Yield curve fitting smoothing

Making comparison between bonds could be difficult and several aspects must be considered. One of these is the bond s maturity. For instance, we know that the yield for a bond that matures in 10 years is not the same compared to the one that matures in 30 years. Therefore, it is important to have a reference yield curve and smooth that for comparison purposes. However, there are other features that affect the bond s comparison such as coupon size and structure, liquidity, embedded options and others. These other features increase the curve fitting and the bond s comparison analysis. In this case, the swap curve represents an objective tool to understand the richness and cheapness in bond market. According to O Kane and Sen (2005), the asset-swap spread is calculated as the difference between the bond s value on the par swap curve and the bond s market value, divided by the sensitivity of 1 bp over the par swap. 

Curve fitting to data is most successhil when the form of the equation used is based on a known theoretical relationship between the variables associated with the data points, eg, use of the Clausius-Clapeyron equation for vapor pressure. In the absence of known theoretical relationships, polynomials are one of the most usehil forms to describe a curve. Polynomials are easy to evaluate the coefficients are linear and the degree, ie, the highest power appearing in the equation, is a convenient measure of smoothness. Lower orders yield smoother fits. 

Our focus is upon the fullerenes. In particular, the stability of C60 and C70 relative to the other fullerenes and to each other is of primary interest because these two species are the fullerenes produced in highest yield in carbon arcs and in combustion. Quite a few calculations mainly using various semi-empirical theories have been carried out to determine the stability of the fullerenes. Figure 2 shows the recent results of Scuseria s group (Strout et al. 1993), who carried out minimal basis set STO-3G SCF calculations on several fullerenes. The smooth curve fitted through... 

The fitted curve is a close approximation to the redemption yield curve, and is also very smooth. However, the fit is inaccurate at the very short end, indicating an underpriced 6-month bond, and also does not approximate the long end of the curve. For this reason, B-spline methods are more commonly used. 

Use of polynomial functions that pass through the observed market data points create a fitted smooth yield curve that does not oscillate wildly between observations. It is possible to either use a single, high order... 

Chapter 3 introduced the basic concepts of bond pricing and analysis. This chapter builds on those concepts and reviews the work conducted in those fields. Term-structure modeling is possibly the most heavily covered subject in the financial economics literature. A comprehensive summary is outside the scope of this book. This chapter, however, attempts to give a solid background that should allow interested readers to deepen their understanding by referring to the accessible texts listed in the References section. This chapter reviews the best-known interest rate models. The following one discusses some of the techniques used to fit a smooth yield curve to market-observed bond yields. 

Two general approaches of smoothing data include curve fitting and digital filtering. Both approaches can yield similar results however, the underlying theory behind each approach is different. 

Adams, K., and D. Van Deventer. 1994. Fitting Yield Curves and Forward Rate Curves with Maximum Smoothness, journal of Fixed Income 4, 52-62. 

X is an acidity function based on the first-order approximation, Eq. (8-92). Values of X have been assigned by an iterative procedure. The data consist of values of Cb/cbh+ as functions of Ch+ for a large number of indicators. For each indicator an initial estimate of pXbh+ and m is made and X is calculated with Eq. (8-94). This yields a large body of X values, which are fitted to a polynomial in acid concentration. From this fitted curve smoothed X values are obtained, and Eq. (8-94), a linear function in X. allows refined values of pXbh + and m to be obtained. This procedure continues until the parameters undergo no further change. Table 8-20 gives X values for sulfuric and perchloric acid solutions. ... 

In this case, fitting the concentration-response data to Equation (5.4) would yield a smooth curve that appears to ht well but with a Hill coefficient much less than unity. 

As shown in Figure 1 data from the viscometer detector and DRI are combined to yield the Intrinsic viscosity as a function of retention volume (la). This curve then was fit to a polynomial and a smoothed curved calculated. At this stage of data reduction the analyst can choose to continue to use the polynomial smoothed values of log [n](V) throughout, or continue to use the unsmoothed values for further data reduction. 

The simplest case, that of two large infrared lines, is shown in Fig. 31(a). A smooth curve was fitted to the base line as shown. A spline-fitting computer program developed by De Boor (1978) was used to obtain this fit very conveniently. After the fit was obtained, the data were adjusted to a flat base line, as shown in Fig. 31(b), and the data field was extended by padding with zeros to yield an overall data field of 28 = 256 points. Taking the Fourier transform, we obtained the interferogram function shown in Fig. 32. (Even though it was not obtained directly from the interferometer as recorded... 

If we plot the ionisation energies of A", A, A+, A2+ etc. against charge, a smooth curve is obtained and can be fitted to a polynomial or other simple function. Where no reliable value for E is available, a curve obtained from the ionisation energies 71 /2 etc. can be extrapolated to yield the ionisation energy of A-, i.e. —Ex for A. Uncertainties of 10-50 kJ mol-1 are estimated for this procedure. 

This is simply an empirical equation for fitting a smooth curve between the data points, where K0 and /[Pg.458]

Figure 31. Representative data showing OH formation following 236 nm CO2-HI excitation. The ordinates in (a) and (b) are the Qi,(l) and Qi,(6) LIF signals, respectively, while the abscissa is the delay time. The dashed curve is the response function of the laser system. The solid curves are the calculated best fits, assuming a two-parameter description for the time dependence of OH formation the best least-squares fits yielded (a) t, = 0.9 ps and Tj = 1.9 ps, and (b) x, = 0.7 ps and Xj = 1.1 ps (see Table 4). The points at the top are the residuals between the experimental points and the smooth fit. From Ref. 43 with permission of the Journal of Chemical Physics.

Fig. 14 Typical flow curves for a concentrated emulsion sheared between two rough surfaces closed circles the line is a fit to the Herschel-Bulkley equation), and between rough and smooth surfaces open circles hydrophilic glass surface open squares hydrophobic polymer surface). 7a denotes the apparent shear rate, and is the value of the apparent shear rate at the yield point (cr = CJy). CJs is the sticking yield stress below which the emulsion adheres to the surface...

The range of relaxation times allowed in the fitting was usually between 0.5 ps and 1 s with a density of 12 points per decade. Relaxation rates are obtained from the moments of the peaks in the relaxation time distribution or, if the peaks overlap, from the peak maximum position. With a broad distribution of relaxation times, these inversion methods yield multiple peaks in the "unsmoothed" analysis. The "smoothing" parameter (P) was selected as 0.5 in all cases, after it was established that the number of peaks did not increase with further increase in smoothing. As a further check, an analysis was made on a simulated correlation function consisting of a broad continuous distribution of relaxation times with noise added equal to the residuals from the analysis of the experimental correlation curve. REPES recovers the original distribution except when a very low smoothing parameter (P 0) is used. 

What should one do about anomalies of this kind, points which do not seem to lie close to a smooth curve If the experiment can be repeated, that should be done, and averaged or best data used. In the present instance, measurement at a few more frequencies between the present lowest and next lowest point would yield intermediate points which would help clarify whether the last point is badly off or not. If the experiment cannot be repeated, then outliers of appreciable magnitude, such as the lowest-frequency point in the present plot, should be omitted (or weighted very low) in subsequent CNLS fitting. 

In mathematics a spline is a piecewise polynomial function, made up of individual polynomial sections or segments that are joined together at (user-selected) points known as knot points. Splines used in term structure modeling are generally made up of cubic polynomials. The reason they are often cubic polynomials, as opposed to polynomials of order, say, two or five, is explained in straightforward fashion by de la Grandville (2001). A cubic spline is a function of order three and a piecewise cubic polynomial that is twice differentiable at each knot point. At each knot point the slope and curvature of the curve on either side must match. The cubic spline approach is employed to fit a smooth curve to bond prices (yields) given by the term discount factors. 

Data at three temperatures are replotted in Figure 6 along with smooth curves showing the results of the fit to the 1-D model. Values of q, the trap concentration obtained from Monte Carlo simulations %/Qm> ratio of intrinsic quantum yields and a, the emission probability parameter, for these three temperatures are as follows ... 

Semi-quantitative analysis facilitates fast and simple multi-element measurements with limited precision. ICP-MS offers excellent semi-quantitative capabilities, as a result of the high ionisation efficiencies achieved for the majority of elements and the simplicity of the resulting mass spectra. Semi-quantitative determinations are mostly based on a comparison of response tables and the actual count rates of the sample. The response (intensity I in counts/s) of an analyte ion depends on the concentration of the analyte element, the isotopic abundance of the observed isotope, the ionisation efficiency, the atomic mass and the efficiencies of nebuUsation, ion transmission and ion detection in the mass spectrometer. In most ICP-MS instraments a plot of the atomic response, Ra, versus atomic mass yields a smooth response curve, which is fitted best by a third order polynomial (Figure 4.3). The atomic response is defined by equation (4.7) and is equivalent to the molar response divided by the Avogadro constant, IVa-... 

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