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Estimating a term structure of interest rates for fuzzy financial pricing by using fuzzy regression methods
Fuzzy Sets and Systems 139 (2003) 313 – 331 www.elsevier.com/locate/fss
Estimating a term structure of interest rates for fuzzy !nancial pricing by using fuzzy regression methods Jorge de Andr'es S'anchez∗ , Antonio Terce˜no G'omez Dept. de Gesti on de Empresas, Universitat Rovira i Virgili, Avinguda de la Universitat 1, 43204, Reus, Spain Received 16 January 2002; received in revised form 4 July 2002; accepted 22 July 2002
Abstract Several papers in the !nancial literature propose using fuzzy numbers to model interest rate uncertainty. However, in our opinion, the !rst problem to be solved is how to estimate these rates with fuzzy numbers. 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 the extension proposed by Ishibuchi and Nii (Fuzzy Sets and Systems 119 (2001) 273) of the fuzzy regression methods exposed in Savic and Predrycz (Fuzzy Regression Analysis, Physica-Verlag, Heidelberg, 1992, pp. 91–100), Tanaka (Fuzzy Sets and Systems 24 (1987) 363–375) and Tanaka and Ishibuchi (Fuzzy Regression Analysis, Physica-Verlag, Heidelberg, 1992, pp. 47). This method will enable to quantify the anticipated rates in the !xed income markets for the future with fuzzy numbers. In particular, we discuss how to estimate the TSIR with triangular fuzzy numbers because of the desirable properties of this kind of fuzzy numbers. c 2002 Elsevier B.V. All rights reserved. Keywords: Economics; Finance; Fuzzy regression; Temporal structure of interest rates
1. Introduction To obtain the !nancial price of any asset we have to discount the cash >ows 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 !nancial analyses are concerned with the medium and long term and, in our opinion, modelling the behaviour of interest rates in the long term by means of a stochastic model is not ∗
Corresponding author. Fax: +349-77759810. E-mail address: [email protected] (J. de Andr'es S'anchez).
c 2002 Elsevier B.V. All rights reserved. 0165-0114/03/$ - see front matter PII: S 0 1 6 5 - 0 1 1 4 ( 0 2 ) 0 0 3 7 3 - 1
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J. de Andr es S anchez, A.T. G omez / Fuzzy Sets and Systems 139 (2003) 313 – 331
very realistic. 1 Gerber [11] points out that there is no commonly accepted stochastic model form making predictions about long-term discount rates. As we have only imprecise data or data related in a vague way with the behaviour of the interest rates along the pricing horizon, data such as the price of !xed income securities, the opinion of “experts” about the future behaviour of macroeconomic magnitudes, etc., many authors think that it is more suitable and realistic to make !nancial analyses in the long term using fuzzy estimates of the discount rates. For !nancial analysis, see [1,12,15,19,29] while for actuarial pricing, see [7,8,18,22,28]. However, the literature has not paid enough attention to how to estimate future discount rates with fuzzy numbers and, consequently, how to construct their characteristic 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. [10] 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 unadequate 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 temporal structure of interest rates (TSIR) with fuzzy subsets, since the TSIR implicitly contains the expectations of the !xed 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 [4]. We would also like to point out that it was in [22] where there is suggested for the !rst time using a fuzzy TSIR. 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 >ow payable in t years with the discount rate associated to that maturity. This rate is named as spot rate and we will symbolise it as it . So, the present value or 1 monetary unit payable in t years is (1 + it )−t . This expression is often called the discount function, which we will denote as ft , i.e. ft = (1 + it )−t , and it is clearly an exponential function of time. The implicit, or forward, rates, which standard !nancial theory interprets as the spot rates that the market expects to prevail in the future, are given implicitly by the TSIR. Then, from the spot rates it and it ∗ where t ∗ ¿t, the forward rate that has to be applied to obtain the price in t-years of a cash >ow that in that moment will be payable in t ∗ –t years (t ∗ −t t ) is obtained by solving: ft · (1 + t ∗ −t t )−(t so
ft t ∗ − t t = ft ∗
∗
−t)
1=(t ∗ −t)
= ft ∗
(1)
− 1;
where the Theory of Rational Expectations interpret t ∗ −t t 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 public 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: 1
The papers that model the 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 diNerential equations.
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(a) The estimates that are obtained after adjusting the coeOcients are not random variables, which are diOcult to manipulate in arithmetical operations, but fuzzy numbers, 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 fuzzy numbers 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 agents’ beliefs and expectations, which are highly subjective and vague. Fuzzy Set Theory, therefore, is a good way of treating this information. For example, the asset prices that are determined in the markets depend on the agents’ expectations of future in>ation, the issuers’ credibility, etc. We think that it is more realistic to consider that the bias between the 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 con!dence intervals. For instance, the price of a !nancial 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). The structure of the paper is as follows. In Section 2, we shall describe some aspects of fuzzy arithmetic and the extension [13] of the fuzzy regression methods described in [25–27]. In Section 3, we shall discuss some basic characteristics of the models that attempt to adjust the TSIR through the discount function de!ned 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 A fuzzy number (FN) is a fuzzy subset A˜ de!ned over real numbers. It is the main instrument used in Fuzzy Set Theory for quantifying uncertain quantities (e.g. in !nancial analysis, the discount rates). Two properties are required for a FN. The !rst one is that it must be a normal fuzzy set, i.e. sup ∀x∈X A˜ (x) = 1. The second is that it must be convex (i.e., its -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 symbolise a TFN A˜ as A˜ = (a; la ; ra ) where a is the centre and la and ra are the left and right spreads, respectively. For example, a sentence by a !nancial analyst such as “I expect for the next 2 years the free risk rate to be around 5% and deviations no longer that the 1%” may be quanti!ed in a very natural way as (0.05, 0.01, 0.01).
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A TFN is characterised by x−a+l a ; la A (x) = a + ra − x ; ra 0
its membership function A (x) or, alternatively, by its -cuts, A , as a − la ¡ x 6 a; a ¡ x 6 a + ra otherwise;
and R = [a − la (1 − ); a + ra (1 − )]: A = A( ); A( ) In many !nancial analyses it is often necessary to evaluate functions (e.g. the net present value), which in a general way we shall name y = f(x1 ; x2 ; : : : ; xn ). Then, if x1 ; x2 ; : : : ; xn are not crisp numbers but are the FNs A˜1 ; A˜2 ; : : : ; A˜n ; f(·) induces a FN B˜ = f(A˜1 ; A˜2 ; : : : ; A˜n ) whose characteristic function must be obtained from Zadeh’s extension principle. Unfortunately, it is often impossible to ˜ although in many cases it is possible obtain a closed expression for the membership function of B, to obtain its -cuts, B , from A1 ; A2 ; : : : ; An , by doing B = f(A˜ 1 ; A˜ 2 ; : : : ; A˜ n ) = f(A1 ; A2 ; : : : ; An ): In !nancial 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 -cuts of B. If the function f(·) that induces B˜ is increasing with respect to the !rst m variables, where m6n, and decreasing with respect to the last n − m variables, in [3] it is demonstrated that B is R B = [B( ); B( )] = f(A1 ( ); : : : ; Am ( ); Am+1 ( ); : : : ; An ( )); f(A1 ( ); : : : ; Am ( ); Am+1 ( ); : : : ; An ( )):
(2)
˜ will be another It is well known that the result of evaluating a 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 approximate B˜ using a TFN that we shall denote by B˜ . If the function that we evaluate, f(·), is increasing with respect to the !rst m variables, where m6n, and decreasing with respect to the others, i.e. the -cuts of B˜ are (2), in [9] it is proposed to approximate B˜ with a FN B˜ = (b; lB ; rB ) by using Taylor’s expansion. Concretely: b = f(aC );
rb =
m n @f(aC ) @f(aC ) lb = lai − ra i ; @xi @xi i=1 i=m+1
m n @f(aC ) @f(aC ) ra i − lai ; @xi @xi i=1 i=m+1
(3)
where aC = (a1 ; : : : ; am ; am+1 ; : : : ; an ). Let us illustrate how the approximating procedure (3) can be used by applying it to the present value, V , of an amount C payable in t years with a pricing interest rate i, i.e: V = C · (1 + i)−t :
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317
Taking the maturity as a !xed parameter, it is easy to check that V is increasing (decreasing) respect to C (i) given that their partial derivatives are: @V = (1 + i)−t @C
and
@V = −t · C · (1 + i)−(t+1) : @i
Suppose now that the amount and the discount rate are given by means of TFNs, being C˜ = (C; lC ; ˜ will be the fuzzy rC ) and i˜ = (i; li ; ri ), respectively. In this case, the present value of the amount, C, number V˜ , which is obtained by evaluating: V˜ = C˜ · (1 + i˜)−t : Clearly, V˜ is not a TFN because it is obtained from a non-linear combination of two TFN. However, by using (3) we can approximate V˜ with the TFN (V; lV ; rV ), where: V = C · (1 + i)−t ;
lV =
lC t · C · ri + (1 + i)t (1 + i)t+1
and
rV =
rC t · C · li + : (1 + i)t (1 + i)t+1
˜ In this paper, to obtain the forward rates we will need to solve fuzzy equations with the structure A· + ˜ ˜ ˜ ˜ X = B where A and B are TFNs with the support contained in − 0. The above equation can be expressed through the -cuts of A;˜ X˜ and B˜ as A · X = B , i.e.: [a − la (1 − ); a + ra (1 − )] · X ( ); XR ( ) = [b − lb (1 − ); b + rb (1 − )]:
(4)
According to [16], the narrowest solution is X ( ) =
b − lb (1 − ) ; a − la (1 − )
b + rb (1 − ) XR ( ) = : a + ra (1 − )
(5)
We must take into account that this solution does not always exist ∀ ∈ [0; 1]. For example, if A0:5 = [1; 2] and B0:5 = [1; 1:5], then X0:5 = [1; 34 ], which is not an -cut of a FN. Nevertheless, in [2] it is demonstrated that the required condition for the existence of (5) ∀ ∈ [0; 1] is: la lb ¿ ; b a
rb ra ¿ : b a
(6)
2.2. Fuzzy regression model with asymmetric coe4cients In this subsection, we will describe Ishibuchi and Nii’s extension of the fuzzy regression models developed in [25–27], which is one of the most widely used models in economic applications (see for example [17,23,24]). 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 or fuzzy numbers. 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 {(Y1 ; X1 ); (Y2 ; X2 ); : : : ; (Yj ; Xj ); : : : ; (Yn ; Xn )} where Xj is the jth observation of the explanatory variable, Xj = (X0j ; X1j ; X2j ; : : : ; Xij ; : : : ; Xmj ). Moreover, X0j = 1 ∀j, and Xij is the observed
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J. de Andr es S anchez, A.T. G omez / Fuzzy Sets and Systems 139 (2003) 313 – 331
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 = 1; 2; : : : ; n and may either be a crisp value or a con!dence interval. In both cases, it can be represented through its centre and its spread as Yj = YjC ; YjR , where YjC is the centre and YjR is the radius. In particular, we must estimate the following fuzzy linear function: Y˜ j = A˜ 0 + A˜ 1 X1j + · · · + A˜ m Xmj ; where Y˜j is the estimation of Yj by a FN after adjusting A˜0 ; A˜1 ; : : : ; A˜m . Likewise, in fuzzy regression models the disturbance is not introduced as a random addend in the linear relation, but it is incorporated into the coeOcients A˜i , i = 0; 1; : : : ; m. Of course, the !nal objective is to adjust the fuzzy numbers A˜i , i = 0; 1; : : : ; m from the available sample. In this paper, we propose to model the parameters A˜i as TFN, i.e., they can be written as A˜i = (ai ; lai ; rai ); i = 0; 1; : : : ; m. Then, the estimate of Yj , Y˜j , is a TFN that we will name Y˜j = (Yj ; lYj ; rYj ), where Y j = a0 +
m
ai Xij ;
lYj = la0 +
i=1
r Y j = ra 0 +
m
lai |Xij | +
i=1
rai |Xij | +
i=1 xij ¿0
m
rai |Xij |;
i=1 xij ¡0
xij ¿0
m
m
lai |Xij |:
i=1 xij ¡0
To !t A˜0 ; A˜1 ; : : : ; A˜m we !rst have to determine the cores a0 , a1 ; : : : ; am by using least squares and taking the centres of the observations Yj = 1; 2; : : : ; n as the realisations of the independent variable. These estimates are named aˆ0 ; aˆ1 ; : : : ; aˆm . Subsequently, we have to determine the parameters lai and rai , which must minimise the uncertainty of the estimates for Yj , Y˜j , (i.e. their spreads), and simultaneously maximise the congruence of the estimates obtained for Yj which will be measured as the level in which Yj is contained within Y˜j ; (Yj ⊆ Y˜j ). In particular, we must solve the following multiple objective programme: Minimise z
=
m n
lai |Xij | +
j=1 i=0
m n
rai |Xij |;
j=1 i=0
Maximise subject to (Yj ⊆ Y˜ j ) ¿ ;
j = 1; 2; : : : ; n; lai ; rai ¿ 0; i = 0; 1; : : : ; m;
∈ [0; 1):
If for the second objective we require a minimum accomplishment level ∗ , j = 1; 2; : : : ; n, the above programme is transformed into the following linear one: Minimise z
=
m n j=1 i=0
lai |Xij | +
m n j=1 i=0
rai |Xij |;
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319
subject to aˆ0 +
m i=1
m m aˆi Xij − la0 + la0 |Xij | + rai |Xij | (1 − ∗ ) 6 YjC − YjR ; j = 1; 2; : : : ; n; i=1 i=1 xij ¿0
xij ¡0
aˆ0 +
m i=1
m m ∗ aˆi Xij + ra0 + rai |Xij | + lai |Xij | (1 − ) ¿ YjC + YjR ; j = 1; 2; : : : ; n; i=1 i=1 xij ¿0
lai ; rai ¿ 0;
xij ¡0
i = 0; 1; : : : ; m:
Note that in the last mathematical programme the !rst two blocks of constraints are consequences of requirement (Yj ⊆ Y˜j )¿ ∗ while the last block ensures that lai ; rai ∀i will be non-negative.
3. Estimating the term structure of interest rates using splines and econometric methods It is fairly straightforward to estimate the TSIR for a speci!c date and a given market if there are many suOciently liquid zero coupon bonds and their price can be observed without perturbations. However, these conditions are rarely found simultaneously in the !xed income markets. This section, then, 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 zero coupon 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 = 1; 2; : : : ; K, and K is the number of available bonds, provides several cash->ows (coupons and principal). These are denoted by (C1k ; t1k ); (C2k ; t2k ); : : : ; (Cnkk ; tnkk ), where Cik is the amount of the ith cash->ow and tik is its maturity expressed in years and nk is the number of cash->ows provided by this bond. If we suppose that the bonds have no embedded option (e.g. convertibility) the price of the kth bond is the sum of the discounted value of every amount with the corresponding spot rate: Pk =
nk i=1
Cik ftik ;
where ftik is the discounted value of one dollar with maturity tik years. Of course, we should then de!ne an analytical form for the discount function in order to specify the 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 ones described in [20,21] (quadratic and cubic splines) and in [31] (cubic exponential splines). These methods assume
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J. de Andr es S anchez, A.T. G omez / Fuzzy Sets and Systems 139 (2003) 313 – 331
that the discount function is a linear combination of m + 1 functions of time, i.e.: ft =
m
aj gj (t):
j=0
We assume that gj (t) are splines and not simply polynomial functions because, with splines, we can determine gj (t) according to the distribution of the maturity dates of the sample and, therefore, !t the discount function better for the most common maturates. Moreover, with splines we can obtain TSIR pro!les that do not >uctuate very much and we ensure that forward rates do not behave explosively. In our paper, we will assume the functions gj (t) are cubic exponential splines. Concretely, gj (·), j = 0; 1; : : : ; m will be cubic splines respect to the argument x, where x = 1 − e−!·t ; where !¿0: Then, x = 0 for t = 0 and x = 1 when t → ∞. Finally, despite the discount function is, in essence, a polynomial function of x, we attain that the discount function to adjust will be approximately an exponential function of time (let us remark that actually ft = (1 + it )−t ): ft = "0 + "1 e−!·t + "2 e−2!·t + "3 e−3!·t : Then, we can deduce that the following equation must be estimated: Pk =
nk i=1
Cik
m
aj gj (xik ) + ”k ;
(7)
j=0 k
where xik = 1 − e−!·ti . The estimation of the parameters (a0 ; a1 ; : : : ; am ) in (7) corresponds to a problem of conventional linear regression if the parameter ! is !xed beforehand. However, the penultimate step will be to determine the optimal value of !, that will permit minimise the sum of the square residuals. As it is pointed out in [31] it can be done by use of numerical procedures, such as the three-point Newton minimisation method. The last step to implement is to obtain, again, the estimate of the vector (a0 ; a1 ; : : : ; am ) with the optimum !. Likewise, a random disturbance is justi!ed because the formation of the prices of !xed income securities is disturbed by several factors. In [5] there are outlined some of them; for example, the coupon bearing of bonds with maturities greater than one contain information about more than one present value coeOcient; the bond portfolios are not continuously rebalanced, so at any moment each bond can deviate by an uncertain 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. In our opinion, these reasons justify better the use of fuzzy regression than conventional regression methods when adjusting the TSIR. 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 the analysed framework in Section 3 and the fuzzy regression method presented
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321
above. Subsequently, we shall discuss how to estimate the spot and the forward rates using fuzzy numbers. We shall end this section by making an empirical application to the Spanish public debt market. 4.1. Estimating the discount factor with Triangular Fuzzy Numbers (TFNs) Fitting the TSIR with a fuzzy regression method, we can take all the prices traded on 1 day as the price of the bonds, not just the average price. So, this instrument is attractive because the observed price in one session for any bond is often not 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 = Pk C ; Pk R i.e. the minimum price negotiated during the analysed session is Pk C − Pk R while the maximum is Pk C + Pk R . For example, if the traded price for one bond on 1 day >uctuates between 100 and 103, it will be expressed as 101:5; 1:5 . To !t the discount function, we assume that it is quanti!ed 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 = (ft ; lft ; lft ); t ¿ 0;
0 6 ft − lft 6 ft 6 ft + rft 6 1:
So, the predicted price of the kth bond with the triangular discount function can be written as P˜ k =
nk i=1
Cik f˜ tik
and this implies that P˜ k = (PCk ; lPk ; rPk ) =
nk i=1
Cik (ftik ; lftk ; rftk ): i
i
Likewise, we will construct the discount function from a linear combination of m + 1 piecewise polynomials gj (x), j = 0; 1; : : : ; m where x = 1 − e−!·t and !¿0, with images in + , continuously diNerentiable, and whose parameters are given by TFNs. In this way, these parameters can be represented as a˜j = (aj ; laj ; raj )laj ;
raj ¿ 0;
j = 0; 1; : : : ; m:
So the discount function is f˜ t =
m
a˜j gj (x) =
m
j=0
(aj ; laj ; raj )gj (x) =
j=0
m
(ajC gj (x); laj gj (x); raj gj (x));
j=0
i.e. the discount function is approximately a fuzzy exponential function of time: f˜t = "˜ 0 + "˜ 1 e−!·t + "˜ 2 e−2!t + "˜ 3 e−3!t where the parameters "˜ 0 ; : : : ; "˜ 3 will be TFNs. Thus, the estimated price of the kth bond, P˜k , is P˜ k = (PCk ; lPk ; rPk ) =
nk i=1
Cik
m j=0
(aj ; laj ; raj )gj (xik ):
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J. de Andr es S anchez, A.T. G omez / Fuzzy Sets and Systems 139 (2003) 313 – 331
Since the value of the discount function for t = 0 (x = 0) should be the crisp number 1, then f˜0 = (f0 ; lf0 ; rf0 ) = (1; 0; 0). This condition is met if a˜0 = (a0 ; la0 ; ra0 ) = (1; 0; 0), g0 (0) = (1 − x) and gj (0) = 0; j = 1; 2; : : : ; m. So, we can express the price of the kth bond as: P˜ k = (PCk ; lPk ; rPk ) =
nk
Cik (1 − xik ) +
i=1
=
nk
Cik (1
−
xik )
i=1
+
m
(aj ; laj ; raj )gj (xik )
j=1
nk
Cik
i=1
m
(aj ; laj ; raj )gj (xik )
j=1
and, using fuzzy arithmetic, we !nally write: (PCk ; lPk ; rPk )
−
nk
Cik (1
−
xik )
i=1
=
m
(aj ; laj ; raj )
j=1
nk
Cik gj (xik ):
i=1
In this way, by identifying Y˜k = (YCk ; lYk ; rYk ) = (PCk ; lPk ; rPk ) − YCk
=
PCk
−
nk
Cik (1 − xik );
lYk = lPk ;
nk
i=1
Cik (1 − xik ), we obtain
rYk = rPk ;
i=1
So, it is easy to that the value of the jth explanatory variable for the kth bond is the verify k crisp value Xjk = ni=1 Cik gj (xik ) . So, the expression to be adjusted by a fuzzy regression method is simply expressed as (YCk ; lYk ; rYk ) = (a1 ; la1 ; ra1 )X1k + (a2 ; la2 ; ra2 )X2k + · · · + (am ; lam ; ram )Xmk since Y˜k is the prediction with fuzzy regression of the observation of price of the kth bond minus the sum of the cash->ows provided by the asset. us remark that the observed value of this variable Let k for the kth bond is Yk = Yk C ; Yk R = Pk − ni=1 Cik (1 − xik ) and so: Y k C = Pk C −
nk
Cik (1 − xik )
and
Yk R = Pk R :
i=1
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 = 1; 2; : : : ; K only Pk C . These estimates are symbolised as aˆ1 ; aˆ2 ; : : : ; aˆm . Moreover, we must to obtain the estimate of !, which will be named !ˆ, by using numerical methods. Subsequently, we have to obtain the spreads of the parameters a˜j ; j = 1; 2; : : : ; m by solving a linear programme. To do it, we must !rst choose a level ∗ of accomplishment of the congruency of the observations Yk with their predicted value Y˜k ; k = 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 (5) of Eq. (4). These constraints will be the consequence of conditions (6). To
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323
conclude, we must solve Minimise laj ;raj j = 1;2;:::;m
subject to m
aˆj
j=1
nk
z=
m j=1
laj
nk K
Cik gj (xik ) +
k=1 i=1
m j=1
raj
nk K
Cik gj (xik )
nk m Cik gj (xik ) − la j Cik gj (xik ) (1 − ∗ ) 6 Yk C − Yk R ;
i=1
j=1
i=1
k = 1; 2; : : : ; K;
m j=1
aˆj
nk
(8a)
k=1 i=1
(8b)
nk m Cik gj (xik ) + ra j Cik gj (xik ) (1 − ∗ ) ¿ Yk C + Yk R ;
i=1
j=1
i=1
k = 1; 2; : : : ; K;
(8c)
m ˆ −!(s+1)P ˆ laj gj (1 − e−!sP ) ) j=1 laj gj (1 − e − 6 0; m m ˆ + −!sP ˆ ) ˆ ˆ e−!sP e−!(s+1)P + j=1 aˆj gj (1 − e−!(s+1)P ) j=1 aˆj gj (1 − e m
j=1
s = 1; : : : ; u − 1;
(8d)
m ˆ −!(s+1)P ˆ raj gj (1 − e−!sP ) ) j=1 raj gj (1 − e − 6 0; m m ˆ + −!sP ˆ ) ˆ ˆ e−!sP e−!(s+1)P + j=1 aˆj gj (1 − e−!(s+1)P ) j=1 aˆj gj (1 − e m
j=1
s = 1; : : : ; u − 1; laj ; raj ¿ 0;
j = 1; 2; : : : ; m:
(8e) (8f)
Constraints (8b), (8c) and (8f) correspond to Ishibuchi and Nii’s regression model. Likewise, (8d) and (8e) ensure that the solution for the forward rates in the sense of (5) exists for an arbitrary periodicity P (in years), which is the periodicity of the forward rates that we would like to obtain. Therefore, uP is the greatest maturity at which we want to determine the spot and forward rates. 4.2. Estimating the spot rates and the forward rates with TFN The value of the discount function in t, ft , is obtained from its corresponding spot rate it making ft = (1 + it )−t . Then, the spot rate can be obtained by evaluating the expression it = (ft )−1=t − 1. So,
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if the discount function is a TFN f˜t = (ft ; lf; rf), the spot rate will be a FN i˜t whose -cuts, it , are, from (2): it = [it ( ); it ( )] = [(ft + rft (1 − ))−1=t − 1; (ft − lft (1 − ))−1=t − 1]: It is easy to check that the spot rate is not a TFN. However, from (3), we can deduce that the following TFN can approximate well the spot rate associated to maturity t: r ft lf t 1 : − 1; ; it ≈ (ft )1=t t · (ft )(t+1)=t t · (ft )(t+1)=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 t , t = 1; : : : ; u. Although annual periods are used to deduce these rates, for other periodicities the discount function that we have !tted can be used because it is de!ned for any value of t. To obtain the forward rate 1 ˜t , we have to solve the fuzzy version of (1) f˜ t −1 (1 +1 ˜t )−1 = f˜ t : To solve this equation, we identify (1+1 ˜t )−1 = G˜ t , where G˜ t is the value in t −1 of one monetary unit payable at t according to the TSIR. The above equation can therefore be written using the -cuts as: [ft −1 − lft−1 (1 − ); ft −1 + rft−1 (1 − )] · [Gt ( ); Gt ( )] = [ft − lft (1 − ); ft + rft (1 − )] and it can be solved using (5) because constraints (8d) and (8e) ensure condition (6): ft + rft (1 − ) ft − lft (1 − ) ; : Gt = [Gt ( ); Gt ( )] = ft −1 − lft−1 (1 − ) ft −1 + rft−1 (1 − ) Then, to obtain the -cuts of the 1 year forward rates for the tth year (1 ˜t ); 1 t , we must only take into account that under non-fuzziness Gt = (1+1t )−1 , i.e. 1t = (Gt )−1 − 1 = (ft −1 =ft ) − 1. Then ft −1 − lft−1 (1 − ) ft −1 + rft−1 (1 − ) − 1; −1 : 1 t = [1 t ( ); 1 t ( )] = ft + rft (1 − ) ft − lft (1 − )
Obviously, 1 ˜t is not a TFN since the extremes of its -cuts are not linear functions of . However, the upper (lower) extreme of 1 t is obtained from the lower (upper) extreme of the discount functions for 1 monetary unit payable in t − 1 years and in t years. Then, we can approximate 1 ˜t with (3), and the result that we obtain is ft −1 · rft − ft · rft−1 ft −1 · lft − ft · lft−1 ft −1 ; ˜t ≈ − 1; ; 1 (ft )2 (ft )2 ft where ft −1 · rf − ft · rf−1 ¿ 0 and ft −1 · lf − ft · lf−1 ¿ 0 due to constraints (8d) and (8e).
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325
Table 1 Fixed income securities negotiated in the Spanish public debt market 29 June 2001 k
Asset
Coupon (annual) (%)
Maturity (days)
Maturity (years)
k Pmin
k Pmax
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
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
18 382 391 521 576 576 576 756 757 1122 1216 1307 1490 1673 1945 2098 2402 2775 2948 3134 3680 3771 4229 4230 4774 10073 11351
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
4.3. Empirical application In this subsection, we use our methodology to estimate the TSIR in the Spanish public debt market on 29 June 2001 with our methodology. The bonds in our sample are shown in Table 1. To !t the TSIR on this date, we de!ned the discount function with Vasicek and Fong’s cubic exponential splines, that we de!ne as the cubic splines of the argument x = 1 − e−!·t de!ned by McCulloch in [20]. We took m = 5, and the knots we used to construct the splines were d1 = 0 years, d2 = 1:5 years, d3 = 5 years and d4 = 1000 years (i.e. ∞ years). Taking this knots, we can distinguish independently the shapes of the TSIR for the cash->ows maturities corresponding to the “short term” (from 0 to 1.5 years), “medium term” (from 1.5 to 5 years) and the “long term” (more than 5 years). Likewise, the knots expressed with the transformed variable x are y1 = 0, y2 = 0:14494, y3 = 0:40664 and y4 = 1 because, we have obtained the optimal ! equal to 0.10439.
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Table 2 Results of fuzzy regression for ∗ = 0:5 and 0.75
aj = aˆj
=∗ 0:5, z = 65:92 l aj raj
a˜1
a˜2
a˜3
a˜4
a˜5
−4:9448
0.3007
−1:0652
−4:6033
0.7870
∗ = 0:75, z = 131:85 l aj raj
0 0
0 0
0.2204 0
0 0.0396
0.0095 0.0151
0 0
0 0
0.4408 0
0 0.0792
0.0190 0.0303
Then, the functions gj (t), j = 1; : : : ; 5 are 0; (x−yj−1 )3 ; 6(yj −yj−1 ) (y −y )(x−yj ) (yj −yj−1 ) + j j−21 gj (x) = 6 (x−y )2 (x−y )3 + 2 j − 6(yj+1 −j yj ) ; 2y −y −y (yj+1 − yj−1 )[ j+1 6j j−1 +
x 6 yj − 1 ; yj−1 ¡ x 6 yj ; j = 1; : : : ; 4: yj ¡ x 6 yj+1 ; x−yj+1 ]; 2
yj+1 ¡ x ¡ 1;
g5 (x) = x ∀x: The result that we obtain when ordinary least squares is used, Pk C is taken as the observed price for the kth bond and with x = 1 − e−0:10439·t , is ft = (1 − x) + a1 g1 (x) + a2 g2 (x) + a3 g3 (x) + a4 g4 (x) + a5 g5 (x); where the values of ai , i = 1; 2; : : : ; 5 are the centres of TFNs in Table 2. Moreover, we point out that the determination coeOcient is close to 100% (exactly, 99.83%). The spot rates and 1 year forward rates for the next 30 years are given in the Fig. 1 of the annex. It is well known that the model of fuzzy regression we are using is very sensitive to outliers. The consequence of its existence is to obtain a fuzzy shape too wide (uncertain) and unrealistic. Of course, the outliers are the data that is around the model’s upper (lower) bound while the great proportion of the data is near of the core of the fuzzy mapping that we have obtained or around of its lower (upper) bound. In [6] there is a wide survey of methods to use in this circumstances. As far as we are concerned, to avoid this problem we have used the procedure proposed in [14], which has been employed, for example, in [30]. When implementing fuzzy regression we have obtained in a previous estimation that the !rst observation (a T-Bill) was clearly an outlier (it is clearly underpriced) because only constraint (8b) corresponding to this T-Bill was active while for !xed income assets 2–27 in Table 1 neither (8b) nor (8c) were active constraints. The consequence is that this !rst observation hinder the accomplishment
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327
0.075 interest rate
0.07 0.065 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
discount function
Fig. 1. Estimations of the spot rates and forward rates 29 June 2001 in Spanish public debt market.
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0
2
4
6
8
10 12 14 16 18 20 22 24 26 28 30 time (years)
lower bound
center
upper bound
Fig. 2. Fuzzy discount function for ∗ = 0:5.
of the fuzzy regression constraints with a reasonable cost, becoming the uncertainty of the system too high. Concretely, when we took * = 0.5, the value of the objective function was z = 709.61. So, for estimating the parameters of the fuzzy discount function in this subsection we have deleted the !rst observation and we have solved the model again. Table 2 shows the results when fuzzy regression is applied for the levels of inclusion ∗ = 0:5 and 0.75. To interpret the value of the objective function z, we should remember that we have taken 100 to be the principal of a bond and that the size of our sample is 27 bonds (26 after erasing the T-Bill). To build constraints (8d) and (8e) we assumed an annual periodicity since we will subsequently obtain only the implicit rates for this periodicity. So, the fuzzy discount function that we have found has the form: f˜ t = (1 − x) + a˜1 g1 (x) + a˜2 g2 (x) + a˜3 g3 (x) + a˜4 g4 (x) + a˜5 g5 (x); where the fuzzy parameters a˜i , i = 1; 2; : : : ; 5 are given by means of the TFNs in Table 2. Figs. 2 and 5 show the discount function that we !tted for the next 30 years. Figs. 3 and 6 show the shapes of the fuzzy TSIR that we obtained with fuzzy regression and Figs. 4 and 7 plot the 1 year forward rates for the next 30 years. We would like to point out that parameter ∗ can be
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interest rate
0.0725 0.065 0.0575 0.05 0.0425 0.035 0
2
4
6
8 10 12 14 16 18 20 22 24 26 28 30 time (years)
lower bound
center
upper bound
interest rate
Fig. 3. Fuzzy TSIR for ∗ = 0:5.
0.08 0.0725 0.065 0.0575 0.05 0.0425 0.035 0
2
4
6
8 10 12 14 16 18 20 22 24 26 28 30 time (years)
lower bound
center
upper bound
discount function
Fig. 4. Forward rates for ∗ = 0:5. 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0
2
4
6
8
10 12 14 16 18 20 22 24 26 28 30 time (years)
lower bound
center
upper bound
Fig. 5. Fuzzy discount function for ∗ = 0:75.
interpreted as an indicator of the perceived uncertainty in the market by the decision maker. If ∗ increases, then the uncertainty of the estimates of the explained variable (the price of the bonds) also increases (see [32]) and consequently the spread of the subsequent estimates of the spot rates and the forward rates will be wider.
interest rate
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0.08 0.0725 0.065 0.0575 0.05 0.0425 0.035 0
2
4
6
8 10 12 14 16 18 20 22 24 26 28 30 time (years)
lower bound
center
upper bound
interest ra
Fig. 6. Fuzzy TSIR for ∗ = 0:75.
0.1025 0.095 0.0875 0.08 0.0725 0.065 0.0575 0.05 0.0425 0.035 0
2
4
6
8 10 12 14 16 18 20 22 24 26 28 30 time (years)
lower bound
center
upper bound
Fig. 7. Forward rates for ∗ = 0:75.
As a conclusion of this subsection we would like to remark some of the advantages in this concrete case of our procedure for !tting the TSIR. Firstly, we have been able to estimate the TSIR using all the observed prices for the bonds of our sample (and so, all the available information). For example, it can be observed that the price of the 26th bond has been negotiated within the interval [103.722, 105.194]. With fuzzy regression we can take as the observed price the whole interval while when using conventional least squares we must take only a representative value of this interval and as a consequence, some information is lost. Secondly, any arithmetical manipulation is easier to implement maintaining the imprecision (and so, the information) of the embedded variables when they are fuzzy numbers (e.g. they are estimated from a fuzzy regression) rather than if they are random (e.g. they derive from a least-squares regression). Suppose that 29 June 2001 an investor wants to buy an Spanish public debt STRIP with facial value $100 payable in 3 years (remember that this asset does not belong to our sample) and wants to know what is the price to accept. This price will depend on the investor’s expectation about the future behaviour of interest rates, in such a way that if he uses the 1 year forward rates ˜ will to quantify this expectation and these rates are given by means of fuzzy numbers, the price, P, be the fuzzy number: P˜ =
100 ; (1 +1 ˜1 )(1 +1 ˜2 )(1 +1 ˜3 )
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which can be also obtained using the corresponding spot rate or the value of the fuzzy discount function: P˜ =
100 = 100f˜ 3 : (1 + i˜3 )3
Taking the parameters of the discount function obtained with the fuzzy regression where ∗ = 0:5 (see Table 2), it is straightforward to obtain f˜3 quanti!ed via the TFN (0.87430766, 0.00282102, 0.00407042). So, the bond price to accept is a TFN that we obtain simply by doing P˜ = 100 (0:87430766; 0:00282102; 0:00407042) = (87:43; 0:28; 0:41). 5. Conclusions In this paper, we have attempted to make up for the lack of explanations in the !nancial and actuarial literature about estimating interest rates with fuzzy numbers. When we price assets with a long maturity, we agree with several authors that it is a suitable way to model the uncertainty of the future interest rates using fuzzy numbers, especially when the !nancial 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 !xed income markets, these subjective estimates are included in the price of the debt instruments and, therefore, in the TSIR of these instruments. Our method quanti!es the experts’ subjective estimates of the spot rates and spot interest rates for the future (the forward rates) with fuzzy numbers. This method is based on a fuzzy regression technique and uses all the prices of the bonds negotiated during one session to !t 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 triangular fuzzy numbers. 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. Finally, let us point out a possible extension of the methodology presented in this article. In this paper, we have used an interval for representing an oscillating variable (the price of !xed income securities). For a future research, an alternative idea is to represent it by an asymmetric triangular fuzzy number. For example, let us assume that we have the following observations for the price: 101.500, 101.500, 101.750, 101, 101.500, 101.654, 101.375. Observe that in this case, the triangular fuzzy number (101.5, 0.5, 0.25) may be more appropriate than the interval 101:375; 0:375 . References [1] J.J Buckley, The fuzzy mathematics of !nance, Fuzzy Sets and Systems 21 (1987) 57–73. [2] J.J Buckley, Y. Qu, Solving linear and quadratic fuzzy equations, Fuzzy Sets and Systems 38 (1990) 43–59. [3] J.J Buckley, Y. Qu, On using -cuts to evaluate fuzzy equations, Fuzzy Sets and Systems 38 (1990) 309–312.
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Estimating a term structure of interest rates for fuzzy !nancial pricing by using fuzzy regression methods Jorge de Andr'es S'anchez∗ , Antonio Terce˜no G'omez Dept. de Gesti on de Empresas, Universitat Rovira i Virgili, Avinguda de la Universitat 1, 43204, Reus, Spain Received 16 January 2002; received in revised form 4 July 2002; accepted 22 July 2002
Abstract Several papers in the !nancial literature propose using fuzzy numbers to model interest rate uncertainty. However, in our opinion, the !rst problem to be solved is how to estimate these rates with fuzzy numbers. 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 the extension proposed by Ishibuchi and Nii (Fuzzy Sets and Systems 119 (2001) 273) of the fuzzy regression methods exposed in Savic and Predrycz (Fuzzy Regression Analysis, Physica-Verlag, Heidelberg, 1992, pp. 91–100), Tanaka (Fuzzy Sets and Systems 24 (1987) 363–375) and Tanaka and Ishibuchi (Fuzzy Regression Analysis, Physica-Verlag, Heidelberg, 1992, pp. 47). This method will enable to quantify the anticipated rates in the !xed income markets for the future with fuzzy numbers. In particular, we discuss how to estimate the TSIR with triangular fuzzy numbers because of the desirable properties of this kind of fuzzy numbers. c 2002 Elsevier B.V. All rights reserved. Keywords: Economics; Finance; Fuzzy regression; Temporal structure of interest rates
1. Introduction To obtain the !nancial price of any asset we have to discount the cash >ows 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 !nancial analyses are concerned with the medium and long term and, in our opinion, modelling the behaviour of interest rates in the long term by means of a stochastic model is not ∗
Corresponding author. Fax: +349-77759810. E-mail address: [email protected] (J. de Andr'es S'anchez).
c 2002 Elsevier B.V. All rights reserved. 0165-0114/03/$ - see front matter PII: S 0 1 6 5 - 0 1 1 4 ( 0 2 ) 0 0 3 7 3 - 1
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very realistic. 1 Gerber [11] points out that there is no commonly accepted stochastic model form making predictions about long-term discount rates. As we have only imprecise data or data related in a vague way with the behaviour of the interest rates along the pricing horizon, data such as the price of !xed income securities, the opinion of “experts” about the future behaviour of macroeconomic magnitudes, etc., many authors think that it is more suitable and realistic to make !nancial analyses in the long term using fuzzy estimates of the discount rates. For !nancial analysis, see [1,12,15,19,29] while for actuarial pricing, see [7,8,18,22,28]. However, the literature has not paid enough attention to how to estimate future discount rates with fuzzy numbers and, consequently, how to construct their characteristic 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. [10] 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 unadequate 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 temporal structure of interest rates (TSIR) with fuzzy subsets, since the TSIR implicitly contains the expectations of the !xed 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 [4]. We would also like to point out that it was in [22] where there is suggested for the !rst time using a fuzzy TSIR. 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 >ow payable in t years with the discount rate associated to that maturity. This rate is named as spot rate and we will symbolise it as it . So, the present value or 1 monetary unit payable in t years is (1 + it )−t . This expression is often called the discount function, which we will denote as ft , i.e. ft = (1 + it )−t , and it is clearly an exponential function of time. The implicit, or forward, rates, which standard !nancial theory interprets as the spot rates that the market expects to prevail in the future, are given implicitly by the TSIR. Then, from the spot rates it and it ∗ where t ∗ ¿t, the forward rate that has to be applied to obtain the price in t-years of a cash >ow that in that moment will be payable in t ∗ –t years (t ∗ −t t ) is obtained by solving: ft · (1 + t ∗ −t t )−(t so
ft t ∗ − t t = ft ∗
∗
−t)
1=(t ∗ −t)
= ft ∗
(1)
− 1;
where the Theory of Rational Expectations interpret t ∗ −t t 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 public 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: 1
The papers that model the 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 diNerential equations.
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315
(a) The estimates that are obtained after adjusting the coeOcients are not random variables, which are diOcult to manipulate in arithmetical operations, but fuzzy numbers, 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 fuzzy numbers 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 agents’ beliefs and expectations, which are highly subjective and vague. Fuzzy Set Theory, therefore, is a good way of treating this information. For example, the asset prices that are determined in the markets depend on the agents’ expectations of future in>ation, the issuers’ credibility, etc. We think that it is more realistic to consider that the bias between the 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 con!dence intervals. For instance, the price of a !nancial 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). The structure of the paper is as follows. In Section 2, we shall describe some aspects of fuzzy arithmetic and the extension [13] of the fuzzy regression methods described in [25–27]. In Section 3, we shall discuss some basic characteristics of the models that attempt to adjust the TSIR through the discount function de!ned 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 A fuzzy number (FN) is a fuzzy subset A˜ de!ned over real numbers. It is the main instrument used in Fuzzy Set Theory for quantifying uncertain quantities (e.g. in !nancial analysis, the discount rates). Two properties are required for a FN. The !rst one is that it must be a normal fuzzy set, i.e. sup ∀x∈X A˜ (x) = 1. The second is that it must be convex (i.e., its -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 symbolise a TFN A˜ as A˜ = (a; la ; ra ) where a is the centre and la and ra are the left and right spreads, respectively. For example, a sentence by a !nancial analyst such as “I expect for the next 2 years the free risk rate to be around 5% and deviations no longer that the 1%” may be quanti!ed in a very natural way as (0.05, 0.01, 0.01).
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A TFN is characterised by x−a+l a ; la A (x) = a + ra − x ; ra 0
its membership function A (x) or, alternatively, by its -cuts, A , as a − la ¡ x 6 a; a ¡ x 6 a + ra otherwise;
and R = [a − la (1 − ); a + ra (1 − )]: A = A( ); A( ) In many !nancial analyses it is often necessary to evaluate functions (e.g. the net present value), which in a general way we shall name y = f(x1 ; x2 ; : : : ; xn ). Then, if x1 ; x2 ; : : : ; xn are not crisp numbers but are the FNs A˜1 ; A˜2 ; : : : ; A˜n ; f(·) induces a FN B˜ = f(A˜1 ; A˜2 ; : : : ; A˜n ) whose characteristic function must be obtained from Zadeh’s extension principle. Unfortunately, it is often impossible to ˜ although in many cases it is possible obtain a closed expression for the membership function of B, to obtain its -cuts, B , from A1 ; A2 ; : : : ; An , by doing B = f(A˜ 1 ; A˜ 2 ; : : : ; A˜ n ) = f(A1 ; A2 ; : : : ; An ): In !nancial 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 -cuts of B. If the function f(·) that induces B˜ is increasing with respect to the !rst m variables, where m6n, and decreasing with respect to the last n − m variables, in [3] it is demonstrated that B is R B = [B( ); B( )] = f(A1 ( ); : : : ; Am ( ); Am+1 ( ); : : : ; An ( )); f(A1 ( ); : : : ; Am ( ); Am+1 ( ); : : : ; An ( )):
(2)
˜ will be another It is well known that the result of evaluating a 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 approximate B˜ using a TFN that we shall denote by B˜ . If the function that we evaluate, f(·), is increasing with respect to the !rst m variables, where m6n, and decreasing with respect to the others, i.e. the -cuts of B˜ are (2), in [9] it is proposed to approximate B˜ with a FN B˜ = (b; lB ; rB ) by using Taylor’s expansion. Concretely: b = f(aC );
rb =
m n @f(aC ) @f(aC ) lb = lai − ra i ; @xi @xi i=1 i=m+1
m n @f(aC ) @f(aC ) ra i − lai ; @xi @xi i=1 i=m+1
(3)
where aC = (a1 ; : : : ; am ; am+1 ; : : : ; an ). Let us illustrate how the approximating procedure (3) can be used by applying it to the present value, V , of an amount C payable in t years with a pricing interest rate i, i.e: V = C · (1 + i)−t :
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317
Taking the maturity as a !xed parameter, it is easy to check that V is increasing (decreasing) respect to C (i) given that their partial derivatives are: @V = (1 + i)−t @C
and
@V = −t · C · (1 + i)−(t+1) : @i
Suppose now that the amount and the discount rate are given by means of TFNs, being C˜ = (C; lC ; ˜ will be the fuzzy rC ) and i˜ = (i; li ; ri ), respectively. In this case, the present value of the amount, C, number V˜ , which is obtained by evaluating: V˜ = C˜ · (1 + i˜)−t : Clearly, V˜ is not a TFN because it is obtained from a non-linear combination of two TFN. However, by using (3) we can approximate V˜ with the TFN (V; lV ; rV ), where: V = C · (1 + i)−t ;
lV =
lC t · C · ri + (1 + i)t (1 + i)t+1
and
rV =
rC t · C · li + : (1 + i)t (1 + i)t+1
˜ In this paper, to obtain the forward rates we will need to solve fuzzy equations with the structure A· + ˜ ˜ ˜ ˜ X = B where A and B are TFNs with the support contained in − 0. The above equation can be expressed through the -cuts of A;˜ X˜ and B˜ as A · X = B , i.e.: [a − la (1 − ); a + ra (1 − )] · X ( ); XR ( ) = [b − lb (1 − ); b + rb (1 − )]:
(4)
According to [16], the narrowest solution is X ( ) =
b − lb (1 − ) ; a − la (1 − )
b + rb (1 − ) XR ( ) = : a + ra (1 − )
(5)
We must take into account that this solution does not always exist ∀ ∈ [0; 1]. For example, if A0:5 = [1; 2] and B0:5 = [1; 1:5], then X0:5 = [1; 34 ], which is not an -cut of a FN. Nevertheless, in [2] it is demonstrated that the required condition for the existence of (5) ∀ ∈ [0; 1] is: la lb ¿ ; b a
rb ra ¿ : b a
(6)
2.2. Fuzzy regression model with asymmetric coe4cients In this subsection, we will describe Ishibuchi and Nii’s extension of the fuzzy regression models developed in [25–27], which is one of the most widely used models in economic applications (see for example [17,23,24]). 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 or fuzzy numbers. 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 {(Y1 ; X1 ); (Y2 ; X2 ); : : : ; (Yj ; Xj ); : : : ; (Yn ; Xn )} where Xj is the jth observation of the explanatory variable, Xj = (X0j ; X1j ; X2j ; : : : ; Xij ; : : : ; Xmj ). Moreover, X0j = 1 ∀j, and Xij is the observed
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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 = 1; 2; : : : ; n and may either be a crisp value or a con!dence interval. In both cases, it can be represented through its centre and its spread as Yj = YjC ; YjR , where YjC is the centre and YjR is the radius. In particular, we must estimate the following fuzzy linear function: Y˜ j = A˜ 0 + A˜ 1 X1j + · · · + A˜ m Xmj ; where Y˜j is the estimation of Yj by a FN after adjusting A˜0 ; A˜1 ; : : : ; A˜m . Likewise, in fuzzy regression models the disturbance is not introduced as a random addend in the linear relation, but it is incorporated into the coeOcients A˜i , i = 0; 1; : : : ; m. Of course, the !nal objective is to adjust the fuzzy numbers A˜i , i = 0; 1; : : : ; m from the available sample. In this paper, we propose to model the parameters A˜i as TFN, i.e., they can be written as A˜i = (ai ; lai ; rai ); i = 0; 1; : : : ; m. Then, the estimate of Yj , Y˜j , is a TFN that we will name Y˜j = (Yj ; lYj ; rYj ), where Y j = a0 +
m
ai Xij ;
lYj = la0 +
i=1
r Y j = ra 0 +
m
lai |Xij | +
i=1
rai |Xij | +
i=1 xij ¿0
m
rai |Xij |;
i=1 xij ¡0
xij ¿0
m
m
lai |Xij |:
i=1 xij ¡0
To !t A˜0 ; A˜1 ; : : : ; A˜m we !rst have to determine the cores a0 , a1 ; : : : ; am by using least squares and taking the centres of the observations Yj = 1; 2; : : : ; n as the realisations of the independent variable. These estimates are named aˆ0 ; aˆ1 ; : : : ; aˆm . Subsequently, we have to determine the parameters lai and rai , which must minimise the uncertainty of the estimates for Yj , Y˜j , (i.e. their spreads), and simultaneously maximise the congruence of the estimates obtained for Yj which will be measured as the level in which Yj is contained within Y˜j ; (Yj ⊆ Y˜j ). In particular, we must solve the following multiple objective programme: Minimise z
=
m n
lai |Xij | +
j=1 i=0
m n
rai |Xij |;
j=1 i=0
Maximise subject to (Yj ⊆ Y˜ j ) ¿ ;
j = 1; 2; : : : ; n; lai ; rai ¿ 0; i = 0; 1; : : : ; m;
∈ [0; 1):
If for the second objective we require a minimum accomplishment level ∗ , j = 1; 2; : : : ; n, the above programme is transformed into the following linear one: Minimise z
=
m n j=1 i=0
lai |Xij | +
m n j=1 i=0
rai |Xij |;
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319
subject to aˆ0 +
m i=1
m m aˆi Xij − la0 + la0 |Xij | + rai |Xij | (1 − ∗ ) 6 YjC − YjR ; j = 1; 2; : : : ; n; i=1 i=1 xij ¿0
xij ¡0
aˆ0 +
m i=1
m m ∗ aˆi Xij + ra0 + rai |Xij | + lai |Xij | (1 − ) ¿ YjC + YjR ; j = 1; 2; : : : ; n; i=1 i=1 xij ¿0
lai ; rai ¿ 0;
xij ¡0
i = 0; 1; : : : ; m:
Note that in the last mathematical programme the !rst two blocks of constraints are consequences of requirement (Yj ⊆ Y˜j )¿ ∗ while the last block ensures that lai ; rai ∀i will be non-negative.
3. Estimating the term structure of interest rates using splines and econometric methods It is fairly straightforward to estimate the TSIR for a speci!c date and a given market if there are many suOciently liquid zero coupon bonds and their price can be observed without perturbations. However, these conditions are rarely found simultaneously in the !xed income markets. This section, then, 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 zero coupon 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 = 1; 2; : : : ; K, and K is the number of available bonds, provides several cash->ows (coupons and principal). These are denoted by (C1k ; t1k ); (C2k ; t2k ); : : : ; (Cnkk ; tnkk ), where Cik is the amount of the ith cash->ow and tik is its maturity expressed in years and nk is the number of cash->ows provided by this bond. If we suppose that the bonds have no embedded option (e.g. convertibility) the price of the kth bond is the sum of the discounted value of every amount with the corresponding spot rate: Pk =
nk i=1
Cik ftik ;
where ftik is the discounted value of one dollar with maturity tik years. Of course, we should then de!ne an analytical form for the discount function in order to specify the 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 ones described in [20,21] (quadratic and cubic splines) and in [31] (cubic exponential splines). These methods assume
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that the discount function is a linear combination of m + 1 functions of time, i.e.: ft =
m
aj gj (t):
j=0
We assume that gj (t) are splines and not simply polynomial functions because, with splines, we can determine gj (t) according to the distribution of the maturity dates of the sample and, therefore, !t the discount function better for the most common maturates. Moreover, with splines we can obtain TSIR pro!les that do not >uctuate very much and we ensure that forward rates do not behave explosively. In our paper, we will assume the functions gj (t) are cubic exponential splines. Concretely, gj (·), j = 0; 1; : : : ; m will be cubic splines respect to the argument x, where x = 1 − e−!·t ; where !¿0: Then, x = 0 for t = 0 and x = 1 when t → ∞. Finally, despite the discount function is, in essence, a polynomial function of x, we attain that the discount function to adjust will be approximately an exponential function of time (let us remark that actually ft = (1 + it )−t ): ft = "0 + "1 e−!·t + "2 e−2!·t + "3 e−3!·t : Then, we can deduce that the following equation must be estimated: Pk =
nk i=1
Cik
m
aj gj (xik ) + ”k ;
(7)
j=0 k
where xik = 1 − e−!·ti . The estimation of the parameters (a0 ; a1 ; : : : ; am ) in (7) corresponds to a problem of conventional linear regression if the parameter ! is !xed beforehand. However, the penultimate step will be to determine the optimal value of !, that will permit minimise the sum of the square residuals. As it is pointed out in [31] it can be done by use of numerical procedures, such as the three-point Newton minimisation method. The last step to implement is to obtain, again, the estimate of the vector (a0 ; a1 ; : : : ; am ) with the optimum !. Likewise, a random disturbance is justi!ed because the formation of the prices of !xed income securities is disturbed by several factors. In [5] there are outlined some of them; for example, the coupon bearing of bonds with maturities greater than one contain information about more than one present value coeOcient; the bond portfolios are not continuously rebalanced, so at any moment each bond can deviate by an uncertain 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. In our opinion, these reasons justify better the use of fuzzy regression than conventional regression methods when adjusting the TSIR. 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 the analysed framework in Section 3 and the fuzzy regression method presented
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321
above. Subsequently, we shall discuss how to estimate the spot and the forward rates using fuzzy numbers. We shall end this section by making an empirical application to the Spanish public debt market. 4.1. Estimating the discount factor with Triangular Fuzzy Numbers (TFNs) Fitting the TSIR with a fuzzy regression method, we can take all the prices traded on 1 day as the price of the bonds, not just the average price. So, this instrument is attractive because the observed price in one session for any bond is often not 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 = Pk C ; Pk R i.e. the minimum price negotiated during the analysed session is Pk C − Pk R while the maximum is Pk C + Pk R . For example, if the traded price for one bond on 1 day >uctuates between 100 and 103, it will be expressed as 101:5; 1:5 . To !t the discount function, we assume that it is quanti!ed 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 = (ft ; lft ; lft ); t ¿ 0;
0 6 ft − lft 6 ft 6 ft + rft 6 1:
So, the predicted price of the kth bond with the triangular discount function can be written as P˜ k =
nk i=1
Cik f˜ tik
and this implies that P˜ k = (PCk ; lPk ; rPk ) =
nk i=1
Cik (ftik ; lftk ; rftk ): i
i
Likewise, we will construct the discount function from a linear combination of m + 1 piecewise polynomials gj (x), j = 0; 1; : : : ; m where x = 1 − e−!·t and !¿0, with images in + , continuously diNerentiable, and whose parameters are given by TFNs. In this way, these parameters can be represented as a˜j = (aj ; laj ; raj )laj ;
raj ¿ 0;
j = 0; 1; : : : ; m:
So the discount function is f˜ t =
m
a˜j gj (x) =
m
j=0
(aj ; laj ; raj )gj (x) =
j=0
m
(ajC gj (x); laj gj (x); raj gj (x));
j=0
i.e. the discount function is approximately a fuzzy exponential function of time: f˜t = "˜ 0 + "˜ 1 e−!·t + "˜ 2 e−2!t + "˜ 3 e−3!t where the parameters "˜ 0 ; : : : ; "˜ 3 will be TFNs. Thus, the estimated price of the kth bond, P˜k , is P˜ k = (PCk ; lPk ; rPk ) =
nk i=1
Cik
m j=0
(aj ; laj ; raj )gj (xik ):
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J. de Andr es S anchez, A.T. G omez / Fuzzy Sets and Systems 139 (2003) 313 – 331
Since the value of the discount function for t = 0 (x = 0) should be the crisp number 1, then f˜0 = (f0 ; lf0 ; rf0 ) = (1; 0; 0). This condition is met if a˜0 = (a0 ; la0 ; ra0 ) = (1; 0; 0), g0 (0) = (1 − x) and gj (0) = 0; j = 1; 2; : : : ; m. So, we can express the price of the kth bond as: P˜ k = (PCk ; lPk ; rPk ) =
nk
Cik (1 − xik ) +
i=1
=
nk
Cik (1
−
xik )
i=1
+
m
(aj ; laj ; raj )gj (xik )
j=1
nk
Cik
i=1
m
(aj ; laj ; raj )gj (xik )
j=1
and, using fuzzy arithmetic, we !nally write: (PCk ; lPk ; rPk )
−
nk
Cik (1
−
xik )
i=1
=
m
(aj ; laj ; raj )
j=1
nk
Cik gj (xik ):
i=1
In this way, by identifying Y˜k = (YCk ; lYk ; rYk ) = (PCk ; lPk ; rPk ) − YCk
=
PCk
−
nk
Cik (1 − xik );
lYk = lPk ;
nk
i=1
Cik (1 − xik ), we obtain
rYk = rPk ;
i=1
So, it is easy to that the value of the jth explanatory variable for the kth bond is the verify k crisp value Xjk = ni=1 Cik gj (xik ) . So, the expression to be adjusted by a fuzzy regression method is simply expressed as (YCk ; lYk ; rYk ) = (a1 ; la1 ; ra1 )X1k + (a2 ; la2 ; ra2 )X2k + · · · + (am ; lam ; ram )Xmk since Y˜k is the prediction with fuzzy regression of the observation of price of the kth bond minus the sum of the cash->ows provided by the asset. us remark that the observed value of this variable Let k for the kth bond is Yk = Yk C ; Yk R = Pk − ni=1 Cik (1 − xik ) and so: Y k C = Pk C −
nk
Cik (1 − xik )
and
Yk R = Pk R :
i=1
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 = 1; 2; : : : ; K only Pk C . These estimates are symbolised as aˆ1 ; aˆ2 ; : : : ; aˆm . Moreover, we must to obtain the estimate of !, which will be named !ˆ, by using numerical methods. Subsequently, we have to obtain the spreads of the parameters a˜j ; j = 1; 2; : : : ; m by solving a linear programme. To do it, we must !rst choose a level ∗ of accomplishment of the congruency of the observations Yk with their predicted value Y˜k ; k = 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 (5) of Eq. (4). These constraints will be the consequence of conditions (6). To
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323
conclude, we must solve Minimise laj ;raj j = 1;2;:::;m
subject to m
aˆj
j=1
nk
z=
m j=1
laj
nk K
Cik gj (xik ) +
k=1 i=1
m j=1
raj
nk K
Cik gj (xik )
nk m Cik gj (xik ) − la j Cik gj (xik ) (1 − ∗ ) 6 Yk C − Yk R ;
i=1
j=1
i=1
k = 1; 2; : : : ; K;
m j=1
aˆj
nk
(8a)
k=1 i=1
(8b)
nk m Cik gj (xik ) + ra j Cik gj (xik ) (1 − ∗ ) ¿ Yk C + Yk R ;
i=1
j=1
i=1
k = 1; 2; : : : ; K;
(8c)
m ˆ −!(s+1)P ˆ laj gj (1 − e−!sP ) ) j=1 laj gj (1 − e − 6 0; m m ˆ + −!sP ˆ ) ˆ ˆ e−!sP e−!(s+1)P + j=1 aˆj gj (1 − e−!(s+1)P ) j=1 aˆj gj (1 − e m
j=1
s = 1; : : : ; u − 1;
(8d)
m ˆ −!(s+1)P ˆ raj gj (1 − e−!sP ) ) j=1 raj gj (1 − e − 6 0; m m ˆ + −!sP ˆ ) ˆ ˆ e−!sP e−!(s+1)P + j=1 aˆj gj (1 − e−!(s+1)P ) j=1 aˆj gj (1 − e m
j=1
s = 1; : : : ; u − 1; laj ; raj ¿ 0;
j = 1; 2; : : : ; m:
(8e) (8f)
Constraints (8b), (8c) and (8f) correspond to Ishibuchi and Nii’s regression model. Likewise, (8d) and (8e) ensure that the solution for the forward rates in the sense of (5) exists for an arbitrary periodicity P (in years), which is the periodicity of the forward rates that we would like to obtain. Therefore, uP is the greatest maturity at which we want to determine the spot and forward rates. 4.2. Estimating the spot rates and the forward rates with TFN The value of the discount function in t, ft , is obtained from its corresponding spot rate it making ft = (1 + it )−t . Then, the spot rate can be obtained by evaluating the expression it = (ft )−1=t − 1. So,
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if the discount function is a TFN f˜t = (ft ; lf; rf), the spot rate will be a FN i˜t whose -cuts, it , are, from (2): it = [it ( ); it ( )] = [(ft + rft (1 − ))−1=t − 1; (ft − lft (1 − ))−1=t − 1]: It is easy to check that the spot rate is not a TFN. However, from (3), we can deduce that the following TFN can approximate well the spot rate associated to maturity t: r ft lf t 1 : − 1; ; it ≈ (ft )1=t t · (ft )(t+1)=t t · (ft )(t+1)=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 t , t = 1; : : : ; u. Although annual periods are used to deduce these rates, for other periodicities the discount function that we have !tted can be used because it is de!ned for any value of t. To obtain the forward rate 1 ˜t , we have to solve the fuzzy version of (1) f˜ t −1 (1 +1 ˜t )−1 = f˜ t : To solve this equation, we identify (1+1 ˜t )−1 = G˜ t , where G˜ t is the value in t −1 of one monetary unit payable at t according to the TSIR. The above equation can therefore be written using the -cuts as: [ft −1 − lft−1 (1 − ); ft −1 + rft−1 (1 − )] · [Gt ( ); Gt ( )] = [ft − lft (1 − ); ft + rft (1 − )] and it can be solved using (5) because constraints (8d) and (8e) ensure condition (6): ft + rft (1 − ) ft − lft (1 − ) ; : Gt = [Gt ( ); Gt ( )] = ft −1 − lft−1 (1 − ) ft −1 + rft−1 (1 − ) Then, to obtain the -cuts of the 1 year forward rates for the tth year (1 ˜t ); 1 t , we must only take into account that under non-fuzziness Gt = (1+1t )−1 , i.e. 1t = (Gt )−1 − 1 = (ft −1 =ft ) − 1. Then ft −1 − lft−1 (1 − ) ft −1 + rft−1 (1 − ) − 1; −1 : 1 t = [1 t ( ); 1 t ( )] = ft + rft (1 − ) ft − lft (1 − )
Obviously, 1 ˜t is not a TFN since the extremes of its -cuts are not linear functions of . However, the upper (lower) extreme of 1 t is obtained from the lower (upper) extreme of the discount functions for 1 monetary unit payable in t − 1 years and in t years. Then, we can approximate 1 ˜t with (3), and the result that we obtain is ft −1 · rft − ft · rft−1 ft −1 · lft − ft · lft−1 ft −1 ; ˜t ≈ − 1; ; 1 (ft )2 (ft )2 ft where ft −1 · rf − ft · rf−1 ¿ 0 and ft −1 · lf − ft · lf−1 ¿ 0 due to constraints (8d) and (8e).
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Table 1 Fixed income securities negotiated in the Spanish public debt market 29 June 2001 k
Asset
Coupon (annual) (%)
Maturity (days)
Maturity (years)
k Pmin
k Pmax
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
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
18 382 391 521 576 576 576 756 757 1122 1216 1307 1490 1673 1945 2098 2402 2775 2948 3134 3680 3771 4229 4230 4774 10073 11351
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
4.3. Empirical application In this subsection, we use our methodology to estimate the TSIR in the Spanish public debt market on 29 June 2001 with our methodology. The bonds in our sample are shown in Table 1. To !t the TSIR on this date, we de!ned the discount function with Vasicek and Fong’s cubic exponential splines, that we de!ne as the cubic splines of the argument x = 1 − e−!·t de!ned by McCulloch in [20]. We took m = 5, and the knots we used to construct the splines were d1 = 0 years, d2 = 1:5 years, d3 = 5 years and d4 = 1000 years (i.e. ∞ years). Taking this knots, we can distinguish independently the shapes of the TSIR for the cash->ows maturities corresponding to the “short term” (from 0 to 1.5 years), “medium term” (from 1.5 to 5 years) and the “long term” (more than 5 years). Likewise, the knots expressed with the transformed variable x are y1 = 0, y2 = 0:14494, y3 = 0:40664 and y4 = 1 because, we have obtained the optimal ! equal to 0.10439.
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Table 2 Results of fuzzy regression for ∗ = 0:5 and 0.75
aj = aˆj
=∗ 0:5, z = 65:92 l aj raj
a˜1
a˜2
a˜3
a˜4
a˜5
−4:9448
0.3007
−1:0652
−4:6033
0.7870
∗ = 0:75, z = 131:85 l aj raj
0 0
0 0
0.2204 0
0 0.0396
0.0095 0.0151
0 0
0 0
0.4408 0
0 0.0792
0.0190 0.0303
Then, the functions gj (t), j = 1; : : : ; 5 are 0; (x−yj−1 )3 ; 6(yj −yj−1 ) (y −y )(x−yj ) (yj −yj−1 ) + j j−21 gj (x) = 6 (x−y )2 (x−y )3 + 2 j − 6(yj+1 −j yj ) ; 2y −y −y (yj+1 − yj−1 )[ j+1 6j j−1 +
x 6 yj − 1 ; yj−1 ¡ x 6 yj ; j = 1; : : : ; 4: yj ¡ x 6 yj+1 ; x−yj+1 ]; 2
yj+1 ¡ x ¡ 1;
g5 (x) = x ∀x: The result that we obtain when ordinary least squares is used, Pk C is taken as the observed price for the kth bond and with x = 1 − e−0:10439·t , is ft = (1 − x) + a1 g1 (x) + a2 g2 (x) + a3 g3 (x) + a4 g4 (x) + a5 g5 (x); where the values of ai , i = 1; 2; : : : ; 5 are the centres of TFNs in Table 2. Moreover, we point out that the determination coeOcient is close to 100% (exactly, 99.83%). The spot rates and 1 year forward rates for the next 30 years are given in the Fig. 1 of the annex. It is well known that the model of fuzzy regression we are using is very sensitive to outliers. The consequence of its existence is to obtain a fuzzy shape too wide (uncertain) and unrealistic. Of course, the outliers are the data that is around the model’s upper (lower) bound while the great proportion of the data is near of the core of the fuzzy mapping that we have obtained or around of its lower (upper) bound. In [6] there is a wide survey of methods to use in this circumstances. As far as we are concerned, to avoid this problem we have used the procedure proposed in [14], which has been employed, for example, in [30]. When implementing fuzzy regression we have obtained in a previous estimation that the !rst observation (a T-Bill) was clearly an outlier (it is clearly underpriced) because only constraint (8b) corresponding to this T-Bill was active while for !xed income assets 2–27 in Table 1 neither (8b) nor (8c) were active constraints. The consequence is that this !rst observation hinder the accomplishment
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0.075 interest rate
0.07 0.065 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
discount function
Fig. 1. Estimations of the spot rates and forward rates 29 June 2001 in Spanish public debt market.
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0
2
4
6
8
10 12 14 16 18 20 22 24 26 28 30 time (years)
lower bound
center
upper bound
Fig. 2. Fuzzy discount function for ∗ = 0:5.
of the fuzzy regression constraints with a reasonable cost, becoming the uncertainty of the system too high. Concretely, when we took * = 0.5, the value of the objective function was z = 709.61. So, for estimating the parameters of the fuzzy discount function in this subsection we have deleted the !rst observation and we have solved the model again. Table 2 shows the results when fuzzy regression is applied for the levels of inclusion ∗ = 0:5 and 0.75. To interpret the value of the objective function z, we should remember that we have taken 100 to be the principal of a bond and that the size of our sample is 27 bonds (26 after erasing the T-Bill). To build constraints (8d) and (8e) we assumed an annual periodicity since we will subsequently obtain only the implicit rates for this periodicity. So, the fuzzy discount function that we have found has the form: f˜ t = (1 − x) + a˜1 g1 (x) + a˜2 g2 (x) + a˜3 g3 (x) + a˜4 g4 (x) + a˜5 g5 (x); where the fuzzy parameters a˜i , i = 1; 2; : : : ; 5 are given by means of the TFNs in Table 2. Figs. 2 and 5 show the discount function that we !tted for the next 30 years. Figs. 3 and 6 show the shapes of the fuzzy TSIR that we obtained with fuzzy regression and Figs. 4 and 7 plot the 1 year forward rates for the next 30 years. We would like to point out that parameter ∗ can be
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interest rate
0.0725 0.065 0.0575 0.05 0.0425 0.035 0
2
4
6
8 10 12 14 16 18 20 22 24 26 28 30 time (years)
lower bound
center
upper bound
interest rate
Fig. 3. Fuzzy TSIR for ∗ = 0:5.
0.08 0.0725 0.065 0.0575 0.05 0.0425 0.035 0
2
4
6
8 10 12 14 16 18 20 22 24 26 28 30 time (years)
lower bound
center
upper bound
discount function
Fig. 4. Forward rates for ∗ = 0:5. 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0
2
4
6
8
10 12 14 16 18 20 22 24 26 28 30 time (years)
lower bound
center
upper bound
Fig. 5. Fuzzy discount function for ∗ = 0:75.
interpreted as an indicator of the perceived uncertainty in the market by the decision maker. If ∗ increases, then the uncertainty of the estimates of the explained variable (the price of the bonds) also increases (see [32]) and consequently the spread of the subsequent estimates of the spot rates and the forward rates will be wider.
interest rate
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0.08 0.0725 0.065 0.0575 0.05 0.0425 0.035 0
2
4
6
8 10 12 14 16 18 20 22 24 26 28 30 time (years)
lower bound
center
upper bound
interest ra
Fig. 6. Fuzzy TSIR for ∗ = 0:75.
0.1025 0.095 0.0875 0.08 0.0725 0.065 0.0575 0.05 0.0425 0.035 0
2
4
6
8 10 12 14 16 18 20 22 24 26 28 30 time (years)
lower bound
center
upper bound
Fig. 7. Forward rates for ∗ = 0:75.
As a conclusion of this subsection we would like to remark some of the advantages in this concrete case of our procedure for !tting the TSIR. Firstly, we have been able to estimate the TSIR using all the observed prices for the bonds of our sample (and so, all the available information). For example, it can be observed that the price of the 26th bond has been negotiated within the interval [103.722, 105.194]. With fuzzy regression we can take as the observed price the whole interval while when using conventional least squares we must take only a representative value of this interval and as a consequence, some information is lost. Secondly, any arithmetical manipulation is easier to implement maintaining the imprecision (and so, the information) of the embedded variables when they are fuzzy numbers (e.g. they are estimated from a fuzzy regression) rather than if they are random (e.g. they derive from a least-squares regression). Suppose that 29 June 2001 an investor wants to buy an Spanish public debt STRIP with facial value $100 payable in 3 years (remember that this asset does not belong to our sample) and wants to know what is the price to accept. This price will depend on the investor’s expectation about the future behaviour of interest rates, in such a way that if he uses the 1 year forward rates ˜ will to quantify this expectation and these rates are given by means of fuzzy numbers, the price, P, be the fuzzy number: P˜ =
100 ; (1 +1 ˜1 )(1 +1 ˜2 )(1 +1 ˜3 )
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which can be also obtained using the corresponding spot rate or the value of the fuzzy discount function: P˜ =
100 = 100f˜ 3 : (1 + i˜3 )3
Taking the parameters of the discount function obtained with the fuzzy regression where ∗ = 0:5 (see Table 2), it is straightforward to obtain f˜3 quanti!ed via the TFN (0.87430766, 0.00282102, 0.00407042). So, the bond price to accept is a TFN that we obtain simply by doing P˜ = 100 (0:87430766; 0:00282102; 0:00407042) = (87:43; 0:28; 0:41). 5. Conclusions In this paper, we have attempted to make up for the lack of explanations in the !nancial and actuarial literature about estimating interest rates with fuzzy numbers. When we price assets with a long maturity, we agree with several authors that it is a suitable way to model the uncertainty of the future interest rates using fuzzy numbers, especially when the !nancial 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 !xed income markets, these subjective estimates are included in the price of the debt instruments and, therefore, in the TSIR of these instruments. Our method quanti!es the experts’ subjective estimates of the spot rates and spot interest rates for the future (the forward rates) with fuzzy numbers. This method is based on a fuzzy regression technique and uses all the prices of the bonds negotiated during one session to !t 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 triangular fuzzy numbers. 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. Finally, let us point out a possible extension of the methodology presented in this article. In this paper, we have used an interval for representing an oscillating variable (the price of !xed income securities). For a future research, an alternative idea is to represent it by an asymmetric triangular fuzzy number. For example, let us assume that we have the following observations for the price: 101.500, 101.500, 101.750, 101, 101.500, 101.654, 101.375. Observe that in this case, the triangular fuzzy number (101.5, 0.5, 0.25) may be more appropriate than the interval 101:375; 0:375 . References [1] J.J Buckley, The fuzzy mathematics of !nance, Fuzzy Sets and Systems 21 (1987) 57–73. [2] J.J Buckley, Y. Qu, Solving linear and quadratic fuzzy equations, Fuzzy Sets and Systems 38 (1990) 43–59. [3] J.J Buckley, Y. Qu, On using -cuts to evaluate fuzzy equations, Fuzzy Sets and Systems 38 (1990) 309–312.
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