estimating the cox, ingersoll and ross model of the term structure: a multivariate approach

Ricehe Economiche (1995) 49,51-74

Estimating the Cox, Ingersoll and Ross model of the term structure: a multivariate approach

ANDREA BERAFDI

London Business School, Sussex Place, Regents Park, London NW1 4SA, U.K., and Banca Commerciale Italianu, Economic Research Department, via Borgonuovo, 2, 20121 Milan, Italy

Summary

In this paper we suggest a new methodology to estimate the Cox, Ingersoll and Ross model of the term structure. The approach is based on a multivariate non-linear least squares procedure, which allows us to simultaneously take into account the cross-sectional relations which exist among bond prices at each instant of time and the dynamics of each bond price over time. The methodology involves the use of a fairly simple econometric specification and is developed to deal with both the case of independently and identically distributed error terms and the case of autocorrelated error terms. We estimate and test the model using nominal prices of Italian Treasury bonds.

J.E.L. Classification: C33, E43, G12. Keywords: Term structure, bond pricing, multivariate non-linear models.

1. Introduction

The most recent contributions in modelling the term structure of interest rates might be grouped into two categories: the no-arbitrage approach and the general equilibrium approach. The for- mert is essentially based on the no-arbitrage argument developed in the continuous time model of option pricing advanced by Black and Scholes (1973). The latter has its grounds in the theory proposed by Cox et aZ. (CIR; 1985a,b).

The CIR model of the term structure is based on an intertemporal general equilibrium framework of asset prices, which allows one

7 In the no-arbitrage category we may include, among the others, Vasicek (1977), Bremmn and Schwartz (1979), Schaefer and Schwartz (19&Q, Ho and Lee (1986) and Heath et al. (1992).

51 0035-5054#95/010051+24 $oS.ocYo 0 1995 University of Venice

52 A. BERARDI

to obtain endogenously the dynamics of the instantaneous riskless interest rate (the only state variable), the functional form of the risk premia, and a closed-form solution for prices of real unit discount bonds. The relation between the price of a unit discount bond, its maturity and the short rate is highly non-linear, a feature which gives rise to many problems in testing the theory. Different econometric techniques have been proposed to estimate the parameters of the model.

A non-linear least squares procedure has been applied to cross- sections of bond prices by Brown and Dybvig (1986), Brown and Schaefer (1983,1994cz), and Barone et al. (1991). The method gives as a result a time-series of estimates of the unknown parameters, Hence, the cross-sectional form does not allow one to obtain a constant relation between the variables over the total observation period, since it is not able to take into account the temporal information contained in bond prices data.

An alternative estimation approach, based on the generalized method of moments (GM&Q, has been used by Gibbons and Ramaswamy (1986, 1993), Longstaff (1989) and Longstaff and Schwartz (1992). The econometric specification of the model is constructed by placing some restrictions on the moments of bond yields. The GMM parameter estimates possess some valuable properties (in particular, they are consistent even if there exist conditional heteroskedasticity and serial or cross correlation in the error terms). However, the procedure might give rise to some difficulties in valuing long-term coupon bearing bonds.

A maximum likelihood approach to estimate the CIR model has been proposed by Pearson and Sun (1988, 1994) and Chen and Scott (1993). The econometric specification combines time series information, derived from the joint conditional density of the unobservable state variables,t and cross-sectional information, given by the disturbances added to the theoretical values of bond yields. The application of this procedure, which seems to be the most efficient among those advanced in the empirical literature, may be rather complicated, since it involves the estimation of a modified Bessel function.

In this paper we suggest a new methodology to estimate the CIR model, which is based on a multivariate non-linear least squares procedure. The multivariate approach allows us to simultaneously take into account both the cross-sectional relations which exist among bond prices at each instant of time and the dynamics of each bond price over the sample period. This feature gives to the method a high degree of efficiency in the estimation of the model

t Both Pearson and Sun (1988, 1994) and Chen and Scott (1993) estimate multifactor versions of the CIR model.

THE COX, INGERSOLL AND ROSS MODEL 53

parameters. The procedure is developed for both the case of independently and identically distributed error terms and the case of autocorrelated error terms.

The paper proceeds as follows. In Section 2 we briefly describe the CIR model of the term structure. In Section 3 we develop the econometric methodology and in Section 4 we apply the estimation procedure to the CIRmodel using nominal prices of Italian Treasury bonds. Finally, Section 5 contains some concluding remarks.

2. Theoretical model

The CIR general equilibrium model is a continuous time framework, which determines the equilibrium price of any financial asset and its dynamics. The economy is composed by a fixed number of individuals, who have identical endowments and preferences. They have rational expectations and maximize the expected value of a Von Neumann-Morgenstern utility function. There exists a single physical good in the economy. ‘I’he good, which represents the numeraire, may be allocated to consumption or investment. The investment takes the form of a set of production processes with stochastic constant returns to scale. Production processes are affected by a vector of state variables. The dynamics of both the state variables and the production processes are governed by a system of stochastic differential equations.

The economy is competitive, with continuous trading and no transactions costs. There exist markets for contingent claims written on the good and for instantaneous borrowing and lending at the riskless interest rate.

Solving the constrained maximization problem of the agents, it is possible to determine optimal values for the level of consumption, the riskless interest rate, and the expected returns on the contingent claims. From the equilibrium conditions is also derived the partial differential equation which the price of any contingent claim must satisfy. The solution of the equation, under the ap- propriate boundary conditions, gives the equilibrium value of the assets in terms of the real variables of the economy.

This general equilibrium framework is “specialized” and applied to the term structure of interest rates through the use of some assumptions about preferences and dynamics. In particular, it is assumed that the individuals have logarithmic utility functions and that there exists only one state variable, which follows a square root mean reverting process.

The model allows us to obtain endogenously the dynamics of the instantaneous riskless interest rate. Since it is proportional to the state variable, the interest rate r also follows a square root mean

54 A. BE-1

reverting process. This process, which admits only non-negative values, has the form:

dr=Ic(B-r) dt+&lz, (1)

where K denotes the coefficient of mean reversion, 8 the long-term value of the interest rate, CT a positive constant, and z a Wiener process. The mean reverting behavior of the process requires that K, DO. The conditional distribution function of the interest rate is a non-central chi-square, whereas its steady state distribution is a gamma function.

The model also allows us to derive endogenously the functional form of the risk premium on real unit discount bonds. The risk premium is obtained through the application of the Ito’s Lemma to the price of a unit discount bond and the use of the no-arbitrage argument. Let P=p(r, t, 7) =p(r, z) denote the price of a unit discount bond, where ‘t is_ the current time, T the maturity date and 2, z = T- t, the time to maturity of the bond. We get the following expression for the risk premium:

(2)

where 1 is defined as a ‘market” risk parameter, which stems from the no-arbitrage condition, and where the term (P/h)(r/P), which denotes the elasticity of bond prices to the interest rate, is negative. Hence, the risk premium will be positive when 1<0.

The fundamental partial differential equation implied by expression (2) is given by:

with the boundary condition:

The term in brackets represents the drift term of the interest rate process under the risk-adjusted probability measure and may be rewritten as:

(5)

‘IFIE COX, INGERSOLL AND ROSS MODEL 55

The following closed-form solution of the partial differential equation gives the equilibrium value of the unit discount bond:

fir, 9=&l exp[--BWl, (6)

where

B(z) 3 2(eF - 1) (IC + A+ y)(eyz- 1) + 27’

The function A(z) corresponds to the maximum value which the price of the unit discount bond can assume, whereas the function B(z) measures the relative basis risk of the bond and determines its risk premium. In fact, using the pricing relation (6), we get:

The yield-to-maturity of a unit function of the interest rate:?

r

discount bond, Y(r, T), is an affine

Y(r, z) E -

As maturity tends to in.Cnity, the yield-to-maturity approaches a limit which does not depend on the interest rate, but only on the risk-adjusted parameters:

Y(co)= 2fce lc+A+y'

t On the aftie yield class of term structure models, see Brown and Schaefer (1994b). A multifactor extension has been provided by Duffie and Kan (1993).

56 A. BERARDI

This limit is relevant to determine the shape of the term structure, whjch is monotone increasing for r<Y(co), monotone decreasing for r>B, humped for Y(W) < r < 8. The forward rate curve at time t is given by:

Ar, t, T) = -(ap’aT) = r + [Ice - (Ic + A)r]B(z) -+3M2(2), p(r, z) (10)

where fir, t, 2’) denotes the instantaneous interest rate that will prevail at time T for a payment an instant later.

In the case of coupon bearing bonds, the pricing relation takes the form: I

where ch is the h-th coupon, H the number of payments, TH the maturity date and zH = TH - t. The risk premium on a coupon bond with maturity zH is given by:?

As regards the pricing of derivatives on bonds, the model allows one to determine endogenously the equilibrium value of a great

t The basis risk of a coupon bond is measured by ita stochastic duration, which is defined as the time to maturity of a unit discount bond having the same risk of the coupon bond. The expression for the stochastic duration, D, of a coupon bond with maturity rH is given by:

D=+log 2+(K+1-7)X [ 1 2+(K+1+)‘)Z ’

where

See Cox et al. (1979).


variety of interest-rate-sensitive contingent claims, such as options and futures.?

3. Econometric method

The empirical analysis of the model requires to assume that actual bond prices deviate from their theoretical values by a disturbance term, which arises from measurement errors. Taking simultaneously into account the cross-sectional and time series di- mensions of the term structure, we may rewrite the CIR pricing equation (6) as a multivariate non-linear regression model, which comprises M individuals (the bonds) and N observations:

K(Tj)=p(rt, Tj; fl)+EXZj) (j=l, . . * 9 M; t=l, * * *, N). (13)

P; denotes the actual price and P the theoretical price of a unit discount bond at time t. The theoretical price of the j-th bond at time t is then a function of the value of the interest rate, r, the parameter vector, j3, and the time to maturity of the bond, Zj, Tj z Tj - t, where Tj is the maturity date. M is the number of bonds at each instant of time, whereas N is the number of temporal observations for each bond.

In the following, we will develop an estimation procedure, based on a multivariate non-linear least squares (MNLLS) tecbnique,$ which is able to combine the time series and cross-sectional information contained in bond prices and to provide estimates of the risk premia on the bonds. The procedure is developed to deal with both the case of independently and identically distributed (IID) error terms and the case of serially correlated error terms.

3.1. THE MNLIS PROCEDURE WITH IID ERROR TERMS

Let us assume that the error terms in model (13) are con- temporaneously correlated and heteroskedastic across bonds, but neither serially correlated nor heteroskedastic across time. There- fore, we have:

~(Zj)=p(r~, Zj; /3)+&t(Tj) G=l, . . . , M; t=l, . . . , N),

Et(rj) - IID(O, Z). (14)

t On the closed-form expressions for these securities, see Cox et al. (1981, 1985b) and Jamshidian (1987). For the equilibrium value of futures and options on bonds in a muItifact.or extension of the CIR model, see Chen and Scott (1992).

$ We extend the methods advanced by Gallant and Goebel (1976), Gallant (1987), and Amemiya (1983).

5s A. BE-1

Using a compact notation, we may rewrite the system of equations (14) as:

p*=P+&, (15)

with P, P, E MN-dimensional vectors. ‘I’he variance-covariance matrix of the error terms in equation (E), 8, which is an (MN x MN) matrix, is then equal to:

where IN is an N-dimensional identity matrix. Let A be an (M x A!) matrix such that E-’ = A’h. Therefore, we ’

may construct the rotated model:

(16)

where the error terms are neither correlated nor heteroskedastic. By applying the non-linear least squares technique to the rotated model, we obtain an estimate for the parameter vector /3:t

3.2. THE ItlNLLS PROCEDURE WITH AUTOCORRELA!I‘ED ERROR TERMS

Let us assume that there exists first order serial correlation in the error terms in model (13). We have:

~(Tj)=p(r~, Tj; /3)+Et(Tj) G=l,. . . , M; t=l, . . .,N),

ot(zj) N IID(O, s).

The disturbances {o,(ri>} are cross-sectionally, but not temporally, correlated and heteroskedastic. We assume that the error terms in one equation do not depend on lagged disturbances in other

t Actually, since the variance-covsriance matrix E is unknown, the estimation procedure involves the use of two stages. The first stage consists in estimating separately the pricing equation for each bond, so that we have M equations composed by N observations each. F’rom this step we get an estimate of the variance-covariance matrix. In the second stage, the estimated variant+ covariance matrix is substituted into equation (17) to derive the MNLLS estimate of the parameter vector.


equations. Let @ denote the (Mx iIf) diagonal matrix, whose elements are given by the autocorrelation coefficients in equations (18). Then, using a compact notation, we may write the system of equations (18) in the form:

p*=P+&,

E = maV~q-1) + 0, (19)

where P*, P, E, E(-~), o are MN-dimensional vectors. We have:

E{uu’} = S@1, and E{EE’} = 0,

where the elements of the variance-covariance matrix Q are such that:

J?3{E~(ZJE8(Zj)} = qj, if t = s, .

=Wjlt-819 ift#s (i,j=l,..., M;t,s=l,..., N),

E{&~(ZJz} =E{&JZj)2} = Cjj # Qii =E{E~(ZJ2} (i #j; t ZS).

CB~,~+, is the autocovariance between the i-th and the j-th error terms I t-s 1 periods apart:

It-” mijlt--aJ = aij@ where: ~q=sJ(l- qiqj).

sij is the (i, I)-th element of the variance-covariance matrix S. Now, let us consider an (M x M) matrix 2 such that S-l =Z’Z

and a transformation matrix Y such that Y’BY’ =SO1,. Hence, we get: D-l= Y’(Z@1,)‘(Z@I~)Y. Using these matrices, we may build the following rotated model:

where the error terms are both serially and cross-sectionally uncorrelated and homoskedastic. The application of the non-linear least squares method to the rotated model gives:

Mjn &(p, !2) = dY’(Z@IN)‘(Z@IN) YE = ~‘&?-~e. Gw

As a result we get the MNLLS estimate for the parameter vector /3. The estimation method we have developed so far for a set of unit

discount bonds applies also to a set of coupon bearing bonds. In the case of coupon bonds, the multivariate non-linear regression model takes the following form:

60 A. BEIWRDI

q(T&zl CJyZ, r,; j.?)+fg+j) o’=l, . . ., M t=l, . . .,A$ (22)

where vi denotes the actual price of the coupon bonds. When the error terms are independently and identically distributed, we have:

When the error terms are autocorrelated, we have:

where ut(zHi) - IID(0, 23). The parameter vector comprises the risk-adjusted coefficients:

#? =(&, K+IZ, a2). Therefore, the model specification does not allow us to separately identify the parameters K, 0 and 1, but only a combination of them.

However, in order to be able to calculate the risk premia on the bonds, we need to separately identify the market risk parameter 1. We derive this value in a second stage, through the direct estimation of the coefficients of the interest rate diffusion process. In particular, we apply the maximum likelihood method to the conditional probability density of r, a function which does not depend on A.?

The distribution function of a linear transformation of the interest rate at time s, a~,, is the non-central chi-square with v degrees of freedom and parameter of non-centrality C, where:

t The logarithm of the conditional probability density of the short rate at time 8, given ita value at time t, is:

where q = (v - 2m, and a, v and 5 are defined below in the text. I,(. ) is a modified Bessel function of order q:

r denotes the gamma function.


In order to simplify the estimation procedure, we adopt Sankaran’s approximation (Sankaran, 1963):

Jf%r,; v, 0 - Kv - W4 ~M,,/~, 0 (23)

This approximation, which is quite accurate when the difference between time t and s tends to zero,? allows us to write the likelihood function using the Normal distribution.

We impose the constraint that K, 0 and 2 must be such that ~0 =(a)- and $3 = (@) -. Hence, the likelihood function may be rewritten in a form which depends only on the mean reversion coefficient IC. Given the maximum likelihood estimate of K, it will then be possible, by substituting this value into the estimated risk- adjusted coefficients, to separately identify the risk premium term 3, and the long-term mean 8.

c

4. Estimation and results

We apply the econometric method to estimate and test the CIR model. In the empirical work we use bond prices expressed in nominal terms, assuming that the theoretical framework holds for a real economy as well as for a nominal economy with neutral inflation. We are forced, in a sense, to take this assumption because of the fact that we can not observe bond prices in real terms.$ We use prices of Italian Treasury bonds (Buoni de1 Tesoro Poliennali; BTP), taken from the MTS (Mercato Telematico Secondario), an automated trading system. The data set consists of daily observations from 5 March 1991 to 2 April 1992. Bond prices are the mean of bid and ask price quotations, plus the accumulated interest as of that date.9 We consider six categories of coupon bonds, which are characterized by different maturities: 2,3,5,6,7 and 10 years. The sample is formed by choosing, day by day, the most traded bonds for each category.

t On the comparison among different approximations to the non-central chi- square distribution, see Johnson and Kotz (1970; chapter 28). A useful reference also is Olver (1965).

$ The same assumption has been adopted, among the others, by Brown and Dybvig (1986), Barone et al. (1991), Longstaff and Schwartz (1992). The only empirical studies which have provided estimates for the real term structure are those of Brown and Schaefer (1988, 1994u), where British government index- linked bonds have been used.

1 In order to partially take into account possible tax effects, which may arise from differences in the characteristics of the bonds, we use after-taxes coupon payments, assuming the existence of homogeneous investors in the bond market.

62 A. BE-1

I? 16

:s

.% Maturity (years)

FIGURE A. Distribution of bond maturities in the sample period (frequency in percentage terms).

50

40

30

20

10

all0 Others (O-2: 3-4,5-6,7-9)

Maturity (years)

FIGURE B. Transactions for different categories of bonds in the sample period (percentage on the total nominal amount of transactions).

Figure A shows the distribution of the maturities of the bonds outstanding at MTS in the sample period, whereas Figure B shows the percentage of the nominal amount of transactions in the

THE COX, INGERSOLL AND ROSS MODEL

TABLE 1 IID error terms specification: (a) parameter values; (b) asymptotic corn&&m matrix

(a)

63

Parameter Estimated value Asymptotic standard errvr

ue 0.0268 0-0007 iC+A o-1539 O-0052 2 o-0529 odO21

(b)

Pammeter Ke

ue K+l 2

1 0.966 o-947 1 O-840

1

different categories of bonds on the total nominal amount of transactions. The fact that there are not many bonds with maturities between 7 and 9 years is due to the fact that the first issue of lo- year maturity bonds took place at the beginning of the sample period (5 March 1991). Until that time, the longest maturity for the BTP’s was 7 years.

We consider the instantaneous interest rate as an unknown variable. We estimate the time series of the interest rate by inverting the bond pricing equation (6) with respect to the price of a short-term zero coupon bond, which is assumed to be observed without measurement errors.? Let P;(zO) denote the actual price at time t of the short-term zero coupon bond and z. its maturity. Therefore, we get the following expression for the interest rate at time t:

r t = hid&o; LOI - h$‘hJl

mo; B) . cw

We use the price of the three-month Treasury bill for e(ro) and estimate the model by imposing expression (24) as a constraint.

In estimating the model we first apply the MNLLS procedure with IID error terms.* The parameters estimated from this specification are shown in Table 1. The coefficients are characterized by a low degree of mean reversion in the interest rate risk-adjusted

t A similar procedure has been used by Pearson and Sun (1988, 1994). $ The estimation routines are written using the SAS System software. The

procedure uses the Marquardt iterative algorithm.

64 A. BERARDI

stochastic process and by a high level of volatility. The implied volatility of the interest rate amounts, on average, to 714 basis points. Moreover, it is also very high the estimated level of the risk-adjusted long-term value of the interest rate, which is about 17.5%. The asymptotic correlation matrix of model parameters reveals the presence of a high degree of positive correlation between the coefficients.

The model does not fit well actual bond prices. The bond pricing errors shown in Table 2 indicate that, on average, theoretical prices tend to over-estimate actual prices by an amount which ranges from about Lit O-03 to Lit O-17 per Lit 100 face value of the bonds. In absolute terms, the mean values of the differences between actual and theoretical prices are quite large, ranging from about 0.43% (a-year bond) to 1.49% (lo-year bond) of the face value.

The high value of the pricing errors in absolute terms is strongly influenced by the size of the errors in a small number of observations. However, eve_n if we would eliminate those outliers, the differences between actual and theoretical prices would still remain too large to be attributable only to the existence of measurement errors and may instead be regarded as the consequence of misspecification errors. In particular, the adjusted Box-Pierce statistic @BP)? shows that the errors exhibit first order serial correlation.

We re-estimate the model by applying the MNLLS procedure to the specification with autocorrelated disturbances. In Table 3 are reported the estimated values of the autoregressive coefficients of the diagonal matrix 45, which are significantly different from 0. Table 4 shows the estimated values and the asymptotic correlation matrix of the risk-adjusted parameters ~0, IC + 1,02. The estimates indicate a relatively high degree of mean reversion in the interest rate risk-adjusted process. As we would expect, the drift terms ~0 and K +L, which are statistically highly significant, are strongly positively correlated.

The risk-adjusted long-term mean of the interest rate, 8, is equal to 1144%, whereas the implied long-term yield is about 11.32%. Hence, the yield curve estimated using these parameters will be upward sloping for r,<ll.32%, downward sloping for r,>ll-44%, humped for 11.32%~ r,< 1144%. The volatility term 2, which is negatively correlated with the other coefficients, shows a relatively high standard error. The mean value of the estimated implied

t The adjusted Box-Pierce statistic tests the hypothesis that a process is serially uncorrelated using a correction for heteroskedasticity. Under the null hypothesis, the ABF(Q) statistic follows a &i-square distribution with Q degrees of freedom, where Q is the maximum number of temporal lags.

“* 8 E2

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66 A. BE-1

TABLE 3 Autocor&ated ermr terms specification: estimated coemients of the diagonal matrix @

Parameter

Pl

cp-2 cps

(P4

cps

cps

Estimated value

0663 o-759

::E

O-936 o-959

TABLE 4 Autocor&ated error terms specification: (a) parameter values; (b) asymptotic corrtzlatio~ matrix

(a)

Pammeter Estimated value Asymptotic stanch& error

ice 0.0905 0-0039 iC+l 0.7908 O-0444 02 0.0139 0.0215

Pammeter KO lC+il 2

ice 1 O-978 - 0.480 lC+1 1 -0.646 a2 1

standard deviation of the short rate is about 366 basis points.t We carry out a test for the structural stability of the coefficients

using a likelihood ratio test statistic (see Gallant, 1987; chapters 1, 5). In order to build the statistic, we re-estimate the model under an “unrestricted” specification, where the parameter vector is allowed to change across bonds, and compare the residual sum of squares with that of the “restricted” specification, where the parameter vector is constant across bonds. The statistic enables us to not reject the null hypothesis of structural stability at the 9% significance level.

t This value is almost twice as much as the value estimated for the Italian Treasury bond market by Barone et al. (1991). However, their estimates refer to a different sample period.

‘ITHE COX, INGERSOLL AND ROSS MODEL 67

We also test for fixed effects, which could be explained by bond- specific dummies. Both for the “unrestricted” and the “restricted” specification we do not reject the null hypothesis of absence of fixed effects (with a p-value of 8%, in the first case, and 40%, in the second case).

Table 5 contains summary statistics on bond pricing errors. In this case, the model seems to fit quite well actual bond prices and the differences between actual and theoretical prices indicate that, on average, the model tends to slightly under-estimate actual prices. The mean of the pricing errors, which are much less volatile than in the IID case, ranges from about Lit O-02 to Lit O-07 per Lit 100 face value. The average value of the pricing errors in absolute terms, which ranges from 0.26% (2-year bond) to 0.33% (lo-year bond) of the face value, is small and may be considered reasonable in view of the existence of measurement errors.

In order to assess whether the differences between actual and theoretical prices are pure measurement errors, we must verify that the pricing errors are white noise. The alternative hypothesis of MA-type error terms affecting coupon bearing bonds would imply the existence of misspecification in the pricing relation for each coupon. However, the ABP test statistic allows us to not reject, at a high significance level for the longer maturity bonds, the hypothesis that bond pricing errors are white noise.

The average values of the bond pricing errors shown in Table 5 tend to increase with maturity. This phenomenon is essentially due to the structure of the CIR model, which involves a constant value over the sample period for the long-term yield. The restriction that the yield curve must converge towards a constant fixed value affects the yields on longer maturity bonds more than those on shorter maturity bonds. As a consequence, longer maturity bonds have less degrees of freedom in fitting actual bond prices and then exhibit higher pricing errors. This fact appears to be evident by comparing the volatility of longer maturity bonds with that of shorter maturity bonds (see Figure C and Table 6). The statistics on yield volatilities presented in Table 6 indicate that, on average, the volatility of the lo-year zero coupon bond is about four times smaller than that of the 5-year zero coupon bond and about thirty times smaller than that of the l-year zero coupon bond.

Table 7 shows summary statistics on yields and yield spreads of zero coupon bonds with different maturities. The values indicate that, over the sample period, the yield curves estimated from the model generally take an upward sloping shape (see Figure D). This evidence is consistent with that obtained by previous studies on the Italian Treasury bond market (see, e.g. Moriconi, 1993). Table 8 contains statistics on the forward rate curves.

Finally, we derive the risk premia on the bonds. In order to do this, we should work out the value of the market risk parameter,

TABL

E 5

Auto

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late

d em

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spec

ificat

ion:

di

ffere

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be

twee

n ac

tual

an

d th

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cal

price

s (p

enze

ntag

e va

lues

)

Bond

m

atur

ity

Mea

n S,

D.

Max

imum

M

inim

um

I-ord

er

seria

l co

rrela

tion

ABp(

20)

test

(p-

valu

e)

Diffe

renc

es

2 ye

ars

0.02

0.

34

3 ye

ars

0.02

03

3 5

year

s 0.

03

0.42

6

year

s 0.

04

0.42

7

year

s 0.

05

0.43

10

yea

rs

0.07

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45

Diffe

renc

es

in a

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ute

term

s 2

year

s 0.

26

0.22

3

year

s 0.

28

o-24

5

year

s 0.

30

0.29

6

year

s 0.

31

0.29

7

year

s 0.

31

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10

yea

rs

0.33

0.

30

1.26

-

1.21

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62

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1 33

.3

(044

) 1.

42

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55

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3 19

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0.48

) 1.

56

- 1.

53

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6 19

.6 (

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) 1.

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9 15

.8 (

0.72

)

THE COX, INGERSOLL AND ROSS MODEL

11.6 11.4 - lo-year 11.2 -

11.0 - 10.8 - 10.6 -

2 10.4 - 2 4) 10.2 - F

10.0 - 9.8 - 9.6 - 9.4 - 9.2 -

69

9.01 ' I I I 5 March 1991 19 September 1991 2 April 1992

. Date

FIGURE C. Yields on zero coupon bonds.

TABLE 6 Yield volatility (values x 100)

Maturity (years) Mean S.D. Maximum Minimum

i 0.0582 0.0024 0.0652 o-0501 O-0176 ox@07 0.0198 O-0152

5 om74 omO3 O-0083 0-0064 10 0~0019 0@001 0.0021 OQO16

TABLE 7 YieZds and yield spnx&

Maturity (years) Mean SD. Maximum Minimum

0 (short rate)

i

1:

14 3-l 5-l

10-l l&5

O-0963 0~1018 0.1070 0.1093 o-1112

04040 04lo28 0.0015 0.0010 omO5

om55 0.0012 0.0052 O-0012 odO75 O-0018 0.0094 0.0023 0.0019 omO5

0.1078 O-1098 o-1114 o-1121 O-1127

om97 odO95 o-0135 o-0171 0.0036

0.0828 0.0924 o-1019 o-1059 o-1095

0.0019 om17 oax!4 om29 omo6

70 A. BERARDI

9 1 4 10

5 March 1991 19 September 1991 .2 ~46~ kieare’ ’ April 1992 ~attitY Date .

FIGURE D. Term structure.

TABLE 8 Forward rates

Maturity (years) Mean S.D. Maximum Minimum

i 0.1059 O-1118 04018 04004 o-1129 o-1111 04998 0.1106 5 o-1130 04001 O-1132 0*1128

10 0.1133 0mOO o-1133 0.1133

A. The MNLLS procedure provides estimates of the risk-adjusted parameters, but it does not allow us to separately identify 1, the mean reversion coefficient IC, and the long-term mean of the short rate 0. We get separate estimates of these parameters in a second stage by applying the maximum likelihood method to Sankaran’s approximation of the conditional probability density of the short rate. In the estimation procedure we impose the constraint that the relation between the three coefficients must satisfy the values of the risk-adjusted parameters previously estimated through the MNLLS method.


TABLE 9 Constmined maximum likelihood estimates of the pammeters from the conditional probability density of the short mte

Parameter Estimated value

ii 04043 0.1001

d -0.1135

3-year

0.8 -

0.7 -

5 March 1991 19 September 1991 2 April 1992 Date

FIGURE E. Risk premia on zero coupon bonds.

Table 9 reports the constrained estimates of the coefficients. The value of the mean reversion coefficient K denotes a relatively high speed of adjustment of the short rate towards its long-term value. The value of L is negative and indicates the existence of positive risk premia. Figure E shows the time series of the risk premia estimated for the l-year and the 3-year zero coupon bonds.

5. Conclusions

In this paper we have proposed a multivariate non-linear least squares procedure to estimate the CIR model of the term structure. The methodology, which has been developed to deal with both the case of independently and identically distributed error terms and the case of autocorrelated error terms, has proved to be able to

72 A. BERARDI

simultaneously take into account the cross-sectional and time series information contained in bond prices and to provide estimates for the risk premium on the bonds.

The empirical work has shown that the estimates of the model obtained from the specification with autocorrelated error terms exhibit a good fit and plausible values for all the parameters. However, the evidence has pointed out that the fit is more accurate in pricing shorter maturity bonds than the bonds on the long end of the term structure. This result is mainly due to the constraints that the theoretical model imposes on the term structure and in particular to the fact that it implies that long-term yields must converge rapidly towards a constant fixed value.

Most of the restrictions implied by the model would be removed in a multifactor framework. An extension of the CIR model-for example, along the lines suggested by Longstaff and Schwartz (1992)-would preserve the equilibrium properties of the model and, in addition, would show a higher degree of flexibility in fitting the term structure.

Acknowledgements

I am particularly grateful to Stephen Schaefer for valuable comments and suggestions. I am also indebted to Guglielmo Weber, Marcello Esposito, Umberto Cherubim, and two anonymous ref- erees for constructive comments. Any remaining errors and in- accuracies are mine. Last but not least, I would like to thank Flavio Addolorato for the implementation of the routines used in the estimation.

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