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# An Orthogonal Polynomial Approach to Estimate the Term Structure of Interest Rates

## Author

Listed:
• Hans-Jürg Büttler

## Abstract

In this paper, we introduce a new algorithm to estimate the term structure of interest rates. It is obtained from a constrained optimization, where the objective is to minimize the integral of squared first derivatives of the instantaneous forward interest rate subject to the condition that the estimated bond prices lie within the range of observed bid and ask prices. We use a finite series of ordinary Laguerre polynomials to approximate the unknown function of the instantaneous forward interest rate. The objective function can be written explicitly as a quadratic form of the Laguerre constants and the nonlinear constraints can be obtained from a recurrence relationship. The estimation error is less than one basis point, given a sufficient number of bonds.

## Suggested Citation

• Hans-Jürg Büttler, 2007. "An Orthogonal Polynomial Approach to Estimate the Term Structure of Interest Rates," Working Papers 2007-08, Swiss National Bank.
• Handle: RePEc:snb:snbwpa:2007-08
as

File URL: https://www.snb.ch/n/mmr/reference/working_paper_2007_08/source/working_paper_2007_08.n.pdf

## References listed on IDEAS

as
1. Shea, Gary S., 1984. "Pitfalls in Smoothing Interest Rate Term Structure Data: Equilibrium Models and Spline Approximations," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 19(03), pages 253-269, September.
2. David Jamieson Bolder & Scott Gusba, 2002. "Exponentials, Polynomials, and Fourier Series: More Yield Curve Modelling at the Bank of Canada," Staff Working Papers 02-29, Bank of Canada.
Full references (including those not matched with items on IDEAS)

### Keywords

Term structure of interest rates; orthogonal polynomial;

### JEL classification:

• C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
• E43 - Macroeconomics and Monetary Economics - - Money and Interest Rates - - - Interest Rates: Determination, Term Structure, and Effects

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