# Examining changes in reserves using stochastic interest models.

Introduction

On balance sheets, assets must equal liabilities plus net worth of the firm. An important component of these liabilities for insurers is the reserve, the portion of a firms' assets set aside to meet future uncertain obligations arising from insurance contracts. Although the obligations of each contract are contingent upon uncertain future events, and thus may be modeled stochastically, the reserve set aside is a single number. There are limitations when using a single number to summarize a stochastic quantity. However, reserves play a prominent role in financial statements and thus these quantities are important to managers of insurance organizations.

Several important problems in actuarial science rely heavily on the determination of a reserve. To illustrate, if a company or a block of business is to be traded on the open market, a value must be determined for the associated set of obligations. Thus, it is useful to think of a reserve as the "value" associated with a set of stochastic obligations. As another application, reserves traditionally have been used as a measure of an organization's financial strength. In this context, the reserve should be larger than the "value" of obligations, because a conservative approach should be taken for assessing potential future obligations.

Life insurance and annuity reserves are calculated by summarizing discounted cash flows, where the discounting is done with respect to investment earnings, as well as decrements due to mortality, disability, policy lapse, and so on, that may be applicable to a particular policy. For brevity, this articles considers only investment earnings and the mortality decrement. Extensions to the multidecrement case are straightforward.

In the traditional insurance literature, as in Jordan (1967), the deterministic assumption dominates the development of the theory of life contingencies. Namely, mortality occurs according to a known mortality table and the interest rate is assumed to have a deterministic value. One step further is to allow the age at death to be a random variable, although the interest rate is assumed to be deterministic--a "semi-stochastic" approach followed in Bowers et al. (1986). Insurance literature has generalized the traditional theory of life contingencies by introducing stochastic variation in interest rates (Boyle, 1976; Waters, 1978; Panjer and Bellhouse, 1980; Bellhouse and Panjer, 1981; Giaccotto, 1986; Dhaene, 1989; Frees, 1990; and Beekman and Fuelling, 1990, 1991). Additional stochastic interest models from financial economics are discussed below.

This article computes reserves as (conditional) expectations of sums of future cash flows. Motivation for this approach can be found in, for example, Bowers et al. (1986) for the semi-stochastic approach and Buhlmann (1992) for models using stochastic interest.

Here, we are primarily concerned with quantifying changes in reserves from one financial period to the next. Changes in reserves could be used to quantify the amount of profit released, as in Ramlau-Hansen (1988), which studied gains and losses emerging from margins built into mortality and other decrements and ignored those arising from investments. To complement that work, we focus on changes arising from stochastic interest rates and do not explicitly consider margins built into other decrement rates.

Changes in value of future obligations due to dynamic models of interest have been considered extensively in the financial economics literature, in particular as part of immunization theory, which deals with instantaneous changes in value. Here, we examine changes in value from one financial period to the next.

The new idea of examining changes in reserves can be illustrated by considering the following simple scenario. Let {[y.sub.s]} represent the random force of interest in the sth period. As argued in Frees (1990), [y.sub.s] can be interpreted as a one-period spot rate. Consider an obligation that can be expressed as a T-year pure discount bond. At time zero, the random present value of one unit payable at time T is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Without loss of generality, it is assumed that the time interval is one year. Suppose an insurer must pay one unit T years later with certainty, but under a stochastic interest rate environment. The reserve at time zero is denoted by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where the expectation is taken at time zero. Because the interest rates prior to time zero are all known at time zero, [V.sup.(T)] [sub.0] is a deterministic value.

After one year, the maturity time of the payment shortens by one and the reserve (at time one) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where the expectation is taken at time one. In general, because future spot rates may depend on the first-period spot rate, the quantity [V.sup.(T-1)] [sub.1] is a function of the random first-period spot rate [y.sub.1].

Although the obligations are random, their expected values define the (nonrandom) reserve. Following standard terminology, the reserve is the portion of liabilities that can be recognized in the insurer's balance sheet. Assets are held to meet the random obligations, where, at time zero, the known value of assets equals the reserve. In the absence of perfect matching, the random obligations and stochastic appreciator of the assets diverge over time. This article is about quantifying one aspect of the asset's adequacy in meeting the obligations.

This article examines the change from the initial reserve [V.sub.0] to the time one reserve [V.sub.1]. To examine this change, we discount the time one reserve [V.sup.(T-1)] [sub.1] back to time zero and then study its distribution. That is, we investigate the distribution and the statistical properties of the random reserve

[e.sup.[-y.sub.1]] [V.sup.(T-1)] [sub.1],

from a time zero point of view. Although [V.sub.0] represents the current expected value of the obligation, [V.sub.1] represents the value at the subsequent time period. Thus, for budgeting and other purposes, [V.sub.1] and its discounted value [e.sup.[-y.sub.1] [V.sub.1], are important quantities for risk and other financial managers.

At time zero, the discounted next-period reserve is stochastic because it is a function of the stochastic first-period interest rate [y.sub.1]. However, it is a deterministic value at time one since y, becomes known at that time. In the case of a T-year pure discount bond and random force of interest {[y.sub.s]},

(1) [V.sup.(T)] [sub.0] = E([e.sup.[-y.sub.1]] [V.sup.(T-1)] [sub.1]),

where the expectation is taken at time zero. However, as shown in the Appendix, the relationship in equation (1) must be modified for a life insurance policy.

The next section of this article describes the model used in the analysis and discusses the continuous-time equilibrium term structure model and the discrete-time ARIMA/linear process. Then, the linear process for interest rates is investigated. The following section considers a nonlinear process for interest rates, the autoregressive conditionally heteroskedastic (ARCH) process, which is widely used in economics. An empirical example on the interest rates is presented before the conclusion. The proofs of all results are given in the Appendix.

The Basic Model

Insurance Model

We consider here the individual risk model for insurance contracts, using the notation of Bowers et al. (1986). Denote the valuation time to be h so that, at the initial valuation, h = 0. Assume that there are n policies in the block of business. For the ith policy, the age at issue is [x.sub.i], the duration is [k.sub.i] when h = 0, the curtate random time of decrement is [K.sub.1], and the curtate-future-lifetime is [J.sub.i] (i.e., [J.sub.i] = [K.sub.i]-[k.sub.i]-h). Suppose that a death benefit [[b.sub.i] [K.sub.i +1]] payable at the end of the year of loss and that the annual premiums [[a.sub.i] [sub.m]] are payable at the beginning of each year up to and including the year of loss. Then, at time point h + [Tau] + 1, the random cash flow of the ith policy is

(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and the probability function of [J.sub.i] is: Prob([J.sub.i] = [Tau]) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [Tau] = 0, 1,...; and Prob([J.sub.i] [is greater than] [Tau]) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Here, [sub.[Tau]]|[q.sub.x] and [sub.[Tau]] [p.sub.x] are the traditional deferred decrement probabilities and survival functions calculated from a mortality table.

Because the definition of cash flow is quite general, it can be used for general insurance as well as combinations of whole life insurance, term life insurance, deferred life insurance, annuities, and pure discount bonds.

Interest Rate Model

Let [y.sub.s] denote the force of interest in the sth year (s = 1, 2,...). It is natural to assume that [y.sub.s] has a parametric form,

[y.sub.s] = f(??,[[Epsilon].sub.1],...,[[Epsilon].sub.2]),

where ?? is a vector of parameters, {[[Epsilon].sub.s]} are independent and identically distributed (i.i.d.), and f is a known function. To illustrate, the next section considers the recursive linear process

[y.sub.t] = a + [[Phi].sub.1] [y.sub.t-1] + ... + [[Phi].sub.p] [y.sub.t-p] + [[Epsilon].sub.t] - [[Iota].sub.1] [[Epsilon].sub.t-1] - ... -[[Iota].sub.q] [[Epsilon].sub.t-q].

The traditional ARMA/ARIMA model belongs to this class of model. In the fourth section, we consider an ARCH model, a nonlinear representation for the force of interest. Before proceeding, the following basic assumption and notations are made.

Assumption

[y.sub.t] is a Borel measurable function of ([[Epsilon].sub.1], [[Epsilon].sub.2],..., [[Epsilon].sub.t]).

Notations

Let M(t) = E([e.sup.t[Epsilon]), the moment-generating function of

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

and,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

E(??X) means taking expectation conditional on the information generated by X.

The Reserves

To introduce reserves, we first define the loss at valuation time h for the ith policy,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The sum of losses for the whole block of business at time h is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The reserve at time h is defined as [V.sub.h] = [E.sub.h] ([S.sup.(h)] [sub.L]), where [E.sub.h] ([multiplied by]) = E(??[[Epsilon].sub.1,...,[[Epsilon].sub.h], so that [V.sub.h] is a function of ([[Epsilon].sub.1],..., [[Epsilon].sub.h]). The discussion below concentrates on [V.sub.0] and [V.sub.1], the reserves at the beginning of valuation and its following period, respectively. The results can be extended to consider further periods.

The following discussion assumes the random cash flows [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are independent of the stochastic interest rates ([y.sub.s]). The following result for the reserves is a basic one that we will use frequently.

Proposition 1. Under the basic assumption above, we have

(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

(4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where

(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

provided the expectations exist.

In fact, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the expected cash flow of the ith policy at time h = 0, and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the expected cash flow of the ith policy at time h = 1. Although [V.sub.0] is a constant, [V.sub.1] is usually stochastic since it is a function of [[Epsilon].sub.1], the disturbance of force of interest generated in period one. When considering the discounted reserve, we simply multiply [V.sub.1] by [e.sup.-[y.sub.i]]. Actuarial science traditionally presents recursive calculations of reserves. The Appendix provides the recursive calculations relating [V.sub.1] to [V.sub.0], which generalize the T-year pure discount bond case described in the introduction.

Other Interest Rate Models

Several models of stochastic interest have been developed in the financial economics literature. These models have been developed from economic frameworks including equilibrium analysis and no arbitrage arguments. Thus, the emphasis differs from the purely statistical models introduced above. Here, we describe some of the similarities and differences of these competing models.

One disadvantage of the statistical models is that they allow only one-period investments, unlike the financial economic models, which consider the entire term structure of interest rates. It is well known (see, e.g., Ingersoll, 1987, chap. 13), in a world with deterministic interest, that precluding arbitrage opportunities means these two economic models are equivalent. However, this has also been shown to be true in certain stochastic interest environments, as follows.

Let P(t,s) denote the price at time t of a pure discount bond paying one at time s, t [is less than or equal to] s. If r(u) is the instantaneous risk-free interest rate at time u, t [is less than or equal to] u [is less than or equal to] s, then the local expectation hypothesis of Cox, Ingersoll, and Ross (1981) is written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

They show that only this proposition (of three considered) can be sustained in a continuous-time rational expectations equilibrium. In their appendix, the logarithmic utility model also supports this form of the expectation hypothesis in discrete time:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [E.sub.t] denotes the expectation taken at time t. Hence, the use of

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

as a price at time zero is justified.

Next, we try to find evidence to support the use of an ARIMA/linear process on the interest rates, as well as the conditional heteroskedasticity aspect for the error variance. A single-factor model for the instantaneous risk-free interest rate r usually is assumed to follow a diffusion process of the form

(6) dr = (a+br)dt+ [Sigma] [r.sup.[Beta]] d[B.sub.t],

where B is a standard Brownian motion, and a, b, [Sigma], and [Beta] are parameters with [Beta] [is greater than or equal to] 0. This specification includes as special cases almost all the models that have appeared in the literature: [Beta] = 0 corresponds to the model proposed by Vasicek (1977); [Beta] = 1/2 was considered by Cox, Ingersoll, and Ross (1985); and [Beta] = 1 was used by Courtadon (1982) and also by Dothan (1978) with an additional restriction that a = b = 0; and others.

Here, we concentrate our discussion on the Vasicek and Cox, Ingersoll, and Ross models. The continuous-time Vasicek process is

dr = [Kappa] ([Theta]-r)dt+ [Sigma] d[B.sub.t].

The change in the level of r between time t and t+1 may be written (see Vasicek, 1977, eqs. [25] and [26]) as

[r.sub.t+1] - [r.sub.t] = (1 - [e.sup.-[Kappa]])([Theta]-[r.sub.t]) + [u.sub.t+1]

or

(7) [r.sub.t+1] = [Theta] (1 - [e.sup.-[Kappa]]) + [e.sup.-[Kappa] [r.sub.t] + [u.sub.t+1],

where

Var([u.sub.t+1]) = [[Sigma].sup.2 / 2[Kappa] (1 - [e.sup.-2[Kappa]]),

the drift (constant term) is [Theta](1 - [e.sup.-[Kappa]]) and the distribution of [u.sub.t] is normal. Obviously, the discrete-time econometric specification shown in equation (7) of the Vasicek model is a traditional AR(1) model.

The other well-known term structure model is the Cox, Ingersoll, and Ross model:

dr = [Kappa] ([Theta]-r)dt+[Sigma] [square root of r] d[B.sub.t].

Note that, although the models are arbitrage-free, they are approximations due to the lack of replication. The discrete-time version may be represented (see Cox, Ingersoll, and Ross, 1985, eq. [19]) as

[r.sub.t+1]-[r.sub.t] = (1 - [e.sup.-[Kappa]])([Theta]-[r.sub.t]) + [u.sub.t+1]

or

[r.sub.t+1] = [Theta] (1 - [e.sup.-[Kappa]]) + [e.sup.-[Kappa]] [r.sub.t] + [u.sub.t+1],

where

[Var.sub.t] ([u.sub.t+1]) = [[Lambda].sub.1]+[[Lambda].sub.2] [r.sub.t],

[[Lambda].sub.1] = [Theta] [[Sigma].sup.2] / 2[Kappa] [(1 - [e.sup.-[Kappa]]).sup.2], [[Lambda].sub.2] = [[Sigma].sup.2] / [Kappa] ([e.sup.-[Kappa] - [e.sup.-2[Kappa]]),

and the drift (constant term) is [Theta](1 - [e.sup.-[Kappa]]). The conditional distribution of [u.sub.t+1], is noncentral [[Chi].sup.2] and converges to normal distribution only when the time interval between two consecutive observations tends to zero. Over short periods of time, the deviation from normality is unlikely to be significant, and we treat [u.sub.t+1], as normal (Chan et al., 1992, and Brown and Schaefer, 1994, also adopt this approach). Write the discrete-time version as the traditional AR(1) format:

[r.sub.t] = a+[Phi] [r.sub.t-1] + [u.sub.t],

where a = [Phi] (1-[e.sup.-[Kappa]]), [Phi] = [e.sup.-[Kappa]. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Substituting into the [Var.sub.t]([u.sub.t+1]) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In order to guarantee the positivity of [Var.sub.t] ([u.sub.t-1]), we use [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] or |[u.sub.t-i| on the right-hand side. The autoregressive conditional heteroskedasticity (ARCH) feature appears. Linear declining weights in the expression of [Var.sub.t] ([u.sub.t]) were used in Engle (1982) and Bollerslev (1986) for weighting the past (squared) residuals. We found exponentially declining weights. Because the weights decay exponentially, only the first few past residuals are effectively useful in the dynamic evolvement of the variance of error term. (Of course, the degree of decay depends upon the magnitude of [Phi].

The ARCH process has been applied to the term structure in finance literature. Longstaff and Schwartz (1992) present a two-factor general equilibrium model of the term structure.(1) The discrete-time approximation of the continuous-time specification is then formulated as a generalized autoregressive conditional heteroskedasticity (GARCH) framework introduced by Bollerslev (1986). Also, Bollerslev, Engle, and Wooldridge (1988) use a trivariate GARCH model to implement a capital asset pricing model with time-varying covariances, assuming the market consists of only bills, bonds, and stocks. In this article, we work in the univariate case; extension to a multivariate aspect will be investigated in future research.

Empirical studies suggest that model (6) does not match the observed data well (see Becker, 1991, and Chan et al., 1992). If the spot rate is assumed to follow a stochastic differential equation (6) in a continuous-time setting, then its instantaneous changes depend only on the current value and an independent random error process. The discrete-time econometric specification of the short-term riskless rate then usually results in the AR(1) formulation only. Extensions to higher-order ARIMA or linear processes can be used to model the short-term rates and may capture more information about the variation of the term structure.

To avoid the creation of arbitrage opportunities, the market is assumed to be efficient. However, it is well known that frictions in the real market place exist. That is, there are transaction costs, taxes, restrictions on short sales, asymmetries in information available to investors, etc. Perhaps due to these frictions, the spot rate curves derived by the theoretical equilibrium/arbitrage-free models do not conform well to observed data on bond yields and prices. In addition, these frictions exist for pricing assets in an efficient, highly active bond market. Further frictions arise in pricing liabilities due to the absence of an efficient secondary market for insurance reserves.

For the concept of changes in values over a period, since no particular model needs to be assumed for the term structure, we can use our idea of changes in reserves for equilibrium models as well as actuarial models. For solvency valuation purposes, we only need to value the net cash flows; that is, the excess of incomes over outgoing liabilities. In this case, the valuation method is less important than the case of valuing assets or liabilities. See Frees (1990) for solvency considerations of a life insurer.

Linear Interest Rate Process

Define a linear interest process so that the force of interest {[y.sub.t]} can be represented as

(8) [y.sub.t] = a+[[Phi].sub.1] [Y.sub.t-1] +...+ [[Phi].sub.p]

[y.sub.t-p] + [[Epsilon].sub.t] [-l.sub.1] [[Epsilon].sub.t-1] -...[-l.sub.q]

[[Epsilon].sub.t-q],

where (a, [[Phi].sub.1],,..., [[Phi].sub.p] [l.sub.t],,...,[l.sub.q]) are parameters, {[[Epsilon].sub.t]) are i.i.d., E([[Epsilon].sub.t]) = 0, ([y.sub.0],..., [Y.sub.1-p], and ([[Epsilon].sub.0] [[Epsilon].sub.-1],..., [[Epsilon].sub.1-q]) are known. The traditional ARMA/ARIMA

model belongs to this class of process. The linear interest rate process also can be written in the form

(9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Where

(10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(11) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(see, e.g., Box and Jenkins, 1976, or Dhaene, 1989). In other words, [y.sub.t] is affine (linear plus constant) in ([[Epsilon].sub.1]...,[[Epsilon].sub.t]).

If the force of interest follows a linear process, then the two discount factors in Proposition 1 can be expressed explicitly by using the next theorem.

Theorem 1. For a one-unit T-year default-free pure discount bond, if {[y.sub.t]} follows a linear interest process, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

And

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where {[[Beta].sub.i] and {[[Gamma].sub.s]} are defined in equations (10) and (11).

Therefore, supposing a linear process for the force of interest, we can combine Proposition 1 and Theorem 1 to write the reserves [V.sub.0] and [V.sub.1]. To illustrate, it is of interest to write the special case of independent interest.

Corollary 1 (Independent Interest Case). If [[Phi].sub.1] = ... = [[Phi].sub.p] = [l.sub.1] = ... = [l.sub.q] = 0 so that {[y.sub.t]} are i.i.d., then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

From equation (4), it is evident that, in general, [V.sub.1] is a function of [[Epsilon].sub.1], and is thus a random variable. However, in the i.i.d. case, [V.sub.1] becomes a constant, which means that it is independent of [[Epsilon].sub.1]. The intuition is that in this i.i.d. case, {[y.sub.t]} are purely random, and thus [[Epsilon].sub.1], does not provide any sequential information about ([y.sub.2], [y.sub.3],...) and hence the next-period reserve [V.sub.1]. Therefore, by employing the expectation approach for pricing and assuming that the force of interest is generated by a white noise series, the measure [V.sub.1] (also for [V.sub.h], h [is greater than or equal to] 1) is a deterministic value, which may not capture the real events. Examining an autocorrelated interest environment may model practical situations more appropriately than the independent interest benchmark case.

Default-Free Pure Discount Bond Example

The simplest special case for the cash flows is a pure discount bond. Suppose that an insurer must pay one unit T years later with certainty. Then, using Theorem 1 and a linear process for the interest rates as in equation (8), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Making the normality assumption for [[Epsilon]' s yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus, the discounted reserve [[e.sup.-y][.sup.1]]. [[V.sup.(T-1)][.sub.1]] has a lognormal distribution in this special case. If [[Sigma].sup.2]] increases, so does the mean and variance of [[e.sup.-y][.sub.1]. [[V.sup.(T-1)][.sub.1]].

For a single payment with fixed payment date as above, the exact distribution of [[e.sup.-y][.sup.1]] [V.sub.1] can be obtained. However, for a general insurance contract such as life insurance or deferred annuity, analytically the exact distribution is extremely difficult to find. In this case, a moment-matching method can be used to approximate the distribution. A simulation example for the distribution under a general insurance contract is provided below.

A Simulation Example for Linear Process

To illustrate the distributions of [V.sub.1] and [[e.sup.-y][.sup.1]] [V.sub.1], we consider a block of whole life business. For simplicity, policies are categorized into three groups of ten so the total size is 30. Assume that, for each category, ages at issue are x = 30, 30, 40, and durations are k = S, 10, 5, respectively. All death benefits are one dollar. The mortality decrements are those in the 1979-1981 U.S. Life Table that appear in Bowers et al. (1986). Here, assume that we have the following stationary AR(2) model for the interest rate:

[y.sub.t] = 0.08 + 0.6([y.sub.t-1] - 0.08) - 0.3([y.sub.t-2] - 0.08) + [[Epsilon].sub.t], t [is greater than or equal to] 1,

where [[Epsilon].sub.t] ~ i.i.d. N(O, [Sigma], = 0.025), [y.sub.0] = 0.06, and [y.sub.-1] = 0.07, as in Giaccotto (1986) and Dhaene (1989). To compute the level premiums for each of three categories, first use constant force of interest 0.08 (=E([y.sub.t])) to get the net level annual premiums, and then add 20 percent as the relative security loading to obtain the final level premiums.

Combining Proposition I and Theorem 1 yields analytic expressions for [V.sub.t] and its discounted version, [[e.sup.-y][.sup.1]] [V.sub.1]. To approximate these expressions, we performed 500 simulations. The resulting frequency histograms are shown in Figures 1a and 1b. Normally distributed curves are superimposed for comparison purposes. The sample mean and standard deviation of the simulated distribution of reserve [V.sub.1] are 0.7875 and 0.009748, respectively. For the discounted reserve [[e.sup.-y][.sup.1]] [V.sub.1], the corresponding measures are 0.7349 and 0.02799. Interestingly, the dispersion for the discounted reserve [[e.sup.-y][.sup.1]] [V.sub.1], is almost three times that for reserve [V.sub.1] (0.02799/0.009748 = 2.9) in this example. The difference originates from the additional random portion--the discount factor [[e.sup.-y]]sup.1].

Because the mean of the force of interest E([y.sub.t]) = 0.08, suppose that we had used [Delta] = 0.08 as constant force of interest to calculate the next-period reserve. Then, [V.sub.1] = 0.765, which underestimates the "true" reserve, as shown in Figure 1. For comparison, choosing another o = 0.079 as constant force of interest gives V, = 0.808, which overestimates the "true" distribution. This example indicates that ignoring the stochastic interest rate environment would easily miss the target distribution.

[Figure 1 ILLUSTRATION OMITTED]

Since [V.sub.0] and discounted reserve [[e.sup.-y][.sup.1] [V.sub.1], are both valued at time zero under the AR(2) interest rate environment, we can compare [V.sub.0] = 0.602 and the distribution of [[e.sup.-y][.sup.1]] [V.sub.1] to see the changes in reserves.(2) To illustrate, we may wish to know how much to add to the current reserve so that the next-period funds can meet future obligations measured by V, with 95 percent probability. That is, we want to find A such that P[([V.sub.0] + A) [[e.sup.-y][.sup.1]] [is greater than] [V.sub.1]] = 0.95. Hence A = (95th percentile of [[e.sup.-y][.sup.1]] [V.sub.1]) - [V.sub.0] ~ 0.782 - 0.602 = 0.18. The value 0.782 is the empirical distribution value. In fact, it can be approximated by using a normal variate.(3)

Nonlinear Interest Rate Process

Introduction to ARCH Process

Under the traditional linear time series setting, the conditional variance of one-step-ahead prediction is time invariant. Applied researchers have recognized the importance of explicitly modeling time-varying second- and higher-order moments. One of the most prominent tools that has emerged for describing such changing variances is the ARCH model of Engle (1982) and its various extensions. Bollerslev et al. (1992) contains an overview of some of the developments in the formulation of ARCH models and a survey of the numerous empirical applications using financial data. In particular, this survey includes a discussion of the modeling of interest rates.

The simplest nontrivial ARCH model is the first-order linear model given by

(12) [[Epsilon].sub.t]|[[Psi].sub.t-1] ~ N(O,[h.sub.t]),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [[Psi].sub.t-1] is the information set ([Sigma]-field) available at time t-1, and [[Delta]sub.0] [is greater than] 0, [[Delta][sub.1] [is greater than or equal to] 0, unknown parameters. The quantity [[Epsilon].sub.t] could be the observation or the disturbance term from the relevant model. The nonlinear ARCH process is serially uncorrelated with nonconstant variances conditional on the past, but constant unconditional variances. This model captures the tendency for volatility clustering, that is, for large changes to be followed by other large changes, and small changes to be followed by other small changes, but of unpredictable sign. The nonlinearity stems from the variance of the disturbance term et. If 6, were equal to zero, then the model would be the conventional Gaussian white noise. However, for [[Delta].sub.1] [Not equal to] 0, the effect of equation (12) is to make the variance of the disturbance terms at time t dependent on the realized value of the disturbance term in the previous period.

We have just seen a simple example of an ARCH process for the innovations. To a certain extent, reserves [V.sub.0] and [V.sub.1], as obtained from equations (3) and (4), involve the moment-generating function of [[Epsilon]'.sub.t] s. Because of the nonexistence of higher moments under the ARCH model of equation (12) and hence its moment-generating function (see Engle, 1982, and Bollerslev, 1986), we consider a model similar to expression (12) but in absolute value form:

(13) [[Epsilon].sub.t]l[[Psi].sub.t-1] ~ N(0,[h.sub.t]),

[h.sub.t] = [[Delta].sub.0] + [[Delta].sub.1]|[[Epsilon].sub.t-1]|,

where [[Delta].sub.0] [is greater than] 0, [[Delta].sub.1] [is greater than or equal to] 0. This model was mentioned in Engle's (1982) seminal article but is seldom used in the ARCH literature, perhaps due to the mathematical tractability difficulty in absolute values. Engle and Bollerslev (1986), Schwert (1990), and Higgins and Bera (1992) discuss the absolute version of the ARCH model. The absolute value form of the model in expression (13) shares the features described above for model (12); and its moment-generating function exists, a desirable characteristic. We note that o, in model (13) depends on the measurement unit.

The discussion below uses the following model for the force of interest:

(14a) [y.sub.t] = a + [[Phi].sub.1] [y.sub.1] + ... + [[Phi].sub.p]

[y.sub.t-p] + [[Epsilon].sub.t] [-l.sub.1] [[Epsilon].sub.t-1]

-...- [l.sub.q] [[Epsilon].sub.t-q],

(14b) [[Epsilon].sub.t]|[Psi].sub.t-1] ~ N(O, [h.sub.t]),

(14c) [h.sub.t] = [[Delta].sub.0] + [[Delta].sub.1]|[[Epsilon].sub.t-1]|,

t [is greater than or equal to] 1].

That is, we allow for conditional (prediction) variance to depend on the absolute value of immediately previous innovation.

Approximate Results for Reserves Under ARCH Processes

We examine the effect of [[Delta].sub.1] on the reserves [V.sub.0] and [V.sub.1] under the model described by expressions (14a), (14b), and (14c), supposing [[Delta].sub.1], to be small. Before stating the result for reserves, a lemma for the finiteness of the moment-generating function of innovations is given first. According to Engle (1982), "The absolute value form...can be shown to have finite variance for any parameter values." In fact, we can prove a stronger, new result that the moment-generating function is finite, hence so are all moments of [[Epsilon]'[.sub.t] s.

Lemma 1. If [[Epsilon]'[.sub.t] s follows model (14b) and (14c), then the moment-generating function of ([[Epsilon].sub.t],,...,[[Epsilon].sub.T]) is finite for all [[Delta].sub.0] [is greater than or equal to] 0, [[Delta].sub.1] [is greater than or equal to] 0.

With the above lemma, we are ready to state the basic main result for a pure discount bond under the ARCH process.

Theorem 2. For a one-unit T-year default-free pure discount bond, under the model described by expressions (14a), (14b), and (14c) for the force of interest, we have that, as [[Delta].sub.1] [right arrow] O,

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[Phi] (x) = distribution function of a standard normal variable, and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We can see that the terms outside the square brackets of [[Chi].sup.(T)][sub.0] and [[Chi].sup.(T)][sub.1] correspond to the results of the linear process case ([[Delta].sub.1] = 0) under normality. It can be shown that both coefficients of the linear- and quadratic-order terms (with respect to [[Delta].sub.1])--[C.sub.1] and [C.sub.2]--are nonnegative. Thus for a pure discount bond, when [[Delta].sub.1] is small, the reserves [V.sub.0] and [V.sub.1] are both larger in the absolute-value form ARCH process case than in the linear process case. Moreover, we expect [V.sub.0] and [V.sub.1] to increase if [[Delta].sub.1] increases.

Once we have Theorem 2 for a pure discount bond, we can have corresponding results for [V.sub.0] and [V.sub.1] in the general block of business case. That is, by retrieving equations (3) and (4) in Proposition 1 and substituting [[Chi].sup.([Tau] +1])][sub.0] and [[Chi].sup.([Tau] + 2][sub.1] for those two discount factors, respectively, the desired results are obtained.

The above analysis studies the case where [[Sigma].sub.1] is assumed to be small. For larger values of [[Sigma].sub.1], we resort to simulation techniques, as described below.

A Simulation Example for an ARCH Process

Unlike the ARMA model, the ARCH model takes into account the conditional heteroscedasticity. The model that we use for an ARCH process resembles that for a linear process. The insurance model is the same as the linear process case. However, the model for interest rate variations becomes:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with [[Delta].sub.1] = 0, 0.001, 0.003, . . ., 0.08, where [y.sub.0] = 0.06, [y.sub.-1] = 0.07, and [[Epsilon].sub.0] = -0.01. We use this ARCH model as a nonlinear time series example to simulate the next period reserve. The largest value of [[Delta].sub.1] chosen to investigate is 0.08 because, in Appendix B, we show that sd([[Epsilon].sub.t]) [is less than or equal to] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. With [[Delta].sub.0] = [(0.025).sup.2] and [[Delta].sub.1] = 0.08, we have sd([[Epsilon].sub.t]) [is less than] 0.072, which is about 2.9 times the standard deviation of [[Epsilon].sub.t] in the linear process case (0.072/0.025 = 2.88).

The valuation algorithm. Part of our simulation ideas comes from Tilley (1993), who presents a simulation algorithm for valuing American-style options. For nonlinear time series models, like ARCH processes, it is difficult to compute analytically [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] or [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and hence the distribution of the reserve, since the error term [Epsilon]'s are no longer independent. However, simulations can provide some insight into the distribution of [V.sub.1] and [[e.sup.-y][.sub.1]] [V.sub.1]. For simplicity, consider first a one-unit T-year pure discount bond. The reserve at time one is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Because it involves an expectation that could not be expressed simply, we need to approximate it by simulation. Figure 2 illustrates how the simulation works.

[Figure 2 ILLUSTRATION OMITTED]

Given a value of [[Epsilon].sub.1] (its 98th percentile, for example), we simulate ([[Epsilon].sub.2], . . ., [[Epsilon].sub.T]) on each path from time 1 to T. Suppose we have B (= 1,200, say) such repetitions. Then the expectation [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] can be approximated by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [y.sup.(k)][sub.s], is the simulated sth period force of interest on path k (k = 1,..., B) for a certain percentile of [[Epsilon].sub.1].

Now extend a pure discount bond to the case of a whole block of business with n insureds. Let [Omega] be the limiting age. From equation (4), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where M = [Omega] - [min.sub.1 [is less than or equal to] i [is less than or equal to] n] ([x.sub.i] + [k.sub.i]) and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], since [f.sub.[Tau] = 0 for [Tau] [is greater than] M - 2. In this case, we use the same ideas as above, replacing T = [Tau] + 2, 0 [is less than or equal to] [Tau] [is less than or equal to] M - 2, then we obtain a value of [V.sub.1] that corresponds to a given percentile of [[Epsilon].sub.1].

Sensitivity of the reserve to the variance autoregression parameter. Our numerical example uses different values of [[Delta].sub.1] in the simulation. Thus, we can study the sensitivity of [V.sub.1] and [e.sup.-y][.sub.1] [V.sub.1] to a change in the value of the variance autoregression parameter [[Delta].sub.1]. For example, we might like to see how large this parameter can be before it matters. Therefore, we include the case [[Delta].sub.1] = 0 for comparison purposes. For each percentile of [[Epsilon].sub.1], we implement the simulation method described above to obtain the values of [V.sub.1] ([[micro].sub.p], [[Delta].sub.1]), where [[micro].sub.p] denotes the 100pth percentile of the distribution of [[Epsilon].sub.1]. By examining many different percentiles of [[Epsilon].sub.1] especially the tails, we can present the whole picture of sensitivity of the variance autoregression parameter on the next-period reserve.

The simulated values of [V.sub.1], and associated simulation standard errors, are summarized in Table 1. Plotting those values of [V.sub.1] together in Figure 3a presents a clearer picture. We see that, for each fixed value of [[Epsilon].sub.1], as [[Delta].sub.1] increases, the reserve [V.sub.1] also increases. This aspect is consistent with the discussion following Theorem 2 for a pure discount bond. Fixing [[Delta].sub.1], a larger value of [[Epsilon].sub.1] reduces the reserve [V.sub.1], as shown in Table 1 and Figure 3a. However, this is not necessarily always true. That is, recall from Theorem 1 that there need not be a monotonic relationship between the value of [[Epsilon].sub.1] and reserve [V.sub.1]. From Figure 3a, it seems that [V.sub.1] increases quadratically for [[Delta].sub.1] [element of] [0, 0.08] so the approximation up to a second-order term appears adequate. When [[Delta].sub.1] [is less than] 0.02, the linear-order term already gives a good approximation to the reserve [V.sub.1]. The plot also indicates that the dispersion of [V.sub.1] increases as [[Delta].sub.1] increases.

[Figure 3a ILLUSTRATION OMITTED]

Table 1 Simulated Values of Reserve [V.sub.1] for Different Combinations of Percentiles of [[Epsilon].sub.1] and Values of [[Delta].sub.1]

Note: The values in parentheses beneath the values of [V.sub.1] correspond to simulation standard errors. The number of simulations is 1,200.

A corresponding graph for the discounted reserve [e.sup.-y][.sub.1] [V.sub.1] is shown in Figure 3b in which similar features appear. In this example, the discounted reserve is less influenced by [[Delta].sub.1] than the reserve, as indicated by the growing rate of curves in the figure. As noted in the linear process case, the dispersion of discounted reserve [e.sup.-y][.sub.1] is larger than that of reserve [V.sub.1], for various values of [[Delta].sub.1]. In addition, the discounted reserve seems to grow linearly for [[Delta].sub.1] up to about 0.03.

[Figure 3b ILLUSTRATION OMITTED]

Next, we compare the values of reserve [V.sub.1] obtained from two different methods: by simulation and by the approximation result in Theorem 2. Table 2 shows the values of [V.sub.1] obtained from approximation approach. The corresponding plot for Table 2 would look like Figure 3a. To compare these two methods let us consider the following ratio:

Standardized Error = Approximation Value - Simulated Value/

Simulation Standard Error.

The standardized error values (reported in Table 3) are all less than 0.34 in absolute value. Theorem 2 therefore provides good approximations to the reserve [V.sub.1] under the absolute-value version ARCH model for the force of interest. All of the ratio values are negative, from which we deduce that the approximation method would slightly underestimate the "true" reserve [V.sub.1].

Table 3 Values of Standardized Error Obtained from Simulated Values of Table 1 and Approximate Values of Table 2

Empirical Example

To illustrate the importance of conditionally heteroskedastic models, we use data from the Citibank Interest Rate data base. Treasury interest rates are fundamental and usually used as a reference to other rates (for example, corporate rates). We consider the three-month Treasury bill yield as the short-term spot rate. This series consists of N = 124 monthly Treasury yields from the first month of 1982 through the fourth month of 1992. The monthly values are averages of daily figures that were originally obtained from the Board of Governors of the Federal Reserve System.

Figure 4 plots the time series and its first differences for three-month U.S. Treasure bills. The stationarity in mean is obtained after first differencing, but the variance appears to be changing over time. The summary statistics of the original and first differenced series are shown in Table 4.

[Figure 4 ILLUSTRATION OMITTED]

Table 4 Summary Statistics for a Three-Month U.S. Treasury Bill Rate Time Series and Its First Differences

For a simple illustration, we consider only an AR(1) model for the differenced series, which corresponds to the Vasicek and Cox, Ingersoll, and Ross models described earlier. It was shown above that the moment-generating function exists for the absolute-value form ARCH process described in equation (14c). Although the moment-generating function of the regular ARCH model described in equation (12) does not exist (except [[Delta].sub.1] = 0), it is widely used in the literature. Hence, we also consider it in the data analysis for comparison. In sum, we fit an ARIMA(1,1,0) model for the three-month Treasury bill series ([y.sub.t]), with the non-, absolute-, and regular-ARCH processes for the conditional variance of error term. Maximum likelihood estimation was used to fit each model. Estimation procedures for non-ARCH and regular ARCH models are well known. Estimation procedures for the absolute ARCH model and the consideration of the other competitive models for this yield rate series can be found in Lai (1995). The fitted models for ARIMA(1,1,0) are presented below. Asymptotic standard errors are in parentheses. Symbols *, [dagger], and [double dagger] indicate statistical significance at 10 percent, 5 percent, and 1 percent, respectively. B denotes the backward shift operator.

ARIMA(1,1,0) model for U.S. three-month Treasury-bill yield rates. Non-ARCH

(1 - 0.45 [double dagger] B) (1-B) [y.sub.t], =-0.0386 + [[Epsilon].sub.t],

(0.084) (0.0335) [h.sub.t] =0.117 [double dagger], [AIC.sub.3] = 0.7894.

(0.0155)

Absolute-ARCH

(1 - 0.365 [double dagger] B) (1-B) [y.sub.t] =-0.0281* + [[Epsilon].sub.t],

(0.078) (0.017) [h.sub.t] =0.0346 [double dagger] + 0.265 [double dagger] | [Epsilon.sub.t-1] |,

(0.010) (0.058) [AIC.sub.4] = 0.4043.

Regular-ARCH

(1 - 0.393 [double dagger] B) (1-B) [y.sub.t] = -0.0166 + [[Epsilon].sub.t],

(0.089) (0.023) [h.sub.t] = 0.0484 [double dagger] + 0.467 [double dagger] +

(0.009) (0.145) [[Epsilon].sup.2] [sub.t-1], [AIC.sub.4] = 0.3150.

Here, the Akaike information model selection criterion (AIC) is given by

[AIC.sub.t] = [-2 log(maximized likelihood) + 2r]/N*,

where r denotes the number of parameters estimated by maximum likelihood, and N* = N - (p+1) is the "effective" sample size. The significance of coefficient [[Delta].sub.1] indicates the usefulness of incorporating the ARCH process in the model.

Conclusion

This article examines the discrete-time short-term consequences on reserves due to changes in the interest rate environment. Generally, when viewed at initial time, the next period reserve is a random variable, which is a function of the disturbance generated in the first period. However, in the special case of a white noise process for the force of interest, the next period reserve is a deterministic value. Linear and nonlinear ARCH process models for the force of interest are considered. Under linear interest rate processes, explicit expressions are given for the changes in reserves. In particular, for a pure discount bond, the next period reserve and its discounted value have a lognormal distribution.

As an extension from linear processes to nonlinear processes, approximation formulas and simulation algorithms are presented. Compared with simulation results, we find that the approximation formulas perform well in the sense that the discrepancies between the approximations and simulations are small relative to the simulation standard error. Although we used only one type of insurance contract for our comparisons, the results are generalizable as long as cash flows are independent of interest rates. This is because, under this assumption, those two factors can be separated, as shown in Proposition 1. Theorems 1 and 2 refer to linear rate and nonlinear rate processes, respectively. A numerical example using U.S. Treasury securities indicates strong evidence of the ARCH process.

We consider only the initial and next period reserves, whose values will evolve with time. To handle this one-period-ahead problem sequentially, stochastic control theory may be used to access not only the dynamic nature of the time development, but also the mobility of an insurer's cash flows (claims, premiums, and expenses). Martin-Lof (1983) and Vandebroek and Dhaene (1990) apply the control theory in an insurance context, where they assume fixed interest rates.

In this article, the cash flows are assumed to be independent of interest rate variations. For many insurance contracts, this assumption appears to be unnecessarily restrictive. We hope to investigate this issue in the future.

(1) The Two factors are the short-time interest rate and the volatility of the short-term interest rate.

(2) proposition 1 and Theorem 1 gives [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where M(u) = exp([[Sigma].sup.2] [u.sup.2]/2), the moment-generating function of [[Epsilon].sub.t].

(3) The sample values of skewness and kurtosis for [V.sub.1] and [[e.sup.-y][[.sup.1]] [V.sub.1] in this simulation example are 0.0643 and 3.19, and 0.155 and 3.21, respectively. The kurtosis defined here is equal to three for a normal variate. Thus, a normal distribution could be used to approximate the reserve [V.sub.1] even though slightly positive skewness and slightly thicker tails feature are exhibited from the sample quantities. The positivity of skewness of [[e.sup.-y][.sup.1]] [V.sub.1] is larger than that of [V.sub.1], but a normal approximation to the distribution of discounted

Appendix A

Expected Value of Discounted Reserve

For a single policy, we have the following result, which can serve as a stepping stone to compute the expected value of discounted reserve for an entire block of business.

Proposition 2

Suppose that the basic assumption that [y.sub.t] is a Borel measurable function of ([[Epsilon].sub.1], [[Epsilon].sub.2],..., [[Epsilon].sub.t]) holds. For single policy, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In particular, for h = 1,

(15) [V.sub.0] + [a.sub.k] = [p.sub.x+k] E([e.sup.[-y.sub.h]] [multiplied by] [V.sub.] + [q.sub.x+k] [multiplied by] [b.sub.k+1] E([e.sup.[-y.sub.1]]

(compare Bowers et al., 1986, 7.8.2). The resources required at the beginning of policy year h equal the actuarial present value of year-end requirements (in expected value sense). Furthermore, we can employ equation (15) to calculate the expected discounted reserve E([e.sup.[-y.sub.1]] [multiplied by] [V.sub.h]) for each policy and then add them to obtain the entire block of business expected discounted reserve.

A recursive expression for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the expected value of discounted period h reserve, is obtained from Proposition 2:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Note that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];its value can therefore be computed by using Theorems 1 and 2 under different models for {[y.sub.t]}.

Proof

For a single policy, let the loss at time h be

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [F.sub.[Tau]] [sup.(h)] is defined similarly to equation (2). By definition,

(16) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

So,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which implies that

(17) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

From equation (16), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

by noting equation (17). The proof is completed.

Appendix B

An Upper Bound for the Variance of Innovation Under the ARCH Model

First, E([[Epsilon].sub.t]) = EE([[Epsilon].sub.e]|[[Psi].sub.t-1]) = 0, by equation (14b). Thus, Var([[Epsilon].sub.t]) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], by equation (14c). Since for X ~ N(0, [[Sigma].sup.2]), E|X| = 2/[square root of [Pi]] [Sigma], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Assuming that the series starts indefinitely far in the past with a finite first absolute moment, the limit of the upper bound on the right-hand side as k goes to infinity tends to L, where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence L = 2/[square root of 2 [Pi]] [square root of [[Delta].sub.0] + [[Delta].sub.1] L], which implies that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Thus,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Appendix C

Proof

Proof of Proposition 1

By definition,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then we have the desired results for [V.sub.0] and [V.sub.1] by utilizing the independence between cash flows and interest rates, the definition of cash flow [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], in equation (2) and the probability function of [J.sub.i].

Proof of Theorem 1

The [y.sub.t] expression from equation (9) implies that

(18) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

(19) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

by using the i.i.d. property of [Epsilon]'s and reindexing.

Proof of Lemma 1

Let ?? = ([t.sub.1],..., [t.sub.T])'. Then the moment-generating function of ?? = ([[Epsilon].sub.1],..., [[Epsilon].sub.T])' is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(by the Holder inequality).

Since for all b [is greater than] 0 and X ~ N(0, [[Sigma].sup.2]), E([e.sup.b|X|) = E[[e.sup.bX]I(X [is less than] 0)] + E[[e.sup.bX]I(X [is greater than or equal to] 0)] (where I is an indicator function) [is less than or equal to] E([e.sup.bX]) + E([e.sup.bX]) = 2 [e.sup.[b.sup.2] [[Sigma].sup.2/2]]

Thus,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

by a recursive argument, where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

by a recursive argument, where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof of Theorem 2 requires the following lemma.

Lemma 2

If the ARCH model of equations (14b) and (14c) governs the innovation process {[[Epsilon].sub.t]}, then for a fixed positive integer T and as [[Delta].sub.0] [arrow right] 0, we have that

(1) the joint density of [[Epsilon].sub.??] = ([[Epsilon].sub.1],...,[[Epsilon].sub.T] is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and

(2) the joint density of ([[Epsilon].sub.2],..., [[Epsilon].sub.T]) given [[Epsilon].sub.1] is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof of Lemma 2

Taylor's series expansion is used to derive the results. From the conditional densities of equations (14b) and (14c), we have

(20) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

When [[Delta].sub.1] = 0, then f([[Epsilon].sub.t],...,[[Epsilon].sub.T]) = [Phi] ([[Epsilon][.sub,??] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

From equation (20),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Differentiating with respect to [[Delta].sub.1] gives

(21) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Differentiating again on equation (21) gives

(22) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus,

(23) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

so,

(24) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

And

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

From Taylor's formula, as [[Delta].sub.1] approaches zero,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

so result (1) is proven.

For f([[Epsilon.sub.2],..., [[Epsilon].sub.T|[[Epsilon].sub.1]) in (2), the derivation is essentially the same as f([[Epsilon].sub.1],,..., [[Epsilon].sub.T]), replacing [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] respec completed.

Proof of Theorem 2

The proof of Theorem 2 is organized as follows. First, we write Taylor's expansion for f([Epsilon]) with an explicit form for the remainder. Second, we derive the linear- and quadratic-order expressions (with respect to [[Delta].sub.1]) for [[V.sup.(T0)[.sub.0]. Third, the boundedness of the remainder term is established.

For the first step, we differentiate equation (22) to give

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

So, by substituting df/d[[Delta].sib.1] and [d.sub.f]/d [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23) and (24), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By Taylor's formula with Lagrange form of the remainder,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For the second step, by using equation (18),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[Phi] is the distribution function of a standard normal variate, Hence,

(25) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By viewing [e.sub.1] ~ i.i.d. N(O,1), the first integral of equation (25) is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] by noting [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and after reindexing. The second intergral (linera-order term with respect to [[Delta].sub.1] is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For i = 0,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For i = 1,..., T,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

since for X ~ N(O,1), E(|X|[e.sup.aX]) = [[e.sup.a][.sup.2/2] (a), where q(a) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] + a[2[Phi](a) -1].

Define [[Pi].sub.0] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then

E(exp([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

A similar technique is used to find that

E[exp([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

by using, for X ~ N(0,1), E([X.sup.2] [e.sup.aX] = [[e.sup.a][sup.2/2]] (1+[a.sup.a]). Hence, the second integral becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

by noting [[Pi].sub.s] = [[-Zeta].sub.T-s], q(a) = q(-a) and reindexing. The linear-order term for [[V.sup.T][sub.0]] is proven. For the quadratic-order term (the third integral of equation [25]), the derivations are more tedious but the ideas are the same as for the linear-order term. That proof is omitted. The third integral can be proven to be

equal to exp [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For the last (remainder) term in equation (25), there exist constants [a.sub.1], [a.sub.2] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore,

(26) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We claim that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Because

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and noting equation (26), it suffices to prove that for i = 1,.., T and all r [is greater than] 0,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It is obviously true since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (by the Holder inequality)

[is less than] [infinity] (by Lemma 1).

Result (1) follows.

The proof for [[V.sup.(T-1)][.sub.1] follows the same steps as for [[V.sup.(T)][sub.0], by noting that, from equation (19),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and using Lemma 2(2).

References

Becker, D. N., 1991, Statistical Tests of the Lognormal Distribution as a Basis for Interest Rate Changes, Transactions of the Society of Actuaries, 43: 7-57, Discusion 59-72.

Beekman, J. A. and C. P. Fuelling, 1990, Interest and Mortality Randomness in Some Annuities, Insurance: Mathematics and Economics, 9: 185-196.

Beekman, J. A. and C. P. Fuelling, 1991, Extra Randomness in Certain Annuity Models, Insurance: Mathematics and Economics, 10: 275-287.

Bellhouse, D. R. and H. H. Panjer, 1981, Stochastic Modelling of Interest Rates with Applications to Life Contingencies-Part II, Journal of Risk and Insurance, 48: 628-637.

Bollerslev, T., 1986, Generalized Autoregressive Conditional Heteroskedasticity, Journal of Econometrics, 31: 307-327.

Bollerslev, T., R. Y. Chou, and K. F. Kroner, 1992, ARCH Modeling in Finance: A Review of the Theory and Empirical Evidence, Journal of Econometrics, 52: 5-59.

Bollerslev, T., R. F. Engle, and J. M. Wooldridge, 1988, A Capital Asset Pricing Model with Time Varying Covariances, Journal of Political Economy, 96: 116-131.

Bowers, N. L., H. U. Gerber, J. C. Hickman, D. A. Jones, and C. J. Nesbitt, 1986, Actuarial Mathematics (Schaumburg, III.: Society of Actuaries).

Box, G. and G. Jenkins, 1976, Time Series Analysis, Forecasting, and Control, Second Edition (San Francisco: Holden-Day).

Boyle, P. P., 1976, Rates of Return as Random Variables, Journal of Risk and Insurance, 43: 693-713.

Brown, R. H. and S. M. Schaefer, 1994, The Term Structure of Real Interest Rates and the Cox, Ingersoll, and Ross Model, Journal of Financial Economics, 35: 3-42.

Buhlmann, H., 1992, Stochastic Discounting, Insurance: Mathematics and Economics, 11: 113-127.

Chan, K. C., G. A. Karolyi, F. A. Longstaff, and A. B. Sanders, 1992, An Empirical Comparison of Alternative Models of the Short-Term Interest Rate, Journal of Finance, 47: 617-630.

Courtadon, G., 1982, The Pricing of Options on Default-Free Bonds, Journal of Financial and Quantitative Analysis, 17: 75-100.

Cox, J. C., J. E. Ingersoll, and S. A. Ross, 1981, A Reexamination of Traditional Hypotheses About the Term Structure of Interest Rates, Journal of Finance, 36: 769-799.

Cox, J. C., J. E. Ingersoll, and S. A. Ross, 1985, A Theory of the Term Structure of Interest Rates, Econometrica, 53: 385-407.

Dhaene, J., 1989, Stochastic Interest Rates and Autoregressive Integrated Moving Average Processes, ASTIN Bulletin, 19: 131-138.

Dothan, L. U., 1978, On the Term Structure of Interest Rates, Journal of Financial Economics, 6: 59-69.

Engle, R. F., 1982, Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of U.K. Inflation, Econometrica, 50: 987-1008.

Engle, R. F. and T. Bollerslev, 1986, Modelling the Persistence of Conditional Variances, Econometric Reviews, 5: 1-50.

Frees, E. W., 1990, Stochastic Life Contingencies with Solvency Considerations, Transactions of the Society of Actuaries, 42: 91-129.

Giacotto, C., 1986, Stochastic Modelling of Interest Rates: Actuarial vs Equilibrium Approach, Journal of Risk and Insurance, 53: 435-453.

Higgins, M. L. and A. K. Bera, 1992, A Class of Nonlinear ARCH Models, International Economic Review, 33: 137-158.

Ingersoll, J. E., 1987, Theory of Financial Decision Making (Rowman & Littlefield).

Jordan, C. W., 1967, Life Contingencies (Chicago: Society of Actuaries).

Lai, S. W., 1995, Stochastic Models of Interest Rates in Actuarial Science, Ph.D. Thesis, University of Wisconsin-Madison.

Longstaff, F. A. and E. S. Schwartz, 1992, Interest Rate Volatility and the Term Structure: A Two-Factor General Equilibrium Model, Journal of Finance, 47: 1259-1282.

Martin-Lof, A., 1983, Premium Control in an Insurance System, An Approach Using Linear Control Theory, Scandinavian Actuarial Journal, 1-27.

Panjer, H. H. and D. R. Bellhouse, 1980, Stochastic Modelling of Interest Rates with Applications to Life Contingencies, Journal of Risk and Insurance, 47: 91-110.

Ramlau-Hansen, H., 1988, The Emergence of Profit in Life Insurance, Insurance: Mathematics and Economics, 7: 225-236.

Schwert, W., 1990, Stock Volatility and the Crash of '87, Review of Financial Studies, 3: 77-102.

Tilley, J. A., 1993, Valuing American Options in a Path Simulation Model, Transactions of the Society of Actuaries, 45.

Vandebroek, M. and J. Dhaene, 1990, Optimal Premium Control in a Non-Life Insurance Business, Scandinavian Actuarial Journal, 3-13.

Vasicek, O., 1977, An Equilibrium Characterization of the Term Structure, Journal of Financial Economics, 5: 177-188.

Waters, H. R., 1978, The Moments and Distributions of Actuarial Functions, Journal of the Institute of Actuaries, 105: 61-75.

Siu-Wai Lai is a Ph.D. student of statistics at the University of Wisconsin-Madison. Edward W. Frees is Professor of Business and Statistics at the University of Wisconsin-Madison. The first author gratefully acknowledges a Ph.D. grant support from the Society of Actuaries. The research of the second author was supported by a research grant from the Society of Actuaries, Actuarial Education and Research Fund, and a professorship funded by Time Insurance Company. The authors wish to thank the Editor, two anonymous reviewers, and James Hickman for their valuable comments.

On balance sheets, assets must equal liabilities plus net worth of the firm. An important component of these liabilities for insurers is the reserve, the portion of a firms' assets set aside to meet future uncertain obligations arising from insurance contracts. Although the obligations of each contract are contingent upon uncertain future events, and thus may be modeled stochastically, the reserve set aside is a single number. There are limitations when using a single number to summarize a stochastic quantity. However, reserves play a prominent role in financial statements and thus these quantities are important to managers of insurance organizations.

Several important problems in actuarial science rely heavily on the determination of a reserve. To illustrate, if a company or a block of business is to be traded on the open market, a value must be determined for the associated set of obligations. Thus, it is useful to think of a reserve as the "value" associated with a set of stochastic obligations. As another application, reserves traditionally have been used as a measure of an organization's financial strength. In this context, the reserve should be larger than the "value" of obligations, because a conservative approach should be taken for assessing potential future obligations.

Life insurance and annuity reserves are calculated by summarizing discounted cash flows, where the discounting is done with respect to investment earnings, as well as decrements due to mortality, disability, policy lapse, and so on, that may be applicable to a particular policy. For brevity, this articles considers only investment earnings and the mortality decrement. Extensions to the multidecrement case are straightforward.

In the traditional insurance literature, as in Jordan (1967), the deterministic assumption dominates the development of the theory of life contingencies. Namely, mortality occurs according to a known mortality table and the interest rate is assumed to have a deterministic value. One step further is to allow the age at death to be a random variable, although the interest rate is assumed to be deterministic--a "semi-stochastic" approach followed in Bowers et al. (1986). Insurance literature has generalized the traditional theory of life contingencies by introducing stochastic variation in interest rates (Boyle, 1976; Waters, 1978; Panjer and Bellhouse, 1980; Bellhouse and Panjer, 1981; Giaccotto, 1986; Dhaene, 1989; Frees, 1990; and Beekman and Fuelling, 1990, 1991). Additional stochastic interest models from financial economics are discussed below.

This article computes reserves as (conditional) expectations of sums of future cash flows. Motivation for this approach can be found in, for example, Bowers et al. (1986) for the semi-stochastic approach and Buhlmann (1992) for models using stochastic interest.

Here, we are primarily concerned with quantifying changes in reserves from one financial period to the next. Changes in reserves could be used to quantify the amount of profit released, as in Ramlau-Hansen (1988), which studied gains and losses emerging from margins built into mortality and other decrements and ignored those arising from investments. To complement that work, we focus on changes arising from stochastic interest rates and do not explicitly consider margins built into other decrement rates.

Changes in value of future obligations due to dynamic models of interest have been considered extensively in the financial economics literature, in particular as part of immunization theory, which deals with instantaneous changes in value. Here, we examine changes in value from one financial period to the next.

The new idea of examining changes in reserves can be illustrated by considering the following simple scenario. Let {[y.sub.s]} represent the random force of interest in the sth period. As argued in Frees (1990), [y.sub.s] can be interpreted as a one-period spot rate. Consider an obligation that can be expressed as a T-year pure discount bond. At time zero, the random present value of one unit payable at time T is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Without loss of generality, it is assumed that the time interval is one year. Suppose an insurer must pay one unit T years later with certainty, but under a stochastic interest rate environment. The reserve at time zero is denoted by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where the expectation is taken at time zero. Because the interest rates prior to time zero are all known at time zero, [V.sup.(T)] [sub.0] is a deterministic value.

After one year, the maturity time of the payment shortens by one and the reserve (at time one) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where the expectation is taken at time one. In general, because future spot rates may depend on the first-period spot rate, the quantity [V.sup.(T-1)] [sub.1] is a function of the random first-period spot rate [y.sub.1].

Although the obligations are random, their expected values define the (nonrandom) reserve. Following standard terminology, the reserve is the portion of liabilities that can be recognized in the insurer's balance sheet. Assets are held to meet the random obligations, where, at time zero, the known value of assets equals the reserve. In the absence of perfect matching, the random obligations and stochastic appreciator of the assets diverge over time. This article is about quantifying one aspect of the asset's adequacy in meeting the obligations.

This article examines the change from the initial reserve [V.sub.0] to the time one reserve [V.sub.1]. To examine this change, we discount the time one reserve [V.sup.(T-1)] [sub.1] back to time zero and then study its distribution. That is, we investigate the distribution and the statistical properties of the random reserve

[e.sup.[-y.sub.1]] [V.sup.(T-1)] [sub.1],

from a time zero point of view. Although [V.sub.0] represents the current expected value of the obligation, [V.sub.1] represents the value at the subsequent time period. Thus, for budgeting and other purposes, [V.sub.1] and its discounted value [e.sup.[-y.sub.1] [V.sub.1], are important quantities for risk and other financial managers.

At time zero, the discounted next-period reserve is stochastic because it is a function of the stochastic first-period interest rate [y.sub.1]. However, it is a deterministic value at time one since y, becomes known at that time. In the case of a T-year pure discount bond and random force of interest {[y.sub.s]},

(1) [V.sup.(T)] [sub.0] = E([e.sup.[-y.sub.1]] [V.sup.(T-1)] [sub.1]),

where the expectation is taken at time zero. However, as shown in the Appendix, the relationship in equation (1) must be modified for a life insurance policy.

The next section of this article describes the model used in the analysis and discusses the continuous-time equilibrium term structure model and the discrete-time ARIMA/linear process. Then, the linear process for interest rates is investigated. The following section considers a nonlinear process for interest rates, the autoregressive conditionally heteroskedastic (ARCH) process, which is widely used in economics. An empirical example on the interest rates is presented before the conclusion. The proofs of all results are given in the Appendix.

The Basic Model

Insurance Model

We consider here the individual risk model for insurance contracts, using the notation of Bowers et al. (1986). Denote the valuation time to be h so that, at the initial valuation, h = 0. Assume that there are n policies in the block of business. For the ith policy, the age at issue is [x.sub.i], the duration is [k.sub.i] when h = 0, the curtate random time of decrement is [K.sub.1], and the curtate-future-lifetime is [J.sub.i] (i.e., [J.sub.i] = [K.sub.i]-[k.sub.i]-h). Suppose that a death benefit [[b.sub.i] [K.sub.i +1]] payable at the end of the year of loss and that the annual premiums [[a.sub.i] [sub.m]] are payable at the beginning of each year up to and including the year of loss. Then, at time point h + [Tau] + 1, the random cash flow of the ith policy is

(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and the probability function of [J.sub.i] is: Prob([J.sub.i] = [Tau]) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [Tau] = 0, 1,...; and Prob([J.sub.i] [is greater than] [Tau]) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Here, [sub.[Tau]]|[q.sub.x] and [sub.[Tau]] [p.sub.x] are the traditional deferred decrement probabilities and survival functions calculated from a mortality table.

Because the definition of cash flow is quite general, it can be used for general insurance as well as combinations of whole life insurance, term life insurance, deferred life insurance, annuities, and pure discount bonds.

Interest Rate Model

Let [y.sub.s] denote the force of interest in the sth year (s = 1, 2,...). It is natural to assume that [y.sub.s] has a parametric form,

[y.sub.s] = f(??,[[Epsilon].sub.1],...,[[Epsilon].sub.2]),

where ?? is a vector of parameters, {[[Epsilon].sub.s]} are independent and identically distributed (i.i.d.), and f is a known function. To illustrate, the next section considers the recursive linear process

[y.sub.t] = a + [[Phi].sub.1] [y.sub.t-1] + ... + [[Phi].sub.p] [y.sub.t-p] + [[Epsilon].sub.t] - [[Iota].sub.1] [[Epsilon].sub.t-1] - ... -[[Iota].sub.q] [[Epsilon].sub.t-q].

The traditional ARMA/ARIMA model belongs to this class of model. In the fourth section, we consider an ARCH model, a nonlinear representation for the force of interest. Before proceeding, the following basic assumption and notations are made.

Assumption

[y.sub.t] is a Borel measurable function of ([[Epsilon].sub.1], [[Epsilon].sub.2],..., [[Epsilon].sub.t]).

Notations

Let M(t) = E([e.sup.t[Epsilon]), the moment-generating function of

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

and,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

E(??X) means taking expectation conditional on the information generated by X.

The Reserves

To introduce reserves, we first define the loss at valuation time h for the ith policy,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The sum of losses for the whole block of business at time h is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The reserve at time h is defined as [V.sub.h] = [E.sub.h] ([S.sup.(h)] [sub.L]), where [E.sub.h] ([multiplied by]) = E(??[[Epsilon].sub.1,...,[[Epsilon].sub.h], so that [V.sub.h] is a function of ([[Epsilon].sub.1],..., [[Epsilon].sub.h]). The discussion below concentrates on [V.sub.0] and [V.sub.1], the reserves at the beginning of valuation and its following period, respectively. The results can be extended to consider further periods.

The following discussion assumes the random cash flows [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are independent of the stochastic interest rates ([y.sub.s]). The following result for the reserves is a basic one that we will use frequently.

Proposition 1. Under the basic assumption above, we have

(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

(4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where

(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

provided the expectations exist.

In fact, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the expected cash flow of the ith policy at time h = 0, and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the expected cash flow of the ith policy at time h = 1. Although [V.sub.0] is a constant, [V.sub.1] is usually stochastic since it is a function of [[Epsilon].sub.1], the disturbance of force of interest generated in period one. When considering the discounted reserve, we simply multiply [V.sub.1] by [e.sup.-[y.sub.i]]. Actuarial science traditionally presents recursive calculations of reserves. The Appendix provides the recursive calculations relating [V.sub.1] to [V.sub.0], which generalize the T-year pure discount bond case described in the introduction.

Other Interest Rate Models

Several models of stochastic interest have been developed in the financial economics literature. These models have been developed from economic frameworks including equilibrium analysis and no arbitrage arguments. Thus, the emphasis differs from the purely statistical models introduced above. Here, we describe some of the similarities and differences of these competing models.

One disadvantage of the statistical models is that they allow only one-period investments, unlike the financial economic models, which consider the entire term structure of interest rates. It is well known (see, e.g., Ingersoll, 1987, chap. 13), in a world with deterministic interest, that precluding arbitrage opportunities means these two economic models are equivalent. However, this has also been shown to be true in certain stochastic interest environments, as follows.

Let P(t,s) denote the price at time t of a pure discount bond paying one at time s, t [is less than or equal to] s. If r(u) is the instantaneous risk-free interest rate at time u, t [is less than or equal to] u [is less than or equal to] s, then the local expectation hypothesis of Cox, Ingersoll, and Ross (1981) is written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

They show that only this proposition (of three considered) can be sustained in a continuous-time rational expectations equilibrium. In their appendix, the logarithmic utility model also supports this form of the expectation hypothesis in discrete time:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [E.sub.t] denotes the expectation taken at time t. Hence, the use of

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

as a price at time zero is justified.

Next, we try to find evidence to support the use of an ARIMA/linear process on the interest rates, as well as the conditional heteroskedasticity aspect for the error variance. A single-factor model for the instantaneous risk-free interest rate r usually is assumed to follow a diffusion process of the form

(6) dr = (a+br)dt+ [Sigma] [r.sup.[Beta]] d[B.sub.t],

where B is a standard Brownian motion, and a, b, [Sigma], and [Beta] are parameters with [Beta] [is greater than or equal to] 0. This specification includes as special cases almost all the models that have appeared in the literature: [Beta] = 0 corresponds to the model proposed by Vasicek (1977); [Beta] = 1/2 was considered by Cox, Ingersoll, and Ross (1985); and [Beta] = 1 was used by Courtadon (1982) and also by Dothan (1978) with an additional restriction that a = b = 0; and others.

Here, we concentrate our discussion on the Vasicek and Cox, Ingersoll, and Ross models. The continuous-time Vasicek process is

dr = [Kappa] ([Theta]-r)dt+ [Sigma] d[B.sub.t].

The change in the level of r between time t and t+1 may be written (see Vasicek, 1977, eqs. [25] and [26]) as

[r.sub.t+1] - [r.sub.t] = (1 - [e.sup.-[Kappa]])([Theta]-[r.sub.t]) + [u.sub.t+1]

or

(7) [r.sub.t+1] = [Theta] (1 - [e.sup.-[Kappa]]) + [e.sup.-[Kappa] [r.sub.t] + [u.sub.t+1],

where

Var([u.sub.t+1]) = [[Sigma].sup.2 / 2[Kappa] (1 - [e.sup.-2[Kappa]]),

the drift (constant term) is [Theta](1 - [e.sup.-[Kappa]]) and the distribution of [u.sub.t] is normal. Obviously, the discrete-time econometric specification shown in equation (7) of the Vasicek model is a traditional AR(1) model.

The other well-known term structure model is the Cox, Ingersoll, and Ross model:

dr = [Kappa] ([Theta]-r)dt+[Sigma] [square root of r] d[B.sub.t].

Note that, although the models are arbitrage-free, they are approximations due to the lack of replication. The discrete-time version may be represented (see Cox, Ingersoll, and Ross, 1985, eq. [19]) as

[r.sub.t+1]-[r.sub.t] = (1 - [e.sup.-[Kappa]])([Theta]-[r.sub.t]) + [u.sub.t+1]

or

[r.sub.t+1] = [Theta] (1 - [e.sup.-[Kappa]]) + [e.sup.-[Kappa]] [r.sub.t] + [u.sub.t+1],

where

[Var.sub.t] ([u.sub.t+1]) = [[Lambda].sub.1]+[[Lambda].sub.2] [r.sub.t],

[[Lambda].sub.1] = [Theta] [[Sigma].sup.2] / 2[Kappa] [(1 - [e.sup.-[Kappa]]).sup.2], [[Lambda].sub.2] = [[Sigma].sup.2] / [Kappa] ([e.sup.-[Kappa] - [e.sup.-2[Kappa]]),

and the drift (constant term) is [Theta](1 - [e.sup.-[Kappa]]). The conditional distribution of [u.sub.t+1], is noncentral [[Chi].sup.2] and converges to normal distribution only when the time interval between two consecutive observations tends to zero. Over short periods of time, the deviation from normality is unlikely to be significant, and we treat [u.sub.t+1], as normal (Chan et al., 1992, and Brown and Schaefer, 1994, also adopt this approach). Write the discrete-time version as the traditional AR(1) format:

[r.sub.t] = a+[Phi] [r.sub.t-1] + [u.sub.t],

where a = [Phi] (1-[e.sup.-[Kappa]]), [Phi] = [e.sup.-[Kappa]. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Substituting into the [Var.sub.t]([u.sub.t+1]) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In order to guarantee the positivity of [Var.sub.t] ([u.sub.t-1]), we use [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] or |[u.sub.t-i| on the right-hand side. The autoregressive conditional heteroskedasticity (ARCH) feature appears. Linear declining weights in the expression of [Var.sub.t] ([u.sub.t]) were used in Engle (1982) and Bollerslev (1986) for weighting the past (squared) residuals. We found exponentially declining weights. Because the weights decay exponentially, only the first few past residuals are effectively useful in the dynamic evolvement of the variance of error term. (Of course, the degree of decay depends upon the magnitude of [Phi].

The ARCH process has been applied to the term structure in finance literature. Longstaff and Schwartz (1992) present a two-factor general equilibrium model of the term structure.(1) The discrete-time approximation of the continuous-time specification is then formulated as a generalized autoregressive conditional heteroskedasticity (GARCH) framework introduced by Bollerslev (1986). Also, Bollerslev, Engle, and Wooldridge (1988) use a trivariate GARCH model to implement a capital asset pricing model with time-varying covariances, assuming the market consists of only bills, bonds, and stocks. In this article, we work in the univariate case; extension to a multivariate aspect will be investigated in future research.

Empirical studies suggest that model (6) does not match the observed data well (see Becker, 1991, and Chan et al., 1992). If the spot rate is assumed to follow a stochastic differential equation (6) in a continuous-time setting, then its instantaneous changes depend only on the current value and an independent random error process. The discrete-time econometric specification of the short-term riskless rate then usually results in the AR(1) formulation only. Extensions to higher-order ARIMA or linear processes can be used to model the short-term rates and may capture more information about the variation of the term structure.

To avoid the creation of arbitrage opportunities, the market is assumed to be efficient. However, it is well known that frictions in the real market place exist. That is, there are transaction costs, taxes, restrictions on short sales, asymmetries in information available to investors, etc. Perhaps due to these frictions, the spot rate curves derived by the theoretical equilibrium/arbitrage-free models do not conform well to observed data on bond yields and prices. In addition, these frictions exist for pricing assets in an efficient, highly active bond market. Further frictions arise in pricing liabilities due to the absence of an efficient secondary market for insurance reserves.

For the concept of changes in values over a period, since no particular model needs to be assumed for the term structure, we can use our idea of changes in reserves for equilibrium models as well as actuarial models. For solvency valuation purposes, we only need to value the net cash flows; that is, the excess of incomes over outgoing liabilities. In this case, the valuation method is less important than the case of valuing assets or liabilities. See Frees (1990) for solvency considerations of a life insurer.

Linear Interest Rate Process

Define a linear interest process so that the force of interest {[y.sub.t]} can be represented as

(8) [y.sub.t] = a+[[Phi].sub.1] [Y.sub.t-1] +...+ [[Phi].sub.p]

[y.sub.t-p] + [[Epsilon].sub.t] [-l.sub.1] [[Epsilon].sub.t-1] -...[-l.sub.q]

[[Epsilon].sub.t-q],

where (a, [[Phi].sub.1],,..., [[Phi].sub.p] [l.sub.t],,...,[l.sub.q]) are parameters, {[[Epsilon].sub.t]) are i.i.d., E([[Epsilon].sub.t]) = 0, ([y.sub.0],..., [Y.sub.1-p], and ([[Epsilon].sub.0] [[Epsilon].sub.-1],..., [[Epsilon].sub.1-q]) are known. The traditional ARMA/ARIMA

model belongs to this class of process. The linear interest rate process also can be written in the form

(9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Where

(10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(11) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(see, e.g., Box and Jenkins, 1976, or Dhaene, 1989). In other words, [y.sub.t] is affine (linear plus constant) in ([[Epsilon].sub.1]...,[[Epsilon].sub.t]).

If the force of interest follows a linear process, then the two discount factors in Proposition 1 can be expressed explicitly by using the next theorem.

Theorem 1. For a one-unit T-year default-free pure discount bond, if {[y.sub.t]} follows a linear interest process, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

And

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where {[[Beta].sub.i] and {[[Gamma].sub.s]} are defined in equations (10) and (11).

Therefore, supposing a linear process for the force of interest, we can combine Proposition 1 and Theorem 1 to write the reserves [V.sub.0] and [V.sub.1]. To illustrate, it is of interest to write the special case of independent interest.

Corollary 1 (Independent Interest Case). If [[Phi].sub.1] = ... = [[Phi].sub.p] = [l.sub.1] = ... = [l.sub.q] = 0 so that {[y.sub.t]} are i.i.d., then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

From equation (4), it is evident that, in general, [V.sub.1] is a function of [[Epsilon].sub.1], and is thus a random variable. However, in the i.i.d. case, [V.sub.1] becomes a constant, which means that it is independent of [[Epsilon].sub.1]. The intuition is that in this i.i.d. case, {[y.sub.t]} are purely random, and thus [[Epsilon].sub.1], does not provide any sequential information about ([y.sub.2], [y.sub.3],...) and hence the next-period reserve [V.sub.1]. Therefore, by employing the expectation approach for pricing and assuming that the force of interest is generated by a white noise series, the measure [V.sub.1] (also for [V.sub.h], h [is greater than or equal to] 1) is a deterministic value, which may not capture the real events. Examining an autocorrelated interest environment may model practical situations more appropriately than the independent interest benchmark case.

Default-Free Pure Discount Bond Example

The simplest special case for the cash flows is a pure discount bond. Suppose that an insurer must pay one unit T years later with certainty. Then, using Theorem 1 and a linear process for the interest rates as in equation (8), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Making the normality assumption for [[Epsilon]' s yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus, the discounted reserve [[e.sup.-y][.sup.1]]. [[V.sup.(T-1)][.sub.1]] has a lognormal distribution in this special case. If [[Sigma].sup.2]] increases, so does the mean and variance of [[e.sup.-y][.sub.1]. [[V.sup.(T-1)][.sub.1]].

For a single payment with fixed payment date as above, the exact distribution of [[e.sup.-y][.sup.1]] [V.sub.1] can be obtained. However, for a general insurance contract such as life insurance or deferred annuity, analytically the exact distribution is extremely difficult to find. In this case, a moment-matching method can be used to approximate the distribution. A simulation example for the distribution under a general insurance contract is provided below.

A Simulation Example for Linear Process

To illustrate the distributions of [V.sub.1] and [[e.sup.-y][.sup.1]] [V.sub.1], we consider a block of whole life business. For simplicity, policies are categorized into three groups of ten so the total size is 30. Assume that, for each category, ages at issue are x = 30, 30, 40, and durations are k = S, 10, 5, respectively. All death benefits are one dollar. The mortality decrements are those in the 1979-1981 U.S. Life Table that appear in Bowers et al. (1986). Here, assume that we have the following stationary AR(2) model for the interest rate:

[y.sub.t] = 0.08 + 0.6([y.sub.t-1] - 0.08) - 0.3([y.sub.t-2] - 0.08) + [[Epsilon].sub.t], t [is greater than or equal to] 1,

where [[Epsilon].sub.t] ~ i.i.d. N(O, [Sigma], = 0.025), [y.sub.0] = 0.06, and [y.sub.-1] = 0.07, as in Giaccotto (1986) and Dhaene (1989). To compute the level premiums for each of three categories, first use constant force of interest 0.08 (=E([y.sub.t])) to get the net level annual premiums, and then add 20 percent as the relative security loading to obtain the final level premiums.

Combining Proposition I and Theorem 1 yields analytic expressions for [V.sub.t] and its discounted version, [[e.sup.-y][.sup.1]] [V.sub.1]. To approximate these expressions, we performed 500 simulations. The resulting frequency histograms are shown in Figures 1a and 1b. Normally distributed curves are superimposed for comparison purposes. The sample mean and standard deviation of the simulated distribution of reserve [V.sub.1] are 0.7875 and 0.009748, respectively. For the discounted reserve [[e.sup.-y][.sup.1]] [V.sub.1], the corresponding measures are 0.7349 and 0.02799. Interestingly, the dispersion for the discounted reserve [[e.sup.-y][.sup.1]] [V.sub.1], is almost three times that for reserve [V.sub.1] (0.02799/0.009748 = 2.9) in this example. The difference originates from the additional random portion--the discount factor [[e.sup.-y]]sup.1].

Because the mean of the force of interest E([y.sub.t]) = 0.08, suppose that we had used [Delta] = 0.08 as constant force of interest to calculate the next-period reserve. Then, [V.sub.1] = 0.765, which underestimates the "true" reserve, as shown in Figure 1. For comparison, choosing another o = 0.079 as constant force of interest gives V, = 0.808, which overestimates the "true" distribution. This example indicates that ignoring the stochastic interest rate environment would easily miss the target distribution.

[Figure 1 ILLUSTRATION OMITTED]

Since [V.sub.0] and discounted reserve [[e.sup.-y][.sup.1] [V.sub.1], are both valued at time zero under the AR(2) interest rate environment, we can compare [V.sub.0] = 0.602 and the distribution of [[e.sup.-y][.sup.1]] [V.sub.1] to see the changes in reserves.(2) To illustrate, we may wish to know how much to add to the current reserve so that the next-period funds can meet future obligations measured by V, with 95 percent probability. That is, we want to find A such that P[([V.sub.0] + A) [[e.sup.-y][.sup.1]] [is greater than] [V.sub.1]] = 0.95. Hence A = (95th percentile of [[e.sup.-y][.sup.1]] [V.sub.1]) - [V.sub.0] ~ 0.782 - 0.602 = 0.18. The value 0.782 is the empirical distribution value. In fact, it can be approximated by using a normal variate.(3)

Nonlinear Interest Rate Process

Introduction to ARCH Process

Under the traditional linear time series setting, the conditional variance of one-step-ahead prediction is time invariant. Applied researchers have recognized the importance of explicitly modeling time-varying second- and higher-order moments. One of the most prominent tools that has emerged for describing such changing variances is the ARCH model of Engle (1982) and its various extensions. Bollerslev et al. (1992) contains an overview of some of the developments in the formulation of ARCH models and a survey of the numerous empirical applications using financial data. In particular, this survey includes a discussion of the modeling of interest rates.

The simplest nontrivial ARCH model is the first-order linear model given by

(12) [[Epsilon].sub.t]|[[Psi].sub.t-1] ~ N(O,[h.sub.t]),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [[Psi].sub.t-1] is the information set ([Sigma]-field) available at time t-1, and [[Delta]sub.0] [is greater than] 0, [[Delta][sub.1] [is greater than or equal to] 0, unknown parameters. The quantity [[Epsilon].sub.t] could be the observation or the disturbance term from the relevant model. The nonlinear ARCH process is serially uncorrelated with nonconstant variances conditional on the past, but constant unconditional variances. This model captures the tendency for volatility clustering, that is, for large changes to be followed by other large changes, and small changes to be followed by other small changes, but of unpredictable sign. The nonlinearity stems from the variance of the disturbance term et. If 6, were equal to zero, then the model would be the conventional Gaussian white noise. However, for [[Delta].sub.1] [Not equal to] 0, the effect of equation (12) is to make the variance of the disturbance terms at time t dependent on the realized value of the disturbance term in the previous period.

We have just seen a simple example of an ARCH process for the innovations. To a certain extent, reserves [V.sub.0] and [V.sub.1], as obtained from equations (3) and (4), involve the moment-generating function of [[Epsilon]'.sub.t] s. Because of the nonexistence of higher moments under the ARCH model of equation (12) and hence its moment-generating function (see Engle, 1982, and Bollerslev, 1986), we consider a model similar to expression (12) but in absolute value form:

(13) [[Epsilon].sub.t]l[[Psi].sub.t-1] ~ N(0,[h.sub.t]),

[h.sub.t] = [[Delta].sub.0] + [[Delta].sub.1]|[[Epsilon].sub.t-1]|,

where [[Delta].sub.0] [is greater than] 0, [[Delta].sub.1] [is greater than or equal to] 0. This model was mentioned in Engle's (1982) seminal article but is seldom used in the ARCH literature, perhaps due to the mathematical tractability difficulty in absolute values. Engle and Bollerslev (1986), Schwert (1990), and Higgins and Bera (1992) discuss the absolute version of the ARCH model. The absolute value form of the model in expression (13) shares the features described above for model (12); and its moment-generating function exists, a desirable characteristic. We note that o, in model (13) depends on the measurement unit.

The discussion below uses the following model for the force of interest:

(14a) [y.sub.t] = a + [[Phi].sub.1] [y.sub.1] + ... + [[Phi].sub.p]

[y.sub.t-p] + [[Epsilon].sub.t] [-l.sub.1] [[Epsilon].sub.t-1]

-...- [l.sub.q] [[Epsilon].sub.t-q],

(14b) [[Epsilon].sub.t]|[Psi].sub.t-1] ~ N(O, [h.sub.t]),

(14c) [h.sub.t] = [[Delta].sub.0] + [[Delta].sub.1]|[[Epsilon].sub.t-1]|,

t [is greater than or equal to] 1].

That is, we allow for conditional (prediction) variance to depend on the absolute value of immediately previous innovation.

Approximate Results for Reserves Under ARCH Processes

We examine the effect of [[Delta].sub.1] on the reserves [V.sub.0] and [V.sub.1] under the model described by expressions (14a), (14b), and (14c), supposing [[Delta].sub.1], to be small. Before stating the result for reserves, a lemma for the finiteness of the moment-generating function of innovations is given first. According to Engle (1982), "The absolute value form...can be shown to have finite variance for any parameter values." In fact, we can prove a stronger, new result that the moment-generating function is finite, hence so are all moments of [[Epsilon]'[.sub.t] s.

Lemma 1. If [[Epsilon]'[.sub.t] s follows model (14b) and (14c), then the moment-generating function of ([[Epsilon].sub.t],,...,[[Epsilon].sub.T]) is finite for all [[Delta].sub.0] [is greater than or equal to] 0, [[Delta].sub.1] [is greater than or equal to] 0.

With the above lemma, we are ready to state the basic main result for a pure discount bond under the ARCH process.

Theorem 2. For a one-unit T-year default-free pure discount bond, under the model described by expressions (14a), (14b), and (14c) for the force of interest, we have that, as [[Delta].sub.1] [right arrow] O,

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[Phi] (x) = distribution function of a standard normal variable, and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We can see that the terms outside the square brackets of [[Chi].sup.(T)][sub.0] and [[Chi].sup.(T)][sub.1] correspond to the results of the linear process case ([[Delta].sub.1] = 0) under normality. It can be shown that both coefficients of the linear- and quadratic-order terms (with respect to [[Delta].sub.1])--[C.sub.1] and [C.sub.2]--are nonnegative. Thus for a pure discount bond, when [[Delta].sub.1] is small, the reserves [V.sub.0] and [V.sub.1] are both larger in the absolute-value form ARCH process case than in the linear process case. Moreover, we expect [V.sub.0] and [V.sub.1] to increase if [[Delta].sub.1] increases.

Once we have Theorem 2 for a pure discount bond, we can have corresponding results for [V.sub.0] and [V.sub.1] in the general block of business case. That is, by retrieving equations (3) and (4) in Proposition 1 and substituting [[Chi].sup.([Tau] +1])][sub.0] and [[Chi].sup.([Tau] + 2][sub.1] for those two discount factors, respectively, the desired results are obtained.

The above analysis studies the case where [[Sigma].sub.1] is assumed to be small. For larger values of [[Sigma].sub.1], we resort to simulation techniques, as described below.

A Simulation Example for an ARCH Process

Unlike the ARMA model, the ARCH model takes into account the conditional heteroscedasticity. The model that we use for an ARCH process resembles that for a linear process. The insurance model is the same as the linear process case. However, the model for interest rate variations becomes:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with [[Delta].sub.1] = 0, 0.001, 0.003, . . ., 0.08, where [y.sub.0] = 0.06, [y.sub.-1] = 0.07, and [[Epsilon].sub.0] = -0.01. We use this ARCH model as a nonlinear time series example to simulate the next period reserve. The largest value of [[Delta].sub.1] chosen to investigate is 0.08 because, in Appendix B, we show that sd([[Epsilon].sub.t]) [is less than or equal to] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. With [[Delta].sub.0] = [(0.025).sup.2] and [[Delta].sub.1] = 0.08, we have sd([[Epsilon].sub.t]) [is less than] 0.072, which is about 2.9 times the standard deviation of [[Epsilon].sub.t] in the linear process case (0.072/0.025 = 2.88).

The valuation algorithm. Part of our simulation ideas comes from Tilley (1993), who presents a simulation algorithm for valuing American-style options. For nonlinear time series models, like ARCH processes, it is difficult to compute analytically [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] or [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and hence the distribution of the reserve, since the error term [Epsilon]'s are no longer independent. However, simulations can provide some insight into the distribution of [V.sub.1] and [[e.sup.-y][.sub.1]] [V.sub.1]. For simplicity, consider first a one-unit T-year pure discount bond. The reserve at time one is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Because it involves an expectation that could not be expressed simply, we need to approximate it by simulation. Figure 2 illustrates how the simulation works.

[Figure 2 ILLUSTRATION OMITTED]

Given a value of [[Epsilon].sub.1] (its 98th percentile, for example), we simulate ([[Epsilon].sub.2], . . ., [[Epsilon].sub.T]) on each path from time 1 to T. Suppose we have B (= 1,200, say) such repetitions. Then the expectation [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] can be approximated by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [y.sup.(k)][sub.s], is the simulated sth period force of interest on path k (k = 1,..., B) for a certain percentile of [[Epsilon].sub.1].

Now extend a pure discount bond to the case of a whole block of business with n insureds. Let [Omega] be the limiting age. From equation (4), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where M = [Omega] - [min.sub.1 [is less than or equal to] i [is less than or equal to] n] ([x.sub.i] + [k.sub.i]) and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], since [f.sub.[Tau] = 0 for [Tau] [is greater than] M - 2. In this case, we use the same ideas as above, replacing T = [Tau] + 2, 0 [is less than or equal to] [Tau] [is less than or equal to] M - 2, then we obtain a value of [V.sub.1] that corresponds to a given percentile of [[Epsilon].sub.1].

Sensitivity of the reserve to the variance autoregression parameter. Our numerical example uses different values of [[Delta].sub.1] in the simulation. Thus, we can study the sensitivity of [V.sub.1] and [e.sup.-y][.sub.1] [V.sub.1] to a change in the value of the variance autoregression parameter [[Delta].sub.1]. For example, we might like to see how large this parameter can be before it matters. Therefore, we include the case [[Delta].sub.1] = 0 for comparison purposes. For each percentile of [[Epsilon].sub.1], we implement the simulation method described above to obtain the values of [V.sub.1] ([[micro].sub.p], [[Delta].sub.1]), where [[micro].sub.p] denotes the 100pth percentile of the distribution of [[Epsilon].sub.1]. By examining many different percentiles of [[Epsilon].sub.1] especially the tails, we can present the whole picture of sensitivity of the variance autoregression parameter on the next-period reserve.

The simulated values of [V.sub.1], and associated simulation standard errors, are summarized in Table 1. Plotting those values of [V.sub.1] together in Figure 3a presents a clearer picture. We see that, for each fixed value of [[Epsilon].sub.1], as [[Delta].sub.1] increases, the reserve [V.sub.1] also increases. This aspect is consistent with the discussion following Theorem 2 for a pure discount bond. Fixing [[Delta].sub.1], a larger value of [[Epsilon].sub.1] reduces the reserve [V.sub.1], as shown in Table 1 and Figure 3a. However, this is not necessarily always true. That is, recall from Theorem 1 that there need not be a monotonic relationship between the value of [[Epsilon].sub.1] and reserve [V.sub.1]. From Figure 3a, it seems that [V.sub.1] increases quadratically for [[Delta].sub.1] [element of] [0, 0.08] so the approximation up to a second-order term appears adequate. When [[Delta].sub.1] [is less than] 0.02, the linear-order term already gives a good approximation to the reserve [V.sub.1]. The plot also indicates that the dispersion of [V.sub.1] increases as [[Delta].sub.1] increases.

[Figure 3a ILLUSTRATION OMITTED]

Table 1 Simulated Values of Reserve [V.sub.1] for Different Combinations of Percentiles of [[Epsilon].sub.1] and Values of [[Delta].sub.1]

Percentile of [[Epsilon].sub.1] [[Delta].sub.1] 1% 2% 5% 10% 0 0.8107 0.8080 0.8041 0.8006 (0.0075) (0.0075) (0.0075) (0.0074) 0.001 0.8120 0.8093 0.8053 0.8017 (0.0077) (0.0076) (0.0076) (0.0076) 0.003 0.8147 0.8119 0.8078 0.8042 (0.0080) (0.0079) (0.0079) (0.0078) 0.005 0.8176 0.8147 0.8105 0.8068 (0.0082) (0.0082) (0.0082) (0.0081) 0.010 0.8253 0.8223 0.8178 0.8138 (0.0090) (0.0090) (0.0089) (0.0089) 0.020 0.8436 0.8402 0.8351 0.8306 (0.0108) (0.0108) (0.0107) (0.0106) 0.030 0.8660 0.8622 0.8565 0.8514 (0.0129) (0.0128) (0.0127) (0.0126) 0.040 0.8930 0.8887 0.8824 0.8767 (0.0152) (0.0151) (0.0149) (0.0148) 0.060 0.9628 0.9574 0.9495 0.9423 (0.0208) (0.0206) (0.0204) (0.0201) 0.080 1.0576 1.0508 1.0408 1.0318 (0.0281) (0.0279) (0.0275) (0.0271) Percentile of [[Epsilon].sub.1] [[Delta].sub.1] 20% 50% 80% 90% 0 0.7963 0.7883 0.7804 0.7762 (0.0074) (0.0073) (0.0073) (0.0072) 0.001 0.7974 0.7893 0.7813 0.7772 (0.0075) (0.0075) (0.0074) (0.0074) 0.003 0.7998 0.7914 0.7834 0.7782 (0.0078) (0.0077) (0.0077) (0.0076) 0.005 0.8022 0.7936 0.7855 0.7813 (0.0081) (0.0080) (0.0080) (0.0079) 0.010 0.8090 0.7998 0.7915 0.7872 (0.0088) (0.0087) (0.0087) (0.0086) 0.020 0.8252 0.8148 0.8063 0.8018 (0.0105) (0.0103) (0.0103) (0.0103) 0.030 0.8453 0.8336 0.8248 0.8202 (0.0124) (0.0122) (0.0122) (0.0122) 0.040 0.8698 0.8565 0.8477 0.8430 (0.0146) (0.0143) (0.0143) (0.0143) 0.060 0.9336 0.9165 0.9079 0.9031 (0.0199) (0.0193) (0.0194) (0.0194) 0.080 1.0207 0.9983 0.9907 0.9859 (0.0267) (0.0257) (0.0260) (0.0260) Percentile of [[Epsilon].sub.1] [[Delta].sub.1] 95% 98% 99% 0 0.7728 0.7690 0.7665 (0.0072) (0.0072) (0.0072) 0.001 0.7737 0.7700 0.7674 (0.0073) (0.0073) (0.0073) 0.003 0.7757 0.7719 0.7693 (0.0076) (0.0076) (0.0076) 0.005 0.7778 0.7740 0.7713 (0.0079) (0.0079) (0.0078) 0.010 0.7837 0.7797 0.7771 (0.0086) (0.0086) (0.0086) 0.020 0.7981 0.7940 0.7912 (0.0103) (0.0102) (0.0102) 0.030 0.8164 0.8122 0.8093 (0.0121) (0.0121) (0.0121) 0.040 0.8391 0.8347 0.8317 (0.0143) (0.0142) (0.0142) 0.060 0.8990 0.8945 0.8914 (0.0194) (0.0194) (0.0194) 0.080 0.9818 0.9771 0.9739 (0.0261) (0.0261) (0.0261)

Note: The values in parentheses beneath the values of [V.sub.1] correspond to simulation standard errors. The number of simulations is 1,200.

A corresponding graph for the discounted reserve [e.sup.-y][.sub.1] [V.sub.1] is shown in Figure 3b in which similar features appear. In this example, the discounted reserve is less influenced by [[Delta].sub.1] than the reserve, as indicated by the growing rate of curves in the figure. As noted in the linear process case, the dispersion of discounted reserve [e.sup.-y][.sub.1] is larger than that of reserve [V.sub.1], for various values of [[Delta].sub.1]. In addition, the discounted reserve seems to grow linearly for [[Delta].sub.1] up to about 0.03.

[Figure 3b ILLUSTRATION OMITTED]

Next, we compare the values of reserve [V.sub.1] obtained from two different methods: by simulation and by the approximation result in Theorem 2. Table 2 shows the values of [V.sub.1] obtained from approximation approach. The corresponding plot for Table 2 would look like Figure 3a. To compare these two methods let us consider the following ratio:

Standardized Error = Approximation Value - Simulated Value/

Simulation Standard Error.

The standardized error values (reported in Table 3) are all less than 0.34 in absolute value. Theorem 2 therefore provides good approximations to the reserve [V.sub.1] under the absolute-value version ARCH model for the force of interest. All of the ratio values are negative, from which we deduce that the approximation method would slightly underestimate the "true" reserve [V.sub.1].

Table 2 Values of [V.sub.1] Obtained from the Approximation Result of Theorem 2 Percentile of [[Epsilon].sub.1] [[Delta].sub.1] 1% 2% 5% 10% 0 0.8088 0.8061 0.8022 0.7987 0.001 0.8099 0.8072 0.8033 0.7998 0.003 0.8124 0.8096 0.8055 0.8019 0.005 0.8150 0.8121 0.8080 0.8043 0.010 0.8223 0.8193 0.8149 0.8110 0.020 0.8406 0.8372 0.8323 0.8278 0.030 0.8637 0.8599 0.8543 0.8493 0.040 0.8918 0.8874 0.8810 0.8754 0.060 0.9628 0.9572 0.9489 0.9416 0.080 1.0541 1.0468 1.0363 1.0269 Percentile of [[Epsilon].sub.1] [[Delta].sub.1] 20% 50% 80% 90% 0 0.7945 0.7865 0.7786 0.7744 0.001 0.7955 0.7874 0.7794 0.7752 0.003 0.7975 0.7893 0.7812 0.7770 0.005 0.7998 0.7913 0.7832 0.7789 0.010 0.8062 0.7972 0.7889 0.7845 0.020 0.8225 0.8123 0.8036 0.7990 0.030 0.8432 0.8318 0.8228 0.8181 0.040 0.8685 0.8555 0.8465 0.8417 0.060 0.9328 0.9162 0.9071 0.9024 0.080 1.0155 0.9942 0.9856 0.9811 Percentile of [[Epsilon].sub.1] [[Delta].sub.1] 95% 98% 99% 0 0.7710 0.7672 0.7647 0.001 0.7718 0.7680 0.7654 0.003 0.7735 0.7697 0.7670 0.005 0.7754 0.7715 0.7689 0.010 0.7809 0.7769 0.7742 0.020 0.7953 0.7912 0.7883 0.030 0.8143 0.8100 0.8071 0.040 0.8378 0.8335 0.8305 0.060 0.8985 0.8942 0.8913 0.080 0.9775 0.9734 0.9706

Table 3 Values of Standardized Error Obtained from Simulated Values of Table 1 and Approximate Values of Table 2

Percentile of [[Epsilon].sub.1] [[Delta].sub.1] 1% 2% 5% 10% 0 -0.247 -0.248 -0.248 -0.247 0.001 -0.267 -0.266 -0.264 -0.264 0.003 -0.297 -0.294 -0.290 -0.288 0.005 -0.316 -0.315 -0.309 -0.305 0.010 -0.334 -0.330 -0.323 -0.317 0.020 -0.279 -0.276 -0.271 -0.265 0.030 -0.178 -0.178 -0.175 -0.173 0.040 -0.083 -0.085 -0.089 -0.089 0.060 -0.002 -0.013 -0.026 -0.034 0.080 -0.127 -0.143 -0.165 -0.181 Percentile of [[Epsilon].sub.1] [[Delta].sub.1] 20% 50% 80% 90% 0 -0.247 -0.248 -0.248 -0.247 0.001 -0.261 -0.258 -0.261 -0.263 0.003 -0.284 -0.276 -0.284 -0.288 0.005 -0.299 -0.287 -0.298 -0.305 0.010 -0.309 -0.291 -0.309 -0.318 0.020 -0.256 -0.235 -0.257 -0.265 0.030 -0.167 -0.147 -0.167 -0.173 0.040 -0.088 -0.068 -0.088 -0.090 0.060 -0.041 -0.019 -0.041 -0.035 0.080 -0.194 -0.161 -0.194 -0.181 Percentile of [[Epsilon].sub.1] [[Delta].sub.1] 95% 98% 99% 0 -0.248 -0.248 -0.247 0.001 -0.264 -0.267 -0.267 0.003 -0.292 -0.294 -0.298 0.005 -0.309 -0.314 -0.318 0.010 -0.324 -0.331 -0.335 0.020 -0.271 -0.277 -0.280 0.030 -0.176 -0.178 -0.179 0.040 -0.089 -0.086 -0.084 0.060 -0.026 -0.013 -0.004 0.080 -0.165 -0.144 -0.128

Empirical Example

To illustrate the importance of conditionally heteroskedastic models, we use data from the Citibank Interest Rate data base. Treasury interest rates are fundamental and usually used as a reference to other rates (for example, corporate rates). We consider the three-month Treasury bill yield as the short-term spot rate. This series consists of N = 124 monthly Treasury yields from the first month of 1982 through the fourth month of 1992. The monthly values are averages of daily figures that were originally obtained from the Board of Governors of the Federal Reserve System.

Figure 4 plots the time series and its first differences for three-month U.S. Treasure bills. The stationarity in mean is obtained after first differencing, but the variance appears to be changing over time. The summary statistics of the original and first differenced series are shown in Table 4.

[Figure 4 ILLUSTRATION OMITTED]

Table 4 Summary Statistics for a Three-Month U.S. Treasury Bill Rate Time Series and Its First Differences

Sample Sample Standard Size Mean Deviation Minimum Maximum Original 124 7.44 1.92 3.75 13.48 Series First Difference 123 -0.07 0.41 -2.67 1.20 Series Autocorrelations Lag 1 2 3 4 5 6 Original 0.937 0.845 0.756 0.667 0.582 0.491 Series First Difference 0.367 0.066 -0.046 0.091 0.091 -0.139 Series Lag 9 12 15 18 Original 0.353 0.283 0.215 0.142 Series First Difference 0.015 -0.003 0.027 0.003 Series

For a simple illustration, we consider only an AR(1) model for the differenced series, which corresponds to the Vasicek and Cox, Ingersoll, and Ross models described earlier. It was shown above that the moment-generating function exists for the absolute-value form ARCH process described in equation (14c). Although the moment-generating function of the regular ARCH model described in equation (12) does not exist (except [[Delta].sub.1] = 0), it is widely used in the literature. Hence, we also consider it in the data analysis for comparison. In sum, we fit an ARIMA(1,1,0) model for the three-month Treasury bill series ([y.sub.t]), with the non-, absolute-, and regular-ARCH processes for the conditional variance of error term. Maximum likelihood estimation was used to fit each model. Estimation procedures for non-ARCH and regular ARCH models are well known. Estimation procedures for the absolute ARCH model and the consideration of the other competitive models for this yield rate series can be found in Lai (1995). The fitted models for ARIMA(1,1,0) are presented below. Asymptotic standard errors are in parentheses. Symbols *, [dagger], and [double dagger] indicate statistical significance at 10 percent, 5 percent, and 1 percent, respectively. B denotes the backward shift operator.

ARIMA(1,1,0) model for U.S. three-month Treasury-bill yield rates. Non-ARCH

(1 - 0.45 [double dagger] B) (1-B) [y.sub.t], =-0.0386 + [[Epsilon].sub.t],

(0.084) (0.0335) [h.sub.t] =0.117 [double dagger], [AIC.sub.3] = 0.7894.

(0.0155)

Absolute-ARCH

(1 - 0.365 [double dagger] B) (1-B) [y.sub.t] =-0.0281* + [[Epsilon].sub.t],

(0.078) (0.017) [h.sub.t] =0.0346 [double dagger] + 0.265 [double dagger] | [Epsilon.sub.t-1] |,

(0.010) (0.058) [AIC.sub.4] = 0.4043.

Regular-ARCH

(1 - 0.393 [double dagger] B) (1-B) [y.sub.t] = -0.0166 + [[Epsilon].sub.t],

(0.089) (0.023) [h.sub.t] = 0.0484 [double dagger] + 0.467 [double dagger] +

(0.009) (0.145) [[Epsilon].sup.2] [sub.t-1], [AIC.sub.4] = 0.3150.

Here, the Akaike information model selection criterion (AIC) is given by

[AIC.sub.t] = [-2 log(maximized likelihood) + 2r]/N*,

where r denotes the number of parameters estimated by maximum likelihood, and N* = N - (p+1) is the "effective" sample size. The significance of coefficient [[Delta].sub.1] indicates the usefulness of incorporating the ARCH process in the model.

Conclusion

This article examines the discrete-time short-term consequences on reserves due to changes in the interest rate environment. Generally, when viewed at initial time, the next period reserve is a random variable, which is a function of the disturbance generated in the first period. However, in the special case of a white noise process for the force of interest, the next period reserve is a deterministic value. Linear and nonlinear ARCH process models for the force of interest are considered. Under linear interest rate processes, explicit expressions are given for the changes in reserves. In particular, for a pure discount bond, the next period reserve and its discounted value have a lognormal distribution.

As an extension from linear processes to nonlinear processes, approximation formulas and simulation algorithms are presented. Compared with simulation results, we find that the approximation formulas perform well in the sense that the discrepancies between the approximations and simulations are small relative to the simulation standard error. Although we used only one type of insurance contract for our comparisons, the results are generalizable as long as cash flows are independent of interest rates. This is because, under this assumption, those two factors can be separated, as shown in Proposition 1. Theorems 1 and 2 refer to linear rate and nonlinear rate processes, respectively. A numerical example using U.S. Treasury securities indicates strong evidence of the ARCH process.

We consider only the initial and next period reserves, whose values will evolve with time. To handle this one-period-ahead problem sequentially, stochastic control theory may be used to access not only the dynamic nature of the time development, but also the mobility of an insurer's cash flows (claims, premiums, and expenses). Martin-Lof (1983) and Vandebroek and Dhaene (1990) apply the control theory in an insurance context, where they assume fixed interest rates.

In this article, the cash flows are assumed to be independent of interest rate variations. For many insurance contracts, this assumption appears to be unnecessarily restrictive. We hope to investigate this issue in the future.

(1) The Two factors are the short-time interest rate and the volatility of the short-term interest rate.

(2) proposition 1 and Theorem 1 gives [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where M(u) = exp([[Sigma].sup.2] [u.sup.2]/2), the moment-generating function of [[Epsilon].sub.t].

(3) The sample values of skewness and kurtosis for [V.sub.1] and [[e.sup.-y][[.sup.1]] [V.sub.1] in this simulation example are 0.0643 and 3.19, and 0.155 and 3.21, respectively. The kurtosis defined here is equal to three for a normal variate. Thus, a normal distribution could be used to approximate the reserve [V.sub.1] even though slightly positive skewness and slightly thicker tails feature are exhibited from the sample quantities. The positivity of skewness of [[e.sup.-y][.sup.1]] [V.sub.1] is larger than that of [V.sub.1], but a normal approximation to the distribution of discounted

Appendix A

Expected Value of Discounted Reserve

For a single policy, we have the following result, which can serve as a stepping stone to compute the expected value of discounted reserve for an entire block of business.

Proposition 2

Suppose that the basic assumption that [y.sub.t] is a Borel measurable function of ([[Epsilon].sub.1], [[Epsilon].sub.2],..., [[Epsilon].sub.t]) holds. For single policy, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In particular, for h = 1,

(15) [V.sub.0] + [a.sub.k] = [p.sub.x+k] E([e.sup.[-y.sub.h]] [multiplied by] [V.sub.] + [q.sub.x+k] [multiplied by] [b.sub.k+1] E([e.sup.[-y.sub.1]]

(compare Bowers et al., 1986, 7.8.2). The resources required at the beginning of policy year h equal the actuarial present value of year-end requirements (in expected value sense). Furthermore, we can employ equation (15) to calculate the expected discounted reserve E([e.sup.[-y.sub.1]] [multiplied by] [V.sub.h]) for each policy and then add them to obtain the entire block of business expected discounted reserve.

A recursive expression for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the expected value of discounted period h reserve, is obtained from Proposition 2:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Note that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];its value can therefore be computed by using Theorems 1 and 2 under different models for {[y.sub.t]}.

Proof

For a single policy, let the loss at time h be

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [F.sub.[Tau]] [sup.(h)] is defined similarly to equation (2). By definition,

(16) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

So,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which implies that

(17) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

From equation (16), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

by noting equation (17). The proof is completed.

Appendix B

An Upper Bound for the Variance of Innovation Under the ARCH Model

First, E([[Epsilon].sub.t]) = EE([[Epsilon].sub.e]|[[Psi].sub.t-1]) = 0, by equation (14b). Thus, Var([[Epsilon].sub.t]) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], by equation (14c). Since for X ~ N(0, [[Sigma].sup.2]), E|X| = 2/[square root of [Pi]] [Sigma], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Assuming that the series starts indefinitely far in the past with a finite first absolute moment, the limit of the upper bound on the right-hand side as k goes to infinity tends to L, where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence L = 2/[square root of 2 [Pi]] [square root of [[Delta].sub.0] + [[Delta].sub.1] L], which implies that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Thus,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Appendix C

Proof

Proof of Proposition 1

By definition,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then we have the desired results for [V.sub.0] and [V.sub.1] by utilizing the independence between cash flows and interest rates, the definition of cash flow [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], in equation (2) and the probability function of [J.sub.i].

Proof of Theorem 1

The [y.sub.t] expression from equation (9) implies that

(18) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

(19) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

by using the i.i.d. property of [Epsilon]'s and reindexing.

Proof of Lemma 1

Let ?? = ([t.sub.1],..., [t.sub.T])'. Then the moment-generating function of ?? = ([[Epsilon].sub.1],..., [[Epsilon].sub.T])' is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(by the Holder inequality).

Since for all b [is greater than] 0 and X ~ N(0, [[Sigma].sup.2]), E([e.sup.b|X|) = E[[e.sup.bX]I(X [is less than] 0)] + E[[e.sup.bX]I(X [is greater than or equal to] 0)] (where I is an indicator function) [is less than or equal to] E([e.sup.bX]) + E([e.sup.bX]) = 2 [e.sup.[b.sup.2] [[Sigma].sup.2/2]]

Thus,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

by a recursive argument, where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

by a recursive argument, where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof of Theorem 2 requires the following lemma.

Lemma 2

If the ARCH model of equations (14b) and (14c) governs the innovation process {[[Epsilon].sub.t]}, then for a fixed positive integer T and as [[Delta].sub.0] [arrow right] 0, we have that

(1) the joint density of [[Epsilon].sub.??] = ([[Epsilon].sub.1],...,[[Epsilon].sub.T] is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and

(2) the joint density of ([[Epsilon].sub.2],..., [[Epsilon].sub.T]) given [[Epsilon].sub.1] is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof of Lemma 2

Taylor's series expansion is used to derive the results. From the conditional densities of equations (14b) and (14c), we have

(20) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

When [[Delta].sub.1] = 0, then f([[Epsilon].sub.t],...,[[Epsilon].sub.T]) = [Phi] ([[Epsilon][.sub,??] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

From equation (20),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Differentiating with respect to [[Delta].sub.1] gives

(21) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Differentiating again on equation (21) gives

(22) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus,

(23) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

so,

(24) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

And

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

From Taylor's formula, as [[Delta].sub.1] approaches zero,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

so result (1) is proven.

For f([[Epsilon.sub.2],..., [[Epsilon].sub.T|[[Epsilon].sub.1]) in (2), the derivation is essentially the same as f([[Epsilon].sub.1],,..., [[Epsilon].sub.T]), replacing [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] respec completed.

Proof of Theorem 2

The proof of Theorem 2 is organized as follows. First, we write Taylor's expansion for f([Epsilon]) with an explicit form for the remainder. Second, we derive the linear- and quadratic-order expressions (with respect to [[Delta].sub.1]) for [[V.sup.(T0)[.sub.0]. Third, the boundedness of the remainder term is established.

For the first step, we differentiate equation (22) to give

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

So, by substituting df/d[[Delta].sib.1] and [d.sub.f]/d [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23) and (24), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By Taylor's formula with Lagrange form of the remainder,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For the second step, by using equation (18),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[Phi] is the distribution function of a standard normal variate, Hence,

(25) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By viewing [e.sub.1] ~ i.i.d. N(O,1), the first integral of equation (25) is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] by noting [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and after reindexing. The second intergral (linera-order term with respect to [[Delta].sub.1] is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For i = 0,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For i = 1,..., T,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

since for X ~ N(O,1), E(|X|[e.sup.aX]) = [[e.sup.a][.sup.2/2] (a), where q(a) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] + a[2[Phi](a) -1].

Define [[Pi].sub.0] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then

E(exp([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

A similar technique is used to find that

E[exp([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

by using, for X ~ N(0,1), E([X.sup.2] [e.sup.aX] = [[e.sup.a][sup.2/2]] (1+[a.sup.a]). Hence, the second integral becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

by noting [[Pi].sub.s] = [[-Zeta].sub.T-s], q(a) = q(-a) and reindexing. The linear-order term for [[V.sup.T][sub.0]] is proven. For the quadratic-order term (the third integral of equation [25]), the derivations are more tedious but the ideas are the same as for the linear-order term. That proof is omitted. The third integral can be proven to be

equal to exp [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For the last (remainder) term in equation (25), there exist constants [a.sub.1], [a.sub.2] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore,

(26) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We claim that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Because

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and noting equation (26), it suffices to prove that for i = 1,.., T and all r [is greater than] 0,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It is obviously true since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (by the Holder inequality)

[is less than] [infinity] (by Lemma 1).

Result (1) follows.

The proof for [[V.sup.(T-1)][.sub.1] follows the same steps as for [[V.sup.(T)][sub.0], by noting that, from equation (19),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and using Lemma 2(2).

References

Becker, D. N., 1991, Statistical Tests of the Lognormal Distribution as a Basis for Interest Rate Changes, Transactions of the Society of Actuaries, 43: 7-57, Discusion 59-72.

Beekman, J. A. and C. P. Fuelling, 1990, Interest and Mortality Randomness in Some Annuities, Insurance: Mathematics and Economics, 9: 185-196.

Beekman, J. A. and C. P. Fuelling, 1991, Extra Randomness in Certain Annuity Models, Insurance: Mathematics and Economics, 10: 275-287.

Bellhouse, D. R. and H. H. Panjer, 1981, Stochastic Modelling of Interest Rates with Applications to Life Contingencies-Part II, Journal of Risk and Insurance, 48: 628-637.

Bollerslev, T., 1986, Generalized Autoregressive Conditional Heteroskedasticity, Journal of Econometrics, 31: 307-327.

Bollerslev, T., R. Y. Chou, and K. F. Kroner, 1992, ARCH Modeling in Finance: A Review of the Theory and Empirical Evidence, Journal of Econometrics, 52: 5-59.

Bollerslev, T., R. F. Engle, and J. M. Wooldridge, 1988, A Capital Asset Pricing Model with Time Varying Covariances, Journal of Political Economy, 96: 116-131.

Bowers, N. L., H. U. Gerber, J. C. Hickman, D. A. Jones, and C. J. Nesbitt, 1986, Actuarial Mathematics (Schaumburg, III.: Society of Actuaries).

Box, G. and G. Jenkins, 1976, Time Series Analysis, Forecasting, and Control, Second Edition (San Francisco: Holden-Day).

Boyle, P. P., 1976, Rates of Return as Random Variables, Journal of Risk and Insurance, 43: 693-713.

Brown, R. H. and S. M. Schaefer, 1994, The Term Structure of Real Interest Rates and the Cox, Ingersoll, and Ross Model, Journal of Financial Economics, 35: 3-42.

Buhlmann, H., 1992, Stochastic Discounting, Insurance: Mathematics and Economics, 11: 113-127.

Chan, K. C., G. A. Karolyi, F. A. Longstaff, and A. B. Sanders, 1992, An Empirical Comparison of Alternative Models of the Short-Term Interest Rate, Journal of Finance, 47: 617-630.

Courtadon, G., 1982, The Pricing of Options on Default-Free Bonds, Journal of Financial and Quantitative Analysis, 17: 75-100.

Cox, J. C., J. E. Ingersoll, and S. A. Ross, 1981, A Reexamination of Traditional Hypotheses About the Term Structure of Interest Rates, Journal of Finance, 36: 769-799.

Cox, J. C., J. E. Ingersoll, and S. A. Ross, 1985, A Theory of the Term Structure of Interest Rates, Econometrica, 53: 385-407.

Dhaene, J., 1989, Stochastic Interest Rates and Autoregressive Integrated Moving Average Processes, ASTIN Bulletin, 19: 131-138.

Dothan, L. U., 1978, On the Term Structure of Interest Rates, Journal of Financial Economics, 6: 59-69.

Engle, R. F., 1982, Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of U.K. Inflation, Econometrica, 50: 987-1008.

Engle, R. F. and T. Bollerslev, 1986, Modelling the Persistence of Conditional Variances, Econometric Reviews, 5: 1-50.

Frees, E. W., 1990, Stochastic Life Contingencies with Solvency Considerations, Transactions of the Society of Actuaries, 42: 91-129.

Giacotto, C., 1986, Stochastic Modelling of Interest Rates: Actuarial vs Equilibrium Approach, Journal of Risk and Insurance, 53: 435-453.

Higgins, M. L. and A. K. Bera, 1992, A Class of Nonlinear ARCH Models, International Economic Review, 33: 137-158.

Ingersoll, J. E., 1987, Theory of Financial Decision Making (Rowman & Littlefield).

Jordan, C. W., 1967, Life Contingencies (Chicago: Society of Actuaries).

Lai, S. W., 1995, Stochastic Models of Interest Rates in Actuarial Science, Ph.D. Thesis, University of Wisconsin-Madison.

Longstaff, F. A. and E. S. Schwartz, 1992, Interest Rate Volatility and the Term Structure: A Two-Factor General Equilibrium Model, Journal of Finance, 47: 1259-1282.

Martin-Lof, A., 1983, Premium Control in an Insurance System, An Approach Using Linear Control Theory, Scandinavian Actuarial Journal, 1-27.

Panjer, H. H. and D. R. Bellhouse, 1980, Stochastic Modelling of Interest Rates with Applications to Life Contingencies, Journal of Risk and Insurance, 47: 91-110.

Ramlau-Hansen, H., 1988, The Emergence of Profit in Life Insurance, Insurance: Mathematics and Economics, 7: 225-236.

Schwert, W., 1990, Stock Volatility and the Crash of '87, Review of Financial Studies, 3: 77-102.

Tilley, J. A., 1993, Valuing American Options in a Path Simulation Model, Transactions of the Society of Actuaries, 45.

Vandebroek, M. and J. Dhaene, 1990, Optimal Premium Control in a Non-Life Insurance Business, Scandinavian Actuarial Journal, 3-13.

Vasicek, O., 1977, An Equilibrium Characterization of the Term Structure, Journal of Financial Economics, 5: 177-188.

Waters, H. R., 1978, The Moments and Distributions of Actuarial Functions, Journal of the Institute of Actuaries, 105: 61-75.

Siu-Wai Lai is a Ph.D. student of statistics at the University of Wisconsin-Madison. Edward W. Frees is Professor of Business and Statistics at the University of Wisconsin-Madison. The first author gratefully acknowledges a Ph.D. grant support from the Society of Actuaries. The research of the second author was supported by a research grant from the Society of Actuaries, Actuarial Education and Research Fund, and a professorship funded by Time Insurance Company. The authors wish to thank the Editor, two anonymous reviewers, and James Hickman for their valuable comments.

Printer friendly Cite/link Email Feedback | |

Title Annotation: | Symposium on Insurance Solvency and Finance |
---|---|

Author: | Lai, Siu-Wai; Frees, Edward W. |

Publication: | Journal of Risk and Insurance |

Date: | Sep 1, 1995 |

Words: | 11591 |

Previous Article: | The valuation of option features in retirement benefits. |

Next Article: | Solvency risk and the tax sheltering behavior of property-liability insurers. |

Topics: |

## Reader Opinion