Subject:

Actuaries
(Analysis)

Regression analysis

Regression analysis

Authors:

Sanchez, Jorge de Andres

Gomez, Antonio Terceno

Gomez, Antonio Terceno

Pub Date:

12/01/2003

Publication:

Name: Journal of Risk and Insurance Publisher: American Risk and Insurance Association, Inc. Audience: Trade Format: Magazine/Journal Subject: Business; Insurance Copyright: COPYRIGHT 2003 American Risk and Insurance Association, Inc. ISSN: 0022-4367

Issue:

Date: Dec, 2003 Source Volume: 70 Source Issue: 4

Geographic:

Geographic Scope: United States Geographic Code: 1USA United States

Accession Number:

111737963

Full Text:

ABSTRACT

In this article, we propose several applications of fuzzy regression techniques for actuarial problems. Our main analysis is motivated, on the one hand, by the fact that several articles in the financial and actuarial literature suggest using fuzzy numbers to model interest rate uncertainty but do not explain how to quantify these rates with fuzzy numbers. Likewise, actuarial literature has recently focused some of its attention in analyzing the Term Structure of Interest Rates (TSIR) because this is a key instrument for pricing insurance contracts. With these two ideas in mind, we show that fuzzy regression is suitable for adjusting the TSIR and discuss how to apply a fuzzy TSIR when pricing life insurance contracts and property-liability policies. Finally, we reflect on other actuarial applications of fuzzy regression and develop with this technique the London Chain Ladder Method for obtaining Incurred But Not Reported Reserves.

INTRODUCTION

To obtain the financial price of an insurance contract and in general the price of any other asset, we have to discount the cash flows that the asset produces throughout its life. We therefore need to know the discount rates that must be applied at each moment. The reference values of these discount rates are those interest rates that are free of default risk, i.e., those that correspond to public debt bonds. Of course, this is especially true in an actuarial pricing context because the profit to the insurer must be in accordance with the return from the insurer's investments of the premiums, and some of the premiums are invested in public debt securities. So, predicting the evolution of the default-free interest rate is a crucial question in actuarial pricing. This explains why yield curve analysis has become an important topic in actuarial science. Babbell and Merrill (1996) and Ang and Sherris (1997) provided a wide survey of Term Structure of Interest Rate (TSIR) models derived from the contingent claims theory, whereas Yao (1999) discussed the asymptotic properties of the rates fitted with some of these models, bearing in mind actuarial pricing. Delbaen and Lorimier (1992) used a nonparametric method based on quadratic programming to fit the short-term yield curve, whereas Carriere (1999) proposed combining bootstrapping and spline functions to estimate long-term yield rates embedded in the TSIR.

However, many actuarial analyses are concerned with the medium and long term and, in our opinion, modeling the behavior of interest rates in the long term by means of a stochastic model is not very realistic. As Gerber (1995) pointed out, there is no commonly accepted stochastic model for predicting long-term discount rates.

Fuzzy Sets Theory (FST) has been used successfully in insurance problems that require much actuarial subjective judgment and those for which measuring the embedded variables is difficult. Lemaire (1990) applied fuzzy logic to underwriting and reinsurance decisions whereas Cummins and Derrig (1993) used fuzzy decision to evaluate several econometric methods of claim cost forecasting. Derrig and Ostaszewski (1995) showed that fuzzy clustering methods are suitable for risk classification and Young (1996) applied fuzzy reasoning to insurance rate decisions. Therefore, and given that there are only vague data or data ill related with the behavior of future discount rates to predict them (e.g., the price of fixed-income securities or the opinions of "experts" about the future behavior of macroeconomic magnitudes), many authors think that it is often more suitable and realistic to make financial analyses in the long term with yields quantified with fuzzy numbers. For financial analysis, see Kaufmann (1986), Buckley (1987), or Li Calzi (1990), whereas in actuarial literature, see Lemaire (1990), Ostaszewski (1993), or Terceno et al. (1996) in a life insurance context, and articles by Cummins and Derrig (1997) and Derrig and Ostaszewski (1997) on the financial analysis of property-liability insurance. However, these articles do not explain in great detail how to estimate the discount rates with fuzzy sets. They usually suggest that "the rates are estimated subjectively by the experts using fuzzy numbers" but offer no more explanation.

In this article, we propose a solution to this problem. This involves estimating the TSIR with fuzzy sets, since the TSIR implicitly contains the expectations of the fixed income market agents (i.e., the experts) regarding the evolution of the future interest rates. Our results will be similar to those of Carriere (1999). His method obtains an estimate of the TSIR with probabilistic confidence intervals, whereas ours describes the yield curve as fuzzy confidence intervals. We also discuss how to use our fuzzy TSIR to price life-insurance contracts and property-liability policies. We would like to point out that using a fuzzy TSIR to price insurance policies was initially suggested in Ostaszewski (1993).

Another aim of this article is to suggest other actuarial applications of fuzzy regression. We have therefore developed the method for obtaining Incurred But Not Reported Reserves proposed by Benjamin and Eagles (1986) with fuzzy regression methods. We also discuss how fuzzy regression can help us with trending claim costs and with premium rating from the CAPM perspective.

The structure of the article is as follows. In the next section we describe some basic aspects of fuzzy arithmetic and fuzzy regression. In "Estimating the TSIR With Fuzzy Methods" we propose a method for obtaining a fuzzy TSIR based on fuzzy regression, apply our method to the Spanish public debt market, and compare our results with those of standard econometric methods. In "Using a Fuzzy TSIR for Financial and Actuarial Pricing" we discuss how our fuzzy TSIR can be used in actuarial pricing. In "Discussing Further Actuarial Applications of Fuzzy Regression" we suggest further applications of fuzzy regression to insurance problems.

FUZZY ARITHMETIC AND FUZZY REGRESSION

Basics of FST and Fuzzy Numbers

FST is constructed from the concept of fuzzy subset. A fuzzy subset A is a subset defined over a reference set X for which the level of membership of an element x [member of] X to A accepts values other than 0 or 1 (absolute nonmembership or absolute membership). A fuzzy subset A can therefore be defined as A = {(x, [[mu].sub.A](x)) | x [member of] X}, where [[mu].sub.A](x) is called the membership function and is a mapping [[mu].sub.A]: X [right arrow] [0, 1]. So, an element x has its image within [0, 1], where 0 indicates nonmembership to the fuzzy subset A and 1 indicates absolute membership. Alternatively, a fuzzy subset A can be represented by its level sets [alpha] or [alpha]-cuts. An [alpha]-cut is an ordinary (crisp) set containing elements whose membership level is at least [alpha]. For a fuzzy subset A, we will name an [alpha]-cut with [A.sub.[alpha]] being its mathematical expression: [A.sub.[alpha] = {x [member of] X | [[mu].sub.A](x) [greater than or equal to] [alpha]}, 0 [less than or equal to] [alpha] [less than or equal to] 1.

A fuzzy number (FN) is a fuzzy subset A defined over the real numbers (X is the set R). It is the main instrument of FST for quantifying uncertain or imprecise magnitudes (e.g., the discount rates in financial mathematics). Two other conditions are required for an FN. First, it must be a normal fuzzy set, i.e., it exists at least one x [member of] X such that [[mu].sub.A](x) = 1. Second, it must be convex (i.e., its [alpha]-cuts must be convex sets in the real numbers). Figure 1 shows the shape of an FN.

[FIGURE 1 OMITTED]

The most widely used FNs are triangular fuzzy numbers (TFNs) because they are easy to use and can be interpreted intuitively. (1) To construct a TFN named A, we must establish its center, a unique value [a.sub.C] (i.e., [a.sub.C] = [a.sub.2] = [a.sub.3] in Figure 1), and the deviations from there that we consider reasonable, i.e., its left spread, [l.sub.A]; and its right spread, [r.sub.A]. A TFN (2) will be denoted as A = ([a.sub.C], [l.sub.A], [r.sub.A]). Its membership function, [[mu].sub.A](x), is given by linear functions and its [alpha]-cuts, [A.sub.[alpha]], are confidence intervals where its extremes are also done by linear functions. So

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In fuzzy regression, the symmetrical TFNs (STFNs) are widely used. These are TFNs where [l.sub.A] = [r.sub.A] = [a.sub.R] and we will denote them as A = ([a.sub.C], [a.sub.R]). The membership function and [alpha]-cuts of an STFN A are

(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Figures 2 and 3 show the shape of a TFN and an STFN, respectively.

[FIGURES 2-3 OMITTED]

To develop our article we need to know the level of inclusion of an FN B within another FN A, [mu] (B [subset or equal to] A). If the [alpha]-cuts of these FNs are [A.sub.[alpha]] = [[A.sup.1]([alpha]), [A.sup.2]([alpha])] and [B.sub.[alpha]] = [[B.sup.1]([alpha]), [B.sup.2]([alpha])], then [mu](B [subset or equal to] A) [greater than or equal to] [alpha], if [B.sub.[alpha]] [subset or equal to] [A.sub.[alpha]], i.e., if

(3) [A.sup.1]([alpha]) [less than or equal to] [B.sup.1]([alpha]) and [A.sup.2]([alpha]) [greater than or equal to] [B.sup.2]([alpha]).

For example, in Figure 4, [mu] (B [subset or equal to] A) [greater than or equal to] 0.4.

[FIGURE 4 OMITTED]

We also need to establish when one FN is greater than another. (3) Ramik and Rimanek (1985) suggested that B is greater or equal to A with a membership level of at least [alpha], [mu](B [greater than or equal to] A) [greater than or equal to] [alpha], if

(4) [A.sup.1]([alpha]) [less than or equal to] [B.sup.1]([alpha]) and [A.sup.2]([alpha]) [less than or equal to] [B.sup.2] ([alpha]).

For example, in Figure 4, [mu] (B [greater than or equal to] A) [greater than or equal to] 1 and [mu](A [greater than or equal to] B) [greater than or equal to] 0.4.

In actuarial analysis we often need to evaluate functions (e.g., the net present value), which in a general way we shall symbolize as y = f ([x.sub.1], [x.sub.2], ..., [x.sub.n])--e.g., [x.sub.1], [x.sub.2], ..., [x.sub.n-1] may be the cash flows and [x.sub.n] the discount rate. Then, if [x.sub.1], [x.sub.2], ..., [x.sub.n] are not given by crisp numbers but by the FNs [A.sub.1], [A.sub.2], ..., [A.sub.n] (i.e., to calculate the net present value we know the cash flows and the discount rate imprecisely), when evaluating f(*) we will obtain an FN B, B = f([A.sub.1], [A.sub.2], ..., [A.sub.n]). To determine the membership function of B, [[mu].sub.B](y), we must apply the Zadeh's extension principle, exposed in the seminal paper by Zadeh (1965). In a similar way as when handling arithmetically random variables, we obtain the membership function of the result when operating with FNs by convoluting membership functions of the FNs [A.sub.1], [A.sub.2], ..., [A.sub.n] (for random variables, the density functions). The difference is that random variables are convoluted using operators sum-product, whereas with FN they are convoluted using operators max-min (max instead of sum and rain instead of product). Mathematically,

(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Unfortunately, it is often impossible to obtain a closed expression for the membership function of B (this problem often arises when handling random variables). However, we may be able to obtain its [alpha]-cuts, [B.sub.[alpha]], from [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] as

(6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In actuarial mathematics, many functional relationships are continuously increasing or decreasing with respect to every variable in such a way that it is easy to evaluate the [alpha]-cuts of B. Buckley and Qu (1990b) demonstrated that if the function f(*) that induces B is increasing with respect to the first m variables, where m [less than or equal to] n, and decreasing with respect to the last n-m variables, [B.sub.[alpha]] is

(7) [B.sub.[alpha]] = [[B.sup.1] ([alpha]), [B.sup.2] ([alpha])] = [f ([A.sup.1.sub.1] ([alpha]), ..., [A.sup.1.sub.m] ([alpha]), [A.sup.2.sub.m+1], ([alpha]), ..., [A.sup.2.sub.n] ([alpha])), f ([A.sup.2.sub.1] ([alpha]), ..., [A.sup.2.sub.m] ([alpha]), [A.sup.1.sub.m+1] ([alpha]), ..., [A.sup.sub.n] ([alpha]))].

Some operations with STFN are easy to solve. For instance, if we multiply A = ([a.sub.C], [a.sub.R]) by a real number k, B = k A, the result is B = ([b.sub.C], [b.sub.R]) = ([ka.sub.C], [absolute value of k] [a.sub.R]). The sum of two (4) STFNs, C = A + B, is also an STFN. Concretely, C =([c.sub.C], [c.sub.R]) = ([a.sub.C], [a.sub.R]) + ([b.sub.C], [b.sub.R]) = ([a.sub.C], + [b.sub.C], [a.sub.R] + [b.sub.R]). So, if the FN B is obtained from a linear combination of the STFNs [A.sub.i] = ([a.sub.iC], [a.sub.iR), i = 1, ..., n, i.e., B = [[summation of].sup.n.sub.(i=1)] [k.sub.i], [A.sub.i] where [k.sub.i] [member of] R, B will be an STFN, B = ([b.sub.C], [b.sub.R]) where

(8) ([b.sub.C], [b.sub.R]) = ([k.sub.1] * [a.sub.1C] + [k.sub.2] * [a.sub.2C] + ... + [k.sub.n] * [a.sub.nC], [[absolute value of k].sub.1] * [a.sub.1R] + [[absolute value of k].sub.2] * [a.sub.2R] + ... + [[absolute value of k].sub.n] * [a.sub.nR]).

Unfortunately, the result of a nonlinear operation with STFNs is not an STFN. So, if we evaluate B = f ([A.sub.1], [A.sub.2], ..., [A.sub.n]), where we suppose that [A.sub.i] = ([a.sub.iC], [a.sub.iR]) [for all] i, B is often not an STFN despite the characteristics of [A.sub.1], [A.sub.2], ..., [A.sub.n]. Even so, Dubois and Prade (1993) showed that if f(*) is increasing with respect to the first m variables, where m [less than or equal to] n, and decreasing with respect to the others, B can be estimated well by B' = ([b.sub.C], [b.sub.R]),

where

(9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

So, for B = [A.sup.k] where k [member of] R, if A = ([a.sub.C], [a.sub.R]), from (9) we obtain:

(10) B [approximately equal to] ([b.sub.C], [b.sub.R]) = [(([a.sub.C].sup.k], [absolute value of k][([a.sub.C]).sup.k-1][a.sub.R])

while for C = A * B where k [member of] R, if A = ([a.sub.C], [a.sub.R]) and B = ([b.sub.C], [b.sub.R]), from (9) we obtain

(11) C [approximately equal to] ([c.sub.C], [c.sub.R]) = ([a.sub.C] * [b.sub.C], [a.sub.C] * [b.sub.R] + [b.sub.C] * [a.sub.R]).

Tanaka and Ishibuchi's Fuzzy Regression Model

The fuzzy regression model developed in Tanaka (1987) and Tanaka and Ishibuchi (1992) is one of the most widely used models in fuzzy literature for economic applications. (5) Like any regression technique, the aim of fuzzy regression is to determine a functional relationship between a dependent variable and a set of independent ones. Fuzzy regression allows to obtain functional relationships when independent variables, dependent variables, or both, are not crisp values but confidence intervals.

As in econometric linear regression, we shall suppose that the explained variable is a linear combination of the explanatory variables. This relationship should be obtained from a sample of n observations {([Y.sub.1], [X.sub.1]), ([Y.sub.2], [X.sub.2]), ..., ([Y.sub.j], [X.sub.j]), ..., ([Y.sub.n], [X.sub.n])} where [X.sub.j] is the jth observation of the explanatory variable, [X.sub.j] = ([X.sub.0j], [X.sub.1j], [X.sub.2j], ..., [X.sub.1j], ..., [X.sub.mj]). Moreover, [X.sub.0j] = 1 [for all] j, and [X.sub.ij] is the observed value for the ith variable in the jth case of the sample. [Y.sub.j] is the jth observation of the explained variable, j = 1, 2, ..., n. The jth observation may either be a crisp value or a confidence interval, in either case, it can be represented through its center and its spread or radius as [Y.sub.j] = <[Y'.sub.jC], [Y'.sub.jR]>, where [Y'.sub.jC] is the center and [Y'.sub.jR] is the radius.

Similarly, we suppose that the jth observation for the dependent variable is an [[alpha].sup.*]-cut of the FN it arises from where [[alpha].sup.*] may be stated previously by the decision maker. Also, the FN that quantifies the jth observation of the dependent variable is an STFN that we will write as [Y.sub.j] = ([Y.sub.jC], [Y.sub.jR]). Therefore, since the [[alpha].sup.*]-cut of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is (see (2)):

(12) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

the center and spread of [Y.sub.j] can be obtained from its [[alpha].sup.*]-cut taking into account (2) as

(13) [Y'.sub.jC] = [Y.sub.jC] and [Y'.sub.jR] = [Y.sub.jR](1 - [[alpha].sup.*]) [??] [Y.sub.jR] = [Y'.sub.jR]/(1 - [[alpha].sup.*]

and we must estimate the following fuzzy linear function

(14) [Y.sub.j] = [A.sub.0] + [A.sub.1][X.sub.1j] + ... + [A.sub.m][X.sub.mj].

In this model of fuzzy regression, the disturbance is not introduced as a random addend in the linear relation but is incorporated into the coefficients [A.sub.i], i = 0, 1, ..., m. Of course, the final objective is to adjust the fuzzy numbers [A.sub.i] which estimate [A.sub.i] from the available sample. Given the characteristics of [Y.sub.j], the parameters [A.sub.i], i = 0, 1, 2, ..., m must be STFNs. These parameters can therefore be written as [A.sub.i] = ([a.sub.iC], [a.sub.iR]), i = 0, 1, ..., m. The main objective is to estimate every FN [A.sub.i], [A.sub.i] = ([a.sub.iC], [a.sub.iR]). When we have obtained [A.sub.i] the estimates of [Y.sub.j] = ([Y.sub.jC], [Y.sub.jR]), will be

(15) [Y.sub.j] = [A.sub.0] + [A.sub.1][X.sub.1j] + ... + [A.sub.m][X.sub.mj].

Therefore, [Y.sub.j] is obtained from (8):

(16) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

whose [alpha]-cuts for a level [[alpha].sup.*] are

(17) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The parameters [a.sub.iC] and [a.sub.iR], must minimize the spreads of [Y.sub.j], and simultaneously maximize the congruence of [Y.sub.j] with [Y.sub.j], which is measured as [mu]([Y.sub.j] [subset or equal to] [Y.sub.j]). Specifically, we must solve the following multiple objective program:

(18a) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

subject to

(18b) [mu]([Y.sub.j] [subset or equal to] [Y.sub.j] [greater than or equal to] [alpha] j = 1, 2, ..., n; [a.sub.iR] [greater than or equal to] 0 i = 0, 1, ..., m, [alpha] [member of] [0, 1].

If for the second objective we require a minimum accomplishment level [[alpha].sup.*], i.e., the level that the decision maker considers that <[Y'.sub.jC], [Y'.sub.jR]>, j = 1, 2, ..., n, has been obtained, the above program is transformed into the following linear one

(19a) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

subject to

(19b) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The first block of constraints in Equation (19b) is a consequence of the requirement that [mu](Y.sub.j] [subset or equal to] [Y.sub.j] [greater than or equal to] [[alpha].sup.*], which must be implemented by taking into account (3), (12), and (17). With the last block of constraints in Equation (19b) we ensure that [a.sub.iR] [for all] i will be nonnegative.

In our opinion, fuzzy regression techniques have a number of advantages over traditional regression techniques:

(a) The estimates obtained after adjusting the coefficients are not random variables, which are difficult to manipulate in arithmetical operations, but fuzzy numbers, which are easier to handle arithmetically using [alpha]-cuts. So, when starting from magnitudes estimated by random variables (e.g., from a least squares regression), these random variables are often reduced to their mathematical expectation (which may or may not be corrected by its variance) to make them easier to handle. As we have already pointed out, this loss of information does not necessarily take place when we operate with FNs.

(b) When investigating economic or social phenomena, the observations are a consequence of the interaction between the economic agents' beliefs and expectations, which are highly subjective and vague. A good way to treat this kind of information is therefore with FST. For example, the asset prices that are determined in the markets are due to the agents' expectations of future inflation and the issuers' credibility. We think that it is more realistic to consider that the bias between the observed value of the dependent variable and its theoretical value (the error) is not random but fuzzy. At least in this way we assume that the analyzed phenomena have a large subjective component.

(c) The observations are often not crisp numbers but confidence intervals. For instance, the price of one financial asset throughout one session often oscillates within an interval and is rarely unique (e.g., it can oscillate within [$100, $105]). To be able to use econometric methods, the observations for the explained variable and/or the explanatory variable must be represented by a single value (e.g., $102.5 for [$100, $105]), which involves losing a great deal of information. However, fuzzy regression does not necessarily reduce each variable to a crisp number, i.e., all the observed values can be used in the regression analysis.

ESTIMATING THE TSIR WITH FUZZY METHODS

Estimating the TSIR With Conventional Econometric Methods

Estimating the TSIR for a concrete date and a given market is fairly straightforward if there are many sufficiently liquid zero coupon bonds and their price can be observed without perturbations. However, fixed-income markets rarely enjoy these conditions simultaneously.

This subsection describes the essence of a family of methods for estimating the discount function associated to the TSIR with econometric methods. They can be used if the sample is made up entirely of zero coupon bonds, entirely of bonds with coupon or, as is usual, if it is made up of both types of bonds.

These methods start from the fact that the rth bond, where r = 1, 2, ..., k, in which k is the number of available bonds, provides several cash flows (coupons and principal). These are denoted by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [C.sup.r.sub.i] is the amount of the ith cash flow and [t.sup.r.sub.i] is its maturity in years. If we suppose that the default-free bonds do not include any option (i.e., they are not convertible or callable, etc.), the price of the rth bond is therefore the sum of the discounted value of every coupon and the principal with the corresponding spot rate

(20) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the discounted value of one dollar with maturity [t.sup.r.sub.i] years.

Of course, subsequently we should define a form for the discount function to specify the econometric equation to be estimated. Our proposal is based on the methods that use splines (piecewise functions) to model the discount function. The best known methods are those in McCulloch's articles (1971, 1975) (quadratic splines and cubic splines) and in Vasicek and Fong's article (1982) (exponential splines). These methods suppose that the discount function is a linear combination of m + 1 functions of time. Therefore

(21) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

From (20) and (21) we can deduce that the following linear equation must be estimated

(22) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We assume that [g.sub.j](t) are splines and not simply polynomial functions because with splines we can determine [g.sub.j](t) according to the distribution of the maturity dates of the sample and, therefore, fit the discount function better for the most common maturities. Moreover, with splines we can obtain TSIR profiles that do not fluctuate very much and forward rates that do not behave explosively.

A random disturbance is justified because several factors disturb the formation of the prices of fixed-income securities. Chambers, Carleton, and Waldman (1984) point out some of them; for example, the coupon-bearing of bonds with maturities greater than one contain information about more than one present value coefficient; the bond portfolios are not continuously rebalanced, so at any moment each bond can deviate by some (presumably random) amount or there is no single price for each bond, which implies that there is some inherent imprecision to the concept of a single price.

We will now present our fuzzy method for estimating the TSIR. We will first establish a hypothesis on which to construct a method for estimating the TSIR that uses fuzzy regression from the analyzed framework. We will then discuss how to estimate the TSIR and the forward rates using fuzzy numbers and make an empirical application. As we stated above, we will suppose that the bonds in our analysis are default-free and only produce a stream of payments (coupons and principal or only principal if they are zero coupon bonds) and that they do not have any embedded option. Also, as the price of one bond we will take all the prices traded on 1 day, rather than an average price.

Hypothesis

Hypothesis 1: The price of the rth bond in a session is an STFN. This price will be written as [P.sup.r] where

(23) [P.sup.r] = ([P.sup.r.sub.C], [P.sup.r.sub.R]) [P.sup.r.sub.C], [P.sup.r.sub.R] [greater than or equal to] 0, r = 1, 2, ..., k.

This hypothesis considers the price of a bond on i day to be "approximately [P.sub.C]," and not exactly [P.sub.C]. We think that this is more suitable because over a session it is usual to negotiate more than one price for the same bond (one for each trade). However, if these prices are unique then [P.sup.r.sub.R] = 0.

Hypothesis 2: The observed price for each security is an [alpha]-cut of the FN that quantifies that price for a predefined [[alpha].sup.*]. Its inferior and upper extremes are the minimum and maximum price of the bond over the session. Therefore, this interval will be expressed through its center and spread as <[P'.sup.r.sub.C], [P'.sup.r.sub.R]> For example, if the traded price for one bond on 1 day has fluctuated between 100 and 103, it will be expressed as <101.5, 1.5>. Similarly, from these parameters we can obtain the center and the spread of its corresponding FN, (23), taking into account (13):

(24) [P'.sup.r.sub.C] = [P.sup.r.sub.C] y [P'.sup.r.sub.R] = (1 - [[alpha].sup.*]) [P.sup.r.sub.R] [??] [P.sup.r.sub.R] = [P'.sup.r.sub.R] /(1 - [[alpha].sup.*]), 0 [less than or equal to] [[alpha].sup.*] [less than or equal to] 1, r = 1, 2, ..., k.

Hypothesis 3: The discount function is quantified via an FN that depends on the time. So, for a given maturity t, the discount function is the following STFN:

(25) [f.sub.t] = ([f.sub.tC], [f.sub.tR]), t > 0, 0 [less than or equal to] [f.sub.tC] - [f.sub.tR] [less than or equal to] [f.sub.tC] + [f.sub.tR] [less than or equal to] 1.

The price of the rth bond, (23), can therefore be written from (20) as

(26) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and combining (25) and (26) we obtain

(27) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

So, the observed [[alpha].sup.*]-cut for the price of the rth bond, is

(28) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hypothesis 4: The discount function, (25), can be approximated from a linear combination of m + 1 functions [g.sub.j](t), j = 0, 1, ..., m with image in [R.sup.+] that are continuously differentiable, and whose parameters are given by STFN. In this way, these parameters can be represented as

(29) [a.sub.j] = ([a.sub.jC], [a.sub.jR]), [a.sub.jR] [greater than or equal to] 0, j = 0, 1, ..., m

and so the discount function is obtained from (21), (29) and using (8):

(30) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then, using (27) and (30), the price of the rth bond can be expressed by

(31) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and the [[alpha].sup.*]-cut of the rth bond price, (28), is now

(32) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with [a'.sub.jC] = [a.sub.jC] and [a'.sub.jR] = (1 - [[alpha].sup.*])[a.sub.jR], 0 [less than or equal to] [[alpha].sup.*] [less than or equal to] 1, r = 1, 2, ..., k.

Adjusting the Discount Function Using a Fuzz Regression Model

Since the value of the discount function for t = 0 should be 1, [f.sub.0] = ([f.sub.0C], [f.sub.0R]) = (1, 0). As McCulloch stated in (1971), this condition is met if [a.sub.0] =([a.sub.0C], [a.sub.0R]) = (1, 0), [g.sub.0](t) = 1, and [g.sub.j](0) = 0, j = 1, 2, ..., m. So, from (31), we can express the price of the rth bond as

(33) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and, using fuzzy arithmetic, we finally write

(34) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In this way, by identifying in Equation (34) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we obtain

(35) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It is easy to verify in Equation (34) that the value of the jth explanatory variable for the rth bond is the crisp value [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Therefore, from (34) and (35) we can write

(36) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then, after obtaining the estimate for every [a.sub.j], [a.sub.j] = [a.sub.jC], [a.sub.jR]), the fitted value of the explained variable of the rth observation, [Y.sup.r], is

(37) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Bearing in mind that the dependent variable and the parameters are actually quantified via their [[alpha].sup.*]-cut, the expression of the [[alpha].sup.*]-cut of [Y.sup.r] (that of FN (37)) is

(38) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where we must estimate the center and the spread of the [[alpha].sup.*]-cut for [a.sub.j], j = 1, ..., m. After estimating these parameters, the center and the radius [a.sub.jC] and [a.sub.jR] are estimated using (2) as

(39) [a.sub.jC] =[a'.sub.jC] and [a.sub.jR] = [a'.sub.jR]/(1 - [[alpha].sup.*]).

Adding certain constraints, which are related to the properties of the discount function, to obtain [a'.sub.jC] and [a'.sub.jR], we have to solve the following linear program

(40a) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

subject to

(40b) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(40c) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(40d) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(40e) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(40f) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(40g) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(40h) [a'.sub.j] R [greater than or equal to] 0 j = 1, 2, ..., m.

The constraints (40b), (40c), and (40h) correspond to Tanaka's regression model (see (18b) or (19b)). Similarly, (40d) and (40e) ensure that the discount function is decreasing, and that it is accomplished for an arbitrary periodicity P (in years). Therefore uP is the greatest maturity that we will use in a later analysis. It is reasonable to suppose that uP is close to the expiration of the bond with the greatest maturity. In Equations (40d) and (40e) we use the Ramik and Rimanek's criteria for ordering FNs (see (4)). Finally, constraints (40f) and (40g) ensure that the discount function is within [0, 1].

Estimating the Spot Rates and the Forward Rates Using Fuzzy Numbers

The discount function in t, [f.sub.t], is obtained from its corresponding spot rate [i.sub.t] by [f.sub.t] = [(1 + [i.sub.t]).sup.-t] and then

(41) [i.sub.t] = [([f.sub.t]).sup.-1/t] - 1

If the discount function in t is an FN, the spot rate will be an FN [i.sub.t]. Its membership function can be obtained applying the extension principle (5) to the relation (41) as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Unfortunately, despite using a discount function quantified via an STFN, the spot rate is not an STFN because it is not a linear function of [f.sub.t]. However, applying (10) in (41) we obtain

(42) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

To obtain the forward rate for the tth year, [r.sub.t], we should solve the following fuzzy equation:

(43) [f.sub.t-1] [(1 + [r.sub.t]).sup.-1] = [f.sub.t].

The [[alpha].sup.*]-cuts of [r.sub.t], [r.sub.t[alpha] are (see Appendix):

(44) [r.sub.t[alpha]] = [[r.sup.1.sub.t]([alpha]), [r.sup.2.sub.t]([alpha])] = [[f.sub.(t-1)C] + [f.sub.(t-1)R](1 - [alpha])/[f.sub.tC] + [f.sub.tR](1 - [alpha]) - 1, [f.sub.(t-1)C] - [f.sub.(t-1)R](1 - [alpha])/[f.sub.tC] - [f.sub.tR](1 - [alpha]) - 1].

Then, although [r.sub.t] is not an STFN, it can be approximated reasonably well by this type of FN. In the Appendix we demonstrate that its approximation by means of an STFN is

(45) [r.sub.t] [approximately equal to] ([r.sub.tC], [r.sub.tR]) = ([f.sub.(t-1)C]/[f.sub.tR] - 1, [f.sub.(t-1)C] * [f.sub.tR] - [f.sub.tC] * [f.sub.(t-1)R]/[([f.sub.tC]).sup.2]).

Notice that, although we take annual periods, calculating implied rates for any other periodicity is not a problem because the discount function is a continuous function of the maturity.

Empirical Application

In this subsection, we used our method to estimate the TSIR in the Spanish public debt market on June 29, 2001. Table 1 shows the bonds included in our sample and their characteristics.

To fit the TSIR in this date, we formalized the discount function using McCulloch's quadratic splines. (6) So we took m = 5, and the knots that we used to construct the splines were [d.sub.1] = 0 years, [d.sub.2] = 1.58 years, [d.sub.3] = 3.83 years, [d.sub.4] = 8.96 years, and [d.sub.5] = 31.1 years. Then, the functions [g.sub.j](t), j = 1, ..., 5 are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We used OLS regression, taking for the price of the kth bond ([P.sup.k.sub.min] + [P.sup.k.sub.max])/2. Its determination coefficient is [R.sup.2] = 99.98% whereas the discount function is

[f.sub.t] = 1 - 0.04394[g.sub.1](t) - 0.03672[g.sub.2](t) - 0.04875[g.sub.3](t) - 0.03572[g.sub.4](t) - 0.00730[g.sub.5](t).

To fit the TSIR by using fuzzy regression we took for the price of the kth bond the interval [P.sup.k] = <[P.sup.k.sub.C], where [P.sup.k.sub.R]>, where [P.sup.k.sub.C] = ([P.sup.k.sub.min] + [P.sup.k.sub.max])/2 and [P.sup.k.sub.R] = ([P.sup.k.sub.max] - [P.sup.k.sub.min])/2. To build the constraints (40d), (40e), (40f), and (40g) in fuzzy regression we assumed an annual periodicity. The final value of the objective function (40a) is z = 23.77. To interpret this value, we should remember that the size of our sample was 28 assets. When taking the level of congruence [[alpha].sup.*] = 0.5, our fuzzy discount function is

[f.sub.t] = (1, 0) + (-0.04280, 0.00422)[g.sub.1](t) + (-0.03874, 0.00043)[g.sub.2](t) + (-0.04675, 0.00397)[g.sub.3](t) + (-0.03841, 0.00069)[g.sub.4](t) + (-0.00255, 0)[g.sub.5](t)

and requiring [[alpha].sup.*] = 0.75, the discount function is

[f.sub.t] = (1, 0) + (-0.04280, 0.00843)[g.sub.1](t) + (-0.03874, 0.00086)[g.sub.2](t) +(-0.04675, 0.00793)[g.sub.3](t) + (-0.03841, 0.00139)[g.sub.4](t) + (-0.00255, 0)[g.sub.5](t).

Table 2 shows the spot and forward rates for the next 15 years with OLS and fuzzy regressions (in the last case, for [[alpha].sup.*] = 0.5, 0.75). We can see that the parameter [[alpha].sup.*] can be interpreted as an indicator of the perceived uncertainty in the market. If [[alpha].sup.*] increases, uncertainty of the observations for the explained variable (price of the bonds) also increases (see Wang and Tsaur, 2000) and the spread of the subsequent estimates of the spot rates and the forward rates will be wider.

To compare the difference between the OLS estimates and the estimates of fuzzy regression for discount rates, Table 2 includes coefficient D = 100 x [absolute value of [V.sup.E] - [V.sup.F]]/[V.sup.E], where [V.sup.E] is the value of a yield rate obtained with econometric methods and [V.sup.F] is the center of the fuzzy estimate for this rate. We can see that the estimates of the spot rates with each method are quite similar. This similarity decreases when estimating forward rates.

USING A FUZZY TSIR FOR FINANCIAL AND ACTUARIAL PRICING

Calculating the Present Value of an Annuity

Buckley (1987) determines the present value of a stream of amounts when these amounts and the discount rate are given by FNs. Buckley supposes that the discount rate to be applied throughout the evaluation horizon was a unique FN i. It implies that the reference TSIR is flat. If we also suppose that amounts are given by means of nonnegative FNs, in which the tth cash flow is the FN [C.sub.t], and that they are an immediate postpayable annuity with an annual periodicity, then the net present value of that annuity is the following FN, V:

(46) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It is often impossible to obtain the membership function of V by means of the extension principle. However, it is fairly straightforward to obtain a closed expression of the [alpha]-cuts of the present value. If we bear in mind that the function of present value is continuously decreasing (increasing) with respect to the discount rate (the amounts), we can calculate the upper and lower extremes of its [alpha]-cuts immediately from (7). If the amounts and the cash flows are given by STFNs, 19 will not be an STFN but it can be approximated by an STFN from (9).

It is well known that it is quite unrealistic to suppose a flat TSIR. Moreover, in the previous section we proposed a method for obtaining an empirical fuzzy TSIR and discussed how to obtain its spot and implied rates. So, (46) can be generalized to any shape of the TSIR, using the spot rates, the forward rates, or the discount function

(47) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Of course, from the [alpha]-cuts of the amounts and the discount rates (or alternatively the discount function), we can easily obtain 19 from (7). Moreover, if the amounts are crisp (e.g., if they are the amounts paid by a bond), and to obtain their present value we use a discount function quantified via an STFN like (25), (47) can be written as

(48) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Example: Suppose an investor is going to buy bonds of the Spanish public debt market on June 29, 2001. The maturity of these bonds is 5 years, and they offer a 5 percent annual coupon. The values of 1 monetary unit with maturities from 1 to 5 years obtained from the regression with [[alpha].sup.*] = 0.5 are given in Table 3. Therefore, the investor obtains the following preliminary price for one of these bonds from (48):

P = 5 * (0.9585, 0.0030) + 5 * (0.9190, 0.0040) + 5 * (0.8770, 0.0059) + 5 * (0.8315, 0.0093) + 105 * (0.7858, 0.0128) = (100.44, 1.46).

Pricing Life-Insurance Contracts

In this subsection, we show how to obtain the net single premium for some life-insurance contracts (n-year pure endowments, n-year term life-insurance contracts, and n-year endowments--the combination of the two first contracts) from our fuzzy TSIR. To simplify the analysis, we will suppose that the insured amounts are fixed beforehand and that they are annual and payable at the end of each year of the contract. Taking into account that standard life-insurance mathematics establishes that the net single premium for a policy is the discounted value of the mathematical expectation of the guaranteed amounts, and naming as [C.sub.n] the amount payable at the end of an n-year pure endowment, the premium for an individual aged x for this contract, [[PI].sub.1], is

(49) [[PI].sub.1] = [C.sub.n] * [(1 + [i.sub.t]).sup.-n] * [sub.n][p.sub.x] = [C.sub.n] * [f.sub.n] * [sub.n][p.sub.x],

where [sub.n] [p.sub.x] stands for the probability that an individual aged x attains at the age x + n.

If we suppose that for an n-year term life insurance the amounts are payable at the end of the year of death, for an insured person aged x, the premium, [[PI].sub.2], is

(50) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [sub. t|][q.sub.x] stands for the probability of death at age x + t and [C.sub.t] the insured amount for this event.

The net single premium for an n-year endowment, [[PI].sub.3], is obtained from (49) and (50) as:

(51) [[PI].sub.1] = [[PI].sub.1] + [[PI].sub.2]

So, if the discount function estimated by an STFN, the net single premium for an n-year pure endowment, (49), is reduced to

(52) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and then, for an n-year term life insurance, (50), we obtain

(53) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then, from (51), (52), and (53) we obtain the price of an n-year endowment, [[PI].sub.3]:

(54) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Table 4 shows the fuzzy pure single premiums of the three types of policies for people of several ages and with the value of the discount function in Table 3. In all cases the duration of the contracts is 5 years. We suppose that the amounts of the insured events are 1,000 monetary units and for the calculations we have taken the Swiss GRM-82 mortality tables.

The maturities of the bonds used to estimate the TSIR, and obviously the temporal horizon covered with this TSIR may not be large enough to discount all the cash flows (e.g., when pricing a whole life annuity). To complete the interest rates for the maturities of the TSIR that are not covered in the regression, de Andres (2000) suggested two solutions:

(1) The first involves estimating subjectively a unique nominal interest rate for the longer maturities by Fisher's relationship. In this way, Devolder (1988) suggested obtaining the discount rate for long-term insurance policies by using Fisher's relationship as: nominal interest rate = real interest rate + [lambda] x anticipated inflation, where 0 [less than or equal to] [lambda] < 1. Regarding the real interest rate, Devolder stated that "generally it must be quantified between the 2 and 3 percent" and the anticipated inflation "must be reasonable in the long term." Clearly, these sentences allow a fuzzy quantification even though this is probably not the aim of the author. If we call i the FN that quantifies the discount rate, we can obtain this, e.g., as i = (0.025, 0.01) + [lambda][pi], where [pi] stands for the anticipated inflation.

(2) From financial logic, the shape of the TSIR must be asymptotic. So, a second solution involves taking the latest forward (or spot) rate of our estimated TSIR as a reference for the rate to discount the amounts whose maturities are not covered by the TSIR.

Fuzzy Financial Pricing of Property-Liability Insurance

In this subsection we will show how to apply our fuzzy TSIR when pricing property-liability insurance. For a wide discussion on this topic, consult Myers and Cohn (1987) (MC) and Cummins (1990) under a nonfuzzy environment or Cummins and Derrig (1997) (CD) under a fuzzy environment. To simplify our explanation, we will suppose only a three-period model. The MC model states that the present value of the premiums must compensate the cost of the liabilities and the taxes for the insurer. Supposing a single premium, only two periods (years) in claiming, but that the TSIR can have any shape, the CD formulation can be transformed into

(55) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where P = pure single premium. The parameter [delta] ([delta] > 0) indicates the proportion of the fair premium corresponding to the surplus and [tau] is the tax rate that we suppose the same for the underwriting profit and the investment income. L is the total amount of the claim cost whereas [c.sub.t] = the proportion of the liabilities payable at the tth year. Therefore [c.sub.1] + [c.sub.2] = 1. Notice that we have supposed that the proportion of the claims cost in the tth year deductible from income taxes is equal to [c.sub.t].

If [f.sub.t] is the present value of 1 unitary unit payable at t with the spot rate free of risk of failure for that maturity ([i.sub.t]), then [f.sub.t] = [(1+ [i.sub.t]).sup.-t]. Similarly, [f.sup.(L).sub.t] is the value of the discount traction for one monetary unit of liability payable at the tth year. This can be obtained from the spot rate of the liabilities at time t, [i.sup.(L).sub.t]. Therefore, [f.sup.(L).sub.t] = [(1 + [i.sup.(L).sub.t]).sup.-t]. To simplify our discussion, we will suppose that [i.sup.(L).sub.t] is obtained by applying over it [i.sub.t] the risk loading [k.sub.t], by doing [i.sup.(L).sub.t] = (1 - [k.sub.t])(1 + [i.sub.t]) - 1, where 1 > [k.sub.t] > 0. From this relation, the following connection between [f.sup.(L).sub.t] and [f.sub.t] arises

(56) [f.sup.(L).sub.t] = [(1 - [k.sub.t]).sup.-t] [f.sub.t].

Finally, [r.sub.t], t = 1, 2 is the return obtained by investing the premium within the tth year of the contract. If we assume, as is usual, that this return corresponds to the risk-free rate, within a pure expectations framework, [r.sub.t] can be quantified by the forward rate for the tth year of the contract. Then, taking into account that [r.sub.t] can be obtained from the values of spot discount function [f.sub.t-1] and [f.sub.t] as

(57) [r.sub.t] = ([f.sub.t-1]/[f.sub.t]) - 1.

Then, (55) can be rewritten from (56), (57) and remembering that [c.sub.1] + [c.sub.2] = 1 as

(58) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and so the value of the pure single premium is

(59) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We wish to show how to introduce fuzziness into the future behavior of interest rates when pricing a property-liability contract. The fuzziness of this behavior is introduced by the discount function, which is fuzzy. Moreover, we will allow the total amount of the liabilities, L, to be estimated by an STFN, i.e., L = ([L.sub.C], [L.sub.R]). This is easy to interpret from an intuitive point of view: the actuary estimates that the total cost of the claims will be "around [L.sub.C]." We will suppose that [c.sub.t], [k.sub.t], [tau], and [delta] are crisp parameters.

If the discount factor and the total cost of the claims are done by FNs, we will actually obtain a fuzzy premium, which must be obtained by solving the fuzzy version of Equation (55):

(60) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Buckley and Qu (1990c) suggested obtaining the solution of a fuzzy equation from the solution of its crisp version. So, we will obtain P from (59) as

(61) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

To determine the [alpha]-cuts of P, [P.sub.[alpha]], we must bear in mind that in Equation (59) the premium is a function of the value of the liabilities and the free risk discount function (or, alternatively a function of the expected evolution of interest rates), i.e., the premium in Equation (59) can be denoted as P = f(L, [f.sub.1], [f.sub.2]). Clearly, f(*) is an increasing function of L. On the other hand, it is not easy to know whether the premium increases (or decreases) when the discount function increases (the interest rates decreases) from the partial derivatives of P(L, [f.sub.1] [f.sub.2]). However, we do know by financial intuition that the price of the insurance is basically related to the present value of the liabilities, and that it is clearly increasing (decreasing) with respect to the values of the discount function (spot rates). Therefore, if we start from the discount function that define our TSIR, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we can obtain [P.sub.[alpha]] by applying (7) as

(62) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

However, from (9), we can approximate P by a STBN, i.e., P [approximately equal to] ([P.sub.C], [P.sub.R]) where

(63) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In the following example we consider the following variables of a property-liability insurance: [k.sub.1] = [k.sub.2] = 1%, [delta] = 5%, and [tau] = 34% and the values of the discount function in Table 3. The duration of the contract is 2 years and the total amount of the liability for one contract is L = (1000, 50). Table 5 shows the fuzzy premiums for several pairs ([c.sub.1], [c.sub.2]) when using the approximating formula (63) of L.

Figure 5 represents the shapes of the true fuzzy premium (obtained from (62) and represented with a solid line) and our approximation (63), for the distribution of the claims [c.sub.1] = 0.5 and [c.sub.2] = 0.5. Clearly, the triangular approximation fits the real value of the premium well and is easier to interpret: the value of the fair premium must be 992.37 but there may be acceptable deviations no longer than 52.95.

[FIGURE 5 OMITTED]

DISCUSSING FURTHER ACTUARIAL APPLICATIONS OF FUZZY REGRESSION

In this section, we reflect on other applications of fuzzy regression to insurance problems. We will focus our discussion on the specific problem of estimating the incurred but not reported claims reserves, but we will also suggest other (in our opinion) promising applications.

Calculating the Incurred But Not Reported Claims Reserves (IBNR reserves) is a classic topic in nonlife-insurance mathematics. Unfortunately, they may not be calculated from a wide statistical database. Straub (1997) states that taking into account experiences that are too far from the present can lead to unrealistic estimates. For example, if the claims are related to bodily injuries, the future losses for the company will depend on the behavior of the wage index grown that will be taken to determine the amount of indemnification, changes in court practices, and public awareness of liability matters.

To calculate the IBNR reserve we begin from the historical data ordered in a triangle like that in Table 6.

In Table 6, [Z.sub.i,j] is the accumulated incurred losses of accident year j at the end of development year i where j = 1 denotes the most recent accident year and j = n the oldest accident year. Obviously, we do not know, for the jth year of occurrence, the accumulated losses in the development years i = j + 1, ..., n and therefore, these losses must be predicted. It is well known that the classical method of predicting these losses is the Chain Ladder (CL). However, as it is pointed out in the survey England and Verrall (2002) claims reserving methods, during the last years greater interest of actuarial literature has been focused not only in calculating the best estimate of claim reserves but also on determining its downside potential from a stochastic perspective. Obviously, the final objective is to provide the actuary a well-founded mathematical tool to determine solvency margins for the reserves.

One way to focus this problem consists of departing from the pure CL method and making statistical refinements over it. In this way, Benjamin and Eagles (1986) propose a slight generalization of the CL method, known as London Chain Ladder (LCL), which is based on the use of OLS regression over the accumulated claims. On the other hand, Mack (1993) and England and Verrall (1999) do not suppose a concrete structure of the underlying data. Concretely, Mack (1993) provides analytical expressions for the prediction errors in claims and reserve estimates whereas England and Verrall (1999) propose combining standard CL estimates and bootstrapping techniques to determine the variance of the error in the predicted reserves.

Another extended way to focus this problem consists of modeling the incremental claims as random variables with a predefined distribution function. So, while Wright (1990) and Renshaw and Verrall (1998) use the Poisson random variables, Kremer (1982), Renshaw (1989), and Verrall (1989) use a log-normal approach. Then, the subsequent question to answer is how to define the associated parameters to these random variables with respect to the year of underwriting and delay period. There are several approaches in the literature to do it. The more extended way consists of using one separate parameter for each development period. However, to avoid over-parameterization, some articles propose using parametric expressions (e.g., the Hoerl curve) or nonparametric smoothing methods like the one given by England and Verrall (2001). It must be remarked that, as it is pointed out in Mack (1993), in spite of the fact that these approaches are based on CL "philosophy," some of them present fundamental differences with respect to the pure CL method.

Our fuzzy sets approach to determine the value and variability of IBNR will be based on combining the generalization of CL by Benjamin and Eagles (1986), the LCL, and fuzzy regression. So, let us briefly expose Benjamin and Eagle's method. This is built from the hypothesis that the evolution of the claims of the accidents occurred in the year j from the ith year to the i+1th year of development can be approximated by the linear relation:

(64) [Z.sub.i+1,j] = [b.sub.i] + [c.sub.i] [Z.sub.i,j] + [[epsilon].sub.I],

where [b.sub.i] is the intercept, [c.sub.i] is the slope, and [[epsilon].sub.i] is the perturbation term. The coefficients [b.sub.i] and [c.sub.i] must be estimated by OLS. Of course, the estimates of [b.sub.i] and [c.sub.i], [b.sub.i] and [c.sub.i], respectively, must be obtained from the observations in the IBNR-triangle, i.e., from the pairs [{([Z.sub.i+1,j]; Z.sub.[i,j])}.sub.j[greater than or equal to]i]. Notice that the CL method is a special case of the LCL method when we consider that [b.sub.i] = 0.

It is easy to check that the amount of the whole claims for the accidents occurred in the jth year at the end of the n years of development, [Z.sub.n,j], is

(65) [Z.sub.n,j] = [b.sub.n-1] + [c.sub.n-1] { ... [b.sub.j+2] + [c.sub.j+2][[b.sub.j+1] + [c.sub.j+1]([b.sub.j] + [c.sub.j][Z.sub.j,j])]}.

If we suppose that the expansion of the claims produced by the accidents in a given year is done within n years, then [Z.sub.n,j] is the estimate of the amount of all the claims corresponding to the year of occurrence j. Therefore, the IBNR reserves corresponding to the accidents of the year j, [R.sub.j], is obtained by calculating the difference between [Z.sub.n,j] and the amount of the claims reported, [Z.sub.j,j], i.e., [R.sub.j] = [Z.sub.n,j] - [Z.sub.j,j] So, the total IBNR reserve (R) (corresponding to all the accidents from year 1 to year n) is (7)

(66) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We think that this approach has several drawbacks. First, OLS is useful when we start from a wide sample, but it is not advisable for calculating IBNR reserves. Similarly, using all the information available in the IBNR-triangle requires estimating [Z.sub.n,j] not by exact values but by means of probabilistic confidence intervals. Unfortunately, this requires a great computational effort. In any case, we think that these considerations can be extended to the statistical methods discussed above.

We will show that adapting LCL method to fuzzy regression can be a suitable alternative. It will allow us to use all the information provided by the IBNR-triangle more efficiently. So, let us assume that the evolution of the accumulated claims of the accidents happened in the year j from the ith to the i+1th developing years can be adjusted using a fuzzy linear relation [Z.sub.i+1,j] = [b.sub.i] + [c.sub.i] [Z.sub.i,j]. If we state that [b.sub.i] and [c.sub.i] are the STFN ([b.sub.iC], [b.sub.iR]), ([C.sub.iC], [C.sub.iR]), respectively, we can write

(67) [Z.sub.i+1,j] = ([Z.sub.(i+1,j)C], [Z.sub.(i+1,j)R]) = ([b.sub.iC], [b.sub.iR]) + ([c.sub.iC], [c.sub.iR]) [Z.sub.i,j] = ([b.sub.iC] + [c.sub.iC] [Z.sub.i,j], [b.sub.iR] + [c.sub.iR] [Z.sub.i,j]).

The estimates of [b.sub.i] and [c.sub.i], are symbolized as [b.sub.i] = ([b.sub.iC], [b.sub.iR]) and [c.sub.i] = ([c.sub.iC], [c.sub.iR]), respectively. Then, the prediction of the final cost of the accidents produced in year j, [Z.sub.n,j], is obtained from (65) as

(68) [Z.sub.n,j] = [b.sub.n-1] + [c.sub.n-1] {... [b.sub.j+2] + [c.sub.j+2] [[b.sub.j+1] + [c.sub.j+1] ([b.sub.j] + [c.sub.j] [Z.sub.j,j])]}.

Clearly, [Z.sub.n,j] is not an STFN, but it can be approximated reasonably well by a TSFN, i.e., [Z.sub.n,j] [approximately equal to] ([Z.sub.(n,j)C], [Z.sub.(n,j)R]). To do this, we must, in the fuzzy recursive calculation (68), use the approximating formula (11) for multiplication between two STFN. Finally, we obtain the IBNR reserve for the jth year of occurrence as the FN [R.sub.j] = ([R.sub.jC], [R.sub.jR]):

[R.sub.j] = [Z.sub.n,j] - [Z.sub.j,j] = ([Z.sub.(n,j)C] - [Z.sub.(n,j)R]). (69)

Therefore, the whole IBNR reserve is the STFN R = ([R.sub.C], [R.sub.R]), which is obtained from (66) as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

To illustrate our proposal, we develop the following example, which is similar to the one in Straub (1997, p. 106). Table 7 shows the IBNR triangle we will use.

Table 8 shows the main results when nonfuzzy methods are used. The development coefficients when we use fuzzy regression with an inclusion level [alpha] = 0.5 and the fuzzy IBNR reserves are given in Table 9.

Let us illustrate how we have used fuzzy regression. If we use the fuzzy LCL with intercept, to obtain the development coefficients from the year i = 1 to the year i = 2, i.e., the STFBs [b.sub.1] = ([b.sub.1C], [b.sub.1R]) and [c.sub.1] = ([c.sub.1C], [c.sub.1R]), and taking a level of inclusion [alpha] [greater than or equal to] [[alpha].sup.*] = 0.5, we must take the data in the second row of Table 7 and solve the following linear program:

Minimize 5 [b.sub.1R] + 690[c.sub.1R]

subject to:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and solving [b.sub.1C] = -12; [b.sub.1R] = 0; [c.sub.1C] = 1.771; [c.sub.1R] = 0.057.

Finally, we would like to mention some other promising applications of fuzzy regression in actuarial science. One useful application may be in trending claims costs. (8) When projecting future claims costs, standard actuarial practice uses time trend models. More academic approaches propose regressing the cost of the claims with respect to the value of an economic index (e.g., the consumer price index). So, to obtain the final prediction of the claims amount at a future moment we need to predict the value of the economic index at that moment. To do this, time trending may again be used. An alternative is to employ ARIMA time-series models.

Several instruments are derived from the fuzzy regression model in this article that may provide suitable solutions. Watada (1992) proposed a fuzzy model for time series analysis based on polynomial time trending with STFNs. Tseng et al. (2001) combined conventional ARIMA models with Tanaka and Ishibuchi's regression method. Obviously, in both cases these fuzzy methods will lead to forecasting with STFNs.

Another way of determining the discount rate of the liabilities is to use CAPM (see, e.g., Taylor, 1994, for a review of CAPM applied to insurance). With CAPM, the beta of the liabilities ([[beta].sub.L]) is obtained from the beta of the asset portfolio ([[beta].sub.A]) and the insurer's equity ([[beta].sub.E]), i.e., [[beta].sub.L] = f([[beta].sub.A], [[beta].sub.E]) and [[beta].sub.L] < 0. Cummins and Derrig (1997, p. 25) suggested fuzzifying the statistical estimates of the betas since, as they point out, there are several sources that disturb their quantification. Fuzzy regression provides a suitable way of doing this fuzzifycation and enables the decision maker to grade the level of uncertainty in the final estimates by choosing the appropriate level of congruency in fuzzy regression ([alpha].sup.*).

CONCLUSIONS

Actuarial pricing requires an estimate of the behavior of future interest rates that are unknown when the valuation is made. In the last few years, the analysis of temporal structure of interest rates has become an important topic in actuarial science. Likewise, several articles in the actuarial literature consider that estimating the discount rates using fuzzy numbers is a good alternative. With this in mind, we have attempted to make up for the uncertainty about estimating interest rates with fuzzy numbers, i.e., to develop the hypothesis "an 'expert' subjectively estimates the yield rates." If we accept that the "experts" are the traders of the fixed income markets, these subjective estimates are implied in the price of the debt instruments and, therefore, in the yield curve of these instruments. Our method quantifies the experts' subjective estimates of the spot rates and spot interest rates for the future (the forward rates) with fuzzy numbers. This method, based on a fuzzy regression technique, uses all the prices of the bonds negotiated throughout one session in such a way that we do not lose any information. On the other hand, when we use an econometric method we must reduce these prices to representative ones, thus losing some information. We would like to remark that the results of our method are similar to those of Carriere (1999), but that he suggests fitting the yield curve using probabilistic confidence interval values whereas we fit the temporal structure of interest rates with fuzzy numbers. So, if we represent these fuzzy numbers by their [alpha]-cuts, we obtain a yield curve described by a direct analog of confidence intervals.

We have adjusted the yield curve with symmetrical TFNs because the arithmetic is easy and interpreting the estimates is intuitive because they are well adapted to the way people make predictions. An interest rate given by (0.03, 0.005) indicates that we expect an interest rate of about 3 percent, and that we do not expect deviations from it to be greater than 50 basis points. We then discussed how to use a fuzzy TSIR to calculate the present value of a stream of nonrandom amounts and the net single premium for some life and property-liability insurance contracts. We showed that to obtain the fuzzy prices is easy because our method fits the TSIR by a discount function that is described with symmetrical TFNs.

Finally, we discussed other actuarial problems where applying fuzzy regression is promising. We concentrated our discussion on calculating the IBNR reserves but also indicated other future areas of research such as trending cost claims and estimating the beta of liabilities.

Notice that when our initial data is given by fuzzy numbers, the estimate of the objective magnitude (the premiums, the IBNR reserve, etc.) is not an exact value but a fuzzy number. For example, in "Fuzzy Financial Pricing of Property-Liability Insurance" the value of the premium for property-liability insurance was (992.37, 52.36). This can be understood as "the premium must be approximately 992.37." To obtain the definitive value of the magnitude, it must be transformed into a crisp value. To do this we need to apply a defuzzifying method--see Zhao and Govind (1990) for a wide discussion of fuzzy mathematics, and Cummins and Derrig (1997) or Terceno et al. (1996) for applications in fuzzy-actuarial analysis. Another way of doing this, which is very consistent with practice in the real world, is to consider the fuzzy quantification as a first approximation that allows a margin for the "actuarial subjective judgment" or upper and lower bounds for acceptable market prices. Finally, the actuary must use his/her intuition and experience to establish the crisp value of the fuzzy estimate. For example, for premium (992.37, 52.36), the actuary might decide that a final price 1017.37 is acceptable but 928 is not.

APPENDIX

Obtaining the Forward Rates From a Fuzzy TSIR

The forward rate for the tth year, [r.sub.t], can be obtained from the fuzzy equation

(A1) [f.sub.t-1] [(1 + [r.sub.t]).sup.-1] = [f.sub.t].

To solve this equation, we identify [(1 + [r.sub.t]).sup.-1] = [G.sub.t], where [G.sub.t] is the value in t-1 of one monetary unit payable at t according to the TSIR. Bearing in mind that the value of the discount function for any t is an STFN [f.sub.t] = ([f.sub.tC], [f.sub.tR]), the above equation can therefore be written using the [alpha]-cuts as:

(A2) [[f.sub.(t-1)C] - [f.sub.(t-1)R] (1 - [alpha]), [f.sub.(t-1)C] + [f.sub.(t-1)R](1 - [alpha])] * [[G.sup.1.sub.t]([alpha]), [G.sup.2.sub.t] ([alpha])] = [[f.sub.tC] - [f.sub.tR](1 - [alpha]), [f.sub.tC] + [f.sub.tR] (1 - [alpha])].

It is easy to see that in Equation (A2):

(A3) [G.sup.1.sub.t]([alpha]) = [(1 + [r.sup.2.sub.t]([alpha]).sup.-1] and [G.sup.2.sub.t]([alpha]) = [(1 + [r.sup.1.sub.t]([alpha]).sup.-1].

The solution of the [alpha]-cut Equation (A3) is: (9)

(A4) [G.sub.t[alpha]] = [[G.sup.1.sub.t]([alpha]), [G.sup.2.sub.t]([alpha])] = [[f.sub.tC] - [f.sub.tR](1 - [alpha]) / [f.sub.(t-1)C] - [f.sub.(t-1)R] (1 - [alpha]), [f.sub.tC] + [f.sub.tR](1 - [alpha]) / [f.sub.(t-1)C] + [f.sub.(t-1)R(1-[alpha])].

Unfortunately, the solution (A4) might not exist (i.e., the expression (A4) does not correspond to a confidence interval). For example, consider that (A4) for a predefined a is given by [0.9, 0.95]. [[G.sup.1.sub.t]([alpha]), [G.sup.2.sub.t]([alpha])] = [0.875, 0.9]. Then [[G.sup.1.sub.t]([alpha]), [G.sup.2.sub.t]([alpha])] = [0.972, 0.947]. Clearly, it is not a confidence interval since 0.972 < 0.947. From Buckley and Qu (1990a, p. 46, Theorem 3), the necessary and sufficient condition for the existence of [G.sub.t] is

(A5) [f.sub.tR] / [f.sub.(t-1)R] [greater than or equal to] [f.sub.tC] / [f.sub.(t-1)C] ?? [f.sub.(t-1)C] * [f.sub.tR] [greater than or equal to] [f.sub.tC] * [f.sub.(t-1)R].

If we define P in Equations (40d), (40e), (400, and (40g) as being lower than or equal to the periodicity used to obtain the spot rates and forward rates, and [g.sub.j(t)] as nondecreasing, it is easy to check that [f.sub.tR] [greater than or equal to] [f.sub.(t-1)R]. Moreover, if we combine the constraints (40d) and (40e), we see that [f.sub.(t-1)C [greater than or equal to] [f.sub.tC]. Therefore, [f.sub.(t-1)C] * [f.sub.tR] [greater than or equal to] [f.sub.tC] * [f.sub.(t-1)R], and we can obtain the [alpha]-cuts of [G.sub.t] with (A4).

Then, from (A3) and (A4), we obtain the [alpha]-cuts of [r.sub.t], [r.sub.t[alpha]] by making:

(A6) [r.sub.t[alpha]] = [[r.sup.1.sub.t]([alpha]), [r.sup.2.sub.t]([alpha])] = [[f.sub.(t-1)C] + [f.sub.(t-1)R](1 - [alpha]) / [f.sub.tC] + [f.sub.tR](1 - [alpha]) - 1, [f.sub.(t-1)C - [f.sub.(t-1)R(1 - [alpha]) / [f.sub.tC] - [f.sub.tR](1 - [alpha]) - 1].

It is easy to check that the extremes of [r.sub.t[alpha]], are not given as a linear expression of [alpha]. However, if we approximate the functions [r.sup.1.sub.t]([alpha]), [r.sup.2.sub.t]([alpha]) using Taylor's expansion to the first grade from [alpha] = 1, then:

(A7) [r.sup.1.sub.t]([alpha]) [approximately equal to] [f.sub.(t-1)C] / [f.sub.tR] - 1 - [f.sub.(t-1)C] * [f.sub.tR] - [f.sub.tC] * [f.sub.(t-1)R] / [([f.sub.tC]).sup.2](1 - [alpha]) and [r.sup.2.sub.t]([alpha]) [approximately equal to] [f.sub.(t-1)C] / [f.sub.tR] - 1 + [f.sub.(t-1)C] * [f.sub.tR] - [f.sub.tC] * [f.sub.(t-1)R] / [([f.sub.tC]).sup.2](1 - [alpha]).

In conclusion, from (A7) it is easy to check that [r.sub.t] can be approximated by an STFN [r.sub.t] [approximately equal to] ([r.sub.tC], [r.sub.tR]) where:

(A8) [r.sub.tC] = [f.sub.(t-1)C] / [f.sub.tR] - 1 and [r.sub.tR] = [f.sub.(t-1)C] * [f.sub.tR] - [f.sub.tC] * [f.sub.(t-1)R] / [([f.sub.tC]).sup.2](1 - [alpha]).

(1) The TFNs are a special case of a wider family of FNs called L-R FNs. For a detailed explanation see Dubois and Prade (1980).

(2) If [l.sub.A] = [r.sub.A] = 0, A is the crisp number [a.sub.C].

(3) Many methods for ordering fuzzy numbers are proposed in the literature. Obviously, the choice of method depends on the problem. For our purpose we have chosen Ramik and Rimanek's criteria.

(4) Clearly, the subtraction of two STFNs is an STFN because the subtraction is the sum of the first FN with the second one multiplied by -1.

(5) Fedrizzi, Fedrizzi, and Ostasiewicz (1993) initially suggested fuzzy regression methods for economics. Some economic applications can be found in Ramenazi and Duckstein (1992) or Profillidis, Papadopoulos, and Botzoris (1999).

(6) For a detailed explanation of how we constructed our [g.sub.j](t) and chose our knots, see McCulloch (1971).

(7) Notice that with this approach, we do not consider investment income (i.e., the investment income is assumed to be 0%). However, this is the traditional approach to IBNR reserves. To simplify our explanation, we will continue with this hypothesis.

(8) For an extensive exposition see Cummins and Derrig (1993).

(9) In this case we apply the so-called solution "classical solution" in fuzzy sets literature. This solution does not always exist but as we will specify the functions [g.sub.j] as nondecreasing, we can ensure that it does, as we will show.

REFERENCES

Ang, A., and M. Sherris, 1997, Term Structure Models: A Perspective from the Long Rate, North American Actuarial Journal, 3(3): 122-138.

Bababbell, D. F., and C. B. Merrill, 1996, Valuation of Interest-Sensitive Financial Instruments (Schaumburg: Society of Actuaries).

Benjamin, S., and L. M. Eagles, 1986, Reserves in Lloyd's and the London Market, Journal of the Institute of Actuaries, 113(2): 197-257.

Buckley, J. J., 1987, The Fuzzy Mathematics of Finance, Fuzzy Sets and Systems, 21: 57-73.

Buckley, J. J., and Y. Qu, 1990a, Solving Linear and Quadratic Fuzzy Equations, Fuzzy Sets and Systems, 38: 43-59.

Buckley, J. J., and Y. Qu, 1990b, On Using [alpha]-Cuts to Evaluate Fuzzy Equations, Fuzzy Sets and Systems, 38: 309-312.

Buckley, J. J., and Y. Qu, 1990c, Solving Fuzzy Equations: A New Solution Concept, Fuzzy Sets and Systems, 39: 291-301.

Carriere, J. F., 1999, Long-Term Yield Rates for Actuarial Valuations, North American Actuarial Journal, 3(3): 13-24.

Chambers, D. R., W. T. Carleton, and D. W. Waldman, 1984, A New Approach to Estimation of the Term Structure of Interest Rates, Journal of Financial and Quantitative Analysis, 19: 233-252.

Cummins, J. D., 1990, Multi-Period Discounted Cash Flow Rate-Making Models in Property-Liability Insurance, Journal of Risk and Insurance, 57(1): 79-109.

Cummins, J. D., and R. A. Derrig, 1993, Fuzzy Trends in Property-Liability Insurance Claim Costs, Journal of Risk and Insurance, 60(3): 429-465.

Cummins, J. D., and R. A Derrig, 1997, Fuzzy Financial Pricing of Property-Liability Insurance, North American Actuarial Journal, 1(4): 21-44.

de Andres, J., 2000, Estimating the Temporal Structure of Interest Rates Using Fuzzy Numbers. Application to Financial and Actuarial Valuation and to the Analysis of the Life Insurer's Solvency, Rovira i Virgili University, Unpublished doctoral thesis (in Spanish).

Delbaen, F., and S. Lorimier, 1992, Estimation of the Yield Curve and the Forward Rate Curve Starting From a Finite Number of Observations, Insurance: Mathematics and Economics, 11: 259-269.

Derrig, R. A., and K. M. Ostaszewski, 1995, Fuzzy Techniques of Pattern Recognition in Risk and Claim Classification, Journal of Risk and Insurance, 62(3): 447-482.

Derrig, R. A., and K. M. Ostaszewski, 1997, Managing the Tax Liability of a Property Liability Insurance Company, Journal of Risk and Insurance, 64: 695-711.

Devolder, P., 1988, Le taux d'actualisation en assurance, The Geneva Papers on Risk and Insurance, 13: 265-272.

Dubois, D., and H. Prade, 1980, Fuzzy Sets and Systems: Theory and Applications (New York: Academic Press).

Dubois, D., and H. Prade, 1993, Fuzzy Numbers: An Overview, in: D. Dubois, H. Prade, and R. R. Yager, eds., Fuzzy Sets for Intelligent Systems (San Mateo, Calif.: Morgan Kaufmann Publishers), pp. 113-148.

England, P. D., and R. J. Verrall, 1999, Analytic and Bootstrap Estimates of Prediction Errors in Claim Reserving, Insurance: Mathematics and Economics, 25: 281-293.

England, P. D., and R. J. Verrall, 2001, A Flexible Framework for Stochastic Claims Reserving, Proceedings of the Casualty Actuarial Society, pp. 1-38.

England, P. D., and R. J. Verrall, 2002, Stochastic Claims Reserving in General Insurance, Paper presented to the Institute of Actuaries, Available at http://www. actuaries.org.uk/sessional/sm0201.pdf.

Fedrizzi, M., M. Fedrizzi, and W. Ostasiewicz, 1993, Towards Fuzzy Modelling in Economics, Fuzzy Sets and Systems, 54: 259-268.

Gerber, H. U., 1995, Life Insurance Mathematics (Heildelberg: Springer).

Kaufmann, A., 1986, Fuzzy Subsets Applications in O.R. and Management, in: A. Jones, A. Kaufmann, and H.-J. Zimmermann, eds., Fuzzy Set Theory and Applications (Dordretch: Reidel), pp. 257-300.

Kremer, E., 1982, IBNR Claims and the Two-Way of ANOVA, Scandinavian Actuarial Journal, I: 47-55.

Lemaire, J., 1990, Fuzzy Insurance, ASTIN Bulletin, 20: 33-55.

Li Calzi, M., 1990, Towards a General Setting for the Fuzzy Mathematics of Finance, Fuzzy Sets and Systems, 35: 265-280.

Mack, T., 1993, Distribution-Free Calculation of the Standard Error of Chain-Ladder Reserve Estimates, ASTIN Bulletin, 23: 213-223.

McCulloch, J. H., 1971, Measuring the Term Structure of Interest Rates, The Journal of Business, 34: 19-31.

McCulloch, J. H., 1975, The Tax-Adjusted Yield Curve, Journal of Finance, 30: 811-829.

Myers, S.C., and R.A. Cohn, 1987, A Discounted Cash-Flow Approach to Property-Liability Insurance Rate Regulation, in: J. D. Cummins and S. E. Harrington, eds., Fair Rate of Return in Property-Liability Insurance (Norwell, Mass.: Kluwer Academic Publishers).

Ostaszewski, K., 1993, An Investigation into Possible Applications of Fuzzy Sets Methods in Actuarial Science (Schaumburg: Society of Actuaries).

Profillidis, V. A., B. K. Papadopoulos, and G. N. Botzoris, 1999, Similarities in Fuzzy Regression and Application on Transportation, Fuzzy Economic Review, 4(1): 83-98.

Ramenazi, R., and L. Duckstein, 1992, Fuzzy Regression Analysis of the Effect of University Research on Regional Technologies, in: J. Kacprzyk and M. Fedrizzi, eds., Fuzzy Regression Analysis (Heildelberg: Physica-Verlag), pp. 237-263.

Ramik, J., and J. Rimanek, 1985, Inequality Relation Between Fuzzy Numbers and its Use in Fuzzy Optimization, Fuzzy Sets and Systems, 16: 123-138.

Renshaw, A. E., 1989, Chain-Ladder Modelling and Interactive Modelling (Claims Reserving and GLIM), Journal of the Institute of Actuaries, 116: 559-587.

Renshaw, A. E., and R. J. Verrall, 1998, A Stochastic Model Underlying the Chain Ladder Technique, British Actuarial Journal, 4: 903-923.

Straub, E., 1997, Non-Life Insurance Mathematics (Berlin: Springer).

Tanaka, H., 1987, Fuzzy Data Analysis by Possibilistic Linear Models, Fuzzy Sets and Systems, 24: 363-375.

Tanaka, H., and H. Ishibuchi, 1992, A Possibilistic Regression Analysis Based on Linear Programming, in: J. Kacprzyk and M. Fedrizzi, eds., Fuzzy Regression Analysis (Heildelberg: Physica-Verlag), pp. 47-60.

Taylor, G. C., 1994, Fair Premium Rating Methods and the Relations Between Them, Journal of Risk and Insurance, 61(4): 592-615.

Terceno, A., G. Barbera, J. de Andres, and C. Belvis, 1996, Fuzzy Methods Incorporated to the Study of Personal Insurances, Fuzzy Economic Review, 1(2): 105-119.

Tseng, F.-M., G.-H. Tzeng, H.-C. Yu, and B. J.-C. Yuan, 2001, Fuzzy ARIMA Model for Forecasting the Foreign Exchange Market, Fuzzy Sets and Systems, 118: 9-19.

Vasicek, O. A., and H. G. Fong, 1982, Term Structure Modelling Using Exponential Splines, Journal of Finance, 37: 339-349.

Verrall, R. J., 1989, A Space State Representation of the Chain Ladder Model, Journal of the Institute of Actuaries, 116: 589-610.

Wang, H.-F., and R.-C. Tsaur, 2000, Insight of a Fuzzy Regression Model, Fuzzy Sets and Systems, 112: 355-369.

Watada, J., 1992, Fuzzy Time-Series Analysis and Forecasting of Sales Volume, in: J. Kacprzyk and M. Fedrizzi, eds., Fuzzy Regression Analysis (Heildelberg: Physica-Verlag), pp. 211-227.

Wright, T. S., 1990, A Stochastic Method for Claims Reserving in General Insurance, Journal of the Institute of Actuaries, 117: 677-731.

Yao, Y., 1999, Term Structure Models: A Perspective From the Long Rate, North American Actuarial Journal, 3(3): 122-138.

Yen, K. K., S. Ghoshray, and G. Roig, 1999, A Linear Regression Model Using Triangular Fuzzy Number Coefficients, Fuzzy Sets and Systems, 106: 167-177.

Young, V. R., 1996, Insurance Rate Changing: A Fuzzy Logic Approach, Journal of Risk and Insurance, 63(3): 461-484.

Zadeh, L. A., 1965, Fuzzy Sets, Information and Control, 8: 338-353.

Zhao, R., and R. Govind, 1991, Defuzzification of Fuzzy Intervals, Fuzzy Sets and Systems, 43: 45-55.

Jorge de Andres Sanchez and Antonio Terceno Gomez are from the Department of Business Administration, Faculty of Economics and Business Studies, Rovira i Virgili University, Spain. The authors wish to thank two anonymous reviewers for their valuable comments.

In this article, we propose several applications of fuzzy regression techniques for actuarial problems. Our main analysis is motivated, on the one hand, by the fact that several articles in the financial and actuarial literature suggest using fuzzy numbers to model interest rate uncertainty but do not explain how to quantify these rates with fuzzy numbers. Likewise, actuarial literature has recently focused some of its attention in analyzing the Term Structure of Interest Rates (TSIR) because this is a key instrument for pricing insurance contracts. With these two ideas in mind, we show that fuzzy regression is suitable for adjusting the TSIR and discuss how to apply a fuzzy TSIR when pricing life insurance contracts and property-liability policies. Finally, we reflect on other actuarial applications of fuzzy regression and develop with this technique the London Chain Ladder Method for obtaining Incurred But Not Reported Reserves.

INTRODUCTION

To obtain the financial price of an insurance contract and in general the price of any other asset, we have to discount the cash flows that the asset produces throughout its life. We therefore need to know the discount rates that must be applied at each moment. The reference values of these discount rates are those interest rates that are free of default risk, i.e., those that correspond to public debt bonds. Of course, this is especially true in an actuarial pricing context because the profit to the insurer must be in accordance with the return from the insurer's investments of the premiums, and some of the premiums are invested in public debt securities. So, predicting the evolution of the default-free interest rate is a crucial question in actuarial pricing. This explains why yield curve analysis has become an important topic in actuarial science. Babbell and Merrill (1996) and Ang and Sherris (1997) provided a wide survey of Term Structure of Interest Rate (TSIR) models derived from the contingent claims theory, whereas Yao (1999) discussed the asymptotic properties of the rates fitted with some of these models, bearing in mind actuarial pricing. Delbaen and Lorimier (1992) used a nonparametric method based on quadratic programming to fit the short-term yield curve, whereas Carriere (1999) proposed combining bootstrapping and spline functions to estimate long-term yield rates embedded in the TSIR.

However, many actuarial analyses are concerned with the medium and long term and, in our opinion, modeling the behavior of interest rates in the long term by means of a stochastic model is not very realistic. As Gerber (1995) pointed out, there is no commonly accepted stochastic model for predicting long-term discount rates.

Fuzzy Sets Theory (FST) has been used successfully in insurance problems that require much actuarial subjective judgment and those for which measuring the embedded variables is difficult. Lemaire (1990) applied fuzzy logic to underwriting and reinsurance decisions whereas Cummins and Derrig (1993) used fuzzy decision to evaluate several econometric methods of claim cost forecasting. Derrig and Ostaszewski (1995) showed that fuzzy clustering methods are suitable for risk classification and Young (1996) applied fuzzy reasoning to insurance rate decisions. Therefore, and given that there are only vague data or data ill related with the behavior of future discount rates to predict them (e.g., the price of fixed-income securities or the opinions of "experts" about the future behavior of macroeconomic magnitudes), many authors think that it is often more suitable and realistic to make financial analyses in the long term with yields quantified with fuzzy numbers. For financial analysis, see Kaufmann (1986), Buckley (1987), or Li Calzi (1990), whereas in actuarial literature, see Lemaire (1990), Ostaszewski (1993), or Terceno et al. (1996) in a life insurance context, and articles by Cummins and Derrig (1997) and Derrig and Ostaszewski (1997) on the financial analysis of property-liability insurance. However, these articles do not explain in great detail how to estimate the discount rates with fuzzy sets. They usually suggest that "the rates are estimated subjectively by the experts using fuzzy numbers" but offer no more explanation.

In this article, we propose a solution to this problem. This involves estimating the TSIR with fuzzy sets, since the TSIR implicitly contains the expectations of the fixed income market agents (i.e., the experts) regarding the evolution of the future interest rates. Our results will be similar to those of Carriere (1999). His method obtains an estimate of the TSIR with probabilistic confidence intervals, whereas ours describes the yield curve as fuzzy confidence intervals. We also discuss how to use our fuzzy TSIR to price life-insurance contracts and property-liability policies. We would like to point out that using a fuzzy TSIR to price insurance policies was initially suggested in Ostaszewski (1993).

Another aim of this article is to suggest other actuarial applications of fuzzy regression. We have therefore developed the method for obtaining Incurred But Not Reported Reserves proposed by Benjamin and Eagles (1986) with fuzzy regression methods. We also discuss how fuzzy regression can help us with trending claim costs and with premium rating from the CAPM perspective.

The structure of the article is as follows. In the next section we describe some basic aspects of fuzzy arithmetic and fuzzy regression. In "Estimating the TSIR With Fuzzy Methods" we propose a method for obtaining a fuzzy TSIR based on fuzzy regression, apply our method to the Spanish public debt market, and compare our results with those of standard econometric methods. In "Using a Fuzzy TSIR for Financial and Actuarial Pricing" we discuss how our fuzzy TSIR can be used in actuarial pricing. In "Discussing Further Actuarial Applications of Fuzzy Regression" we suggest further applications of fuzzy regression to insurance problems.

FUZZY ARITHMETIC AND FUZZY REGRESSION

FST is constructed from the concept of fuzzy subset. A fuzzy subset A is a subset defined over a reference set X for which the level of membership of an element x [member of] X to A accepts values other than 0 or 1 (absolute nonmembership or absolute membership). A fuzzy subset A can therefore be defined as A = {(x, [[mu].sub.A](x)) | x [member of] X}, where [[mu].sub.A](x) is called the membership function and is a mapping [[mu].sub.A]: X [right arrow] [0, 1]. So, an element x has its image within [0, 1], where 0 indicates nonmembership to the fuzzy subset A and 1 indicates absolute membership. Alternatively, a fuzzy subset A can be represented by its level sets [alpha] or [alpha]-cuts. An [alpha]-cut is an ordinary (crisp) set containing elements whose membership level is at least [alpha]. For a fuzzy subset A, we will name an [alpha]-cut with [A.sub.[alpha]] being its mathematical expression: [A.sub.[alpha] = {x [member of] X | [[mu].sub.A](x) [greater than or equal to] [alpha]}, 0 [less than or equal to] [alpha] [less than or equal to] 1.

A fuzzy number (FN) is a fuzzy subset A defined over the real numbers (X is the set R). It is the main instrument of FST for quantifying uncertain or imprecise magnitudes (e.g., the discount rates in financial mathematics). Two other conditions are required for an FN. First, it must be a normal fuzzy set, i.e., it exists at least one x [member of] X such that [[mu].sub.A](x) = 1. Second, it must be convex (i.e., its [alpha]-cuts must be convex sets in the real numbers). Figure 1 shows the shape of an FN.

[FIGURE 1 OMITTED]

The most widely used FNs are triangular fuzzy numbers (TFNs) because they are easy to use and can be interpreted intuitively. (1) To construct a TFN named A, we must establish its center, a unique value [a.sub.C] (i.e., [a.sub.C] = [a.sub.2] = [a.sub.3] in Figure 1), and the deviations from there that we consider reasonable, i.e., its left spread, [l.sub.A]; and its right spread, [r.sub.A]. A TFN (2) will be denoted as A = ([a.sub.C], [l.sub.A], [r.sub.A]). Its membership function, [[mu].sub.A](x), is given by linear functions and its [alpha]-cuts, [A.sub.[alpha]], are confidence intervals where its extremes are also done by linear functions. So

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In fuzzy regression, the symmetrical TFNs (STFNs) are widely used. These are TFNs where [l.sub.A] = [r.sub.A] = [a.sub.R] and we will denote them as A = ([a.sub.C], [a.sub.R]). The membership function and [alpha]-cuts of an STFN A are

(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Figures 2 and 3 show the shape of a TFN and an STFN, respectively.

[FIGURES 2-3 OMITTED]

To develop our article we need to know the level of inclusion of an FN B within another FN A, [mu] (B [subset or equal to] A). If the [alpha]-cuts of these FNs are [A.sub.[alpha]] = [[A.sup.1]([alpha]), [A.sup.2]([alpha])] and [B.sub.[alpha]] = [[B.sup.1]([alpha]), [B.sup.2]([alpha])], then [mu](B [subset or equal to] A) [greater than or equal to] [alpha], if [B.sub.[alpha]] [subset or equal to] [A.sub.[alpha]], i.e., if

(3) [A.sup.1]([alpha]) [less than or equal to] [B.sup.1]([alpha]) and [A.sup.2]([alpha]) [greater than or equal to] [B.sup.2]([alpha]).

For example, in Figure 4, [mu] (B [subset or equal to] A) [greater than or equal to] 0.4.

[FIGURE 4 OMITTED]

We also need to establish when one FN is greater than another. (3) Ramik and Rimanek (1985) suggested that B is greater or equal to A with a membership level of at least [alpha], [mu](B [greater than or equal to] A) [greater than or equal to] [alpha], if

(4) [A.sup.1]([alpha]) [less than or equal to] [B.sup.1]([alpha]) and [A.sup.2]([alpha]) [less than or equal to] [B.sup.2] ([alpha]).

For example, in Figure 4, [mu] (B [greater than or equal to] A) [greater than or equal to] 1 and [mu](A [greater than or equal to] B) [greater than or equal to] 0.4.

In actuarial analysis we often need to evaluate functions (e.g., the net present value), which in a general way we shall symbolize as y = f ([x.sub.1], [x.sub.2], ..., [x.sub.n])--e.g., [x.sub.1], [x.sub.2], ..., [x.sub.n-1] may be the cash flows and [x.sub.n] the discount rate. Then, if [x.sub.1], [x.sub.2], ..., [x.sub.n] are not given by crisp numbers but by the FNs [A.sub.1], [A.sub.2], ..., [A.sub.n] (i.e., to calculate the net present value we know the cash flows and the discount rate imprecisely), when evaluating f(*) we will obtain an FN B, B = f([A.sub.1], [A.sub.2], ..., [A.sub.n]). To determine the membership function of B, [[mu].sub.B](y), we must apply the Zadeh's extension principle, exposed in the seminal paper by Zadeh (1965). In a similar way as when handling arithmetically random variables, we obtain the membership function of the result when operating with FNs by convoluting membership functions of the FNs [A.sub.1], [A.sub.2], ..., [A.sub.n] (for random variables, the density functions). The difference is that random variables are convoluted using operators sum-product, whereas with FN they are convoluted using operators max-min (max instead of sum and rain instead of product). Mathematically,

(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Unfortunately, it is often impossible to obtain a closed expression for the membership function of B (this problem often arises when handling random variables). However, we may be able to obtain its [alpha]-cuts, [B.sub.[alpha]], from [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] as

(6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In actuarial mathematics, many functional relationships are continuously increasing or decreasing with respect to every variable in such a way that it is easy to evaluate the [alpha]-cuts of B. Buckley and Qu (1990b) demonstrated that if the function f(*) that induces B is increasing with respect to the first m variables, where m [less than or equal to] n, and decreasing with respect to the last n-m variables, [B.sub.[alpha]] is

(7) [B.sub.[alpha]] = [[B.sup.1] ([alpha]), [B.sup.2] ([alpha])] = [f ([A.sup.1.sub.1] ([alpha]), ..., [A.sup.1.sub.m] ([alpha]), [A.sup.2.sub.m+1], ([alpha]), ..., [A.sup.2.sub.n] ([alpha])), f ([A.sup.2.sub.1] ([alpha]), ..., [A.sup.2.sub.m] ([alpha]), [A.sup.1.sub.m+1] ([alpha]), ..., [A.sup.sub.n] ([alpha]))].

Some operations with STFN are easy to solve. For instance, if we multiply A = ([a.sub.C], [a.sub.R]) by a real number k, B = k A, the result is B = ([b.sub.C], [b.sub.R]) = ([ka.sub.C], [absolute value of k] [a.sub.R]). The sum of two (4) STFNs, C = A + B, is also an STFN. Concretely, C =([c.sub.C], [c.sub.R]) = ([a.sub.C], [a.sub.R]) + ([b.sub.C], [b.sub.R]) = ([a.sub.C], + [b.sub.C], [a.sub.R] + [b.sub.R]). So, if the FN B is obtained from a linear combination of the STFNs [A.sub.i] = ([a.sub.iC], [a.sub.iR), i = 1, ..., n, i.e., B = [[summation of].sup.n.sub.(i=1)] [k.sub.i], [A.sub.i] where [k.sub.i] [member of] R, B will be an STFN, B = ([b.sub.C], [b.sub.R]) where

(8) ([b.sub.C], [b.sub.R]) = ([k.sub.1] * [a.sub.1C] + [k.sub.2] * [a.sub.2C] + ... + [k.sub.n] * [a.sub.nC], [[absolute value of k].sub.1] * [a.sub.1R] + [[absolute value of k].sub.2] * [a.sub.2R] + ... + [[absolute value of k].sub.n] * [a.sub.nR]).

Unfortunately, the result of a nonlinear operation with STFNs is not an STFN. So, if we evaluate B = f ([A.sub.1], [A.sub.2], ..., [A.sub.n]), where we suppose that [A.sub.i] = ([a.sub.iC], [a.sub.iR]) [for all] i, B is often not an STFN despite the characteristics of [A.sub.1], [A.sub.2], ..., [A.sub.n]. Even so, Dubois and Prade (1993) showed that if f(*) is increasing with respect to the first m variables, where m [less than or equal to] n, and decreasing with respect to the others, B can be estimated well by B' = ([b.sub.C], [b.sub.R]),

where

(9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

So, for B = [A.sup.k] where k [member of] R, if A = ([a.sub.C], [a.sub.R]), from (9) we obtain:

(10) B [approximately equal to] ([b.sub.C], [b.sub.R]) = [(([a.sub.C].sup.k], [absolute value of k][([a.sub.C]).sup.k-1][a.sub.R])

while for C = A * B where k [member of] R, if A = ([a.sub.C], [a.sub.R]) and B = ([b.sub.C], [b.sub.R]), from (9) we obtain

(11) C [approximately equal to] ([c.sub.C], [c.sub.R]) = ([a.sub.C] * [b.sub.C], [a.sub.C] * [b.sub.R] + [b.sub.C] * [a.sub.R]).

Tanaka and Ishibuchi's Fuzzy Regression Model

The fuzzy regression model developed in Tanaka (1987) and Tanaka and Ishibuchi (1992) is one of the most widely used models in fuzzy literature for economic applications. (5) Like any regression technique, the aim of fuzzy regression is to determine a functional relationship between a dependent variable and a set of independent ones. Fuzzy regression allows to obtain functional relationships when independent variables, dependent variables, or both, are not crisp values but confidence intervals.

As in econometric linear regression, we shall suppose that the explained variable is a linear combination of the explanatory variables. This relationship should be obtained from a sample of n observations {([Y.sub.1], [X.sub.1]), ([Y.sub.2], [X.sub.2]), ..., ([Y.sub.j], [X.sub.j]), ..., ([Y.sub.n], [X.sub.n])} where [X.sub.j] is the jth observation of the explanatory variable, [X.sub.j] = ([X.sub.0j], [X.sub.1j], [X.sub.2j], ..., [X.sub.1j], ..., [X.sub.mj]). Moreover, [X.sub.0j] = 1 [for all] j, and [X.sub.ij] is the observed value for the ith variable in the jth case of the sample. [Y.sub.j] is the jth observation of the explained variable, j = 1, 2, ..., n. The jth observation may either be a crisp value or a confidence interval, in either case, it can be represented through its center and its spread or radius as [Y.sub.j] = <[Y'.sub.jC], [Y'.sub.jR]>, where [Y'.sub.jC] is the center and [Y'.sub.jR] is the radius.

Similarly, we suppose that the jth observation for the dependent variable is an [[alpha].sup.*]-cut of the FN it arises from where [[alpha].sup.*] may be stated previously by the decision maker. Also, the FN that quantifies the jth observation of the dependent variable is an STFN that we will write as [Y.sub.j] = ([Y.sub.jC], [Y.sub.jR]). Therefore, since the [[alpha].sup.*]-cut of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is (see (2)):

(12) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

the center and spread of [Y.sub.j] can be obtained from its [[alpha].sup.*]-cut taking into account (2) as

(13) [Y'.sub.jC] = [Y.sub.jC] and [Y'.sub.jR] = [Y.sub.jR](1 - [[alpha].sup.*]) [??] [Y.sub.jR] = [Y'.sub.jR]/(1 - [[alpha].sup.*]

and we must estimate the following fuzzy linear function

(14) [Y.sub.j] = [A.sub.0] + [A.sub.1][X.sub.1j] + ... + [A.sub.m][X.sub.mj].

In this model of fuzzy regression, the disturbance is not introduced as a random addend in the linear relation but is incorporated into the coefficients [A.sub.i], i = 0, 1, ..., m. Of course, the final objective is to adjust the fuzzy numbers [A.sub.i] which estimate [A.sub.i] from the available sample. Given the characteristics of [Y.sub.j], the parameters [A.sub.i], i = 0, 1, 2, ..., m must be STFNs. These parameters can therefore be written as [A.sub.i] = ([a.sub.iC], [a.sub.iR]), i = 0, 1, ..., m. The main objective is to estimate every FN [A.sub.i], [A.sub.i] = ([a.sub.iC], [a.sub.iR]). When we have obtained [A.sub.i] the estimates of [Y.sub.j] = ([Y.sub.jC], [Y.sub.jR]), will be

(15) [Y.sub.j] = [A.sub.0] + [A.sub.1][X.sub.1j] + ... + [A.sub.m][X.sub.mj].

Therefore, [Y.sub.j] is obtained from (8):

(16) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

whose [alpha]-cuts for a level [[alpha].sup.*] are

(17) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The parameters [a.sub.iC] and [a.sub.iR], must minimize the spreads of [Y.sub.j], and simultaneously maximize the congruence of [Y.sub.j] with [Y.sub.j], which is measured as [mu]([Y.sub.j] [subset or equal to] [Y.sub.j]). Specifically, we must solve the following multiple objective program:

(18a) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

subject to

(18b) [mu]([Y.sub.j] [subset or equal to] [Y.sub.j] [greater than or equal to] [alpha] j = 1, 2, ..., n; [a.sub.iR] [greater than or equal to] 0 i = 0, 1, ..., m, [alpha] [member of] [0, 1].

If for the second objective we require a minimum accomplishment level [[alpha].sup.*], i.e., the level that the decision maker considers that <[Y'.sub.jC], [Y'.sub.jR]>, j = 1, 2, ..., n, has been obtained, the above program is transformed into the following linear one

(19a) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

subject to

(19b) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The first block of constraints in Equation (19b) is a consequence of the requirement that [mu](Y.sub.j] [subset or equal to] [Y.sub.j] [greater than or equal to] [[alpha].sup.*], which must be implemented by taking into account (3), (12), and (17). With the last block of constraints in Equation (19b) we ensure that [a.sub.iR] [for all] i will be nonnegative.

In our opinion, fuzzy regression techniques have a number of advantages over traditional regression techniques:

(a) The estimates obtained after adjusting the coefficients are not random variables, which are difficult to manipulate in arithmetical operations, but fuzzy numbers, which are easier to handle arithmetically using [alpha]-cuts. So, when starting from magnitudes estimated by random variables (e.g., from a least squares regression), these random variables are often reduced to their mathematical expectation (which may or may not be corrected by its variance) to make them easier to handle. As we have already pointed out, this loss of information does not necessarily take place when we operate with FNs.

(b) When investigating economic or social phenomena, the observations are a consequence of the interaction between the economic agents' beliefs and expectations, which are highly subjective and vague. A good way to treat this kind of information is therefore with FST. For example, the asset prices that are determined in the markets are due to the agents' expectations of future inflation and the issuers' credibility. We think that it is more realistic to consider that the bias between the observed value of the dependent variable and its theoretical value (the error) is not random but fuzzy. At least in this way we assume that the analyzed phenomena have a large subjective component.

(c) The observations are often not crisp numbers but confidence intervals. For instance, the price of one financial asset throughout one session often oscillates within an interval and is rarely unique (e.g., it can oscillate within [$100, $105]). To be able to use econometric methods, the observations for the explained variable and/or the explanatory variable must be represented by a single value (e.g., $102.5 for [$100, $105]), which involves losing a great deal of information. However, fuzzy regression does not necessarily reduce each variable to a crisp number, i.e., all the observed values can be used in the regression analysis.

ESTIMATING THE TSIR WITH FUZZY METHODS

Estimating the TSIR With Conventional Econometric Methods

Estimating the TSIR for a concrete date and a given market is fairly straightforward if there are many sufficiently liquid zero coupon bonds and their price can be observed without perturbations. However, fixed-income markets rarely enjoy these conditions simultaneously.

This subsection describes the essence of a family of methods for estimating the discount function associated to the TSIR with econometric methods. They can be used if the sample is made up entirely of zero coupon bonds, entirely of bonds with coupon or, as is usual, if it is made up of both types of bonds.

These methods start from the fact that the rth bond, where r = 1, 2, ..., k, in which k is the number of available bonds, provides several cash flows (coupons and principal). These are denoted by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [C.sup.r.sub.i] is the amount of the ith cash flow and [t.sup.r.sub.i] is its maturity in years. If we suppose that the default-free bonds do not include any option (i.e., they are not convertible or callable, etc.), the price of the rth bond is therefore the sum of the discounted value of every coupon and the principal with the corresponding spot rate

(20) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the discounted value of one dollar with maturity [t.sup.r.sub.i] years.

Of course, subsequently we should define a form for the discount function to specify the econometric equation to be estimated. Our proposal is based on the methods that use splines (piecewise functions) to model the discount function. The best known methods are those in McCulloch's articles (1971, 1975) (quadratic splines and cubic splines) and in Vasicek and Fong's article (1982) (exponential splines). These methods suppose that the discount function is a linear combination of m + 1 functions of time. Therefore

(21) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

From (20) and (21) we can deduce that the following linear equation must be estimated

(22) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We assume that [g.sub.j](t) are splines and not simply polynomial functions because with splines we can determine [g.sub.j](t) according to the distribution of the maturity dates of the sample and, therefore, fit the discount function better for the most common maturities. Moreover, with splines we can obtain TSIR profiles that do not fluctuate very much and forward rates that do not behave explosively.

A random disturbance is justified because several factors disturb the formation of the prices of fixed-income securities. Chambers, Carleton, and Waldman (1984) point out some of them; for example, the coupon-bearing of bonds with maturities greater than one contain information about more than one present value coefficient; the bond portfolios are not continuously rebalanced, so at any moment each bond can deviate by some (presumably random) amount or there is no single price for each bond, which implies that there is some inherent imprecision to the concept of a single price.

We will now present our fuzzy method for estimating the TSIR. We will first establish a hypothesis on which to construct a method for estimating the TSIR that uses fuzzy regression from the analyzed framework. We will then discuss how to estimate the TSIR and the forward rates using fuzzy numbers and make an empirical application. As we stated above, we will suppose that the bonds in our analysis are default-free and only produce a stream of payments (coupons and principal or only principal if they are zero coupon bonds) and that they do not have any embedded option. Also, as the price of one bond we will take all the prices traded on 1 day, rather than an average price.

Hypothesis

Hypothesis 1: The price of the rth bond in a session is an STFN. This price will be written as [P.sup.r] where

(23) [P.sup.r] = ([P.sup.r.sub.C], [P.sup.r.sub.R]) [P.sup.r.sub.C], [P.sup.r.sub.R] [greater than or equal to] 0, r = 1, 2, ..., k.

This hypothesis considers the price of a bond on i day to be "approximately [P.sub.C]," and not exactly [P.sub.C]. We think that this is more suitable because over a session it is usual to negotiate more than one price for the same bond (one for each trade). However, if these prices are unique then [P.sup.r.sub.R] = 0.

Hypothesis 2: The observed price for each security is an [alpha]-cut of the FN that quantifies that price for a predefined [[alpha].sup.*]. Its inferior and upper extremes are the minimum and maximum price of the bond over the session. Therefore, this interval will be expressed through its center and spread as <[P'.sup.r.sub.C], [P'.sup.r.sub.R]> For example, if the traded price for one bond on 1 day has fluctuated between 100 and 103, it will be expressed as <101.5, 1.5>. Similarly, from these parameters we can obtain the center and the spread of its corresponding FN, (23), taking into account (13):

(24) [P'.sup.r.sub.C] = [P.sup.r.sub.C] y [P'.sup.r.sub.R] = (1 - [[alpha].sup.*]) [P.sup.r.sub.R] [??] [P.sup.r.sub.R] = [P'.sup.r.sub.R] /(1 - [[alpha].sup.*]), 0 [less than or equal to] [[alpha].sup.*] [less than or equal to] 1, r = 1, 2, ..., k.

Hypothesis 3: The discount function is quantified via an FN that depends on the time. So, for a given maturity t, the discount function is the following STFN:

(25) [f.sub.t] = ([f.sub.tC], [f.sub.tR]), t > 0, 0 [less than or equal to] [f.sub.tC] - [f.sub.tR] [less than or equal to] [f.sub.tC] + [f.sub.tR] [less than or equal to] 1.

The price of the rth bond, (23), can therefore be written from (20) as

(26) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and combining (25) and (26) we obtain

(27) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

So, the observed [[alpha].sup.*]-cut for the price of the rth bond, is

(28) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hypothesis 4: The discount function, (25), can be approximated from a linear combination of m + 1 functions [g.sub.j](t), j = 0, 1, ..., m with image in [R.sup.+] that are continuously differentiable, and whose parameters are given by STFN. In this way, these parameters can be represented as

(29) [a.sub.j] = ([a.sub.jC], [a.sub.jR]), [a.sub.jR] [greater than or equal to] 0, j = 0, 1, ..., m

and so the discount function is obtained from (21), (29) and using (8):

(30) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then, using (27) and (30), the price of the rth bond can be expressed by

(31) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and the [[alpha].sup.*]-cut of the rth bond price, (28), is now

(32) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with [a'.sub.jC] = [a.sub.jC] and [a'.sub.jR] = (1 - [[alpha].sup.*])[a.sub.jR], 0 [less than or equal to] [[alpha].sup.*] [less than or equal to] 1, r = 1, 2, ..., k.

Adjusting the Discount Function Using a Fuzz Regression Model

Since the value of the discount function for t = 0 should be 1, [f.sub.0] = ([f.sub.0C], [f.sub.0R]) = (1, 0). As McCulloch stated in (1971), this condition is met if [a.sub.0] =([a.sub.0C], [a.sub.0R]) = (1, 0), [g.sub.0](t) = 1, and [g.sub.j](0) = 0, j = 1, 2, ..., m. So, from (31), we can express the price of the rth bond as

(33) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and, using fuzzy arithmetic, we finally write

(34) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In this way, by identifying in Equation (34) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we obtain

(35) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It is easy to verify in Equation (34) that the value of the jth explanatory variable for the rth bond is the crisp value [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Therefore, from (34) and (35) we can write

(36) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then, after obtaining the estimate for every [a.sub.j], [a.sub.j] = [a.sub.jC], [a.sub.jR]), the fitted value of the explained variable of the rth observation, [Y.sup.r], is

(37) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Bearing in mind that the dependent variable and the parameters are actually quantified via their [[alpha].sup.*]-cut, the expression of the [[alpha].sup.*]-cut of [Y.sup.r] (that of FN (37)) is

(38) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where we must estimate the center and the spread of the [[alpha].sup.*]-cut for [a.sub.j], j = 1, ..., m. After estimating these parameters, the center and the radius [a.sub.jC] and [a.sub.jR] are estimated using (2) as

(39) [a.sub.jC] =[a'.sub.jC] and [a.sub.jR] = [a'.sub.jR]/(1 - [[alpha].sup.*]).

Adding certain constraints, which are related to the properties of the discount function, to obtain [a'.sub.jC] and [a'.sub.jR], we have to solve the following linear program

(40a) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

subject to

(40b) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(40c) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(40d) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(40e) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(40f) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(40g) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(40h) [a'.sub.j] R [greater than or equal to] 0 j = 1, 2, ..., m.

The constraints (40b), (40c), and (40h) correspond to Tanaka's regression model (see (18b) or (19b)). Similarly, (40d) and (40e) ensure that the discount function is decreasing, and that it is accomplished for an arbitrary periodicity P (in years). Therefore uP is the greatest maturity that we will use in a later analysis. It is reasonable to suppose that uP is close to the expiration of the bond with the greatest maturity. In Equations (40d) and (40e) we use the Ramik and Rimanek's criteria for ordering FNs (see (4)). Finally, constraints (40f) and (40g) ensure that the discount function is within [0, 1].

Estimating the Spot Rates and the Forward Rates Using Fuzzy Numbers

The discount function in t, [f.sub.t], is obtained from its corresponding spot rate [i.sub.t] by [f.sub.t] = [(1 + [i.sub.t]).sup.-t] and then

(41) [i.sub.t] = [([f.sub.t]).sup.-1/t] - 1

If the discount function in t is an FN, the spot rate will be an FN [i.sub.t]. Its membership function can be obtained applying the extension principle (5) to the relation (41) as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Unfortunately, despite using a discount function quantified via an STFN, the spot rate is not an STFN because it is not a linear function of [f.sub.t]. However, applying (10) in (41) we obtain

(42) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

To obtain the forward rate for the tth year, [r.sub.t], we should solve the following fuzzy equation:

(43) [f.sub.t-1] [(1 + [r.sub.t]).sup.-1] = [f.sub.t].

The [[alpha].sup.*]-cuts of [r.sub.t], [r.sub.t[alpha] are (see Appendix):

(44) [r.sub.t[alpha]] = [[r.sup.1.sub.t]([alpha]), [r.sup.2.sub.t]([alpha])] = [[f.sub.(t-1)C] + [f.sub.(t-1)R](1 - [alpha])/[f.sub.tC] + [f.sub.tR](1 - [alpha]) - 1, [f.sub.(t-1)C] - [f.sub.(t-1)R](1 - [alpha])/[f.sub.tC] - [f.sub.tR](1 - [alpha]) - 1].

Then, although [r.sub.t] is not an STFN, it can be approximated reasonably well by this type of FN. In the Appendix we demonstrate that its approximation by means of an STFN is

(45) [r.sub.t] [approximately equal to] ([r.sub.tC], [r.sub.tR]) = ([f.sub.(t-1)C]/[f.sub.tR] - 1, [f.sub.(t-1)C] * [f.sub.tR] - [f.sub.tC] * [f.sub.(t-1)R]/[([f.sub.tC]).sup.2]).

Notice that, although we take annual periods, calculating implied rates for any other periodicity is not a problem because the discount function is a continuous function of the maturity.

Empirical Application

In this subsection, we used our method to estimate the TSIR in the Spanish public debt market on June 29, 2001. Table 1 shows the bonds included in our sample and their characteristics.

To fit the TSIR in this date, we formalized the discount function using McCulloch's quadratic splines. (6) So we took m = 5, and the knots that we used to construct the splines were [d.sub.1] = 0 years, [d.sub.2] = 1.58 years, [d.sub.3] = 3.83 years, [d.sub.4] = 8.96 years, and [d.sub.5] = 31.1 years. Then, the functions [g.sub.j](t), j = 1, ..., 5 are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We used OLS regression, taking for the price of the kth bond ([P.sup.k.sub.min] + [P.sup.k.sub.max])/2. Its determination coefficient is [R.sup.2] = 99.98% whereas the discount function is

[f.sub.t] = 1 - 0.04394[g.sub.1](t) - 0.03672[g.sub.2](t) - 0.04875[g.sub.3](t) - 0.03572[g.sub.4](t) - 0.00730[g.sub.5](t).

To fit the TSIR by using fuzzy regression we took for the price of the kth bond the interval [P.sup.k] = <[P.sup.k.sub.C], where [P.sup.k.sub.R]>, where [P.sup.k.sub.C] = ([P.sup.k.sub.min] + [P.sup.k.sub.max])/2 and [P.sup.k.sub.R] = ([P.sup.k.sub.max] - [P.sup.k.sub.min])/2. To build the constraints (40d), (40e), (40f), and (40g) in fuzzy regression we assumed an annual periodicity. The final value of the objective function (40a) is z = 23.77. To interpret this value, we should remember that the size of our sample was 28 assets. When taking the level of congruence [[alpha].sup.*] = 0.5, our fuzzy discount function is

[f.sub.t] = (1, 0) + (-0.04280, 0.00422)[g.sub.1](t) + (-0.03874, 0.00043)[g.sub.2](t) + (-0.04675, 0.00397)[g.sub.3](t) + (-0.03841, 0.00069)[g.sub.4](t) + (-0.00255, 0)[g.sub.5](t)

and requiring [[alpha].sup.*] = 0.75, the discount function is

[f.sub.t] = (1, 0) + (-0.04280, 0.00843)[g.sub.1](t) + (-0.03874, 0.00086)[g.sub.2](t) +(-0.04675, 0.00793)[g.sub.3](t) + (-0.03841, 0.00139)[g.sub.4](t) + (-0.00255, 0)[g.sub.5](t).

Table 2 shows the spot and forward rates for the next 15 years with OLS and fuzzy regressions (in the last case, for [[alpha].sup.*] = 0.5, 0.75). We can see that the parameter [[alpha].sup.*] can be interpreted as an indicator of the perceived uncertainty in the market. If [[alpha].sup.*] increases, uncertainty of the observations for the explained variable (price of the bonds) also increases (see Wang and Tsaur, 2000) and the spread of the subsequent estimates of the spot rates and the forward rates will be wider.

To compare the difference between the OLS estimates and the estimates of fuzzy regression for discount rates, Table 2 includes coefficient D = 100 x [absolute value of [V.sup.E] - [V.sup.F]]/[V.sup.E], where [V.sup.E] is the value of a yield rate obtained with econometric methods and [V.sup.F] is the center of the fuzzy estimate for this rate. We can see that the estimates of the spot rates with each method are quite similar. This similarity decreases when estimating forward rates.

USING A FUZZY TSIR FOR FINANCIAL AND ACTUARIAL PRICING

Calculating the Present Value of an Annuity

Buckley (1987) determines the present value of a stream of amounts when these amounts and the discount rate are given by FNs. Buckley supposes that the discount rate to be applied throughout the evaluation horizon was a unique FN i. It implies that the reference TSIR is flat. If we also suppose that amounts are given by means of nonnegative FNs, in which the tth cash flow is the FN [C.sub.t], and that they are an immediate postpayable annuity with an annual periodicity, then the net present value of that annuity is the following FN, V:

(46) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It is often impossible to obtain the membership function of V by means of the extension principle. However, it is fairly straightforward to obtain a closed expression of the [alpha]-cuts of the present value. If we bear in mind that the function of present value is continuously decreasing (increasing) with respect to the discount rate (the amounts), we can calculate the upper and lower extremes of its [alpha]-cuts immediately from (7). If the amounts and the cash flows are given by STFNs, 19 will not be an STFN but it can be approximated by an STFN from (9).

It is well known that it is quite unrealistic to suppose a flat TSIR. Moreover, in the previous section we proposed a method for obtaining an empirical fuzzy TSIR and discussed how to obtain its spot and implied rates. So, (46) can be generalized to any shape of the TSIR, using the spot rates, the forward rates, or the discount function

(47) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Of course, from the [alpha]-cuts of the amounts and the discount rates (or alternatively the discount function), we can easily obtain 19 from (7). Moreover, if the amounts are crisp (e.g., if they are the amounts paid by a bond), and to obtain their present value we use a discount function quantified via an STFN like (25), (47) can be written as

(48) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Example: Suppose an investor is going to buy bonds of the Spanish public debt market on June 29, 2001. The maturity of these bonds is 5 years, and they offer a 5 percent annual coupon. The values of 1 monetary unit with maturities from 1 to 5 years obtained from the regression with [[alpha].sup.*] = 0.5 are given in Table 3. Therefore, the investor obtains the following preliminary price for one of these bonds from (48):

P = 5 * (0.9585, 0.0030) + 5 * (0.9190, 0.0040) + 5 * (0.8770, 0.0059) + 5 * (0.8315, 0.0093) + 105 * (0.7858, 0.0128) = (100.44, 1.46).

Pricing Life-Insurance Contracts

In this subsection, we show how to obtain the net single premium for some life-insurance contracts (n-year pure endowments, n-year term life-insurance contracts, and n-year endowments--the combination of the two first contracts) from our fuzzy TSIR. To simplify the analysis, we will suppose that the insured amounts are fixed beforehand and that they are annual and payable at the end of each year of the contract. Taking into account that standard life-insurance mathematics establishes that the net single premium for a policy is the discounted value of the mathematical expectation of the guaranteed amounts, and naming as [C.sub.n] the amount payable at the end of an n-year pure endowment, the premium for an individual aged x for this contract, [[PI].sub.1], is

(49) [[PI].sub.1] = [C.sub.n] * [(1 + [i.sub.t]).sup.-n] * [sub.n][p.sub.x] = [C.sub.n] * [f.sub.n] * [sub.n][p.sub.x],

where [sub.n] [p.sub.x] stands for the probability that an individual aged x attains at the age x + n.

If we suppose that for an n-year term life insurance the amounts are payable at the end of the year of death, for an insured person aged x, the premium, [[PI].sub.2], is

(50) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [sub. t|][q.sub.x] stands for the probability of death at age x + t and [C.sub.t] the insured amount for this event.

The net single premium for an n-year endowment, [[PI].sub.3], is obtained from (49) and (50) as:

(51) [[PI].sub.1] = [[PI].sub.1] + [[PI].sub.2]

So, if the discount function estimated by an STFN, the net single premium for an n-year pure endowment, (49), is reduced to

(52) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and then, for an n-year term life insurance, (50), we obtain

(53) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then, from (51), (52), and (53) we obtain the price of an n-year endowment, [[PI].sub.3]:

(54) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Table 4 shows the fuzzy pure single premiums of the three types of policies for people of several ages and with the value of the discount function in Table 3. In all cases the duration of the contracts is 5 years. We suppose that the amounts of the insured events are 1,000 monetary units and for the calculations we have taken the Swiss GRM-82 mortality tables.

The maturities of the bonds used to estimate the TSIR, and obviously the temporal horizon covered with this TSIR may not be large enough to discount all the cash flows (e.g., when pricing a whole life annuity). To complete the interest rates for the maturities of the TSIR that are not covered in the regression, de Andres (2000) suggested two solutions:

(1) The first involves estimating subjectively a unique nominal interest rate for the longer maturities by Fisher's relationship. In this way, Devolder (1988) suggested obtaining the discount rate for long-term insurance policies by using Fisher's relationship as: nominal interest rate = real interest rate + [lambda] x anticipated inflation, where 0 [less than or equal to] [lambda] < 1. Regarding the real interest rate, Devolder stated that "generally it must be quantified between the 2 and 3 percent" and the anticipated inflation "must be reasonable in the long term." Clearly, these sentences allow a fuzzy quantification even though this is probably not the aim of the author. If we call i the FN that quantifies the discount rate, we can obtain this, e.g., as i = (0.025, 0.01) + [lambda][pi], where [pi] stands for the anticipated inflation.

(2) From financial logic, the shape of the TSIR must be asymptotic. So, a second solution involves taking the latest forward (or spot) rate of our estimated TSIR as a reference for the rate to discount the amounts whose maturities are not covered by the TSIR.

Fuzzy Financial Pricing of Property-Liability Insurance

In this subsection we will show how to apply our fuzzy TSIR when pricing property-liability insurance. For a wide discussion on this topic, consult Myers and Cohn (1987) (MC) and Cummins (1990) under a nonfuzzy environment or Cummins and Derrig (1997) (CD) under a fuzzy environment. To simplify our explanation, we will suppose only a three-period model. The MC model states that the present value of the premiums must compensate the cost of the liabilities and the taxes for the insurer. Supposing a single premium, only two periods (years) in claiming, but that the TSIR can have any shape, the CD formulation can be transformed into

(55) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where P = pure single premium. The parameter [delta] ([delta] > 0) indicates the proportion of the fair premium corresponding to the surplus and [tau] is the tax rate that we suppose the same for the underwriting profit and the investment income. L is the total amount of the claim cost whereas [c.sub.t] = the proportion of the liabilities payable at the tth year. Therefore [c.sub.1] + [c.sub.2] = 1. Notice that we have supposed that the proportion of the claims cost in the tth year deductible from income taxes is equal to [c.sub.t].

If [f.sub.t] is the present value of 1 unitary unit payable at t with the spot rate free of risk of failure for that maturity ([i.sub.t]), then [f.sub.t] = [(1+ [i.sub.t]).sup.-t]. Similarly, [f.sup.(L).sub.t] is the value of the discount traction for one monetary unit of liability payable at the tth year. This can be obtained from the spot rate of the liabilities at time t, [i.sup.(L).sub.t]. Therefore, [f.sup.(L).sub.t] = [(1 + [i.sup.(L).sub.t]).sup.-t]. To simplify our discussion, we will suppose that [i.sup.(L).sub.t] is obtained by applying over it [i.sub.t] the risk loading [k.sub.t], by doing [i.sup.(L).sub.t] = (1 - [k.sub.t])(1 + [i.sub.t]) - 1, where 1 > [k.sub.t] > 0. From this relation, the following connection between [f.sup.(L).sub.t] and [f.sub.t] arises

(56) [f.sup.(L).sub.t] = [(1 - [k.sub.t]).sup.-t] [f.sub.t].

Finally, [r.sub.t], t = 1, 2 is the return obtained by investing the premium within the tth year of the contract. If we assume, as is usual, that this return corresponds to the risk-free rate, within a pure expectations framework, [r.sub.t] can be quantified by the forward rate for the tth year of the contract. Then, taking into account that [r.sub.t] can be obtained from the values of spot discount function [f.sub.t-1] and [f.sub.t] as

(57) [r.sub.t] = ([f.sub.t-1]/[f.sub.t]) - 1.

Then, (55) can be rewritten from (56), (57) and remembering that [c.sub.1] + [c.sub.2] = 1 as

(58) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and so the value of the pure single premium is

(59) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We wish to show how to introduce fuzziness into the future behavior of interest rates when pricing a property-liability contract. The fuzziness of this behavior is introduced by the discount function, which is fuzzy. Moreover, we will allow the total amount of the liabilities, L, to be estimated by an STFN, i.e., L = ([L.sub.C], [L.sub.R]). This is easy to interpret from an intuitive point of view: the actuary estimates that the total cost of the claims will be "around [L.sub.C]." We will suppose that [c.sub.t], [k.sub.t], [tau], and [delta] are crisp parameters.

If the discount factor and the total cost of the claims are done by FNs, we will actually obtain a fuzzy premium, which must be obtained by solving the fuzzy version of Equation (55):

(60) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Buckley and Qu (1990c) suggested obtaining the solution of a fuzzy equation from the solution of its crisp version. So, we will obtain P from (59) as

(61) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

To determine the [alpha]-cuts of P, [P.sub.[alpha]], we must bear in mind that in Equation (59) the premium is a function of the value of the liabilities and the free risk discount function (or, alternatively a function of the expected evolution of interest rates), i.e., the premium in Equation (59) can be denoted as P = f(L, [f.sub.1], [f.sub.2]). Clearly, f(*) is an increasing function of L. On the other hand, it is not easy to know whether the premium increases (or decreases) when the discount function increases (the interest rates decreases) from the partial derivatives of P(L, [f.sub.1] [f.sub.2]). However, we do know by financial intuition that the price of the insurance is basically related to the present value of the liabilities, and that it is clearly increasing (decreasing) with respect to the values of the discount function (spot rates). Therefore, if we start from the discount function that define our TSIR, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we can obtain [P.sub.[alpha]] by applying (7) as

(62) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

However, from (9), we can approximate P by a STBN, i.e., P [approximately equal to] ([P.sub.C], [P.sub.R]) where

(63) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In the following example we consider the following variables of a property-liability insurance: [k.sub.1] = [k.sub.2] = 1%, [delta] = 5%, and [tau] = 34% and the values of the discount function in Table 3. The duration of the contract is 2 years and the total amount of the liability for one contract is L = (1000, 50). Table 5 shows the fuzzy premiums for several pairs ([c.sub.1], [c.sub.2]) when using the approximating formula (63) of L.

Figure 5 represents the shapes of the true fuzzy premium (obtained from (62) and represented with a solid line) and our approximation (63), for the distribution of the claims [c.sub.1] = 0.5 and [c.sub.2] = 0.5. Clearly, the triangular approximation fits the real value of the premium well and is easier to interpret: the value of the fair premium must be 992.37 but there may be acceptable deviations no longer than 52.95.

[FIGURE 5 OMITTED]

DISCUSSING FURTHER ACTUARIAL APPLICATIONS OF FUZZY REGRESSION

In this section, we reflect on other applications of fuzzy regression to insurance problems. We will focus our discussion on the specific problem of estimating the incurred but not reported claims reserves, but we will also suggest other (in our opinion) promising applications.

Calculating the Incurred But Not Reported Claims Reserves (IBNR reserves) is a classic topic in nonlife-insurance mathematics. Unfortunately, they may not be calculated from a wide statistical database. Straub (1997) states that taking into account experiences that are too far from the present can lead to unrealistic estimates. For example, if the claims are related to bodily injuries, the future losses for the company will depend on the behavior of the wage index grown that will be taken to determine the amount of indemnification, changes in court practices, and public awareness of liability matters.

To calculate the IBNR reserve we begin from the historical data ordered in a triangle like that in Table 6.

In Table 6, [Z.sub.i,j] is the accumulated incurred losses of accident year j at the end of development year i where j = 1 denotes the most recent accident year and j = n the oldest accident year. Obviously, we do not know, for the jth year of occurrence, the accumulated losses in the development years i = j + 1, ..., n and therefore, these losses must be predicted. It is well known that the classical method of predicting these losses is the Chain Ladder (CL). However, as it is pointed out in the survey England and Verrall (2002) claims reserving methods, during the last years greater interest of actuarial literature has been focused not only in calculating the best estimate of claim reserves but also on determining its downside potential from a stochastic perspective. Obviously, the final objective is to provide the actuary a well-founded mathematical tool to determine solvency margins for the reserves.

One way to focus this problem consists of departing from the pure CL method and making statistical refinements over it. In this way, Benjamin and Eagles (1986) propose a slight generalization of the CL method, known as London Chain Ladder (LCL), which is based on the use of OLS regression over the accumulated claims. On the other hand, Mack (1993) and England and Verrall (1999) do not suppose a concrete structure of the underlying data. Concretely, Mack (1993) provides analytical expressions for the prediction errors in claims and reserve estimates whereas England and Verrall (1999) propose combining standard CL estimates and bootstrapping techniques to determine the variance of the error in the predicted reserves.

Another extended way to focus this problem consists of modeling the incremental claims as random variables with a predefined distribution function. So, while Wright (1990) and Renshaw and Verrall (1998) use the Poisson random variables, Kremer (1982), Renshaw (1989), and Verrall (1989) use a log-normal approach. Then, the subsequent question to answer is how to define the associated parameters to these random variables with respect to the year of underwriting and delay period. There are several approaches in the literature to do it. The more extended way consists of using one separate parameter for each development period. However, to avoid over-parameterization, some articles propose using parametric expressions (e.g., the Hoerl curve) or nonparametric smoothing methods like the one given by England and Verrall (2001). It must be remarked that, as it is pointed out in Mack (1993), in spite of the fact that these approaches are based on CL "philosophy," some of them present fundamental differences with respect to the pure CL method.

Our fuzzy sets approach to determine the value and variability of IBNR will be based on combining the generalization of CL by Benjamin and Eagles (1986), the LCL, and fuzzy regression. So, let us briefly expose Benjamin and Eagle's method. This is built from the hypothesis that the evolution of the claims of the accidents occurred in the year j from the ith year to the i+1th year of development can be approximated by the linear relation:

(64) [Z.sub.i+1,j] = [b.sub.i] + [c.sub.i] [Z.sub.i,j] + [[epsilon].sub.I],

where [b.sub.i] is the intercept, [c.sub.i] is the slope, and [[epsilon].sub.i] is the perturbation term. The coefficients [b.sub.i] and [c.sub.i] must be estimated by OLS. Of course, the estimates of [b.sub.i] and [c.sub.i], [b.sub.i] and [c.sub.i], respectively, must be obtained from the observations in the IBNR-triangle, i.e., from the pairs [{([Z.sub.i+1,j]; Z.sub.[i,j])}.sub.j[greater than or equal to]i]. Notice that the CL method is a special case of the LCL method when we consider that [b.sub.i] = 0.

It is easy to check that the amount of the whole claims for the accidents occurred in the jth year at the end of the n years of development, [Z.sub.n,j], is

(65) [Z.sub.n,j] = [b.sub.n-1] + [c.sub.n-1] { ... [b.sub.j+2] + [c.sub.j+2][[b.sub.j+1] + [c.sub.j+1]([b.sub.j] + [c.sub.j][Z.sub.j,j])]}.

If we suppose that the expansion of the claims produced by the accidents in a given year is done within n years, then [Z.sub.n,j] is the estimate of the amount of all the claims corresponding to the year of occurrence j. Therefore, the IBNR reserves corresponding to the accidents of the year j, [R.sub.j], is obtained by calculating the difference between [Z.sub.n,j] and the amount of the claims reported, [Z.sub.j,j], i.e., [R.sub.j] = [Z.sub.n,j] - [Z.sub.j,j] So, the total IBNR reserve (R) (corresponding to all the accidents from year 1 to year n) is (7)

(66) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We think that this approach has several drawbacks. First, OLS is useful when we start from a wide sample, but it is not advisable for calculating IBNR reserves. Similarly, using all the information available in the IBNR-triangle requires estimating [Z.sub.n,j] not by exact values but by means of probabilistic confidence intervals. Unfortunately, this requires a great computational effort. In any case, we think that these considerations can be extended to the statistical methods discussed above.

We will show that adapting LCL method to fuzzy regression can be a suitable alternative. It will allow us to use all the information provided by the IBNR-triangle more efficiently. So, let us assume that the evolution of the accumulated claims of the accidents happened in the year j from the ith to the i+1th developing years can be adjusted using a fuzzy linear relation [Z.sub.i+1,j] = [b.sub.i] + [c.sub.i] [Z.sub.i,j]. If we state that [b.sub.i] and [c.sub.i] are the STFN ([b.sub.iC], [b.sub.iR]), ([C.sub.iC], [C.sub.iR]), respectively, we can write

(67) [Z.sub.i+1,j] = ([Z.sub.(i+1,j)C], [Z.sub.(i+1,j)R]) = ([b.sub.iC], [b.sub.iR]) + ([c.sub.iC], [c.sub.iR]) [Z.sub.i,j] = ([b.sub.iC] + [c.sub.iC] [Z.sub.i,j], [b.sub.iR] + [c.sub.iR] [Z.sub.i,j]).

The estimates of [b.sub.i] and [c.sub.i], are symbolized as [b.sub.i] = ([b.sub.iC], [b.sub.iR]) and [c.sub.i] = ([c.sub.iC], [c.sub.iR]), respectively. Then, the prediction of the final cost of the accidents produced in year j, [Z.sub.n,j], is obtained from (65) as

(68) [Z.sub.n,j] = [b.sub.n-1] + [c.sub.n-1] {... [b.sub.j+2] + [c.sub.j+2] [[b.sub.j+1] + [c.sub.j+1] ([b.sub.j] + [c.sub.j] [Z.sub.j,j])]}.

Clearly, [Z.sub.n,j] is not an STFN, but it can be approximated reasonably well by a TSFN, i.e., [Z.sub.n,j] [approximately equal to] ([Z.sub.(n,j)C], [Z.sub.(n,j)R]). To do this, we must, in the fuzzy recursive calculation (68), use the approximating formula (11) for multiplication between two STFN. Finally, we obtain the IBNR reserve for the jth year of occurrence as the FN [R.sub.j] = ([R.sub.jC], [R.sub.jR]):

[R.sub.j] = [Z.sub.n,j] - [Z.sub.j,j] = ([Z.sub.(n,j)C] - [Z.sub.(n,j)R]). (69)

Therefore, the whole IBNR reserve is the STFN R = ([R.sub.C], [R.sub.R]), which is obtained from (66) as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

To illustrate our proposal, we develop the following example, which is similar to the one in Straub (1997, p. 106). Table 7 shows the IBNR triangle we will use.

Table 8 shows the main results when nonfuzzy methods are used. The development coefficients when we use fuzzy regression with an inclusion level [alpha] = 0.5 and the fuzzy IBNR reserves are given in Table 9.

Let us illustrate how we have used fuzzy regression. If we use the fuzzy LCL with intercept, to obtain the development coefficients from the year i = 1 to the year i = 2, i.e., the STFBs [b.sub.1] = ([b.sub.1C], [b.sub.1R]) and [c.sub.1] = ([c.sub.1C], [c.sub.1R]), and taking a level of inclusion [alpha] [greater than or equal to] [[alpha].sup.*] = 0.5, we must take the data in the second row of Table 7 and solve the following linear program:

Minimize 5 [b.sub.1R] + 690[c.sub.1R]

subject to:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and solving [b.sub.1C] = -12; [b.sub.1R] = 0; [c.sub.1C] = 1.771; [c.sub.1R] = 0.057.

Finally, we would like to mention some other promising applications of fuzzy regression in actuarial science. One useful application may be in trending claims costs. (8) When projecting future claims costs, standard actuarial practice uses time trend models. More academic approaches propose regressing the cost of the claims with respect to the value of an economic index (e.g., the consumer price index). So, to obtain the final prediction of the claims amount at a future moment we need to predict the value of the economic index at that moment. To do this, time trending may again be used. An alternative is to employ ARIMA time-series models.

Several instruments are derived from the fuzzy regression model in this article that may provide suitable solutions. Watada (1992) proposed a fuzzy model for time series analysis based on polynomial time trending with STFNs. Tseng et al. (2001) combined conventional ARIMA models with Tanaka and Ishibuchi's regression method. Obviously, in both cases these fuzzy methods will lead to forecasting with STFNs.

Another way of determining the discount rate of the liabilities is to use CAPM (see, e.g., Taylor, 1994, for a review of CAPM applied to insurance). With CAPM, the beta of the liabilities ([[beta].sub.L]) is obtained from the beta of the asset portfolio ([[beta].sub.A]) and the insurer's equity ([[beta].sub.E]), i.e., [[beta].sub.L] = f([[beta].sub.A], [[beta].sub.E]) and [[beta].sub.L] < 0. Cummins and Derrig (1997, p. 25) suggested fuzzifying the statistical estimates of the betas since, as they point out, there are several sources that disturb their quantification. Fuzzy regression provides a suitable way of doing this fuzzifycation and enables the decision maker to grade the level of uncertainty in the final estimates by choosing the appropriate level of congruency in fuzzy regression ([alpha].sup.*).

CONCLUSIONS

Actuarial pricing requires an estimate of the behavior of future interest rates that are unknown when the valuation is made. In the last few years, the analysis of temporal structure of interest rates has become an important topic in actuarial science. Likewise, several articles in the actuarial literature consider that estimating the discount rates using fuzzy numbers is a good alternative. With this in mind, we have attempted to make up for the uncertainty about estimating interest rates with fuzzy numbers, i.e., to develop the hypothesis "an 'expert' subjectively estimates the yield rates." If we accept that the "experts" are the traders of the fixed income markets, these subjective estimates are implied in the price of the debt instruments and, therefore, in the yield curve of these instruments. Our method quantifies the experts' subjective estimates of the spot rates and spot interest rates for the future (the forward rates) with fuzzy numbers. This method, based on a fuzzy regression technique, uses all the prices of the bonds negotiated throughout one session in such a way that we do not lose any information. On the other hand, when we use an econometric method we must reduce these prices to representative ones, thus losing some information. We would like to remark that the results of our method are similar to those of Carriere (1999), but that he suggests fitting the yield curve using probabilistic confidence interval values whereas we fit the temporal structure of interest rates with fuzzy numbers. So, if we represent these fuzzy numbers by their [alpha]-cuts, we obtain a yield curve described by a direct analog of confidence intervals.

We have adjusted the yield curve with symmetrical TFNs because the arithmetic is easy and interpreting the estimates is intuitive because they are well adapted to the way people make predictions. An interest rate given by (0.03, 0.005) indicates that we expect an interest rate of about 3 percent, and that we do not expect deviations from it to be greater than 50 basis points. We then discussed how to use a fuzzy TSIR to calculate the present value of a stream of nonrandom amounts and the net single premium for some life and property-liability insurance contracts. We showed that to obtain the fuzzy prices is easy because our method fits the TSIR by a discount function that is described with symmetrical TFNs.

Finally, we discussed other actuarial problems where applying fuzzy regression is promising. We concentrated our discussion on calculating the IBNR reserves but also indicated other future areas of research such as trending cost claims and estimating the beta of liabilities.

Notice that when our initial data is given by fuzzy numbers, the estimate of the objective magnitude (the premiums, the IBNR reserve, etc.) is not an exact value but a fuzzy number. For example, in "Fuzzy Financial Pricing of Property-Liability Insurance" the value of the premium for property-liability insurance was (992.37, 52.36). This can be understood as "the premium must be approximately 992.37." To obtain the definitive value of the magnitude, it must be transformed into a crisp value. To do this we need to apply a defuzzifying method--see Zhao and Govind (1990) for a wide discussion of fuzzy mathematics, and Cummins and Derrig (1997) or Terceno et al. (1996) for applications in fuzzy-actuarial analysis. Another way of doing this, which is very consistent with practice in the real world, is to consider the fuzzy quantification as a first approximation that allows a margin for the "actuarial subjective judgment" or upper and lower bounds for acceptable market prices. Finally, the actuary must use his/her intuition and experience to establish the crisp value of the fuzzy estimate. For example, for premium (992.37, 52.36), the actuary might decide that a final price 1017.37 is acceptable but 928 is not.

APPENDIX

Obtaining the Forward Rates From a Fuzzy TSIR

The forward rate for the tth year, [r.sub.t], can be obtained from the fuzzy equation

(A1) [f.sub.t-1] [(1 + [r.sub.t]).sup.-1] = [f.sub.t].

To solve this equation, we identify [(1 + [r.sub.t]).sup.-1] = [G.sub.t], where [G.sub.t] is the value in t-1 of one monetary unit payable at t according to the TSIR. Bearing in mind that the value of the discount function for any t is an STFN [f.sub.t] = ([f.sub.tC], [f.sub.tR]), the above equation can therefore be written using the [alpha]-cuts as:

(A2) [[f.sub.(t-1)C] - [f.sub.(t-1)R] (1 - [alpha]), [f.sub.(t-1)C] + [f.sub.(t-1)R](1 - [alpha])] * [[G.sup.1.sub.t]([alpha]), [G.sup.2.sub.t] ([alpha])] = [[f.sub.tC] - [f.sub.tR](1 - [alpha]), [f.sub.tC] + [f.sub.tR] (1 - [alpha])].

It is easy to see that in Equation (A2):

(A3) [G.sup.1.sub.t]([alpha]) = [(1 + [r.sup.2.sub.t]([alpha]).sup.-1] and [G.sup.2.sub.t]([alpha]) = [(1 + [r.sup.1.sub.t]([alpha]).sup.-1].

The solution of the [alpha]-cut Equation (A3) is: (9)

(A4) [G.sub.t[alpha]] = [[G.sup.1.sub.t]([alpha]), [G.sup.2.sub.t]([alpha])] = [[f.sub.tC] - [f.sub.tR](1 - [alpha]) / [f.sub.(t-1)C] - [f.sub.(t-1)R] (1 - [alpha]), [f.sub.tC] + [f.sub.tR](1 - [alpha]) / [f.sub.(t-1)C] + [f.sub.(t-1)R(1-[alpha])].

Unfortunately, the solution (A4) might not exist (i.e., the expression (A4) does not correspond to a confidence interval). For example, consider that (A4) for a predefined a is given by [0.9, 0.95]. [[G.sup.1.sub.t]([alpha]), [G.sup.2.sub.t]([alpha])] = [0.875, 0.9]. Then [[G.sup.1.sub.t]([alpha]), [G.sup.2.sub.t]([alpha])] = [0.972, 0.947]. Clearly, it is not a confidence interval since 0.972 < 0.947. From Buckley and Qu (1990a, p. 46, Theorem 3), the necessary and sufficient condition for the existence of [G.sub.t] is

(A5) [f.sub.tR] / [f.sub.(t-1)R] [greater than or equal to] [f.sub.tC] / [f.sub.(t-1)C] ?? [f.sub.(t-1)C] * [f.sub.tR] [greater than or equal to] [f.sub.tC] * [f.sub.(t-1)R].

If we define P in Equations (40d), (40e), (400, and (40g) as being lower than or equal to the periodicity used to obtain the spot rates and forward rates, and [g.sub.j(t)] as nondecreasing, it is easy to check that [f.sub.tR] [greater than or equal to] [f.sub.(t-1)R]. Moreover, if we combine the constraints (40d) and (40e), we see that [f.sub.(t-1)C [greater than or equal to] [f.sub.tC]. Therefore, [f.sub.(t-1)C] * [f.sub.tR] [greater than or equal to] [f.sub.tC] * [f.sub.(t-1)R], and we can obtain the [alpha]-cuts of [G.sub.t] with (A4).

Then, from (A3) and (A4), we obtain the [alpha]-cuts of [r.sub.t], [r.sub.t[alpha]] by making:

(A6) [r.sub.t[alpha]] = [[r.sup.1.sub.t]([alpha]), [r.sup.2.sub.t]([alpha])] = [[f.sub.(t-1)C] + [f.sub.(t-1)R](1 - [alpha]) / [f.sub.tC] + [f.sub.tR](1 - [alpha]) - 1, [f.sub.(t-1)C - [f.sub.(t-1)R(1 - [alpha]) / [f.sub.tC] - [f.sub.tR](1 - [alpha]) - 1].

It is easy to check that the extremes of [r.sub.t[alpha]], are not given as a linear expression of [alpha]. However, if we approximate the functions [r.sup.1.sub.t]([alpha]), [r.sup.2.sub.t]([alpha]) using Taylor's expansion to the first grade from [alpha] = 1, then:

(A7) [r.sup.1.sub.t]([alpha]) [approximately equal to] [f.sub.(t-1)C] / [f.sub.tR] - 1 - [f.sub.(t-1)C] * [f.sub.tR] - [f.sub.tC] * [f.sub.(t-1)R] / [([f.sub.tC]).sup.2](1 - [alpha]) and [r.sup.2.sub.t]([alpha]) [approximately equal to] [f.sub.(t-1)C] / [f.sub.tR] - 1 + [f.sub.(t-1)C] * [f.sub.tR] - [f.sub.tC] * [f.sub.(t-1)R] / [([f.sub.tC]).sup.2](1 - [alpha]).

In conclusion, from (A7) it is easy to check that [r.sub.t] can be approximated by an STFN [r.sub.t] [approximately equal to] ([r.sub.tC], [r.sub.tR]) where:

(A8) [r.sub.tC] = [f.sub.(t-1)C] / [f.sub.tR] - 1 and [r.sub.tR] = [f.sub.(t-1)C] * [f.sub.tR] - [f.sub.tC] * [f.sub.(t-1)R] / [([f.sub.tC]).sup.2](1 - [alpha]).

(1) The TFNs are a special case of a wider family of FNs called L-R FNs. For a detailed explanation see Dubois and Prade (1980).

(2) If [l.sub.A] = [r.sub.A] = 0, A is the crisp number [a.sub.C].

(3) Many methods for ordering fuzzy numbers are proposed in the literature. Obviously, the choice of method depends on the problem. For our purpose we have chosen Ramik and Rimanek's criteria.

(4) Clearly, the subtraction of two STFNs is an STFN because the subtraction is the sum of the first FN with the second one multiplied by -1.

(5) Fedrizzi, Fedrizzi, and Ostasiewicz (1993) initially suggested fuzzy regression methods for economics. Some economic applications can be found in Ramenazi and Duckstein (1992) or Profillidis, Papadopoulos, and Botzoris (1999).

(6) For a detailed explanation of how we constructed our [g.sub.j](t) and chose our knots, see McCulloch (1971).

(7) Notice that with this approach, we do not consider investment income (i.e., the investment income is assumed to be 0%). However, this is the traditional approach to IBNR reserves. To simplify our explanation, we will continue with this hypothesis.

(8) For an extensive exposition see Cummins and Derrig (1993).

(9) In this case we apply the so-called solution "classical solution" in fuzzy sets literature. This solution does not always exist but as we will specify the functions [g.sub.j] as nondecreasing, we can ensure that it does, as we will show.

REFERENCES

Ang, A., and M. Sherris, 1997, Term Structure Models: A Perspective from the Long Rate, North American Actuarial Journal, 3(3): 122-138.

Bababbell, D. F., and C. B. Merrill, 1996, Valuation of Interest-Sensitive Financial Instruments (Schaumburg: Society of Actuaries).

Benjamin, S., and L. M. Eagles, 1986, Reserves in Lloyd's and the London Market, Journal of the Institute of Actuaries, 113(2): 197-257.

Buckley, J. J., 1987, The Fuzzy Mathematics of Finance, Fuzzy Sets and Systems, 21: 57-73.

Buckley, J. J., and Y. Qu, 1990a, Solving Linear and Quadratic Fuzzy Equations, Fuzzy Sets and Systems, 38: 43-59.

Buckley, J. J., and Y. Qu, 1990b, On Using [alpha]-Cuts to Evaluate Fuzzy Equations, Fuzzy Sets and Systems, 38: 309-312.

Buckley, J. J., and Y. Qu, 1990c, Solving Fuzzy Equations: A New Solution Concept, Fuzzy Sets and Systems, 39: 291-301.

Carriere, J. F., 1999, Long-Term Yield Rates for Actuarial Valuations, North American Actuarial Journal, 3(3): 13-24.

Chambers, D. R., W. T. Carleton, and D. W. Waldman, 1984, A New Approach to Estimation of the Term Structure of Interest Rates, Journal of Financial and Quantitative Analysis, 19: 233-252.

Cummins, J. D., 1990, Multi-Period Discounted Cash Flow Rate-Making Models in Property-Liability Insurance, Journal of Risk and Insurance, 57(1): 79-109.

Cummins, J. D., and R. A. Derrig, 1993, Fuzzy Trends in Property-Liability Insurance Claim Costs, Journal of Risk and Insurance, 60(3): 429-465.

Cummins, J. D., and R. A Derrig, 1997, Fuzzy Financial Pricing of Property-Liability Insurance, North American Actuarial Journal, 1(4): 21-44.

de Andres, J., 2000, Estimating the Temporal Structure of Interest Rates Using Fuzzy Numbers. Application to Financial and Actuarial Valuation and to the Analysis of the Life Insurer's Solvency, Rovira i Virgili University, Unpublished doctoral thesis (in Spanish).

Delbaen, F., and S. Lorimier, 1992, Estimation of the Yield Curve and the Forward Rate Curve Starting From a Finite Number of Observations, Insurance: Mathematics and Economics, 11: 259-269.

Derrig, R. A., and K. M. Ostaszewski, 1995, Fuzzy Techniques of Pattern Recognition in Risk and Claim Classification, Journal of Risk and Insurance, 62(3): 447-482.

Derrig, R. A., and K. M. Ostaszewski, 1997, Managing the Tax Liability of a Property Liability Insurance Company, Journal of Risk and Insurance, 64: 695-711.

Devolder, P., 1988, Le taux d'actualisation en assurance, The Geneva Papers on Risk and Insurance, 13: 265-272.

Dubois, D., and H. Prade, 1980, Fuzzy Sets and Systems: Theory and Applications (New York: Academic Press).

Dubois, D., and H. Prade, 1993, Fuzzy Numbers: An Overview, in: D. Dubois, H. Prade, and R. R. Yager, eds., Fuzzy Sets for Intelligent Systems (San Mateo, Calif.: Morgan Kaufmann Publishers), pp. 113-148.

England, P. D., and R. J. Verrall, 1999, Analytic and Bootstrap Estimates of Prediction Errors in Claim Reserving, Insurance: Mathematics and Economics, 25: 281-293.

England, P. D., and R. J. Verrall, 2001, A Flexible Framework for Stochastic Claims Reserving, Proceedings of the Casualty Actuarial Society, pp. 1-38.

England, P. D., and R. J. Verrall, 2002, Stochastic Claims Reserving in General Insurance, Paper presented to the Institute of Actuaries, Available at http://www. actuaries.org.uk/sessional/sm0201.pdf.

Fedrizzi, M., M. Fedrizzi, and W. Ostasiewicz, 1993, Towards Fuzzy Modelling in Economics, Fuzzy Sets and Systems, 54: 259-268.

Gerber, H. U., 1995, Life Insurance Mathematics (Heildelberg: Springer).

Kaufmann, A., 1986, Fuzzy Subsets Applications in O.R. and Management, in: A. Jones, A. Kaufmann, and H.-J. Zimmermann, eds., Fuzzy Set Theory and Applications (Dordretch: Reidel), pp. 257-300.

Kremer, E., 1982, IBNR Claims and the Two-Way of ANOVA, Scandinavian Actuarial Journal, I: 47-55.

Lemaire, J., 1990, Fuzzy Insurance, ASTIN Bulletin, 20: 33-55.

Li Calzi, M., 1990, Towards a General Setting for the Fuzzy Mathematics of Finance, Fuzzy Sets and Systems, 35: 265-280.

Mack, T., 1993, Distribution-Free Calculation of the Standard Error of Chain-Ladder Reserve Estimates, ASTIN Bulletin, 23: 213-223.

McCulloch, J. H., 1971, Measuring the Term Structure of Interest Rates, The Journal of Business, 34: 19-31.

McCulloch, J. H., 1975, The Tax-Adjusted Yield Curve, Journal of Finance, 30: 811-829.

Myers, S.C., and R.A. Cohn, 1987, A Discounted Cash-Flow Approach to Property-Liability Insurance Rate Regulation, in: J. D. Cummins and S. E. Harrington, eds., Fair Rate of Return in Property-Liability Insurance (Norwell, Mass.: Kluwer Academic Publishers).

Ostaszewski, K., 1993, An Investigation into Possible Applications of Fuzzy Sets Methods in Actuarial Science (Schaumburg: Society of Actuaries).

Profillidis, V. A., B. K. Papadopoulos, and G. N. Botzoris, 1999, Similarities in Fuzzy Regression and Application on Transportation, Fuzzy Economic Review, 4(1): 83-98.

Ramenazi, R., and L. Duckstein, 1992, Fuzzy Regression Analysis of the Effect of University Research on Regional Technologies, in: J. Kacprzyk and M. Fedrizzi, eds., Fuzzy Regression Analysis (Heildelberg: Physica-Verlag), pp. 237-263.

Ramik, J., and J. Rimanek, 1985, Inequality Relation Between Fuzzy Numbers and its Use in Fuzzy Optimization, Fuzzy Sets and Systems, 16: 123-138.

Renshaw, A. E., 1989, Chain-Ladder Modelling and Interactive Modelling (Claims Reserving and GLIM), Journal of the Institute of Actuaries, 116: 559-587.

Renshaw, A. E., and R. J. Verrall, 1998, A Stochastic Model Underlying the Chain Ladder Technique, British Actuarial Journal, 4: 903-923.

Straub, E., 1997, Non-Life Insurance Mathematics (Berlin: Springer).

Tanaka, H., 1987, Fuzzy Data Analysis by Possibilistic Linear Models, Fuzzy Sets and Systems, 24: 363-375.

Tanaka, H., and H. Ishibuchi, 1992, A Possibilistic Regression Analysis Based on Linear Programming, in: J. Kacprzyk and M. Fedrizzi, eds., Fuzzy Regression Analysis (Heildelberg: Physica-Verlag), pp. 47-60.

Taylor, G. C., 1994, Fair Premium Rating Methods and the Relations Between Them, Journal of Risk and Insurance, 61(4): 592-615.

Terceno, A., G. Barbera, J. de Andres, and C. Belvis, 1996, Fuzzy Methods Incorporated to the Study of Personal Insurances, Fuzzy Economic Review, 1(2): 105-119.

Tseng, F.-M., G.-H. Tzeng, H.-C. Yu, and B. J.-C. Yuan, 2001, Fuzzy ARIMA Model for Forecasting the Foreign Exchange Market, Fuzzy Sets and Systems, 118: 9-19.

Vasicek, O. A., and H. G. Fong, 1982, Term Structure Modelling Using Exponential Splines, Journal of Finance, 37: 339-349.

Verrall, R. J., 1989, A Space State Representation of the Chain Ladder Model, Journal of the Institute of Actuaries, 116: 589-610.

Wang, H.-F., and R.-C. Tsaur, 2000, Insight of a Fuzzy Regression Model, Fuzzy Sets and Systems, 112: 355-369.

Watada, J., 1992, Fuzzy Time-Series Analysis and Forecasting of Sales Volume, in: J. Kacprzyk and M. Fedrizzi, eds., Fuzzy Regression Analysis (Heildelberg: Physica-Verlag), pp. 211-227.

Wright, T. S., 1990, A Stochastic Method for Claims Reserving in General Insurance, Journal of the Institute of Actuaries, 117: 677-731.

Yao, Y., 1999, Term Structure Models: A Perspective From the Long Rate, North American Actuarial Journal, 3(3): 122-138.

Yen, K. K., S. Ghoshray, and G. Roig, 1999, A Linear Regression Model Using Triangular Fuzzy Number Coefficients, Fuzzy Sets and Systems, 106: 167-177.

Young, V. R., 1996, Insurance Rate Changing: A Fuzzy Logic Approach, Journal of Risk and Insurance, 63(3): 461-484.

Zadeh, L. A., 1965, Fuzzy Sets, Information and Control, 8: 338-353.

Zhao, R., and R. Govind, 1991, Defuzzification of Fuzzy Intervals, Fuzzy Sets and Systems, 43: 45-55.

Jorge de Andres Sanchez and Antonio Terceno Gomez are from the Department of Business Administration, Faculty of Economics and Business Studies, Rovira i Virgili University, Spain. The authors wish to thank two anonymous reviewers for their valuable comments.

TABLE 1 Prices of the Bonds Negotiated in the Spanish Debt Market on June 29, 2001 Coupon Maturity Maturity k Asset (annual) (days) (years) 1 T-Bill 0.00% 18 0.05 2 BOND 5.35% 163 0.45 3 T-Bill 0.00% 382 1.05 4 BOND 4.25% 391 1.07 5 T-Bill 0.00% 521 1.43 6 BOND 5.25% 576 1.58 7 STRIP 0.00% 576 1.58 8 BOND 3.00% 576 1.58 9 STRIP 0.00% 756 2.07 10 BOND 4.60% 757 2.07 11 BOND 4.50% 1,122 3.07 12 BOND 4.65% 1,216 3.33 13 BOND 3.25% 1,307 3.58 14 BOND 4.95% 1,490 4.08 15 BOND 10.15% 1,673 4.58 16 BOND 4.80% 1,945 5.33 17 BOND 7.35% 2,098 5.75 18 BOND 6.00% 2,402 6.58 19 STRIP 0.00% 2,775 7.60 20 BOND 5.15% 2,948 8.08 21 BOND 4.00% 3,134 8.59 22 BOND 5.40% 3,680 10.08 23 BOND 5.35% 3,771 10.33 24 BOND 6.05% 4,229 11.59 25 STRIP 0.00% 4,230 11.59 26 BOND 4.75% 4,774 13.08 27 BOND 6.00% 10,073 27.60 28 BOND 5.75% 11,351 31.10 [P.sup.k. [P.sup.k. k Asset sub.min] sub.max] 1 T-Bill 99.779 99.779 2 BOND 103.258 103.313 3 T-Bill 95.758 95.758 4 BOND 103.907 103.947 5 T-Bill 94.220 94.220 6 BOND 103.555 103.669 7 STRIP 93.579 93.749 8 BOND 99.337 99.376 9 STRIP 91.540 91.540 10 BOND 104.670 104.917 11 BOND 104.017 104.166 12 BOND 98.466 98.702 13 BOND 97.026 97.200 14 BOND 105.407 105.918 15 BOND 126.340 126.340 16 BOND 97.785 98.385 17 BOND 113.539 113.539 18 BOND 107.400 108.206 19 STRIP 68.412 68.412 20 BOND 104.101 104.307 21 BOND 92.679 93.473 22 BOND 97.716 98.923 23 BOND 96.966 97.749 24 BOND 108.098 108.168 25 STRIP 53.357 53.357 26 BOND 96.506 97.567 27 BOND 103.722 105.194 28 BOND 93.954 94.777 TABLE 2 Comparison of the Estimates of OLS With the Fuzzy Estimates of the TSIR in the Spanish Public Debt Market on June 29, 2001 Econometric Fuzzy Regression Methods [[alpha].sup.*] = 0.5 T [i.sub.t] [r.sub.t] [i.sub.t] [r.sub.t] 1 0.0435 0.0435 (0.0433, 0.0033) (0.0433, 0.0033) 2 0.0423 0.0412 (0.0431, 0.0023) (0.0430, 0.0012) 3 0.0440 0.0473 (0.0447, 0.0023) (0.0479, 0.0025) 4 0.0470 0.0563 (0.0472, 0.0029) (0.0547, 0.0047) 5 0.0496 0.0599 (0.0494, 0.0034) (0.0581, 0.0055) 6 0.0513 0.0600 (0.0510, 0.0037) (0.0594, 0.0052) 7 0.0525 0.0599 (0.0524, 0.0039) (0.0606, 0.0048) 8 0.0534 0.0595 (0.0536, 0.0039) (0.0619, 0.0043) 9 0.0540 0.0589 (0.0547, 0.0039) (0.0632, 0.0037) 10 0.0545 0.0592 (0.0556, 0.0039) (0.0645, 0.0035) 11 0.0551 0.0606 (0.0566, 0.0039) (0.0658, 0.0038) 12 0.0557 0.0619 (0.0574, 0.0039) (0.0670, 0.0042) 13 0.0562 0.0633 (0.0582, 0.0039) (0.0682, 0.0045) 14 0.0568 0.0647 (0.0590, 0.0040) (0.0693, 0.0049) 15 0.0574 0.0660 (0.0598, 0.0041) (0.0703, 0.0053) Fuzzy Regression D [[alpha].sup.*] = 0.75 T [i.sub.t] [r.sub.t] [i.sub.t] [r.sub.t] 1 (0.0433, 0.0066) (0.0433, 0.0066) 0.44% 0.44% 2 (0.0431, 0.0045) (0.0430, 0.0025) 1.92% 4.41% 3 (0.0447, 0.0047) (0.0479, 0.0049) 1.70% 1.31% 4 (0.0472, 0.0058) (0.0547, 0.0094) 0.38% 2.75% 5 (0.0494, 0.0068) (0.0581, 0.0109) 0.41% 2.92% 6 (0.0510, 0.0074) (0.0594, 0.0103) 0.52% 1.01% 7 (0.0524, 0.0077) (0.0606, 0.0096) 0.23% 1.27% 8 (0.0536, 0.0078) (0.0619, 0.0086) 0.35% 4.01% 9 (0.0547, 0.0078) (0.0632, 0.0074) 1.19% 7.30% 10 (0.0556, 0.0077) (0.0645, 0.0070) 2.02% 8.88% 11 (0.0566, 0.0077) (0.0658, 0.0076) 2.67% 8.61% 12 (0.0574, 0.0078) (0.0670, 0.0083) 3.19% 8.25% 13 (0.0582, 0.0079) (0.0682, 0.0090) 3.58% 7.79% 14 (0.0590, 0.0080) (0.0693, 0.0098) 3.87% 7.22% 15 (0.0598, 0.0082) (0.0703, 0.0106) 4.07% 6.52% TABLE 3 Free Default Discount Factors for the Next 5 Years [f.sub.1] [f.sub.2] [f.sub.3] [f.sub.4] [f.sub.5] (0.9585, (0.9190, (0.8770, (0.8315, (0.7858, 0.0030) 0.0040) 0.0059) 0.0093) 0.0128) TABLE 4 Fuzzy Pure Single Premiums for Several Kinds of Policies 35 Years 45 Years [[PI].sub. 1] (779.69, 12.72) (770.86, 12.58) [[PI].sub. 2] (6.76, 0.06) (16.50, 0.14) [[PI].sub. 3] (786.46, 12.78) (787.37, 12.72) 55 Years 65 Years [[PI].sub. 1] (752.42, 12.27) (711.67, 11.61) [[PI].sub. 2] (36.92, 0.31) (81.92, 0.69) [[PI].sub. 3] (789.34, 12.59) (793.59, 12.30) Table 5 Premiums for Several Payments of Losses Pair ([c.sub.1], (0.75, 0.25) (0.5, 0.5) (0.25, 0.75) [c.sub.2]) Premium (1005.91, 53.54) (992.37, 52.96) (978.99, 52.35) TABLE 6 IBNR-Triangle Accident Years Development Years 1 2 j 1 [Z.sub.1,1] [Z.sub.1,2] ... [Z.sub.1,j] 2 [Z.sub.2,2] ... [Z.sub.2,j] . . . i . . . n - 1 n Accident Years Development Years n - 1 n 1 ... [Z.sub.1,n-1] [Z.sub.1,n] 2 ... [Z.sub.2,n-1] [Z.sub.2,n] . . . . . . . . . i . . . . . . . . . n - 1 [Z.sub.n-1,n-1] [Z.sub.n-1,n] n [Z.sub.n,n] TABLE 7 IBNR Triangle in Our Analysis Occurrence Year Develop- 1 2 3 4 5 6 ment Year (Y-6) (Y-5) (Y-4) (Y-3) (Y-2) (Y-1) 1 130 125 150 150 140 125 2 213 252 258 232 211 3 413 392 374 340 4 549 449 490 5 539 613 6 613 TABLE 8 Determining IBNR Reserves by Nonfuzzy Methods Development Coefficients LCL with [b.sub.i] (A) LCL Year of without Development [b.sub.i] (B) i [b.sub.i] [c.sub.i] [c.sub.i] 1 -1.619 1.702 1.690 2 61.398 1.336 1.592 3 46.844 1.233 1.359 4 158.854 0.927 1.228 5 0 1 1 Development Coefficients IBNR Reserves Year of Development Year of [R.sub.j] [R.sub.j] i Occurrence from (A) from (B) 1 Y-6 477.67 453.97 2 Y-5 384.62 353.18 3 Y-4 261.16 276.41 4 Y-3 118.68 125.14 5 Y-2/Y-1 0.00 0.00 R 1242.13 1208.70 TABLE 9 Determining IBNR Reserves With Fuzzy Regression Development Coefficients LCL with [b.sub.i] (A) Year of LCL without Development [b.sub.i] (B) i [b.sub.i] [c.sub.i] [c.sub.i] 1 (-12, 0) (1.771, 0.57) (1.689, 0.063) 2 (106.553, 0) (1.161, 0.11) (1.579, 0.12) 3 (-1.308, 0) (1.344, 0.12) (1.341, 0.12) 4 (158.85, 0) (0.927, 0) (1.226, 0.051) 5 (0, 0) (1, 0) (1, 0) Development Coefficients IBNR Reserves Year of Development Year of [R.sub.j] [R.sub.j] i Occurrence from (A) from (B) 1 Y-6 (476.13, 80.48) (439.62, 138.70) 2 Y-5 (385.49, 68.33) (339.72, 114.01) 3 Y-4 (259.12, 45.78) (256.66, 88.63) 4 Y-3 (118.68, 0) (123.93, 27.77) 5 Y-2/Y-1 (0, 0) (0, 0) R (1239.42, 194.59) (1168.93, 369.1)

Gale Copyright:

Copyright 2003 Gale, Cengage Learning. All rights
reserved.