THE FRAMEWORK

2.1. The equity-linked savings contract

Our goal is to price an equity-linked savings contract. The contract is not exactly any of the yet existing products, but it has many features covering a large scale of different types of policies.

The contract consists of two parts. The first part is a guaranteed interest and the second part a bonus depending on the yield of some total return equity index. Thus, our product resembles equity-indexed annuities in the United States and equity-linked insurance con-tracts in Germany. On the other hand, in some equity-linked concon-tracts the bonus is linked to a fund or combinaton of funds, for example in variable annuities in the United States or segregated fund contracts in Canada (see Hardy, 2003).

We denote the amount of savings in the insurance contract at timetibyY(ti). Then its growth during a time interval of lengthδ=t_i+1−tiis given by gis a guaranteed rate andbis a bonus rate, the proportion of the excessive equity index yield which is returned to the customer. In this study we use the time intervalδ=1/255, where 255 is approximately the number of the days in a year on which the index is quoted, and the lag length of the moving average is chosen to beq=125 (i.e., half a year). The use of a moving average decreases the volatility of the contract value, and thus facilitates hedging.

The model also incorporates a surrender (early exercise) option and possibility for a penaltypwhich occurs if the customer reclaims the contract before the final expiration

4 LUOMA, PUUSTELLI, KOSKINEN

We follow Bunnin et al. (2002) who use Bayesian numerical techniques to price a European Call option on a share index. The two major benefits from using Bayesian techniques are that we can explicitly acknowledge the risks associated to model choice and parameter estimation. In order to value American-style options we use the Tsitsiklis and Van Roy (1999, 2001) regression approach, which approximates the value of the option against a set of basic functions.

From the methodological point of view we address questions about: (i) implementa-tion of MCMC to estimate the underlying diffusion processes, (ii) implementation of the regression method to determine the fair price of the insurance contract and its confidence interval, and (iii) solution of an inverse problem to determine the fair bonus rate and its confidence interval. From a more applied point of view we investigate the effect of the contract conditions, such as contract length, guaranteed interest rate and penalty rate, on the fair bonus rate and its estimation accuracy. Furthermore, we compare the results obtained in constant and stochastic interest rate environments, and briefly note the dif-ferences in model fit and valuation results between the used CEV model and the simpler GBM model.

The paper is organized as follows. Section 2 introduces the framework and model, Section 3 presents the estimation and evaluation procedures and Section 4 the empirical results. The final Section 5 concludes.

2. THE FRAMEWORK

2.1. The equity-linked savings contract

Our goal is to price an equity-linked savings contract. The contract is not exactly any of the yet existing products, but it has many features covering a large scale of different types of policies.

The contract consists of two parts. The first part is a guaranteed interest and the second part a bonus depending on the yield of some total return equity index. Thus, our product resembles equity-indexed annuities in the United States and equity-linked insurance con-tracts in Germany. On the other hand, in some equity-linked concon-tracts the bonus is linked to a fund or combinaton of funds, for example in variable annuities in the United States or segregated fund contracts in Canada (see Hardy, 2003).

We denote the amount of savings in the insurance contract at timetibyY(ti). Then its growth during a time interval of lengthδ=t_i+1−tiis given by gis a guaranteed rate andbis a bonus rate, the proportion of the excessive equity index yield which is returned to the customer. In this study we use the time intervalδ=1/255, where 255 is approximately the number of the days in a year on which the index is quoted, and the lag length of the moving average is chosen to beq =125 (i.e., half a year). The use of a moving average decreases the volatility of the contract value, and thus facilitates hedging.

The model also incorporates a surrender (early exercise) option and possibility for a penaltypwhich occurs if the customer reclaims the contract before the final expiration

company can better hedge its liabilities, and on the other hand, the customer will have a better idea of the guaranteed growth rate.

In this framework the penaltyp for early exercise and the parametersk,gandbare predefined by the insurance company. However, in the case of a stochastic interest rate,g is reset annually. We will determine a fair bonus ratebso that the risk-neutral price of the contract is equal to the initial investment. This gives the contract a simple structure and makes its costs and returns visible and predictable for the insurer and the customer. Our main interest is to study the effects of the expiration date, guaranteed rate and penalty rate on the fair bonus rate in both constant and stochastic interest rate cases.

The equity-indexed annuity contract has a modification called an annual ratchet in which the index participation is evaluated year by year. Each year the amount of savings is increased by the greater of the floor rate, which is usually 0 percent, and the increase in the underlying index, multiplied by the participation rate. Our contract is similar to this apart from being evaluated on a daily basis. Our contract type is better linked to the dynamics of the financial markets, since the customer may follow the growth of the savings daily and also exercise the contract at market value.

2.2. Model with constant interest rate

The constant elasticity of variance (CEV) process introduced by Cox and Ross (1976) is used to model the equity index process. It is a nonnegative diffusion process, defined by the stochastic differential equation (SDE)

dSt=μStdt+νS^1−α_t dWt, (2)

whereμ,ν(>0) andαare fixed parameters andWtis a standard Brownian motion under a real-world probability measureP. Ifα=0, the process (2) becomes a geometric Brow-nian motion. In the estimation, we assume thatα >0, which means that the volatility is smaller for larger values ofSt. Ifα > ¹₂, there is a positive probability that the process converges to zero. The model may also be written in the form

dSt=rStdt+νS^1−αt dZt, (3) whereris the riskless short-term interest rate andZta standard Brownian motion under a risk-neutral probability measureQ. The parametersμ,νandαare unknown and will be estimated.

The transition densities of the process (2) have closed form solutions which use the modified Bessel function of the first kind (see Bunnin et al., 2002). However, we found

company can better hedge its liabilities, and on the other hand, the customer will have a better idea of the guaranteed growth rate.

In this framework the penaltyp for early exercise and the parametersk,gandbare predefined by the insurance company. However, in the case of a stochastic interest rate,g is reset annually. We will determine a fair bonus ratebso that the risk-neutral price of the contract is equal to the initial investment. This gives the contract a simple structure and makes its costs and returns visible and predictable for the insurer and the customer. Our main interest is to study the effects of the expiration date, guaranteed rate and penalty rate on the fair bonus rate in both constant and stochastic interest rate cases.

The equity-indexed annuity contract has a modification called an annual ratchet in which the index participation is evaluated year by year. Each year the amount of savings is increased by the greater of the floor rate, which is usually 0 percent, and the increase in the underlying index, multiplied by the participation rate. Our contract is similar to this apart from being evaluated on a daily basis. Our contract type is better linked to the dynamics of the financial markets, since the customer may follow the growth of the savings daily and also exercise the contract at market value.

2.2. Model with constant interest rate

The constant elasticity of variance (CEV) process introduced by Cox and Ross (1976) is used to model the equity index process. It is a nonnegative diffusion process, defined by the stochastic differential equation (SDE)

dSt=μStdt+νS^1−α_t dWt, (2)

whereμ,ν(>0) andαare fixed parameters andWtis a standard Brownian motion under a real-world probability measureP. Ifα=0, the process (2) becomes a geometric Brow-nian motion. In the estimation, we assume thatα >0, which means that the volatility is smaller for larger values ofSt. Ifα > ¹₂, there is a positive probability that the process converges to zero. The model may also be written in the form

dSt=rStdt+νS^1−αt dZt, (3) whereris the riskless short-term interest rate andZta standard Brownian motion under a risk-neutral probability measureQ. The parametersμ,νandαare unknown and will be estimated.

The transition densities of the process (2) have closed form solutions which use the modified Bessel function of the first kind (see Bunnin et al., 2002). However, we found

6 LUOMA, PUUSTELLI, KOSKINEN

the Euler discretization of the process to be accurate enough for estimation and simulation purposes, since our discretization interval is only one working day. The Euler scheme is the simplest standard method for approximate simulation of stochastic differential equa-tions; for further details, see Iacus (2008) or Glasserman (2004).

Assuming that the discretized process (2) has been observed at equally-spaced time points 0, δ, ...Nδ, the likelihood function can be written in the form

p(y|θ)=

2.3. Model with stochastic interest rate

In our second set-up, we assume that the dynamics of riskless short-term ratert and stock indexStare described by the following system of SDEs:

drt=κ(ξ−rt)dt+σr_t^γdW_t⁽¹⁾, (4a) dSt=μStdt + νS^1−α_t dW_t⁽²⁾, (4b) withW_t⁽¹⁾ andW_t⁽²⁾two standard Brownian motions, correlated throughW_t⁽²⁾ =ρW_t⁽¹⁾+ 1−ρ²W_t⁽³⁾, whereW_t⁽¹⁾ andW_t⁽³⁾are independent standard Brownian motions under a real-world probability measure. Thus the correlation ofW_t⁽¹⁾andW_t⁽²⁾isρ.

The short-term interest rate model (4a) was introduced by Chan et al. (1992), who provide a useful summary of short-term interest rate models in their paper. The two most commonly used models which may be derived from this model by parameter restriction are the following: Ifγ=0, the model becomes the Ornstein-Uhlenbeck process, proposed by Vasiˇcek (1977) as a model of the short rate, and, if γ = ¹₂, it becomes a square-root diffusion referred to as the Cox-Ingersoll-Ross (CIR) model (Cox et al., 1985). In estimation, the parametersκ, ξ, σandγare assumed to be positive.

SubstitutingZ⁽¹⁾_t =W_t⁽¹⁾andZ⁽³⁾_t =W_t⁽³⁾+(μ−rt)ν⁻¹(1−ρ²)^−1/2S^α_tdt, the system of introduced by assuming thatZ_t⁽¹⁾andZ_t⁽³⁾are two independent standard Brownian motions under this measure. It can then be shown that the discounted price ˜St=Stexp(−_t

0 rsds) is a martingale underQ.

The transition densities of the bivariate process described by (4a) and (4b) do not have a closed form solution, and we will use its Euler discretization to estimate the unknown parametersκ,ξ,σ,γ,μ,νandα. Accordingly, we will simulate the risk-neutral process using the Euler discretization of (5a) and (5b).

In order to obtain numerical stability in estimation, we reparametrize model (4a) as dx_t=(β−κx_t)dt+τx^γ_tdW_t⁽¹⁾,

6 LUOMA, PUUSTELLI, KOSKINEN

the Euler discretization of the process to be accurate enough for estimation and simulation purposes, since our discretization interval is only one working day. The Euler scheme is the simplest standard method for approximate simulation of stochastic differential equa-tions; for further details, see Iacus (2008) or Glasserman (2004).

Assuming that the discretized process (2) has been observed at equally-spaced time points 0, δ, ...Nδ, the likelihood function can be written in the form

p(y|θ)=

2.3. Model with stochastic interest rate

In our second set-up, we assume that the dynamics of riskless short-term ratert and stock indexStare described by the following system of SDEs:

drt=κ(ξ−rt)dt+σr_t^γdW_t⁽¹⁾, (4a) dSt=μStdt + νS^1−α_t dW_t⁽²⁾, (4b) withW_t⁽¹⁾ andW_t⁽²⁾two standard Brownian motions, correlated throughW_t⁽²⁾ =ρW_t⁽¹⁾+ 1−ρ²W_t⁽³⁾, whereW_t⁽¹⁾ andW_t⁽³⁾are independent standard Brownian motions under a real-world probability measure. Thus the correlation ofW_t⁽¹⁾andW_t⁽²⁾isρ.

The short-term interest rate model (4a) was introduced by Chan et al. (1992), who provide a useful summary of short-term interest rate models in their paper. The two most commonly used models which may be derived from this model by parameter restriction are the following: Ifγ=0, the model becomes the Ornstein-Uhlenbeck process, proposed by Vasiˇcek (1977) as a model of the short rate, and, if γ = ¹₂, it becomes a square-root diffusion referred to as the Cox-Ingersoll-Ross (CIR) model (Cox et al., 1985). In estimation, the parametersκ, ξ, σandγare assumed to be positive.

SubstitutingZ⁽¹⁾_t =W_t⁽¹⁾andZ⁽³⁾_t =W_t⁽³⁾+(μ−rt)ν⁻¹(1−ρ²)^−1/2S^α_tdt, the system of introduced by assuming thatZ_t⁽¹⁾andZ_t⁽³⁾are two independent standard Brownian motions under this measure. It can then be shown that the discounted price ˜St=Stexp(−_t

0 rsds) is a martingale underQ.

The transition densities of the bivariate process described by (4a) and (4b) do not have a closed form solution, and we will use its Euler discretization to estimate the unknown parametersκ,ξ,σ,γ,μ,νandα. Accordingly, we will simulate the risk-neutral process using the Euler discretization of (5a) and (5b).

In order to obtain numerical stability in estimation, we reparametrize model (4a) as dx_t=(β−κx_t)dt+τx^γ_tdW_t⁽¹⁾,

exp

⎛⎜⎜⎜⎜⎜

⎜⎜⎜⎝−

ΔSiδ−μS(i−1)δδ−νS¹_(i−1)δ^−α ρΔW_iδ⁽¹⁾2

2ν²S_(i−1)δ^2(1−α)(1−ρ²)δ

⎞⎟⎟⎟⎟⎟

⎟⎟⎟⎠, (6)

whereyis data,θ=(μ, ν, α, β, κ, τ, γ, ρ),Δxiδ=xiδ−x(i−1)δ,ΔSiδ=Siδ−S(i−1)δand ΔW_i⁽¹⁾_δ = x_iδ−x_(i−1)δ−(β−κx_(i−1)δ)δ

τx^γ_(i−1)δ .

3. ESTIMATION AND EVALUATION PROCEDURES