Smooth interpolation of zero curves

Insurance Mathematics & Economics, 2010

In this article we apply the constrained smoothing b-splines (COBS) to interpolate and construct measures associated with the term structure of interest rates. The COBS method has the useful advantage of incorporating important constraints observed in the term structure such as: monotonicity, non-negative values and robustness related to quantile regression methods. We compare COBS with some usual methods utilized in statistical term structure tting: linear interpolation, smoothing splines and the parametric Nelson-Siegel and Svensson methods.

Economic Research-Ekonomska Istraživanja

The zero-coupon yield curve is a common input for most financial purposes. We consider three popular yield curve datasets and explore the extent to which the decision as to what dataset to use for a particular application may have an impact on the results. Many term structure papers evaluate alternative models for estimating zero coupon bonds based on their ability to replicate bond prices. However, in this paper we take a step forward by analyzing the consequences of using these alternative datasets in estimates of other moments and variables such as interest rate volatilities or the resulting forward rates and their correlations. After finding significant differences, we also explore the existence of volatility spillover effects among these three datasets. Finally, we illustrate the relevance of the choice of one particular dataset by examining the differences that may arise when testing the expectations hypothesis. In the conclusions, we provide guidance to end users in selecting a particular dataset.

2011

We present a simple and fast iterative, linear algorithm for simultaneously stripping the coupon payments from and smoothing the yield curve of the term structure of interest rates. The method minimizes pricing errors, constrains initial and terminal conditions of the curves and produces maximally smooth forward rate curves.

Data Science and Applications, Lecture Notes in Networks and Systems, 2023

Traditional yield curve models such as Nelson-Sigeal parsimonious model represent the interest rate term structure at high accuracy with smoothed yield curve data. The model proposed by Nelson-Sigeal has been adopted by many countries across the world due to the economic interpretability of the parameters. Since developed capital markets do not experience turbulent economic conditions very often, the smoothness of the yield curve data does not bring challenges to modeling the yield curves. Countries like Sri Lanka used to face economic downturns frequently and the impact of such conditions influence the country' substantially. Adopting the most flexible and interpretable yield curve model such Dynamic Nelson-Sigeal model is a challenge under such conditions. The yield curve data from January 2010 to May 2022 was examined and clustered into steady-state and non-steady-state data based on inflation and exchange rate movement. It was found the accuracy of the Nelson-Sigeal model deteriorates substantially in the non-steady state compared to the steady state based on the R-Squared and MAD. Further, the smoothness of the data in these two statuses is different, where smoothness is higher in a steady state over a non-steady state based on the autocorrelation function. Several smoothing techniques such as Spline Smoothing, Super Smoothing, Loess Smoothing, Lowess Smoothing, and approx Smoothing were employed for both statuses. Lowess smoothing seems to do a better job of smoothing for steady-state data, while the Super smoothing tool does a better job in non-steady states. Spline smoothing performed better than other smoothing tools on model accuracy tests (R-Squared, MAD, MAPD, Correlation of original vs predicted, Nash-Sutcliffe model efficiency, and Index of Agreement).

SSRN Electronic Journal, 2013

After the credit and liquidity crisis started in summer 2007 the market has recognized that multiple yield curves are required for estimation of both discount and FRA rates with different tenors (e.g. Overnight, Libor 3 months, etc.), consistently with the large basis spreads and the wide diffusion of bilateral collateral agreements and central counterparties for derivatives transactions observed on the market. This paper recovers and extends our previous work to the modern singlecurrency multiple-curve bootstrapping of both discounting and FRA yield curves, consistently with the funding of market instruments. The theoretical pricing framework is introduced and modern pricing formulas for plain vanilla interest rate derivatives, such as Deposits, Forward Rate Agreements (FRA), Futures, Swaps, OIS, and Basis Swaps, are derived from scratch. The concrete EUR market case is worked out, and many details are discussed regarding the selection of market instruments, synthetic market quotes,, smooth interpolation, effect of OIS discounting, possible negative rates, turn of year effect, local vs non local delta sensitivities, performance and yield curve sanity checks. The implementation of the proposed algorithms is available open source within the QuantLib framework.

1998

Polynomial functions of the term to maturity have long been used to provide a general functional form for zero-coupon yield curves. The polynomial form has many advantages over alternative functional forms such as Laguerre, when using non-linear least squares to estimate zero-coupon yield curves with coupon bond data. Most importantly the polynomial form invariably enables convergence of the estimation process. Unfortunately, the simple polynomial form results in estimated models of zero-coupon yield curves that approach either plus or minus infinity as the term increases. This unsatisfactory aspect of the simple polynomial is inconsistent with both theoretical considerations and observational reality. We propose a new zero-coupon yield curve functional form consisting not of simple polynomials of term, τ, but rather constructed from polynomials of 1/(1+τ). This form has the desirable property that longterm yields approach a constant value. Further, we model zero-coupon yields as a linear function of the first k principal components of p polynomials of 1/(1+τ), k<p. Using a small number of principal components produces zero-coupon yield curves with a simple "parsimonious" structure while retaining the flexibility of larger polynomial form with its k-1 potential turning points. Moreover estimating with principal components exploits the principal explanatory variable's lack of colinearity.

Finance Research Letters, 2018

Accounting standards require one to minimize the use of unobservable inputs when calculating fair values of financial assets and liabilities. In emerging markets and less developed countries, zero curves are not as readily observable over the longer term, as data are often more sparse than in developed countries. A proxy for the extended zero curve, calculated from other observable inputs, is found through a simulation approach by incorporating two new techniques, namely permuted integer multiple linear regression and aggregate standardized model scoring. A Nelson Siegel fit, with a mixture of average forward rates as proxies for the long term zero point, and some discarding of initial data points, was found to perform relatively well in the training and testing data sets.

2018

The paper addresses the problem of zero-coupon yield curve (YC) estimation from a portfolio of coupon-bearing instruments, primarily coupon bonds. A fast and stable iterative procedure is proposed and implemented. The optimization problem is formulated in a matrix form, the principal cashflow matrix having the dimension given by the number of instruments (bonds) and the number of knots on the time axis. The number of instruments is arbitrary, as well as the number of time knots. In our concept of "equivalent cashflows", each future cashflow at a time t is replaced by two cashflows, one at the left, the other at the right knot respective to the time t. We solve then a simplified problem of estimating the YC from a portfolio of instruments whose future cashflows occur only at predefined times. The method allows for further additional constraints, e.g., an ultimate forward rate fixing or predefined discount factor at some time. We touch also on the asymptotic case of a very dense partitioning of the time axis. The optimization is carried out in the space of forward rate functions of the simplest form-zero-degree splines, i.e., piecewise constant functions. Our approach is thus a generalization of the bootstrap method with no requirements on the bond maturity ladder and, at the same time, with optional smoothing. This work relates to our previous article [4], where the idea of equivalent cashflows was introduced. Czech bond yields estimated by the proposed method can be found in [5].

Journal of Emerging Market Finance

The aim of this article is to develop a methodology to estimate the interest rates yield curve and its dynamics in the Tunisian bond market, which is considered as an illiquid market with a low trading volume. To achieve this, first, we apply the cubic spline interpolation method to deal with the missing observation problem. Second, we focus on the work of Vasicek (1977) and Cox-Ingersoll and Ross (CIR) (1985), we estimate each model and discuss its performance in predicting the dynamics of the interest rates yield curve using ordinary least squares (OLS) and maximum likelihood estimation (MLE) methods. The data sample consists of Treasury bond prices for the period from 14 July 2004 to 10 September 2012 collected from over the counter market. This assessment is done by Matlab software using the specified algorithms. The results suggest that the cubic spline method is accurate for construction the average yield curve of interest rates. Then, the estimation of Vasicek (1977) and CIR (1985) models generate an upward sloping yield curves and the Vasicek (1977) model shows more ability to replicate the stylised facts of the short-term interest rates in the