J. Comp. & Math. Sci. Vol.2 (5), 693-698 (2011)
Modeling the Term Structure of Nigeria Treasury Bill Interest Rate with the Cox, Ingersol and Ross Model MICHAEL C. ANYANWU and B. I. ORUH Department of Mathematics, Michael Okpara University of Agriculture, Umudike- Abia State, Nigeria ABSTRACT This paper presents the Cox, Ingersoll and Ross term structure model for Nigeria Treasury bill rate. The aim is to obtain an appropriate yield curve that could be used to price any security that depends on the Nigeria Treasury bill rate. The data used is the yearly interest rate on the Nigeria Treasury bill from 1970 to 2005. With the help of the SDE toolbox the interest rate process is simulated using the Milstein numerical scheme. It is observed that the yield curves so obtained are hump-shaped initially, before leveling out as the time to maturity lengthens. Keywords: SDE toolbox, default-free pure discount bond, term structure of interest rates.
1. INTRODUCTION Nowadays, interest rate and its dynamics play important roles in all transactions Nigeria money and financial markets which include simple lending, borrowing and the trading of interest rate derivative securities. An interest rate derivative security is a contingent claim whose underlying financial asset is the interest rate. A good example of interest rate derivative is the default-free pure discount bond or zero coupon bond in which the borrower does not pay anything until the maturity date. To deal on interest rate
derivatives, it is pertinent to know what these derivatives worth in the market and how they can be priced. Since the value of these instruments depends on the movement of the underlying i.e the interest rate it is therefore necessary to determine how the evolution of the interest rate over the term of the security can be modeled. The relationship between the yield on the security and the tern of the security is called the term structure of interest rate. There are many models that have been proposed and used by economist and researchers alike to model the term structure of interest rates. These models
Journal of Computer and Mathematical Sciences Vol. 2, Issue 5, 31 October, 2011 Pages (693-779)
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Michael C. Anyanwu, et al., J. Comp. & Math. Sci. Vol.2 (5), 693-698 (2011)
can be categorized as spline based models, function based models and stochastic models. The spline models focus on interpolating a spline function from the preknown points of the form (time to maturity, yield of zero coupon bonds). The models in this category include McCulloch (1971, 1975), Vasicek and Fong (1982) exponential splines, semi parametric penalized spline model of Jarrow, Ruppert and Yu (2004), Bayesian Penalized spline model of Min and Yu (2005). The famous function based models are that of Nelson-Siegel (1987) and Svensson (1999). The stochastic modeling of term structure of interest rate which is the main focus of this article depends on the interest rate modeling especially on the short rate modeling. The models in this category include Merton (1973), Vasicek (1977) and Cox, Ingersol and Ross (1985) thereafter reffered to as CIR (1985), which is an improvement on the Vasicek (1977) model. The idea of stochastic modeling is to obtain an explicit formula for the price of the zero coupon bond and then extracting the yield from that formula by using the equation ,
ଵ ŕŻ?ି௧
ŕŻ˜ , ,
(1.0)
where Y (t , T ) is the yield of zero coupon bonds on the interval [t, T ]. The CIR (1985) model provides a complete specification of the term structure of zero coupon bond prices in general equilibrium setting. The CIR (1985) assumes amongst other features that the short term interest rate moves randomly around a path that leads to its
long-run equilibrium level. The remaining part of the paper is organized as follows; section 2 describes the relevant features of CIR (1985) model. In section 3 we present the implementation of the model. Section 4 concludes paper. 2.
THE COX, INGERSOL AND ROSS (1985) MODEL
The Cox, Ingersol and Ross (1985) ଵ Model amounts to substituting ଶ in the general mean reverting interest rate process ŕ°‰
(2.1)
The one factor form of the CIR (1985) model assumes that i. the evolution of the instantaneous interest rate follows the stochastic differential equation: √
(2.2)
where Îą is the speed of reverting of the interest rate r towards its long- run mean , √ is the volatility of the variation in r and is the standard Brownian motion. ii. the expected rate of return on bonds of any maturity is equal to the interest rate r plus the a risk premium. That is, ௗ௉
௉
ௗ௉ ௉
(2.3) ௗ௉
Where is the market price of risk, and ௉ is elasticity of the price P of the bond with respect to interest rate r. iii. the market under consideration is perfect.
Journal of Computer and Mathematical Sciences Vol. 2, Issue 5, 31 October, 2011 Pages (693-779)
Michael C. Anyanwu, et al., J. Comp. & Math. Sci. Vol.2 (5), 693-698 (2011)
Under these assumptions, with the application of Ito lemma it can be proved that the price , , of any pure discount bond depending on the current term structure of interest rates satisfies the partial differential equation 1 ଶ ଶ 0, 2 ଶ
with the condition , , 1
(2.4)
It can be shown that the solution to equation (2.4) subject to the boundary condition is given by
, , , , (2.5) Where
ଶ 2 2 , , ଶ ŕ°ŽáˆşŕŻ?ŕŹżŕŻ§áˆť 1 2 áˆşŕ°ˆŕŹžŕ°ŽáˆťáˆşŕŻ?ŕŹżŕŻ§áˆť
ŕŹśáˆşŕŻ˜ ŕ´†áˆşŕł…ŕ°ˇŕł&#x;áˆť ŕŹżŕŹľáˆť and ିଵ൯ାଶమ
, áˆşŕ°ˆŕŹžŕ°ŽáˆťŕľŤŕŻ˜ ŕ´†áˆşŕł…ŕ°ˇŕł&#x;áˆť ! " ଶ 2 ଶ
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3. DATA AND ESTIMATION OF MODEL PARAMETERS The data used in the implementation of the model contain 36 observations of the Nigeria Treasury bill rate from 1970 to 2005 available in the Central Bank of Nigeria Statistical Bulletin. The Treasury bill is a security with maturity of one year issued by the federal government of Nigeria treasury through the central bank of Nigeria. The purchase price of the Treasury bill serves as a temporary loan to the Nigerian government which it returns when it matures. The parameters of the interest rate process to estimated are , and Ďƒ. In this paper we have used the method of NonParametric Simulated Maximum Likelihood (NPSML) in the estimation process. A stochastic differential equation (SDE) toolbox made available by Umberto Picchni was used to implement the NPSML estimation process. The result of the estimation is 0.1558, 0.1006 )* + 0.0962
Fig.1: simulation of CIR interest rate process Journal of Computer and Mathematical Sciences Vol. 2, Issue 5, 31 October, 2011 Pages (693-779)
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Michael C. Anyanwu, et al., J. Comp. & Math. Sci. Vol.2 (5), 693-698 (2011)
Fig.2: Discount functions of the Treasury bill
Fig.3: the yield curves of the Treasury bill
3.1 MODEL IMPLEMENTATION In the implementation of this model, we used the Milstein numerical scheme to simulate the CIR interest rate process.
The estimated parameters and the initial interest rates r0=0.04, 0.05, 0.175, 0.12 for four different years were used in the simulation process and the result is shown in figure 1. The discount functions and the
Journal of Computer and Mathematical Sciences Vol. 2, Issue 5, 31 October, 2011 Pages (693-779)
Michael C. Anyanwu, et al., J. Comp. & Math. Sci. Vol.2 (5), 693-698 (2011)
yield curves for the different years were also calculated from (2.5) and (1.0) respectively and the results are shown in fig 2 and fig 3 respectively. 4. SUMMARY AND CONCLUSION In figure 2, we observe that the discount function for the Treasury bill is downward sloping. This indicates that the price (value) of the Treasury bill bond increases as the time to maturity decreases. It is also observed in figure 3 that the yield curves appear to be humped initially and slopes upwards as the term of the bond increases. The shape of these yield curves could be explained using the expectation hypothesis. The expectation hypothesis implies that the slope of the yield curve at any point in time is determined by the expectation of the market’s participants on the level of the short term interest rates. The expectation hypothesis predicts that when interest rates are high, the market expect them to fall in future and this will make the yield curve to slope downwards participants and when interest rates are low, the markets participants expect them to rise and this will make the yield curve to slope upwards. Therefore the shape of the yield curves in figure 3 is because the treasury bill market participants expect short interest rates to remain constant for a short period and fall sharply. These make the yield curves to appear humped, sloping upwards at very short periods, peaking near periods corresponding to the dates at which interest rates are expected to fall.
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REFERENCES 1. Cox, J.C., Ingersoll, J. E., and Ross, S.A. A theory of the Term Structure of Interest Rates. Econometrica 53:385407 (1985). 2. David Jamieson Bolder. Affine Term Structure Models: Theory and Implementation. Bank of Canada Working Paper 15 (2001). 3. Des Mc Manus and David Watt. Estimating One Factor Models of Short Term Interest Rate,” Bank of Canada Working Paper 18 (1998). 4. Desmond J. Higham. An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations. Siam Review. 3(3) 525-546 (2001). 5. E. Barone, D. Cuoco and E.Zautzik. The Term Structure of Interest Rates- A Test of the Cox, Ingersoll and Ross Model on the Italian Treasury Bond,” Banca d’Italian, Temi di Discussione del Servizio Studio. No. 128 (1989). 6. Mark Fisher. Modeling the Term Structure of Interest rate: An Introduction. Federal Reserve Bank of Atlanta Economic Review Third Quarter (2004). 7. Min Li and Yan Yu. Estimating the Interest Rate Term Structure of Treasury and Corporate Dept with Bayesian Penalized splines. Journal of Data Science 3:223-240 (2005). 8. Robert Jarrow, David Ruppert, and Yan Yu. Estimating the Interest Rate Term Structure of Corporate Dept with a Semiparametric Penalized spline Model. Journal of the American Statistical Association. 99: 465 (2004).
Journal of Computer and Mathematical Sciences Vol. 2, Issue 5, 31 October, 2011 Pages (693-779)
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Michael C. Anyanwu, et al., J. Comp. & Math. Sci. Vol.2 (5), 693-698 (2011)
9. Umberto Picchini. Simulation and Estimation of Stochastic Differential Equations with MATLAB. http://sdetoolbox.sourceforge.net (2007). 10. Vasicek, O. An Equilibrium Characterization of the Term Structure. Journal
of the Financial Economics 5:177-188 (1997). 11. Vasicek, O. and H. Gifford Fong. Term structure Modeling Using Exponential splines. The Journal of Finance Vol. xxxvii 2:339-356 (1982).
Journal of Computer and Mathematical Sciences Vol. 2, Issue 5, 31 October, 2011 Pages (693-779)