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Estimating a fuzzy term structure of interest rates using fuzzy regression techniques
European Journal of Operational Research 154 (2004) 804–818 www.elsevier.com/locate/dsw
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Estimating a fuzzy term structure of interest rates using fuzzy regression techniques ~ o Go mez Jorge de Andres S anchez *, Antonio Tercen Department of Business Administration, Faculty of Economics and Business Studies, Rovira i Virgili University, Av. de la Universitat 1, 43204 Reus, Spain Received 11 January 2002; accepted 12 November 2002
Abstract Several papers in the financial literature propose using fuzzy numbers (FNs) to model interest rate uncertainty. However, in our opinion, the first problem to be solved is how to estimate these rates with FNs. In this paper, we attempt to provide a solution to this question with a method for adjusting the temporal structure of interest rates (TSIR) that is based on fuzzy regression techniques. This method will enable to quantify the anticipated rates in the fixed income markets for the future with FNs. In particular, we discuss how to estimate the TSIR with triangular fuzzy numbers because of their desirable properties. Ó 2003 Elsevier B.V. All rights reserved. Keywords: Finance; Fuzzy sets; Fuzzy regression; Investment analysis; Temporal structure of interest rates
1. Introduction To obtain the financial price of any asset we have to discount the cash flows that the asset produces along its life. So, the discount rates that must be applied throughout the life of the asset must be estimated. The reference values of these discount rates are the interest rates that are free of default risk, i.e. those that correspond to public debt bonds. Many financial analyses are concerned with the medium and long term and, in our opinion,
* Corresponding author. Tel.: +34-977-75-98-32; fax: +34977-75-98-10. E-mail addresses: [email protected] (J. de Andres Sanchez), [email protected] (A. Terce~ no G omez).
modelling the behaviour of interest rates in the long term by means of a stochastic model is not very realistic. 1 Gerber (1995) points out that there is no commonly accepted stochastic model for making predictions about long term discount rates. So, many authors think that it is more suitable and realistic to make financial analyses, in the middle and long term, using estimates of discount rates by means of fuzzy numbers (FNs), because in these circumstances often it is only
1 The papers that model the temporal structure of interest rates (TSIR) in a stochastic way suppose that its dynamics can be synthesised in a few variables (the short term spot rate, the spread of the short term spot rate compared to the long term spot rate, etc.) whose behaviour is given by a set of stochastic differential equations.
0377-2217/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0377-2217(02)00854-8
J. de Andres Sanchez, A. Terce~no Gomez / European Journal of Operational Research 154 (2004) 804–818
available imprecise data or data related in a vague way with the future behaviour of interest rates (for example, the price of fixed income securities, the opinion of ‘‘experts’’ about the future behaviour of macroeconomic magnitudes, etc.). For applications of Fuzzy Sets Theory in financial pricing, see Kaufmann (1986), Buckley (1987), Li Calzi (1990), Gil (1999), Kutcha (2000) or Terce~ no et al. (2001) while for actuarial pricing, see Lemaire (1990), Ostaszewski (1993), Cummins and Derrig (1997), Derrig and Ostaszewski (1997) or Terce~ no et al. (1996). However, the literature has not paid enough attention to how to estimate future discount rates with FNs and, consequently, how to construct their membership function. It is usually suggested that ‘‘the rates are estimated subjectively by the experts using fuzzy numbers’’ with no further explanation. In this respect, Fedrizzi et al. (1993) point out that, the problem of constructing the membership function is a crucial problem for the whole theory of fuzzy sets. Therefore, it seems to be inadequate to start considerations with a proposition like: ‘‘suppose the observations are given as fuzzy numbers’’, which is probably the most often used expression in literature on fuzzy sets. In this paper, we propose to solve the problem of constructing the membership function of future yield rates by estimating the TSIR with fuzzy subsets, since the TSIR implicitly contains the expectations of the fixed income market agents (i.e. the experts) regarding the evolution of future interest rates. This TSIR can then be taken as a reference when pricing any asset––for this topic, see Carleton and Cooper (1976). We would also like to point out that it was Ostaszewski (1993) who first suggested using a fuzzy TSIR in the field of life insurance pricing. The TSIR is a relation between interest rates and maturity (which we will measure in years) in such a way that the market prices a cash flow with the discount rate associated to its maturity (t years). This rate is named as spot rate, and we will symbolise it as it . So, the present value or one t monetary unit payable in t years is ð1 þ it Þ . This expression is often called the discount function, t which we will denote as ft i.e. ft ¼ ð1 þ it Þ . The implicit or forward rates, which standard financial theory interprets as the spot rates that the
805
market expects to prevail in the future, are given implicitly by the TSIR. Then, from the spot rates it and it where t > t, we can obtain the forward rate to apply when calculating the price in t-years of a cash flow that in this moment will be payable in t t years (t t qt ). So, we must solve ft ð1 þ t t qt Þ
ðt tÞ
¼ ft ;
ð1Þ
so
ft t t qt ¼ ft
1=ðt tÞ 1
ð2Þ
where the Theory of Rational Expectations interprets t t qt obtained with (2), as the discount spot rate that the market expects to prevail in t years for the next t t years. In this paper, the key instrument we use to obtain the TSIR for the market of debt bond instruments is fuzzy regression. In our opinion, fuzzy regression techniques have a number of advantages over traditional regression techniques. For our purposes we shall mention the following: (a) The estimates that are obtained after adjusting the coefficients are not random variables, which are difficult to manipulate in arithmetical operations, but FNs, which are easier to handle arithmetically. So, when starting from magnitudes estimated by random variables (e.g. as a result of its prediction from a least squares regression) these random variables are often reduced to their mathematical expectation (which may or may not be corrected by their variance) so that they are easier to handle. If FNs are used, this loss of information is not needed. (b) If the phenomena investigated are economic or social, the observations are a consequence of the interaction between the economic agentsÕ beliefs and expectations, which are highly subjective and vague. Fuzzy Sets Theory, therefore, is a good way of treating this information. For example, the asset prices that are determined in the markets depend on the agentsÕ expectations of future inflation, the issuersÕ credibility, etc. We think that it is more realistic to consider that the bias between the
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observed values for the dependent variable and its theoretical value (the error) is not random but fuzzy. At least in this way we assume that the subjectivity inherent in the phenomena analysed is high. (c) Observations are often not crisp numbers; they are confidence intervals. For instance, the price of a financial asset throughout one session often oscillates within an interval, rarely does it remain the same (e.g. it can oscillate in the interval [$100, $105]). If econometric methods are to be used, the observations for the explained variable and/or the explanatory variable must be represented by a single value (e.g. in the above interval this value would be $102.5) which involves losing a great deal of information. Fuzzy regression, however, does not necessarily reduce the values of each variable to a crisp number, i.e., all the observed values can be used in the regression analysis ([$100, $105] in our example).
sup8x2X lA ðxÞ ¼ 1. The second is that it must be convex (i.e., its a-cuts must be convex sets). For practical purposes, some of the most widely used FNs are triangular fuzzy numbers (TFNs) because they are easy to handle arithmetically and they have an intuitive interpretation. We shall e as A e ¼ ða; la ; ra Þ, where a is the symbolise a TFN A centre and la and ra are the left and right spreads respectively. For example, a sentence by a financial analyst such as ‘‘I expect for the next two years the free risk rate to be around 5% and deviations no longer that the 1%’’ may be quantified in a very natural way as (0.05, 0.01, 0.01). A TFN is characterised by its membership function lA ðxÞ or, alternatively, by its a-cuts, Aa , as 8 x a þ la > > ; > < la lA ðxÞ ¼ a þ ra x ; > > > ra : 0;
a la < x 6 a; a < x 6 a þ ra ;
ð3Þ
otherwise;
and The structure of the paper is as follows. In the next section, we shall describe some aspects of fuzzy arithmetic and Ishibuchi and NiiÕs extension (2001) of the fuzzy regression method described in Tanaka et al. (1989) and Tanaka and Ishibuchi (1992). In Section 3, we shall discuss some basic characteristics of the models that attempt to adjust the TSIR through the discount function defined by the spot rates. In Section 4, we show how a fuzzy TSIR can be obtained by using fuzzy regression and we apply this methodology to the Spanish public debt market. In Section 5, we state the most important conclusions of the paper.
2. Fuzzy arithmetic and fuzzy regression 2.1. Some aspects of fuzzy numbers e defined over A fuzzy number is a fuzzy subset A real numbers. It is the main instrument used in Fuzzy Set Theory for quantifying uncertain quantities (e.g. in financial analysis, the discount rates). Two properties are required by a FN. The first one is that it must be a normal fuzzy set, i.e.
Aa ¼ ½AðaÞ; AðaÞ ¼ ½a la ð1 aÞ; a þ ra ð1 aÞ: ð4Þ In many financial analyses it is often necessary to evaluate functions (e.g. the net present value), which in a general way we shall name y ¼ f ðx1 ; x2 ; . . . ; xn Þ. Then, if x1 ; x2 ; . . . ; xn are not e 1; A e 2; . . . ; A e n , f ðÞ crisp numbers but are the FNs A e 1; A e 2; . . . ; A e n Þ whose meme ¼ f ðA induces a FN B bership function must be obtained from Zadeh’s extension principle. Unfortunately, it is often impossible to obtain a closed expression for the e , although in many cases membership function of B it is possible to obtain its a-cuts, Ba , from A1a ; A2a ; . . . ; Ana , by doing e 1; A e 2; . . . ; A e n Þ ¼ f ðA1a ; A2a ; . . . ; Ana Þ: Ba ¼ f ð A a ð5Þ In financial mathematics, many functional relationships are continuously increasing or decreasing with respect to all the variables involved, in such a way that it is easy to evaluate the a-cuts e . If the function f ðÞ that induces B e is inof B creasing with respect to the first m variables, where m 6 n, and decreasing with respect to the last n m
J. de Andres Sanchez, A. Terce~no Gomez / European Journal of Operational Research 154 (2004) 804–818
variables, Buckley and Qu (1990b) demonstrate that Ba , i.e. (5), is Ba ¼ ½BðaÞ; BðaÞ ¼ ½f ðA1 ðaÞ; . . . ; Am ðaÞ; Amþ1 ðaÞ; . . . ; An ðaÞÞ; f ðA1 ðaÞ; . . . ; Am ðaÞ; Amþ1 ðaÞ; . . . ; An ðaÞÞ: ð6Þ It is well known that the result of evaluating a e , will be another linear function with TFNs, B TFN. On the other hand, the result of evaluating non-linear functions with TFNs will not be another TFN. In this case it will be interesting to e using a TFN that we shall denote approximate B 0 e by B because of the desirable properties of TFNs. If the function that we evaluate, f ðÞ, is increasing with respect to the first m variables, where m 6 n, and decreasing with respect to the others (i.e. the e are (6)), Dubois and Prade (1993) a-cuts of B e with a FN B e0 ¼ propose to approximate B ðb; lB ; rB Þ whose characteristics are b ¼ f ðaC Þ; m n X X of ðaC Þ of ðaC Þ lb ¼ lai rai ; ox oxi i i¼1 i¼mþ1 rb ¼
m n X X of ðaC Þ of ðaC Þ rai lai ; ox oxi i i¼1 i¼mþ1
ð7Þ
where aC ¼ ða1 ; . . . ; am ; amþ1 ; . . . ; an Þ. In this paper, to obtain the forward rates we will need to solve fuzzy equations with the structure eX e and B e ¼B e , where A e are TFNs with the supA port contained in Rþ . The above equation can be e, X e and B e as expressed through the a-cuts of A Aa Xa ¼ Ba , i.e., ½a la ð1 aÞ; a þ ra ð1 aÞ½X ðaÞ; X ðaÞ ¼ ½b lb ð1 aÞ; b þ rb ð1 aÞ:
ð8Þ
According to Kaufmann and Gupta (1985), the narrowest solution is b lb ð1 aÞ b þ rb ð1 aÞ ; X ðaÞ ¼ : X ðaÞ ¼ a la ð1 aÞ a þ ra ð1 aÞ ð9Þ We must take into account that this solution does not always exist 8a 2 ½0; 1. For example, if A0:5 ¼ ½1; 2 and B0:5 ¼ ½1; 1:5, Eq. (8) is ½1; 2½X ð0:5Þ;
807
X ð0:5Þ ¼ ½1; 1:5. Applying (9) we obtain X ð0:5Þ ¼ 1 and X ð0:5Þ ¼ 1:5=2 ¼ 3=4, i.e. X0:5 ¼ ½1; 3=4, which is not an a-cut of a FN. Nevertheless, Buckley and Qu (1990a) demonstrate that the required condition for the existence of (5) 8a 2 ½0; 1 is lb la P ; b a
rb ra P : b a
ð10Þ
2.2. Fuzzy regression model with asymmetric coefficients In this subsection, we will describe Ishibuchi and NiiÕs extension (2001) of the fuzzy regression model developed in Tanaka et al. (1989) and Tanaka and Ishibuchi (1992), which is one of the most widely used models in economic applications. 2 Like any regression technique, the objective of fuzzy regression is to determine a functional relationship between a dependent variable and a set of independent ones. As we will show, fuzzy regression is in many aspects more versatile than conventional linear regression because functional relationships can be obtained when independent variables, dependent variables, or both, are not only crisp values but intervals. As in econometric linear regression, we shall assume that the explained variable is a linear combination of the explanatory variables. This relationship should be obtained from a sample of n observations fðY1 ; X1 Þ; ðY2 ; X2 Þ; . . . ; ðYj ; Xj Þ; . . . ; ðYn ; Xn Þg, where Xj is the jth observation of the explanatory variable, Xj ¼ ðX0j ; X1j ; X2j ; . . . ; Xij ; . . . ; Xmj Þ. Moreover, X0j ¼ 1 8j, and Xij is the observed value for the ith variable in the jth case of the sample and is always crisp. Yj is the jth observation of the explained variable, j ¼ 1; 2; . . . ; n, and may either be a crisp value or a confidence interval. In both cases, it can be represented through its centre and its spread or radius as Yj ¼ hYjC ; YjR i, where YjC is the centre and YjR is the radius. In particular, we must estimate the following fuzzy linear function: 2
Examples of economic applications of fuzzy regression are the papers by Ramenazi and Duckstein (1992), Profillidis et al. (1999), and Lee and Chen (2001).
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e0 þ A e 1 X1j þ þ A e m Xmj ; Yej ¼ A
ð11Þ
where Yej is the estimation of Yj by a FN after e 0; A e 1; . . . ; A e m. adjusting A Likewise, in fuzzy regression models the disturbance is not introduced as a random addend in the linear relation, but it is incorporated into the e i , i ¼ 0; 1; . . . ; m. Of course, the final coefficients A e i , i ¼ 0; 1; . . . ; m, objective is to adjust the FNs A from the available sample. In this paper we proe i as TFN, i.e., they pose to model the parameters A e i ¼ ðai ; lai ; rai Þ, i ¼ 0; 1; . . . ; m. can be written as A Then, the estimate of Yj , Yej is a TFN that we will name Yej ¼ ðYj ; lYj ; rYj Þ where m X ai Xij ; Y j ¼ a0 þ i¼1
lYj ¼ la0 þ
m X
lai jXij j þ
i¼1 xij P 0
rYj ¼ ra0 þ
m X
m X
i¼1 xij P 0
m X
¼
m X
lai jXij j:
ð12Þ
i¼1 xij <0
i¼0
i¼0
lai
Xij þ
j¼1
m X i¼0
i¼0
rai
n X Xij ; j¼1
ð13cÞ
i ¼ 0; 1; . . . ; m;
ð13dÞ
a 2 ½0; 1Þ;
ð13eÞ
where X0j ¼ 1, j ¼ 1; 2; . . . ; n. If for the second objective, (13b), we require a minimum accomplishment level a 2 ½0; 1Þ, the above programme is transformed into the following linear one: n X m n X m X X Minimise z ¼ lai jXij j þ rai jXij j j¼1
i¼0
j¼1
i¼0
ð14aÞ subject to m X a^0 þ a^i Xij 1
B C m m B C X X B C Bla0 þ la0 jXij j þ rai jXij jCð1 a Þ B C @ A i¼1 i¼1 xij P 0
6 YjC YjR ;
j¼1
n X
j ¼ 1; 2; . . . ; n;
i¼1 0
According to Ishibuchi and Nii (2001), to fit e 0; A e 1; . . . ; A e m we first have to determine the cores A a0 ; a1 ; . . . ; am by using least squares and taking the centres of the observations Yj ¼ 1; 2; . . . ; n as the realisations of the independent variable. These estimates are named a^0 ; a^1 ; . . . ; a^m . Subsequently, we have to determine the parameters lai and rai , which must minimise the uncertainty of the estimates for Yj ; Yej (i.e. their spreads), and simultaneously maximise the congruence of the estimates obtained for Yj which will be measured as lðYj Yej Þ. In particular, we must solve the following multiobjective programme: n X m n X m X X lai Xij þ rai Xij Minimise z ¼ j¼1
lai ; rai P 0;
rai jXij j;
i¼1 xij <0
rai jXij j þ
subject to lðYj Yej Þ P a;
a^0 þ
m X
xij <0
j ¼ 1; 2; . . . ; n;
ð14bÞ
a^i Xij
i¼1
0
1
B C m m B C X X B C þ Bra0 þ rai jXij j þ lai jXij jCð1 a Þ B C @ A i¼1 i¼1 xij P 0
P YjC þ YjR ; lai ; rai P 0;
xij <0
j ¼ 1; 2; . . . ; n;
i ¼ 0; 1; . . . ; m:
ð14cÞ ð14dÞ
Notice that in the last mathematical programme the constraints (14b) and (14c) are consequences of requirement lðYj Yej Þ P a in (13c) while the last block ensures that lai ; rai 8i will be non-negative. 3. Estimating the term structure of interest rates using splines and econometric methods
ð13aÞ Maximise a;
ð13bÞ
It is fairly straightforward to estimate the TSIR for a specific date and a given market if there are
J. de Andres Sanchez, A. Terce~no Gomez / European Journal of Operational Research 154 (2004) 804–818
many sufficiently liquid zero-coupon bonds and their price can be observed without perturbations. However, these conditions are rarely found simultaneously in the fixed income markets. This section describes how to estimate the discount function with econometric methods rather than by adjusting the TSIR directly. These methods can be used if the sample is made up entirely of zerocoupon bonds, entirely of bonds with coupons or, as is usual, if it is made up of both types of bonds. Our starting point is the fact that the kth bond, where k ¼ 1; 2; . . . ; K, and K is the number of available bonds, provides several cash flows (coupons and principal). These are denoted by fðC1k ; t1k Þ; ðC2k ; t2k Þ; ; ðCnkr ; tnkk Þg, where Cik is the amount of the ith cash flow and tik is its maturity expressed in years. If we suppose that the bonds have no embedded option (e.g. convertibility) the price of the kth bond is therefore the sum of the discounted value of every amount with the corresponding spot rate Pk ¼
nk X
Cik ftik
ð15Þ
i¼1
where ftik is the discounted value of one dollar with maturity tik years. Of course, we should then define an analytical form for the discount function in order 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 most popular methods are the one described in McCullochÕs papers (1971, 1975) (quadratic splines and cubic splines) and in Vasicek and FongÕs article (1982) (exponential splines). These methods assume that the discount function is a linear combination of m þ 1 functions of time. Therefore ft ¼
m X
aj gj ðtÞ:
ð16Þ
j¼0
Then, we can deduce that the following linear equation must be estimated: Pk ¼
nk X i¼1
Cik
m X j¼0
aj gj ðtik Þ
þ ek :
ð17Þ
809
We assume that gj ðtÞ are splines and not simply polynomial functions because, with splines, we can determine gj ðtÞ 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 then we ensure that forward rates do not behave explosively. A random disturbance is justified because the formation of the prices of fixed income securities is disturbed by several factors. Chambers et al. (1984) discuss 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 a (presumably random) amount; or there is no single price for each bond, which implies that there is an inherent imprecision to the concept of a single price.
4. Estimating the term structure of interest rates using splines and fuzzy methods In this section we shall develop a method for estimating the discount function associated to the TSIR. It is based on using the fuzzy regression method described above and the analysed framework in Section 3. Subsequently we shall discuss how to estimate the spot and the forward rates using FNs. We shall end this section by making an empirical application to the Spanish public debt market. The bonds we consider in our analysis have no embedded option (i.e. they are not callable, convertible, etc.). In any case, this is not a restrictive condition in public debt markets because practically all the securities that are traded have these characteristics. 4.1. Estimating the discount factor with triangular fuzzy numbers Fitting the TSIR with a fuzzy regression method, we can take all the prices traded on one day as the price of the bonds, not just the average price.
810
J. de Andres Sanchez, A. Terce~no Gomez / European Journal of Operational Research 154 (2004) 804–818
This instrument is attractive because the observed price in one session for any bond is often not only single because it is negotiated within an interval. So, the price of the kth bond of a sample, Pk , will be expressed with its centre and its spread as Pk ¼ hPkC ; PkR i i.e. the minimum price negotiated during the analysed session is PkC PkR while the maximum is PkC þ PkR . For example, if the traded price for one bond on one day fluctuates between 100 and 103, it will be expressed as h101:5; 1:5i. To fit the discount function, we assume that it is quantified via a TFN that depends on time. So, for a given maturity t, the actual value of one monetary unit payable at that moment is the TFN f~t ¼ ðft ; lft ; lft Þ; ð18Þ
So, the predicted price of the kth bond with the fuzzy discount function can be written adapting (15) as Pek ¼
Cik f~tik
ð19Þ
i¼1
and then, using (18), we can rewrite (19) as nk X Cik ftik ; lftk ; rftk : Pek ¼ ðPCk ; lPk ; rPk Þ ¼ i
i¼1
ð20Þ
laj ; raj P 0; ð21Þ
So, taking into account the coefficients in (21), the discount function, (18), is obtained by m X
a~j gj ðtÞ ¼
j¼0
¼
m X
Since the value of the discount function for t ¼ 0 should be the crisp number 1 (alternatively, the TFN (1,0,0)), then f~0 ¼ ðf0 ; lf0 ; rf0 Þ ¼ ð1; 0; 0Þ. This condition is met if a~0 ¼ ða0 ; la0 ; ra0 Þ ¼ ð1; 0; 0Þ, g0 ðtÞ ¼ 1 and gj ð0Þ ¼ 0, j ¼ 1; 2; . . . ; m. So, we can express the price of the kth bond, (23), as (24): Pek ¼ ðPCk ; lPk ; rPk Þ " # nk m X X Cik ð1; 0; 0Þ þ aj ; laj ; raj gj ðtik Þ ¼
m X
j¼1
Cik ð1; 0; 0Þ þ
i¼1
nk X
Cik
i¼1
m X
aj ; laj ; raj gj ðtik Þ
j¼1
ð24Þ and rearranging terms in (24), we obtain ðPCk ; lPk ; rPk Þ
nk X
Cik ð1; 0; 0Þ
¼
m X
aj ; laj ; raj
nk X
Cik gj ðtik Þ:
ð25Þ
i¼1
j¼1
k e In this way, by identifying P k k in (25) Y k ¼ ðYC ; lYk ; rYk Þ ¼ ðPCk ; lPk ; rPk Þ ni¼1 Ci ð1; 0; 0Þ, we obtain
YCk ¼ PCk
nk X
Cik ;
lYk ¼ lPk ; rYk ¼ rPk :
ð26Þ
i¼1
So, it is easy to verify that the value of the jth explanatory variable bond in (25) is the P kfor kthe kth Ci gj ðtik Þ. So, the expression crisp value Xjk ¼ ni¼1 to be adjusted by a fuzzy regression method, (25), is simply expressed as ðYCk ; lYk ; rYk Þ ¼ ða1 ; la1 ; ra1 ÞX1k þ ða2 ; la2 ; ra2 ÞX2k þ
aj ; laj ; raj gj ðtÞ
þ ðam ; lam ; ram ÞXmk
j¼0
ajC gj ðtÞ; laj gj ðtÞ; raj gj ðtÞ :
¼
nk X
i
j ¼ 0; 1; . . . ; m:
f~t ¼
ð23Þ
j¼0
i¼1
Likewise, we will construct the discount function from a linear combination of m þ 1 functions gj ðtÞ, j ¼ 0; 1; . . . ; m, with images in Rþ that are continuously differentiable, and whose parameters are given by TFNs. In this way, these parameters can be represented as a~j ¼ ðaj ; laj ; raj Þ;
i¼1
i¼1
t > 0; 0 6 ft lft 6 ft 6 ft þ rft 6 1:
nk X
Pek ¼ ðPCk ; lPk ; rPk Þ nk m X X ¼ Cik aj ; laj ; raj gj ðtik Þ:
ð22Þ
j¼0
Thus, the estimated price of the kth bond, Pek , can be rewritten by substituting (22) in (20) as
ð27Þ
since Yek is the prediction with fuzzy regression of the observation of price of the kth bond minus the sum of the cash flows provided by the asset whose centre and spreads are obtained in (26). Let us remark that the observed value of this variable for
J. de Andres Sanchez, A. Terce~no Gomez / European Journal of Operational Research 154 (2004) 804–818
the kth bond is Yk ¼ hYkC ; YkR i ¼ Pk so nk X YkC ¼ PkC Cik and YkR ¼ PkR :
Pnk
i¼1
Cik , and ð28Þ
811
Pm Pm j¼1 raj gj ðsP Þ j¼1 raj gj ððs þ 1ÞP Þ Pm P 6 0; m 1 þ j¼1 a^j gj ðsP Þ 1 þ j¼1 a^j gj ððs þ 1ÞP Þ s ¼ 1; . . . ; u 1;
ð29eÞ
i¼1
Let us remind you that the centres of the TFNs of the parameters, a1 ; a2 ; . . . ; am will be adjusted by least squares taking as the observations for Pk , k ¼ 1; 2; . . . ; K, only PkC . These estimates are symbolised as a^1 ; a^2 ; . . . ; a^m . Subsequently, we have to obtain the spreads of the parameters a~j , j ¼ 1; 2; . . . ; m, by solving a linear programme. To do so, we must first choose a level a of accomplishment of the congruency of the observations Yk with their predicted value Yek , k ¼ 1; 2; . . . ; K. We will also add certain constraints that will assure that the forward rates can be obtained from the discount function by using solution (9) of Eq. (8). These constraints will be the consequence of conditions in (10). To conclude, we must solve nk m K X X X Minimise z ¼ laj Cik gj ðtik Þ laj ;raj
j¼1
i¼1
k¼1
j¼1;2;...;m
þ
m X j¼1
raj
nk K X X k¼1
Cik gj ðtik Þ
ð29aÞ
i¼1
subject to nk m X X Cik gj ðtik Þ a^j j¼1
"
i¼1 m X
laj
nk X
ita ¼ ½it ðaÞ; it ðaÞ #
Cik gj ðtik Þ
1=t
¼ ½ðft þ rft ð1 aÞÞ
ð1 a Þ 6 YkC YkR ;
k ¼ 1; 2; . . . ; K; m X
a^j
j¼1
þ
nk X
"
i¼1
j¼1
ð29bÞ
Cik gj ðtik Þ
m X
raj
nk X
# Cik gj ðtik Þ
ðft lft ð1 aÞÞ
1:
ð30Þ
~it ðit ; lit ; rit Þ ð1 a Þ P YkC þ YkR ; ¼ ð29cÞ
Pm Pm j¼1 laj gj ðsP Þ j¼1 laj gj ððs þ 1ÞP Þ Pm P 6 0; m 1 þ j¼1 a^j gj ðsP Þ 1 þ j¼1 a^j gj ððs þ 1ÞP Þ s ¼ 1; . . . ; u 1;
1;
1=t
It is easy to check that the spot rate is not a TFN. However, from (7), we can deduce that the following TFN can approximate well the spot rate associated to the maturity t:
i¼1
k ¼ 1; 2; . . . ; K;
ð29fÞ
We remark that (29b), (29c) and (29f) correspond to the constraints (14b), (14c) and (14d) of Ishibuchi and NiiÕs estimation system, respectively. The objective function (29a) (the uncertainty of the system) must be minimised and it corresponds to the objectives (13a) or (14a) in Section 2.2. Likewise, (29d) and (29e) ensure that the solution for the forward rates in the sense of (9) exists for an arbitrary periodicity P (in years), which is the periodicity of the forward rates that we would like to obtain. So, these constraints are consequence of (10). Therefore, uP is the greatest maturity at which we want to determine the spot and forward rates and this will be close to the maturity of the bond with the greatest expiration. The value of the discount function in t, ft , is obtained from its corresponding spot rate it making ft ¼ ð1 þ it Þt . Then, the spot rate can be obtained from the discount by evaluating the expression it ¼ ðft Þ1=t 1. So, if the discount function is a TFN f~t ¼ ðft ; lft ; rft Þ, the spot rate will be a FN ~it whose a-cuts, ita , are from (6),
i¼1
j¼1
j ¼ 1; 2; . . . ; m:
laj ; raj P 0;
ð29dÞ
1 ðft Þ1=t
1;
rft
lf t
; tðft Þðtþ1Þ=t tðft Þðtþ1Þ=t
! :
ð31Þ
Notice that to obtain the left and right spreads of the triangular approximation to ~it (lit and rit respectively), we must take into account that the first derivative of the spot rate respect to its corresponding discount function evaluated in the centre
812
J. de Andres Sanchez, A. Terce~no Gomez / European Journal of Operational Research 154 (2004) 804–818
of f~t is 1=tðft Þðtþ1Þ=t . Moreover, having in mind that the spot rate is decreasing respect to its corresponding discount function, we can identify immediately lit ¼
r ft tðft Þ
ðtþ1Þ=t
and rit ¼
lf t tðft Þ
ðtþ1Þ=t
from Eqs. (7). Finally, the centre of the approximation of ~it , is obtained directly from the centre of f~t . Now we are going to obtain the forward rates, i.e. the spot rates that the market expects for the future. In particular, we are going to obtain the one year forward rates for integer years, that is to say, the forward rates 1 qt , t ¼ 1; . . . ; u. Although annual periods are used to deduce these rates, for other periodicities, the discount function that we have fitted can be used, because it is defined for any value of t. To obtain the forward rate 1 q~t , we have to solve the fuzzy version of (1): 1 f~t1 ð1 þ 1 q~t Þ ¼ f~t :
1 qta
¼ ½1 qt ðaÞ; 1 qt ðaÞ 1 1 ¼ 1 1; Gt ðaÞ Gt ðaÞ ft1 þ rft1 ð1 aÞ ft1 lft1 ð1 aÞ ¼ 1 : 1; ft þ rft ð1 aÞ ft lft ð1 aÞ ð36Þ
Obviously, 1 q~t is not a TFN since the extremes of its a-cuts, (36), are not linear functions of a. However, we can approximate 1 q~t with a TFN. To obtain that approximation we will develop the extremes of 1 qta , 1 qt ðaÞ and 1 qt ðaÞ, with a Taylor expansion to the first grade from a ¼ 1:
ð32Þ 1
To solve this equation, we identify ð1 þ 1 q~t Þ ¼ e t , where G e t is the value in t 1 of one monetary G unit payable at t according to the TSIR. The above equation can therefore be written using the a-cuts, reminding that the embedded discount functions are TFNÕs (see (18) or (22)) and that the expression of the a-cuts of a TFN are given in (4): ½ft1 lft1 ð1 aÞ; ft1 þ rft1 ð1 aÞ½Gt ðaÞ; Gt ðaÞ ¼ ½ft lft ð1 aÞ; ft þ rft ð1 aÞ:
It is easy to check that 1 qt is decreasing respect to Gt because Gt is always a positive number. Then, by using (6), we can obtain the a-cuts of the forward rate for the tth year ð1 q~t Þ, 1 qta . The lower (upper) bound of 1 qta is obtained by calculating (35) with Gt ðaÞ ðGt ðaÞÞ:
ð33Þ
So, (33) can be solved using (9) since the constraints (29d) and (29e) ensure the condition (10): Gta ¼ ½Gt ðaÞ; Gt ðaÞ ft lft ð1 aÞ ft þ rft ð1 aÞ ; ¼ : ft1 lft1 ð1 aÞ ft1 þ rft1 ð1 aÞ
1 qt ðaÞ
¼ 1 qt ðaÞ
1 qt
¼
1:
ð35Þ
ð37aÞ
ft1 ft1 lft ft lft1 1þ ð1 aÞ; 2 ft ðft Þ
ð37bÞ
where 1 q0t ð1Þ and 1 q0t ð1Þ are the values of the first derivatives of 1 qt ðaÞ and 1 qt ðaÞ for a ¼ 1. Taking into account (4), it is straightforward to check in (37a) and (37b) that the centre of the triangular approximation 1 q~t is 1 qt ¼ ðft1 ft Þ 1. Likewise, from (4) and (37a) we can deduce that the left spread of that triangular approximation is
ð34Þ
G1 t
ft1 ft1 rft ft rft1 1 ð1 aÞ; 2 ft ðft Þ
1 qt ð1Þ þ 1 q0t ð1Þða 1Þ ¼
l1 qt ¼
Now, let us remark that under crispness, the value in t 1 of 1 monetary unit payable at t, Gt is obtained from the tth forward rate, 1 qt , as 1 Gt ¼ ð1 þ 1 qt Þ . Then, immediately follows:
1 qt ð1Þ þ 1 q0t ð1Þða 1Þ
ft1 rft ft rft1 ðft Þ
2
:
Finally, using (4) and (37b) we immediately find that the right spread of the triangular approximation is r1 qt ¼
ft1 lft ft lft1 ðft Þ2
:
J. de Andres Sanchez, A. Terce~no Gomez / European Journal of Operational Research 154 (2004) 804–818
Then, the triangular approximation of 1 q~t is ~t 1q
ð1 qt ; l1 qt ; r1 qt Þ ¼
ft1 ft1 rft ft rft1 ft1 lft ft lft1 ; 1; ft ðft Þ2 ðft Þ2
! : ð38Þ
Let us remark that always ft1 rft ft rft1 P 0 and ft1 lft ft lft1 P 0 due to the constraints (29d) and (29e). 4.2. Empirical application In this subsection we use our methodology to estimate the TSIR in the Spanish public debt market on June 29, 2001. The bonds in our sample are shown in Table 1. We can see that in the sample there are zero-coupon bonds (the Strips and T-Bills) and bonds with annual coupon. The
first type of fixed income securities only pays the facial value (1 0 0) at maturity. For example, the 6th asset (a strip) provides 100 at its maturity (1.58 years). An example of bond with annual coupon is the 5th asset. It offers a post-payable annual coupon (5.25% on the facial value) and its facial value (1 0 0) at the expiration date (1.58 years). Then, that bond provides a stream of payments that can be represented by fð5:25; 0:58Þ; ð5:25 þ 100; 1:58Þg. Notice that in Table 1, the price of the kth fixed income asset is quantified through its minimum k and maximum negotiated price, i.e. Pk ¼ ½Pmin ; k Pmax . So, centre and radius of Pk , PkC and PkR k respectively, are obtained by doing PkC ¼ ðPmin þ k k k Pmax Þ=2 and PkR ¼ ðPmax Pmin Þ=2. The price of the 5th bond is P5 ¼ ½103:555; 103:669 and it can be represented equivalently by its centre and spread as P5 ¼ hP5C ; P5R i ¼ h103:612; 0:057i. To fit the TSIR corresponding to our sample, we built the discount function with McCullochÕs
Table 1 Bonds negotiated in the Spanish debt market on June 29, 2001
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
813
Asset
Coupon (annual) (%)
Number of coupons
Maturity (years)
k Pmin
k Pmax
T-Bill T-Bill BOND T-Bill BOND STRIP BOND STRIP BOND BOND BOND BOND BOND BOND BOND BOND BOND STRIP BOND BOND BOND BOND BOND STRIP BOND BOND BOND
0.00 0.00 4.25 0.00 5.25 0.00 3.00 0.00 4.60 4.50 4.65 3.25 4.95 10.15 4.80 7.35 6.00 0.00 5.15 4.00 5.40 5.35 6.15 0.00 4.75 6.00 5.75
0 0 2 0 2 0 2 0 3 4 3 4 5 5 5 6 7 0 9 9 10 10 12 0 14 28 31
0.05 1.05 1.07 1.43 1.58 1.58 1.58 2.07 2.07 3.07 3.33 3.58 4.08 4.58 5.33 5.75 6.58 7.60 8.08 8.59 10.08 10.33 11.59 11.59 13.08 27.60 31.10
99.779 95.758 103.907 94.220 103.555 93.579 99.337 91.540 104.670 104.017 98.466 97.026 105.407 126.340 97.785 113.539 107.400 68.412 104.101 92.679 97.716 96.966 108.098 53.357 96.506 103.722 93.954
99.779 95.758 103.947 94.220 103.669 93.749 99.376 91.540 104.917 104.166 98.702 97.200 105.918 126.340 98.385 113.539 108.206 68.412 104.307 93.473 98.923 97.749 108.168 53.357 97.567 105.194 94.777
814
J. de Andres Sanchez, A. Terce~no Gomez / European Journal of Operational Research 154 (2004) 804–818
quadratic splines that have into account the number of bonds and the structure of maturities in the sample. So, we took m ¼ 5, and the knots we used to construct the splines were d1 ¼ 0 years, d2 ¼ 1:58 years, d3 ¼ 3:83 years, d4 ¼ 8:96 years and d5 ¼ 31:1 years. Concretely, the functions gj ðtÞ, j ¼ 1; . . . ; 5 are 3 8 1 > > t2 þ t; 0 6 t < 1:58; < 2 1:58 ð39Þ g1 ðtÞ ¼ > 1:58 > : ; 1:58 6 t < 31:1; 2
gj ðtÞ ¼
g5 ðtÞ ¼
X15 ¼ 5:25g1 ð0:58Þ þ ð5:25 þ 100Þg1 ð1:58Þ ¼ 85:53; X25 ¼ 5:25g2 ð0:58Þ þ ð5:25 þ 100Þg2 ð1:58Þ ¼ 83:60; .. . X55 ¼ 5:25g5 ð0:58Þ þ ð5:25 þ 100Þg5 ð1:58Þ ¼ 0: The final objective is to obtain a fuzzy discount function like (22) that in our empirical application is
8 0; > > > > > 1 > > ðt dj1 Þ2 ; > > > < 2ðdj dj1 Þ
0 6 t < dj1 ; dj1 6 t 6 dj ;
1 1 > ðt dj Þ2 þ ðt dj Þ þ ðdj dj1 Þ; > > > 2ðdjþ1 dj Þ 2 > > > > > > : 1 ðdjþ1 dj1 Þ; 2 8 < 0;
0 < t 6 8:96; 1 2 ðt 8:96Þ ; 8:96 6 t 6 31:1: : 2ð31:1 8:96Þ ð41Þ
To obtain the dependent variable observations Yk ¼ hYkC ; YkR i; k ¼ 1; 2; . . . ; 27, we need to transform the bond prices by using (28). For example, for the 5th bond we obtain Y5C ¼ P5C
n5 X
Ci5
i¼1
¼ 103:612 ½5:25 þ ð5:25 þ 100Þ ¼ 6:888
and
Y5R ¼ P5R ¼ 0:057:
Moreover, we need to quantify the observations of the explanatory variables in Eq. (27), X1k ; X2k ; . . .P ; X5k , k ¼ 1; 2; . . . ; 27. Let us remind that k k Xj ¼ ni¼1 Cik gj ðtik Þ and then, e.g., the values corresponding to the 5th bond are
dj 6 t 6 djþ1 ;
j ¼ 2; 3; 4;
djþ1 6 t 6 d5 ;
f~t ¼
5 X
a~j gj ðtÞ ¼
j¼0
5 X
aj ; laj ; raj gj ðtÞ
For a detailed discussion about the construction of our gj ðtÞ and the knots we have chosen, see McCulloch (1971).
ð42Þ
j¼0
and given that a~0 ¼ ð1; 0; 0Þ, then (42) is equivalent to 5 5 X X aj ; laj ; raj gj ðtÞ: a~j gj ðtÞ ¼ 1 þ f~t ¼ 1 þ j¼1
j¼1
ð43Þ The first step is to estimate the discount function with ordinary least squares (OLS). In this phase we fit the centres of the parameters a~j , j ¼ 1; 2; . . . ; 5, in (43) whose estimated values are a^1 ; a^2 ; . . . ; a^5 . Since we need to quantify the observations of the dependent variable with crisp numbers, we must take the centre value YkC as a representative value of Yk . Notice that, in doing so, we are considering only PkC as the observed price for the kth bond. In our application the OLS estimation has a determination coefficient close to 100% (exactly, 99.98%) and provides the following crisp discount function: ft ¼ 1 0:04404282g1 ðtÞ 0:03661593g2 ðtÞ
3
ð40Þ
0:04870935g3 ðtÞ 0:03550896g4 ðtÞ 0:00776513g5 ðtÞ:
J. de Andres Sanchez, A. Terce~no Gomez / European Journal of Operational Research 154 (2004) 804–818
815
0.075 0.07 0.065
rate
0.06 0.055 0.05 0.045 0.04 0
2
4
6
8
10 12 14 16 18 20 22 24 26 28 30 Maturity (years) spot rates
forward rates
Fig. 1. Estimations of the spot rates and forward rates 6/29/2002 in Spanish public debt market.
The spot rates and one-year forward rates for the next 30 years that are obtained from the crisp discount function are given in Fig. 1. The second step consists in implementing fuzzy regression by solving the estimation system (29a)–
(29f) taking as the centres of the fuzzy parameters a~j , those obtained in the OLS regression a^j , j ¼ 1; 2; . . . ; 5. The result of solving that linear programme is the estimates of laj , raj , j ¼ 1; 2; . . . ; 5, in (43). Table 2 shows the results for the
Table 2 Results of fuzzy regression for a ¼ 0:5 and 0.75 a~1
a~2
a~3
a~4
a~5
)0.04404282
)0.03661593
)0.04870935
)0.03550896
)0.00776513
a ¼ 0:5, z ¼ 109:62 laj raj
0.00176058 0.00312284
0.00290595 0
0.00661728 0.00459031
0 0
0 0.00073984
a ¼ 0:75, z ¼ 219:25 laj raj
0.00352116 0.00624569
0.00581189 0
0.01323457 0.00918063
0 0
0 0.00147968
aj ¼ a^j
discount function
discount function
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
times (years) (a)
lower bound
centre
times (years)
upper bound
(b)
lower bound
centre
upper bound
Fig. 2. (a) Fuzzy discount function for a ¼ 0:5. (b) Fuzzy discount function for a ¼ 0:75.
816
J. de Andres Sanchez, A. Terce~no Gomez / European Journal of Operational Research 154 (2004) 804–818 0.085 0.075
0.065
spot rate
spot rate
0.075 0.07 0.06 0.055 0.05
0.065 0.055 0.045
0.045 0.04
0.035 0 2
4 6 8 10 1 2 14 16 18 20 22 2 4 26 28 30
0 2
4 6 8 10 12 14 1 6 18 20 22 24 26 28 30
time (in years) centre
lower bound
(a)
time (in years) upper bound
centre
lower bound
(b)
upper bound
Fig. 3. (a) Fuzzy TSIR for a ¼ 0:5. (b) Fuzzy TSIR for a ¼ 0:75.
0.115 0.105 0.095 0.085 0.075 0.065 0.055 0.045 0.035
0.085
forward rate
forward rate
0.095
0.075 0.065 0.055 0.045 0.035 0
2
4
6
8 10 12 14 16 18 20 22 24 26 28 30
0
2
4
6
8
time (in years)
(a)
lower bound
centre
1 0 12 14 1 6 18 20 22 2 4 26 28 3 0 time (in years)
upper bound
(b)
lower bound
centre
upper bound
Fig. 4. (a) Forward rates a ¼ 0:5. (b) Forward rates a ¼ 0:75.
levels of inclusion a ¼ 0:5 and 0.75. We want to point out that, in order to build constraints (29d) and (29e), we assumed an annual periodicity, since we will subsequently obtain only the implicit rates for that periodicity. Fig. 2a and b show the discount function that we fitted for the next 30 years. Fig. 3a and b show the shapes of the fuzzy TSIR that we obtained with fuzzy regression and Fig. 4a and b plot the one-year forward rates for the next 30 years. We would like to point out that parameter a can be interpreted as an indicator of the perceived uncertainty in the market by the decision maker. If a increases, then the uncertainty of the estimates of the explained variable (the price of the bonds) also increases (see Wang and Tsaur, 2000) and consequently the spread of the subsequent estimates of the spot rates and the forward rates will be wider.
5. Conclusions In this paper, we have attempted to make up for the lack of explanations in the financial and actuarial literature about estimating interest rates with FNs. When we price assets with a long maturity, we agree with the authors mentioned above that it is a suitable way to model the uncertainty of the future interest rates using FNs, especially when the financial analysis is concerned with the long term. Unfortunately, it is usual to read in many papers that the subjacent hypothesis is that an ‘‘expert’’ subjectively estimates these rates with no more explanation, i.e. there is no serious analysis about how the ‘‘expert’’ is supposed to express these predictions about future yield rates, using, in our case, fuzzy numbers. If we accept that the ‘‘experts’’ are traders in the fixed income markets,
J. de Andres Sanchez, A. Terce~no Gomez / European Journal of Operational Research 154 (2004) 804–818
these subjective estimates are included in the price of the debt instruments and, therefore, in the TSIR of these instruments. Our method quantifies the expertsÕ subjective estimates of the spot rates and spot interest rates for the future (the forward rates) with FNs. This method is based on a fuzzy regression technique and uses all the prices of the bonds negotiated during one session to fit the TSIR, in such a way that no information provided by the negotiated prices is lost. When econometric methods are used, however, these prices must be reduced to representative ones and then, some of the information provided by the bond prices is lost. We have proposed adjusting the interest rates with TFNs. This is because the arithmetic is easy to implement and the interpretation of the estimations is intuitive because they are well adapted to the way humans make predictions. For example, an interest rate given by (0.03, 0.005, 0.005) indicates that we expect an interest rate of about 3%, and we do not expect deviations to be greater than 50 basis points.
Acknowledgements The authors wish to thank the valuable comments of two anonymous referees.
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Interfaces with Other Disciplines
Estimating a fuzzy term structure of interest rates using fuzzy regression techniques ~ o Go mez Jorge de Andres S anchez *, Antonio Tercen Department of Business Administration, Faculty of Economics and Business Studies, Rovira i Virgili University, Av. de la Universitat 1, 43204 Reus, Spain Received 11 January 2002; accepted 12 November 2002
Abstract Several papers in the financial literature propose using fuzzy numbers (FNs) to model interest rate uncertainty. However, in our opinion, the first problem to be solved is how to estimate these rates with FNs. In this paper, we attempt to provide a solution to this question with a method for adjusting the temporal structure of interest rates (TSIR) that is based on fuzzy regression techniques. This method will enable to quantify the anticipated rates in the fixed income markets for the future with FNs. In particular, we discuss how to estimate the TSIR with triangular fuzzy numbers because of their desirable properties. Ó 2003 Elsevier B.V. All rights reserved. Keywords: Finance; Fuzzy sets; Fuzzy regression; Investment analysis; Temporal structure of interest rates
1. Introduction To obtain the financial price of any asset we have to discount the cash flows that the asset produces along its life. So, the discount rates that must be applied throughout the life of the asset must be estimated. The reference values of these discount rates are the interest rates that are free of default risk, i.e. those that correspond to public debt bonds. Many financial analyses are concerned with the medium and long term and, in our opinion,
* Corresponding author. Tel.: +34-977-75-98-32; fax: +34977-75-98-10. E-mail addresses: [email protected] (J. de Andres Sanchez), [email protected] (A. Terce~ no G omez).
modelling the behaviour of interest rates in the long term by means of a stochastic model is not very realistic. 1 Gerber (1995) points out that there is no commonly accepted stochastic model for making predictions about long term discount rates. So, many authors think that it is more suitable and realistic to make financial analyses, in the middle and long term, using estimates of discount rates by means of fuzzy numbers (FNs), because in these circumstances often it is only
1 The papers that model the temporal structure of interest rates (TSIR) in a stochastic way suppose that its dynamics can be synthesised in a few variables (the short term spot rate, the spread of the short term spot rate compared to the long term spot rate, etc.) whose behaviour is given by a set of stochastic differential equations.
0377-2217/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0377-2217(02)00854-8
J. de Andres Sanchez, A. Terce~no Gomez / European Journal of Operational Research 154 (2004) 804–818
available imprecise data or data related in a vague way with the future behaviour of interest rates (for example, the price of fixed income securities, the opinion of ‘‘experts’’ about the future behaviour of macroeconomic magnitudes, etc.). For applications of Fuzzy Sets Theory in financial pricing, see Kaufmann (1986), Buckley (1987), Li Calzi (1990), Gil (1999), Kutcha (2000) or Terce~ no et al. (2001) while for actuarial pricing, see Lemaire (1990), Ostaszewski (1993), Cummins and Derrig (1997), Derrig and Ostaszewski (1997) or Terce~ no et al. (1996). However, the literature has not paid enough attention to how to estimate future discount rates with FNs and, consequently, how to construct their membership function. It is usually suggested that ‘‘the rates are estimated subjectively by the experts using fuzzy numbers’’ with no further explanation. In this respect, Fedrizzi et al. (1993) point out that, the problem of constructing the membership function is a crucial problem for the whole theory of fuzzy sets. Therefore, it seems to be inadequate to start considerations with a proposition like: ‘‘suppose the observations are given as fuzzy numbers’’, which is probably the most often used expression in literature on fuzzy sets. In this paper, we propose to solve the problem of constructing the membership function of future yield rates by estimating the TSIR with fuzzy subsets, since the TSIR implicitly contains the expectations of the fixed income market agents (i.e. the experts) regarding the evolution of future interest rates. This TSIR can then be taken as a reference when pricing any asset––for this topic, see Carleton and Cooper (1976). We would also like to point out that it was Ostaszewski (1993) who first suggested using a fuzzy TSIR in the field of life insurance pricing. The TSIR is a relation between interest rates and maturity (which we will measure in years) in such a way that the market prices a cash flow with the discount rate associated to its maturity (t years). This rate is named as spot rate, and we will symbolise it as it . So, the present value or one t monetary unit payable in t years is ð1 þ it Þ . This expression is often called the discount function, t which we will denote as ft i.e. ft ¼ ð1 þ it Þ . The implicit or forward rates, which standard financial theory interprets as the spot rates that the
805
market expects to prevail in the future, are given implicitly by the TSIR. Then, from the spot rates it and it where t > t, we can obtain the forward rate to apply when calculating the price in t-years of a cash flow that in this moment will be payable in t t years (t t qt ). So, we must solve ft ð1 þ t t qt Þ
ðt tÞ
¼ ft ;
ð1Þ
so
ft t t qt ¼ ft
1=ðt tÞ 1
ð2Þ
where the Theory of Rational Expectations interprets t t qt obtained with (2), as the discount spot rate that the market expects to prevail in t years for the next t t years. In this paper, the key instrument we use to obtain the TSIR for the market of debt bond instruments is fuzzy regression. In our opinion, fuzzy regression techniques have a number of advantages over traditional regression techniques. For our purposes we shall mention the following: (a) The estimates that are obtained after adjusting the coefficients are not random variables, which are difficult to manipulate in arithmetical operations, but FNs, which are easier to handle arithmetically. So, when starting from magnitudes estimated by random variables (e.g. as a result of its prediction from a least squares regression) these random variables are often reduced to their mathematical expectation (which may or may not be corrected by their variance) so that they are easier to handle. If FNs are used, this loss of information is not needed. (b) If the phenomena investigated are economic or social, the observations are a consequence of the interaction between the economic agentsÕ beliefs and expectations, which are highly subjective and vague. Fuzzy Sets Theory, therefore, is a good way of treating this information. For example, the asset prices that are determined in the markets depend on the agentsÕ expectations of future inflation, the issuersÕ credibility, etc. We think that it is more realistic to consider that the bias between the
806
J. de Andres Sanchez, A. Terce~no Gomez / European Journal of Operational Research 154 (2004) 804–818
observed values for the dependent variable and its theoretical value (the error) is not random but fuzzy. At least in this way we assume that the subjectivity inherent in the phenomena analysed is high. (c) Observations are often not crisp numbers; they are confidence intervals. For instance, the price of a financial asset throughout one session often oscillates within an interval, rarely does it remain the same (e.g. it can oscillate in the interval [$100, $105]). If econometric methods are to be used, the observations for the explained variable and/or the explanatory variable must be represented by a single value (e.g. in the above interval this value would be $102.5) which involves losing a great deal of information. Fuzzy regression, however, does not necessarily reduce the values of each variable to a crisp number, i.e., all the observed values can be used in the regression analysis ([$100, $105] in our example).
sup8x2X lA ðxÞ ¼ 1. The second is that it must be convex (i.e., its a-cuts must be convex sets). For practical purposes, some of the most widely used FNs are triangular fuzzy numbers (TFNs) because they are easy to handle arithmetically and they have an intuitive interpretation. We shall e as A e ¼ ða; la ; ra Þ, where a is the symbolise a TFN A centre and la and ra are the left and right spreads respectively. For example, a sentence by a financial analyst such as ‘‘I expect for the next two years the free risk rate to be around 5% and deviations no longer that the 1%’’ may be quantified in a very natural way as (0.05, 0.01, 0.01). A TFN is characterised by its membership function lA ðxÞ or, alternatively, by its a-cuts, Aa , as 8 x a þ la > > ; > < la lA ðxÞ ¼ a þ ra x ; > > > ra : 0;
a la < x 6 a; a < x 6 a þ ra ;
ð3Þ
otherwise;
and The structure of the paper is as follows. In the next section, we shall describe some aspects of fuzzy arithmetic and Ishibuchi and NiiÕs extension (2001) of the fuzzy regression method described in Tanaka et al. (1989) and Tanaka and Ishibuchi (1992). In Section 3, we shall discuss some basic characteristics of the models that attempt to adjust the TSIR through the discount function defined by the spot rates. In Section 4, we show how a fuzzy TSIR can be obtained by using fuzzy regression and we apply this methodology to the Spanish public debt market. In Section 5, we state the most important conclusions of the paper.
2. Fuzzy arithmetic and fuzzy regression 2.1. Some aspects of fuzzy numbers e defined over A fuzzy number is a fuzzy subset A real numbers. It is the main instrument used in Fuzzy Set Theory for quantifying uncertain quantities (e.g. in financial analysis, the discount rates). Two properties are required by a FN. The first one is that it must be a normal fuzzy set, i.e.
Aa ¼ ½AðaÞ; AðaÞ ¼ ½a la ð1 aÞ; a þ ra ð1 aÞ: ð4Þ In many financial analyses it is often necessary to evaluate functions (e.g. the net present value), which in a general way we shall name y ¼ f ðx1 ; x2 ; . . . ; xn Þ. Then, if x1 ; x2 ; . . . ; xn are not e 1; A e 2; . . . ; A e n , f ðÞ crisp numbers but are the FNs A e 1; A e 2; . . . ; A e n Þ whose meme ¼ f ðA induces a FN B bership function must be obtained from Zadeh’s extension principle. Unfortunately, it is often impossible to obtain a closed expression for the e , although in many cases membership function of B it is possible to obtain its a-cuts, Ba , from A1a ; A2a ; . . . ; Ana , by doing e 1; A e 2; . . . ; A e n Þ ¼ f ðA1a ; A2a ; . . . ; Ana Þ: Ba ¼ f ð A a ð5Þ In financial mathematics, many functional relationships are continuously increasing or decreasing with respect to all the variables involved, in such a way that it is easy to evaluate the a-cuts e . If the function f ðÞ that induces B e is inof B creasing with respect to the first m variables, where m 6 n, and decreasing with respect to the last n m
J. de Andres Sanchez, A. Terce~no Gomez / European Journal of Operational Research 154 (2004) 804–818
variables, Buckley and Qu (1990b) demonstrate that Ba , i.e. (5), is Ba ¼ ½BðaÞ; BðaÞ ¼ ½f ðA1 ðaÞ; . . . ; Am ðaÞ; Amþ1 ðaÞ; . . . ; An ðaÞÞ; f ðA1 ðaÞ; . . . ; Am ðaÞ; Amþ1 ðaÞ; . . . ; An ðaÞÞ: ð6Þ It is well known that the result of evaluating a e , will be another linear function with TFNs, B TFN. On the other hand, the result of evaluating non-linear functions with TFNs will not be another TFN. In this case it will be interesting to e using a TFN that we shall denote approximate B 0 e by B because of the desirable properties of TFNs. If the function that we evaluate, f ðÞ, is increasing with respect to the first m variables, where m 6 n, and decreasing with respect to the others (i.e. the e are (6)), Dubois and Prade (1993) a-cuts of B e with a FN B e0 ¼ propose to approximate B ðb; lB ; rB Þ whose characteristics are b ¼ f ðaC Þ; m n X X of ðaC Þ of ðaC Þ lb ¼ lai rai ; ox oxi i i¼1 i¼mþ1 rb ¼
m n X X of ðaC Þ of ðaC Þ rai lai ; ox oxi i i¼1 i¼mþ1
ð7Þ
where aC ¼ ða1 ; . . . ; am ; amþ1 ; . . . ; an Þ. In this paper, to obtain the forward rates we will need to solve fuzzy equations with the structure eX e and B e ¼B e , where A e are TFNs with the supA port contained in Rþ . The above equation can be e, X e and B e as expressed through the a-cuts of A Aa Xa ¼ Ba , i.e., ½a la ð1 aÞ; a þ ra ð1 aÞ½X ðaÞ; X ðaÞ ¼ ½b lb ð1 aÞ; b þ rb ð1 aÞ:
ð8Þ
According to Kaufmann and Gupta (1985), the narrowest solution is b lb ð1 aÞ b þ rb ð1 aÞ ; X ðaÞ ¼ : X ðaÞ ¼ a la ð1 aÞ a þ ra ð1 aÞ ð9Þ We must take into account that this solution does not always exist 8a 2 ½0; 1. For example, if A0:5 ¼ ½1; 2 and B0:5 ¼ ½1; 1:5, Eq. (8) is ½1; 2½X ð0:5Þ;
807
X ð0:5Þ ¼ ½1; 1:5. Applying (9) we obtain X ð0:5Þ ¼ 1 and X ð0:5Þ ¼ 1:5=2 ¼ 3=4, i.e. X0:5 ¼ ½1; 3=4, which is not an a-cut of a FN. Nevertheless, Buckley and Qu (1990a) demonstrate that the required condition for the existence of (5) 8a 2 ½0; 1 is lb la P ; b a
rb ra P : b a
ð10Þ
2.2. Fuzzy regression model with asymmetric coefficients In this subsection, we will describe Ishibuchi and NiiÕs extension (2001) of the fuzzy regression model developed in Tanaka et al. (1989) and Tanaka and Ishibuchi (1992), which is one of the most widely used models in economic applications. 2 Like any regression technique, the objective of fuzzy regression is to determine a functional relationship between a dependent variable and a set of independent ones. As we will show, fuzzy regression is in many aspects more versatile than conventional linear regression because functional relationships can be obtained when independent variables, dependent variables, or both, are not only crisp values but intervals. As in econometric linear regression, we shall assume that the explained variable is a linear combination of the explanatory variables. This relationship should be obtained from a sample of n observations fðY1 ; X1 Þ; ðY2 ; X2 Þ; . . . ; ðYj ; Xj Þ; . . . ; ðYn ; Xn Þg, where Xj is the jth observation of the explanatory variable, Xj ¼ ðX0j ; X1j ; X2j ; . . . ; Xij ; . . . ; Xmj Þ. Moreover, X0j ¼ 1 8j, and Xij is the observed value for the ith variable in the jth case of the sample and is always crisp. Yj is the jth observation of the explained variable, j ¼ 1; 2; . . . ; n, and may either be a crisp value or a confidence interval. In both cases, it can be represented through its centre and its spread or radius as Yj ¼ hYjC ; YjR i, where YjC is the centre and YjR is the radius. In particular, we must estimate the following fuzzy linear function: 2
Examples of economic applications of fuzzy regression are the papers by Ramenazi and Duckstein (1992), Profillidis et al. (1999), and Lee and Chen (2001).
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J. de Andres Sanchez, A. Terce~no Gomez / European Journal of Operational Research 154 (2004) 804–818
e0 þ A e 1 X1j þ þ A e m Xmj ; Yej ¼ A
ð11Þ
where Yej is the estimation of Yj by a FN after e 0; A e 1; . . . ; A e m. adjusting A Likewise, in fuzzy regression models the disturbance is not introduced as a random addend in the linear relation, but it is incorporated into the e i , i ¼ 0; 1; . . . ; m. Of course, the final coefficients A e i , i ¼ 0; 1; . . . ; m, objective is to adjust the FNs A from the available sample. In this paper we proe i as TFN, i.e., they pose to model the parameters A e i ¼ ðai ; lai ; rai Þ, i ¼ 0; 1; . . . ; m. can be written as A Then, the estimate of Yj , Yej is a TFN that we will name Yej ¼ ðYj ; lYj ; rYj Þ where m X ai Xij ; Y j ¼ a0 þ i¼1
lYj ¼ la0 þ
m X
lai jXij j þ
i¼1 xij P 0
rYj ¼ ra0 þ
m X
m X
i¼1 xij P 0
m X
¼
m X
lai jXij j:
ð12Þ
i¼1 xij <0
i¼0
i¼0
lai
Xij þ
j¼1
m X i¼0
i¼0
rai
n X Xij ; j¼1
ð13cÞ
i ¼ 0; 1; . . . ; m;
ð13dÞ
a 2 ½0; 1Þ;
ð13eÞ
where X0j ¼ 1, j ¼ 1; 2; . . . ; n. If for the second objective, (13b), we require a minimum accomplishment level a 2 ½0; 1Þ, the above programme is transformed into the following linear one: n X m n X m X X Minimise z ¼ lai jXij j þ rai jXij j j¼1
i¼0
j¼1
i¼0
ð14aÞ subject to m X a^0 þ a^i Xij 1
B C m m B C X X B C Bla0 þ la0 jXij j þ rai jXij jCð1 a Þ B C @ A i¼1 i¼1 xij P 0
6 YjC YjR ;
j¼1
n X
j ¼ 1; 2; . . . ; n;
i¼1 0
According to Ishibuchi and Nii (2001), to fit e 0; A e 1; . . . ; A e m we first have to determine the cores A a0 ; a1 ; . . . ; am by using least squares and taking the centres of the observations Yj ¼ 1; 2; . . . ; n as the realisations of the independent variable. These estimates are named a^0 ; a^1 ; . . . ; a^m . Subsequently, we have to determine the parameters lai and rai , which must minimise the uncertainty of the estimates for Yj ; Yej (i.e. their spreads), and simultaneously maximise the congruence of the estimates obtained for Yj which will be measured as lðYj Yej Þ. In particular, we must solve the following multiobjective programme: n X m n X m X X lai Xij þ rai Xij Minimise z ¼ j¼1
lai ; rai P 0;
rai jXij j;
i¼1 xij <0
rai jXij j þ
subject to lðYj Yej Þ P a;
a^0 þ
m X
xij <0
j ¼ 1; 2; . . . ; n;
ð14bÞ
a^i Xij
i¼1
0
1
B C m m B C X X B C þ Bra0 þ rai jXij j þ lai jXij jCð1 a Þ B C @ A i¼1 i¼1 xij P 0
P YjC þ YjR ; lai ; rai P 0;
xij <0
j ¼ 1; 2; . . . ; n;
i ¼ 0; 1; . . . ; m:
ð14cÞ ð14dÞ
Notice that in the last mathematical programme the constraints (14b) and (14c) are consequences of requirement lðYj Yej Þ P a in (13c) while the last block ensures that lai ; rai 8i will be non-negative. 3. Estimating the term structure of interest rates using splines and econometric methods
ð13aÞ Maximise a;
ð13bÞ
It is fairly straightforward to estimate the TSIR for a specific date and a given market if there are
J. de Andres Sanchez, A. Terce~no Gomez / European Journal of Operational Research 154 (2004) 804–818
many sufficiently liquid zero-coupon bonds and their price can be observed without perturbations. However, these conditions are rarely found simultaneously in the fixed income markets. This section describes how to estimate the discount function with econometric methods rather than by adjusting the TSIR directly. These methods can be used if the sample is made up entirely of zerocoupon bonds, entirely of bonds with coupons or, as is usual, if it is made up of both types of bonds. Our starting point is the fact that the kth bond, where k ¼ 1; 2; . . . ; K, and K is the number of available bonds, provides several cash flows (coupons and principal). These are denoted by fðC1k ; t1k Þ; ðC2k ; t2k Þ; ; ðCnkr ; tnkk Þg, where Cik is the amount of the ith cash flow and tik is its maturity expressed in years. If we suppose that the bonds have no embedded option (e.g. convertibility) the price of the kth bond is therefore the sum of the discounted value of every amount with the corresponding spot rate Pk ¼
nk X
Cik ftik
ð15Þ
i¼1
where ftik is the discounted value of one dollar with maturity tik years. Of course, we should then define an analytical form for the discount function in order 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 most popular methods are the one described in McCullochÕs papers (1971, 1975) (quadratic splines and cubic splines) and in Vasicek and FongÕs article (1982) (exponential splines). These methods assume that the discount function is a linear combination of m þ 1 functions of time. Therefore ft ¼
m X
aj gj ðtÞ:
ð16Þ
j¼0
Then, we can deduce that the following linear equation must be estimated: Pk ¼
nk X i¼1
Cik
m X j¼0
aj gj ðtik Þ
þ ek :
ð17Þ
809
We assume that gj ðtÞ are splines and not simply polynomial functions because, with splines, we can determine gj ðtÞ 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 then we ensure that forward rates do not behave explosively. A random disturbance is justified because the formation of the prices of fixed income securities is disturbed by several factors. Chambers et al. (1984) discuss 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 a (presumably random) amount; or there is no single price for each bond, which implies that there is an inherent imprecision to the concept of a single price.
4. Estimating the term structure of interest rates using splines and fuzzy methods In this section we shall develop a method for estimating the discount function associated to the TSIR. It is based on using the fuzzy regression method described above and the analysed framework in Section 3. Subsequently we shall discuss how to estimate the spot and the forward rates using FNs. We shall end this section by making an empirical application to the Spanish public debt market. The bonds we consider in our analysis have no embedded option (i.e. they are not callable, convertible, etc.). In any case, this is not a restrictive condition in public debt markets because practically all the securities that are traded have these characteristics. 4.1. Estimating the discount factor with triangular fuzzy numbers Fitting the TSIR with a fuzzy regression method, we can take all the prices traded on one day as the price of the bonds, not just the average price.
810
J. de Andres Sanchez, A. Terce~no Gomez / European Journal of Operational Research 154 (2004) 804–818
This instrument is attractive because the observed price in one session for any bond is often not only single because it is negotiated within an interval. So, the price of the kth bond of a sample, Pk , will be expressed with its centre and its spread as Pk ¼ hPkC ; PkR i i.e. the minimum price negotiated during the analysed session is PkC PkR while the maximum is PkC þ PkR . For example, if the traded price for one bond on one day fluctuates between 100 and 103, it will be expressed as h101:5; 1:5i. To fit the discount function, we assume that it is quantified via a TFN that depends on time. So, for a given maturity t, the actual value of one monetary unit payable at that moment is the TFN f~t ¼ ðft ; lft ; lft Þ; ð18Þ
So, the predicted price of the kth bond with the fuzzy discount function can be written adapting (15) as Pek ¼
Cik f~tik
ð19Þ
i¼1
and then, using (18), we can rewrite (19) as nk X Cik ftik ; lftk ; rftk : Pek ¼ ðPCk ; lPk ; rPk Þ ¼ i
i¼1
ð20Þ
laj ; raj P 0; ð21Þ
So, taking into account the coefficients in (21), the discount function, (18), is obtained by m X
a~j gj ðtÞ ¼
j¼0
¼
m X
Since the value of the discount function for t ¼ 0 should be the crisp number 1 (alternatively, the TFN (1,0,0)), then f~0 ¼ ðf0 ; lf0 ; rf0 Þ ¼ ð1; 0; 0Þ. This condition is met if a~0 ¼ ða0 ; la0 ; ra0 Þ ¼ ð1; 0; 0Þ, g0 ðtÞ ¼ 1 and gj ð0Þ ¼ 0, j ¼ 1; 2; . . . ; m. So, we can express the price of the kth bond, (23), as (24): Pek ¼ ðPCk ; lPk ; rPk Þ " # nk m X X Cik ð1; 0; 0Þ þ aj ; laj ; raj gj ðtik Þ ¼
m X
j¼1
Cik ð1; 0; 0Þ þ
i¼1
nk X
Cik
i¼1
m X
aj ; laj ; raj gj ðtik Þ
j¼1
ð24Þ and rearranging terms in (24), we obtain ðPCk ; lPk ; rPk Þ
nk X
Cik ð1; 0; 0Þ
¼
m X
aj ; laj ; raj
nk X
Cik gj ðtik Þ:
ð25Þ
i¼1
j¼1
k e In this way, by identifying P k k in (25) Y k ¼ ðYC ; lYk ; rYk Þ ¼ ðPCk ; lPk ; rPk Þ ni¼1 Ci ð1; 0; 0Þ, we obtain
YCk ¼ PCk
nk X
Cik ;
lYk ¼ lPk ; rYk ¼ rPk :
ð26Þ
i¼1
So, it is easy to verify that the value of the jth explanatory variable bond in (25) is the P kfor kthe kth Ci gj ðtik Þ. So, the expression crisp value Xjk ¼ ni¼1 to be adjusted by a fuzzy regression method, (25), is simply expressed as ðYCk ; lYk ; rYk Þ ¼ ða1 ; la1 ; ra1 ÞX1k þ ða2 ; la2 ; ra2 ÞX2k þ
aj ; laj ; raj gj ðtÞ
þ ðam ; lam ; ram ÞXmk
j¼0
ajC gj ðtÞ; laj gj ðtÞ; raj gj ðtÞ :
¼
nk X
i
j ¼ 0; 1; . . . ; m:
f~t ¼
ð23Þ
j¼0
i¼1
Likewise, we will construct the discount function from a linear combination of m þ 1 functions gj ðtÞ, j ¼ 0; 1; . . . ; m, with images in Rþ that are continuously differentiable, and whose parameters are given by TFNs. In this way, these parameters can be represented as a~j ¼ ðaj ; laj ; raj Þ;
i¼1
i¼1
t > 0; 0 6 ft lft 6 ft 6 ft þ rft 6 1:
nk X
Pek ¼ ðPCk ; lPk ; rPk Þ nk m X X ¼ Cik aj ; laj ; raj gj ðtik Þ:
ð22Þ
j¼0
Thus, the estimated price of the kth bond, Pek , can be rewritten by substituting (22) in (20) as
ð27Þ
since Yek is the prediction with fuzzy regression of the observation of price of the kth bond minus the sum of the cash flows provided by the asset whose centre and spreads are obtained in (26). Let us remark that the observed value of this variable for
J. de Andres Sanchez, A. Terce~no Gomez / European Journal of Operational Research 154 (2004) 804–818
the kth bond is Yk ¼ hYkC ; YkR i ¼ Pk so nk X YkC ¼ PkC Cik and YkR ¼ PkR :
Pnk
i¼1
Cik , and ð28Þ
811
Pm Pm j¼1 raj gj ðsP Þ j¼1 raj gj ððs þ 1ÞP Þ Pm P 6 0; m 1 þ j¼1 a^j gj ðsP Þ 1 þ j¼1 a^j gj ððs þ 1ÞP Þ s ¼ 1; . . . ; u 1;
ð29eÞ
i¼1
Let us remind you that the centres of the TFNs of the parameters, a1 ; a2 ; . . . ; am will be adjusted by least squares taking as the observations for Pk , k ¼ 1; 2; . . . ; K, only PkC . These estimates are symbolised as a^1 ; a^2 ; . . . ; a^m . Subsequently, we have to obtain the spreads of the parameters a~j , j ¼ 1; 2; . . . ; m, by solving a linear programme. To do so, we must first choose a level a of accomplishment of the congruency of the observations Yk with their predicted value Yek , k ¼ 1; 2; . . . ; K. We will also add certain constraints that will assure that the forward rates can be obtained from the discount function by using solution (9) of Eq. (8). These constraints will be the consequence of conditions in (10). To conclude, we must solve nk m K X X X Minimise z ¼ laj Cik gj ðtik Þ laj ;raj
j¼1
i¼1
k¼1
j¼1;2;...;m
þ
m X j¼1
raj
nk K X X k¼1
Cik gj ðtik Þ
ð29aÞ
i¼1
subject to nk m X X Cik gj ðtik Þ a^j j¼1
"
i¼1 m X
laj
nk X
ita ¼ ½it ðaÞ; it ðaÞ #
Cik gj ðtik Þ
1=t
¼ ½ðft þ rft ð1 aÞÞ
ð1 a Þ 6 YkC YkR ;
k ¼ 1; 2; . . . ; K; m X
a^j
j¼1
þ
nk X
"
i¼1
j¼1
ð29bÞ
Cik gj ðtik Þ
m X
raj
nk X
# Cik gj ðtik Þ
ðft lft ð1 aÞÞ
1:
ð30Þ
~it ðit ; lit ; rit Þ ð1 a Þ P YkC þ YkR ; ¼ ð29cÞ
Pm Pm j¼1 laj gj ðsP Þ j¼1 laj gj ððs þ 1ÞP Þ Pm P 6 0; m 1 þ j¼1 a^j gj ðsP Þ 1 þ j¼1 a^j gj ððs þ 1ÞP Þ s ¼ 1; . . . ; u 1;
1;
1=t
It is easy to check that the spot rate is not a TFN. However, from (7), we can deduce that the following TFN can approximate well the spot rate associated to the maturity t:
i¼1
k ¼ 1; 2; . . . ; K;
ð29fÞ
We remark that (29b), (29c) and (29f) correspond to the constraints (14b), (14c) and (14d) of Ishibuchi and NiiÕs estimation system, respectively. The objective function (29a) (the uncertainty of the system) must be minimised and it corresponds to the objectives (13a) or (14a) in Section 2.2. Likewise, (29d) and (29e) ensure that the solution for the forward rates in the sense of (9) exists for an arbitrary periodicity P (in years), which is the periodicity of the forward rates that we would like to obtain. So, these constraints are consequence of (10). Therefore, uP is the greatest maturity at which we want to determine the spot and forward rates and this will be close to the maturity of the bond with the greatest expiration. The value of the discount function in t, ft , is obtained from its corresponding spot rate it making ft ¼ ð1 þ it Þt . Then, the spot rate can be obtained from the discount by evaluating the expression it ¼ ðft Þ1=t 1. So, if the discount function is a TFN f~t ¼ ðft ; lft ; rft Þ, the spot rate will be a FN ~it whose a-cuts, ita , are from (6),
i¼1
j¼1
j ¼ 1; 2; . . . ; m:
laj ; raj P 0;
ð29dÞ
1 ðft Þ1=t
1;
rft
lf t
; tðft Þðtþ1Þ=t tðft Þðtþ1Þ=t
! :
ð31Þ
Notice that to obtain the left and right spreads of the triangular approximation to ~it (lit and rit respectively), we must take into account that the first derivative of the spot rate respect to its corresponding discount function evaluated in the centre
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of f~t is 1=tðft Þðtþ1Þ=t . Moreover, having in mind that the spot rate is decreasing respect to its corresponding discount function, we can identify immediately lit ¼
r ft tðft Þ
ðtþ1Þ=t
and rit ¼
lf t tðft Þ
ðtþ1Þ=t
from Eqs. (7). Finally, the centre of the approximation of ~it , is obtained directly from the centre of f~t . Now we are going to obtain the forward rates, i.e. the spot rates that the market expects for the future. In particular, we are going to obtain the one year forward rates for integer years, that is to say, the forward rates 1 qt , t ¼ 1; . . . ; u. Although annual periods are used to deduce these rates, for other periodicities, the discount function that we have fitted can be used, because it is defined for any value of t. To obtain the forward rate 1 q~t , we have to solve the fuzzy version of (1): 1 f~t1 ð1 þ 1 q~t Þ ¼ f~t :
1 qta
¼ ½1 qt ðaÞ; 1 qt ðaÞ 1 1 ¼ 1 1; Gt ðaÞ Gt ðaÞ ft1 þ rft1 ð1 aÞ ft1 lft1 ð1 aÞ ¼ 1 : 1; ft þ rft ð1 aÞ ft lft ð1 aÞ ð36Þ
Obviously, 1 q~t is not a TFN since the extremes of its a-cuts, (36), are not linear functions of a. However, we can approximate 1 q~t with a TFN. To obtain that approximation we will develop the extremes of 1 qta , 1 qt ðaÞ and 1 qt ðaÞ, with a Taylor expansion to the first grade from a ¼ 1:
ð32Þ 1
To solve this equation, we identify ð1 þ 1 q~t Þ ¼ e t , where G e t is the value in t 1 of one monetary G unit payable at t according to the TSIR. The above equation can therefore be written using the a-cuts, reminding that the embedded discount functions are TFNÕs (see (18) or (22)) and that the expression of the a-cuts of a TFN are given in (4): ½ft1 lft1 ð1 aÞ; ft1 þ rft1 ð1 aÞ½Gt ðaÞ; Gt ðaÞ ¼ ½ft lft ð1 aÞ; ft þ rft ð1 aÞ:
It is easy to check that 1 qt is decreasing respect to Gt because Gt is always a positive number. Then, by using (6), we can obtain the a-cuts of the forward rate for the tth year ð1 q~t Þ, 1 qta . The lower (upper) bound of 1 qta is obtained by calculating (35) with Gt ðaÞ ðGt ðaÞÞ:
ð33Þ
So, (33) can be solved using (9) since the constraints (29d) and (29e) ensure the condition (10): Gta ¼ ½Gt ðaÞ; Gt ðaÞ ft lft ð1 aÞ ft þ rft ð1 aÞ ; ¼ : ft1 lft1 ð1 aÞ ft1 þ rft1 ð1 aÞ
1 qt ðaÞ
¼ 1 qt ðaÞ
1 qt
¼
1:
ð35Þ
ð37aÞ
ft1 ft1 lft ft lft1 1þ ð1 aÞ; 2 ft ðft Þ
ð37bÞ
where 1 q0t ð1Þ and 1 q0t ð1Þ are the values of the first derivatives of 1 qt ðaÞ and 1 qt ðaÞ for a ¼ 1. Taking into account (4), it is straightforward to check in (37a) and (37b) that the centre of the triangular approximation 1 q~t is 1 qt ¼ ðft1 ft Þ 1. Likewise, from (4) and (37a) we can deduce that the left spread of that triangular approximation is
ð34Þ
G1 t
ft1 ft1 rft ft rft1 1 ð1 aÞ; 2 ft ðft Þ
1 qt ð1Þ þ 1 q0t ð1Þða 1Þ ¼
l1 qt ¼
Now, let us remark that under crispness, the value in t 1 of 1 monetary unit payable at t, Gt is obtained from the tth forward rate, 1 qt , as 1 Gt ¼ ð1 þ 1 qt Þ . Then, immediately follows:
1 qt ð1Þ þ 1 q0t ð1Þða 1Þ
ft1 rft ft rft1 ðft Þ
2
:
Finally, using (4) and (37b) we immediately find that the right spread of the triangular approximation is r1 qt ¼
ft1 lft ft lft1 ðft Þ2
:
J. de Andres Sanchez, A. Terce~no Gomez / European Journal of Operational Research 154 (2004) 804–818
Then, the triangular approximation of 1 q~t is ~t 1q
ð1 qt ; l1 qt ; r1 qt Þ ¼
ft1 ft1 rft ft rft1 ft1 lft ft lft1 ; 1; ft ðft Þ2 ðft Þ2
! : ð38Þ
Let us remark that always ft1 rft ft rft1 P 0 and ft1 lft ft lft1 P 0 due to the constraints (29d) and (29e). 4.2. Empirical application In this subsection we use our methodology to estimate the TSIR in the Spanish public debt market on June 29, 2001. The bonds in our sample are shown in Table 1. We can see that in the sample there are zero-coupon bonds (the Strips and T-Bills) and bonds with annual coupon. The
first type of fixed income securities only pays the facial value (1 0 0) at maturity. For example, the 6th asset (a strip) provides 100 at its maturity (1.58 years). An example of bond with annual coupon is the 5th asset. It offers a post-payable annual coupon (5.25% on the facial value) and its facial value (1 0 0) at the expiration date (1.58 years). Then, that bond provides a stream of payments that can be represented by fð5:25; 0:58Þ; ð5:25 þ 100; 1:58Þg. Notice that in Table 1, the price of the kth fixed income asset is quantified through its minimum k and maximum negotiated price, i.e. Pk ¼ ½Pmin ; k Pmax . So, centre and radius of Pk , PkC and PkR k respectively, are obtained by doing PkC ¼ ðPmin þ k k k Pmax Þ=2 and PkR ¼ ðPmax Pmin Þ=2. The price of the 5th bond is P5 ¼ ½103:555; 103:669 and it can be represented equivalently by its centre and spread as P5 ¼ hP5C ; P5R i ¼ h103:612; 0:057i. To fit the TSIR corresponding to our sample, we built the discount function with McCullochÕs
Table 1 Bonds negotiated in the Spanish debt market on June 29, 2001
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
813
Asset
Coupon (annual) (%)
Number of coupons
Maturity (years)
k Pmin
k Pmax
T-Bill T-Bill BOND T-Bill BOND STRIP BOND STRIP BOND BOND BOND BOND BOND BOND BOND BOND BOND STRIP BOND BOND BOND BOND BOND STRIP BOND BOND BOND
0.00 0.00 4.25 0.00 5.25 0.00 3.00 0.00 4.60 4.50 4.65 3.25 4.95 10.15 4.80 7.35 6.00 0.00 5.15 4.00 5.40 5.35 6.15 0.00 4.75 6.00 5.75
0 0 2 0 2 0 2 0 3 4 3 4 5 5 5 6 7 0 9 9 10 10 12 0 14 28 31
0.05 1.05 1.07 1.43 1.58 1.58 1.58 2.07 2.07 3.07 3.33 3.58 4.08 4.58 5.33 5.75 6.58 7.60 8.08 8.59 10.08 10.33 11.59 11.59 13.08 27.60 31.10
99.779 95.758 103.907 94.220 103.555 93.579 99.337 91.540 104.670 104.017 98.466 97.026 105.407 126.340 97.785 113.539 107.400 68.412 104.101 92.679 97.716 96.966 108.098 53.357 96.506 103.722 93.954
99.779 95.758 103.947 94.220 103.669 93.749 99.376 91.540 104.917 104.166 98.702 97.200 105.918 126.340 98.385 113.539 108.206 68.412 104.307 93.473 98.923 97.749 108.168 53.357 97.567 105.194 94.777
814
J. de Andres Sanchez, A. Terce~no Gomez / European Journal of Operational Research 154 (2004) 804–818
quadratic splines that have into account the number of bonds and the structure of maturities in the sample. So, we took m ¼ 5, and the knots we used to construct the splines were d1 ¼ 0 years, d2 ¼ 1:58 years, d3 ¼ 3:83 years, d4 ¼ 8:96 years and d5 ¼ 31:1 years. Concretely, the functions gj ðtÞ, j ¼ 1; . . . ; 5 are 3 8 1 > > t2 þ t; 0 6 t < 1:58; < 2 1:58 ð39Þ g1 ðtÞ ¼ > 1:58 > : ; 1:58 6 t < 31:1; 2
gj ðtÞ ¼
g5 ðtÞ ¼
X15 ¼ 5:25g1 ð0:58Þ þ ð5:25 þ 100Þg1 ð1:58Þ ¼ 85:53; X25 ¼ 5:25g2 ð0:58Þ þ ð5:25 þ 100Þg2 ð1:58Þ ¼ 83:60; .. . X55 ¼ 5:25g5 ð0:58Þ þ ð5:25 þ 100Þg5 ð1:58Þ ¼ 0: The final objective is to obtain a fuzzy discount function like (22) that in our empirical application is
8 0; > > > > > 1 > > ðt dj1 Þ2 ; > > > < 2ðdj dj1 Þ
0 6 t < dj1 ; dj1 6 t 6 dj ;
1 1 > ðt dj Þ2 þ ðt dj Þ þ ðdj dj1 Þ; > > > 2ðdjþ1 dj Þ 2 > > > > > > : 1 ðdjþ1 dj1 Þ; 2 8 < 0;
0 < t 6 8:96; 1 2 ðt 8:96Þ ; 8:96 6 t 6 31:1: : 2ð31:1 8:96Þ ð41Þ
To obtain the dependent variable observations Yk ¼ hYkC ; YkR i; k ¼ 1; 2; . . . ; 27, we need to transform the bond prices by using (28). For example, for the 5th bond we obtain Y5C ¼ P5C
n5 X
Ci5
i¼1
¼ 103:612 ½5:25 þ ð5:25 þ 100Þ ¼ 6:888
and
Y5R ¼ P5R ¼ 0:057:
Moreover, we need to quantify the observations of the explanatory variables in Eq. (27), X1k ; X2k ; . . .P ; X5k , k ¼ 1; 2; . . . ; 27. Let us remind that k k Xj ¼ ni¼1 Cik gj ðtik Þ and then, e.g., the values corresponding to the 5th bond are
dj 6 t 6 djþ1 ;
j ¼ 2; 3; 4;
djþ1 6 t 6 d5 ;
f~t ¼
5 X
a~j gj ðtÞ ¼
j¼0
5 X
aj ; laj ; raj gj ðtÞ
For a detailed discussion about the construction of our gj ðtÞ and the knots we have chosen, see McCulloch (1971).
ð42Þ
j¼0
and given that a~0 ¼ ð1; 0; 0Þ, then (42) is equivalent to 5 5 X X aj ; laj ; raj gj ðtÞ: a~j gj ðtÞ ¼ 1 þ f~t ¼ 1 þ j¼1
j¼1
ð43Þ The first step is to estimate the discount function with ordinary least squares (OLS). In this phase we fit the centres of the parameters a~j , j ¼ 1; 2; . . . ; 5, in (43) whose estimated values are a^1 ; a^2 ; . . . ; a^5 . Since we need to quantify the observations of the dependent variable with crisp numbers, we must take the centre value YkC as a representative value of Yk . Notice that, in doing so, we are considering only PkC as the observed price for the kth bond. In our application the OLS estimation has a determination coefficient close to 100% (exactly, 99.98%) and provides the following crisp discount function: ft ¼ 1 0:04404282g1 ðtÞ 0:03661593g2 ðtÞ
3
ð40Þ
0:04870935g3 ðtÞ 0:03550896g4 ðtÞ 0:00776513g5 ðtÞ:
J. de Andres Sanchez, A. Terce~no Gomez / European Journal of Operational Research 154 (2004) 804–818
815
0.075 0.07 0.065
rate
0.06 0.055 0.05 0.045 0.04 0
2
4
6
8
10 12 14 16 18 20 22 24 26 28 30 Maturity (years) spot rates
forward rates
Fig. 1. Estimations of the spot rates and forward rates 6/29/2002 in Spanish public debt market.
The spot rates and one-year forward rates for the next 30 years that are obtained from the crisp discount function are given in Fig. 1. The second step consists in implementing fuzzy regression by solving the estimation system (29a)–
(29f) taking as the centres of the fuzzy parameters a~j , those obtained in the OLS regression a^j , j ¼ 1; 2; . . . ; 5. The result of solving that linear programme is the estimates of laj , raj , j ¼ 1; 2; . . . ; 5, in (43). Table 2 shows the results for the
Table 2 Results of fuzzy regression for a ¼ 0:5 and 0.75 a~1
a~2
a~3
a~4
a~5
)0.04404282
)0.03661593
)0.04870935
)0.03550896
)0.00776513
a ¼ 0:5, z ¼ 109:62 laj raj
0.00176058 0.00312284
0.00290595 0
0.00661728 0.00459031
0 0
0 0.00073984
a ¼ 0:75, z ¼ 219:25 laj raj
0.00352116 0.00624569
0.00581189 0
0.01323457 0.00918063
0 0
0 0.00147968
aj ¼ a^j
discount function
discount function
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
times (years) (a)
lower bound
centre
times (years)
upper bound
(b)
lower bound
centre
upper bound
Fig. 2. (a) Fuzzy discount function for a ¼ 0:5. (b) Fuzzy discount function for a ¼ 0:75.
816
J. de Andres Sanchez, A. Terce~no Gomez / European Journal of Operational Research 154 (2004) 804–818 0.085 0.075
0.065
spot rate
spot rate
0.075 0.07 0.06 0.055 0.05
0.065 0.055 0.045
0.045 0.04
0.035 0 2
4 6 8 10 1 2 14 16 18 20 22 2 4 26 28 30
0 2
4 6 8 10 12 14 1 6 18 20 22 24 26 28 30
time (in years) centre
lower bound
(a)
time (in years) upper bound
centre
lower bound
(b)
upper bound
Fig. 3. (a) Fuzzy TSIR for a ¼ 0:5. (b) Fuzzy TSIR for a ¼ 0:75.
0.115 0.105 0.095 0.085 0.075 0.065 0.055 0.045 0.035
0.085
forward rate
forward rate
0.095
0.075 0.065 0.055 0.045 0.035 0
2
4
6
8 10 12 14 16 18 20 22 24 26 28 30
0
2
4
6
8
time (in years)
(a)
lower bound
centre
1 0 12 14 1 6 18 20 22 2 4 26 28 3 0 time (in years)
upper bound
(b)
lower bound
centre
upper bound
Fig. 4. (a) Forward rates a ¼ 0:5. (b) Forward rates a ¼ 0:75.
levels of inclusion a ¼ 0:5 and 0.75. We want to point out that, in order to build constraints (29d) and (29e), we assumed an annual periodicity, since we will subsequently obtain only the implicit rates for that periodicity. Fig. 2a and b show the discount function that we fitted for the next 30 years. Fig. 3a and b show the shapes of the fuzzy TSIR that we obtained with fuzzy regression and Fig. 4a and b plot the one-year forward rates for the next 30 years. We would like to point out that parameter a can be interpreted as an indicator of the perceived uncertainty in the market by the decision maker. If a increases, then the uncertainty of the estimates of the explained variable (the price of the bonds) also increases (see Wang and Tsaur, 2000) and consequently the spread of the subsequent estimates of the spot rates and the forward rates will be wider.
5. Conclusions In this paper, we have attempted to make up for the lack of explanations in the financial and actuarial literature about estimating interest rates with FNs. When we price assets with a long maturity, we agree with the authors mentioned above that it is a suitable way to model the uncertainty of the future interest rates using FNs, especially when the financial analysis is concerned with the long term. Unfortunately, it is usual to read in many papers that the subjacent hypothesis is that an ‘‘expert’’ subjectively estimates these rates with no more explanation, i.e. there is no serious analysis about how the ‘‘expert’’ is supposed to express these predictions about future yield rates, using, in our case, fuzzy numbers. If we accept that the ‘‘experts’’ are traders in the fixed income markets,
J. de Andres Sanchez, A. Terce~no Gomez / European Journal of Operational Research 154 (2004) 804–818
these subjective estimates are included in the price of the debt instruments and, therefore, in the TSIR of these instruments. Our method quantifies the expertsÕ subjective estimates of the spot rates and spot interest rates for the future (the forward rates) with FNs. This method is based on a fuzzy regression technique and uses all the prices of the bonds negotiated during one session to fit the TSIR, in such a way that no information provided by the negotiated prices is lost. When econometric methods are used, however, these prices must be reduced to representative ones and then, some of the information provided by the bond prices is lost. We have proposed adjusting the interest rates with TFNs. This is because the arithmetic is easy to implement and the interpretation of the estimations is intuitive because they are well adapted to the way humans make predictions. For example, an interest rate given by (0.03, 0.005, 0.005) indicates that we expect an interest rate of about 3%, and we do not expect deviations to be greater than 50 basis points.
Acknowledgements The authors wish to thank the valuable comments of two anonymous referees.
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