Yield Curve Smoothing

Carleton and Cooper (1976) describes an approach to estimating tetm structure that assumes default-free bond cash flows, payable on specified discrete dates, to each of which a set of unrelated discount factors are applied. These discount factors are estimated as regression coefficients, with the bond cash flows beir the independent variables and the bond price at each payment date the dependent variable. This type of simple linear regression produces a discrete discount fimction, not a continuous one. The forward-rate curves estimated from this fimction are accordir ly very ja ed. 

McCulloch (1971) proposes a more practical approach, using polynomial splines. This method produces a fimction that is both continuous and linear, so the ordinary least squares regression technique can be employed. A 1981 study by James Langetieg and Wilson Smoot, cited in Vasicek and Fong (1982), describes an extended McCulloch method that fits cubic splines to zero-coupon rates instead of the discount fimction and uses nonlinear methods of estimation. 

These considerations introduce what is known in statistics as error or noise into market prices. To handle this, smoothing techniques are used in the derivation of the discount function. 

FIGURE 5.2 Discount Function Derived from U.S. Treasury Prices on December P.r P003  

FIGURE 5.2 is the graph of the discount function derived by bootstrapping from the U.S. Treasury prices as of December 23, 2003- FIGURE 5.3 shows the zero-coupon yield and forward-rate curves corresponding to this discount function. Compare these to the yield curve in FIGURE 5.1 

FiRURE 5.1 II S. Treasury Yields to Maturity on December .% P003 

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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. 

It is commendable experimental procedure to repeat each run in duplicate and to be satisfied if the two results agree, but this is expensive in terms of the labor costs involved. Moreover, repetition of each run is not always necessary. For example, if one is studying the effect on the reaction rate of a variable such as temperature or reactant concentration, a series of experiments in which the parameter under investigation is systematically varied may be planned. If a plot of the results versus this parameter yields a smooth curve, one generally assumes that the reproducibility of the data is satisfactory. 

For an infinite data set (in which the symbols ft. and o as defined in Section 1.7.2 apply), a plot of frequency of occurrence vs. the measurement value yields a smooth bell-shaped curve. It is referred to as bell-shaped because there is equal drop-off on both sides of a peak value, resulting in a shape that resembles a bell. The peak value corresponds to /l, the population mean. This curve is called the normal distribution curve because it represents a normal distribution of values for any infinitely repeated measurement. This curve is shown in Figure 1.3. 

The value of this branching ratio measured directly from j8 -decay data is 0.30 (11) whereas a direct measure of the fission yields of Kr86 and Rb86 using mass spectrometry and isotope dilution gave a value of 0.28 (13). The assumption that the Kr88 yield falls on a smooth mass yield curve in the case of the thermal neutron fission of U236 therefore appears to be justified. 

From empirical investigations we know that the correlation should converge to unity as the difference in its maturities approaches zero. One the other hand, the correlation should vanish as the difference in the maturities goes to infinity. Another empirical implication is the relative smoothness of the observed forward rate curved Hence, we are able to separate the class of RF models according to the existence or absence of this smoothness property. Obviously, the non-differentiable class leads to non-smoothed forward rate curves, whereas the T-differentiable Random Fields enforces smoothed yield curves. Even if we restrict the number of admissible RF models to the non-differentiable Field dZ t,T) and the r-differentiable counterpart dU we obtain a new degree of freedom to improve the possible fluctuations of the entire term structure. 

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. 

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. 

For a smoothed spline, the level of oscillation is cmitrolled by setting a roughness penalty in the fimction, and not by reducing the number of node points. The yield curve / is chosen that minimises the objective function (5.15) ... 

In deriving the swap curve, the inputs should cover the complete term structure (i.e., short-, middle-, and long-term parts). The inputs should be observable, liquid, and with similar credit properties. Using an interpolation methodology, the inputs should form a complete, consistent, and smooth yield curve that closely tracks observed market data. Once the complete swap term structure is derived, an instrument is marked to market by extracting the appropriate rates off the derived curve. 

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. 

A common smoothing technique is linear interpolation. This approach fills in gaps in the market-observed yield curve caused by associated gaps in the set of observed bond prices by interpolating missing yields from actual yields. 

Cubic spline interpolation assumes that there is a cubic polynomial that can estimate the yield curve at each maturity gap. A spline can be thought of as a number of separate polynomials of the form y = f(X), where X is the complete range of the maturity term divided into user-specified segments that are joined smoothly at the knot points. Given a set... 

Stress yield strength It is the unit stress at which a material exhibits a specified permanent deformation. It is a measure of the useful limit of materials, particularly of those whose stress-strain curve in the region of yield is smooth and gradually curved. 

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