections remain unchanged.
The distribution of K x can be approximated by a linear combination of geometric distributions: where j c j = 1 and γ j follows a geometric distribution with parameter 0 < π j < 1. We note that π j 's are not necessarily equal to each other and some of coefficients c 1 , c 2 , · · · can be negative. The approximation (32) is closely related to (6). To see this, readers are referred to Section 5 in [15]. Different from Section 3 and Section 4, we implement a curtate-future-lifetime random variable and the time to distribute death benefits is altered. The approximation (32) again indicates that the valuation problem can be simplified, as in (7). Therefore, we assume a geometrically distributed random variable γ, provided that it is independent from S(t), and its probability mass function is We are interested in valuing the following modified expressions: and where 0 < v < 1 is a constant discount factor.
We discuss (33) first. Due to the independence assumption between γ and S(t), we have E v n R X n 1 (γ≥n) =P(γ ≥ n)E v n R X n γ ≥ n =(vπ) n E R X n , and E R X n is obtained in (13). The second term in (33) equals to Similar derivations for (34) are obtained: where E R M n and E R M i follow from (27). 6. Numerical examples. This section presents some illustrative numerical examples regarding Section 3 and Section 4. For this purpose, additional assumptions and parameter settings are required. For the distribution of T x , we follow the ideas in [25], in which they propose a calibrated mortality model by fitting empirical survival data with a linear combination of exponential functions. By solving a nonlinear least-square system, one can obtain the density function of T x in the form of a linear combination of exponential densities. In our numerical realization, we fit the survival function of T x via the life table from Appendix 2A of [2]. Suppose the current age x = 30, the number of data points used for fitting is 25, we adapt techniques in [25] and obtain the fitted densities with 3, 5, 10 exponential density terms. Fitted densities are denoted by f M3 Tx , f M5 Tx , f M10 Tx , and displayed as follows, We refer (35),(36) and (37) as mortality model M 3 , M 5 and M 10 in the sequel. Applying formulas (20) and (31), we calculate values for V X (δ, T x ∧ n) and V M (δ, T x ∧ n) under these models.
As the product is linked with an equity, we use the risk-neutral probability Q for valuation. Let r = 0.05 be a constant risk-free rate. Under measure Q, for any t > 0, the return rate of the stock, X(t), follows a normal distribution, The numerical results are reported in Table 1-Table 4, rounded at the fourth decimal. Since two different types of increments X j and M j are considered, we list corresponding results below BM and RM columns. Numerical results under model M 3 , M 5 and M 10 are close to each other if compared column-by-column. Under each mortality model, results in RM columns are obviously greater than those in BM columns. This is intuitive and easy to see.
To examine sensitivities, we investigate parameters g, α, σ and T one by one, ceteris paribus. In Table 1, we observe valuation results increase as g increases since a higher g results in higher payoff values. Since we assume g is a constant, the increase in g does not cause drastic growth in outcomes. Table 2 shows that the increase of α brings higher valuation results, and the increase of α has a greater boosting effect for the RM case. Since for j = 1, 2, 3, ...n, M j ≥ X j , then a higher α credits greater returns to policyholder's account in each period, which yields higher returns after the accumulation phase.
Our intuition is that when the volatility σ increases, the value of the product will also increase. This is confirmed in Table 3 for both BM and RM cases. A higher σ increases the probability of the asset price moving upward (and downward), thus the increment is more likely to exceed g. Although a higher σ can also push the asset price to move in a downward direction, the guarantee structure assures that the account value will not decrease, but continue to grow at a speed of, at least, g.
In Table 4, valuation results increase as maturity n increases. The reason is twofold. The probability that an insured will survive longer is higher, thus contracts are more expensive to cover this risk. In addition, as maturity becomes longer, the accumulated returns in the account grow consistently.  To check our valuation formulas, as an example, we conduct Monte Carlo (MC) simulations and compare MC results with those computed using formula (20). The simulation experiment sets a sample size of 10 6 . For illustration purpose, we use the same setting in Table 2. Both valuation results and running time (displayed in seconds) via two approaches are reported in Table 5. It is obvious that the running time of MC simulation is much longer than that using closed-form formulas. Conclusion. This work studies the valuation of a cliquet-style equity-linked insurance contract that pays death benefits immediately following the insured's death. The contract is equity-linked, meaning that the performance of policyholder's investment depends on the performance (i.e., price) of the linked financial instrument. Two accumulation methods are considered. The first one is that we directly credit a certain portion of the return rate increment in a year while for the other one, the part of the highest return rate increment ever reached is taken into calculation. Using the fact that the density of the time-until-death variable can be approximated by a linear combination of exponential densities, we simplify our valuation formulae to (8) and (9) respectively. Closed-form valuation formulas are obtained, see (20) and (31). In addition, we also shed light on a discrete-time framework and study the modified valuation problem in Section 5. Closed-form valuation formulae are obtained as well.
In this paper, we assume the stock price is driven by a geometric Brownian motion, which is not fully compatible with real market data. We envisage the adoption of exponential Lévy models for future exploration. Besides, it will be interesting to consider surrender behaviour (or dynamic lapsation) of policyholders, which is also left for future research.