University of Nebraska-Lincoln College of Business

University of Nebraska-Lincoln College of Business Administration The Estimation of Corporate Bond Yield Curves as a Function of Term to Maturity and Coupon Author(s): Tom Barnes and David A. Burnie Source: Quarterly Journal of Business and Economics, Vol. 26, No. 4 (Autumn, 1987), pp. 50-64 Published by: University of Nebraska-Lincoln College of Business Administration Stable URL: http://www.jstor.org/stable/40472900 Accessed: 03-08-2015 16:43 UTC

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THE ESTIMATION OF CORPORATE BOND YIELD CURVES AS A FUNCTION OF TERM TO MATURITY AND COUPON* Tom Barnes Brock University David A. Burnie Universityof Tennessee Abstract This paper builds before tax corporate bond yield curves. The buildingof yield curves is an importantfirststep for many types of financial research. To date, this research has focused on governmentbonds and, with the exception of works by Durand and Johnson,no recent attempt has been made to include corporates. Six regressionmodels are tested. The first model is naive and is based only on the asymptotic propertiesof upward and downwardsloping curves. The other five models are prominentin the publishedliterature on governmentbond yield curves. I. INTRODUCTION Term structurerefers to a vector of spot interest rates on bonds that differonly in the timing of interest paymentsand final maturity. A yield curve is a curve that is fittedthrougha samplingof observationsof yield to maturity(YTM) versus maturityfor any set of homogeneousbonds. Many consider these two definitions to be -synonymouswith each other; most testing of term structuretheoryhas been with yield curves developed from governmentbonds.

Several authors have pointed out that because of a coupon effect, yields to maturityare equivalent to spot rates only under certain restrictive conditions. Carr, Halpern,and McCallum [7] argue that

the conventionalyield curve is biased because it ignores duration,and that duration should replace maturityin the constructionof the yield curve. McCulloch [20], Schaefer [23], and Carleton and Cooper [6] use various proceduresto obtain spot rates directly. Even if yield curves are not perfect representationsof term structure, Chambers [8] concludes that they are close approximationsand, therefore,are usefulin financialresearch.

♦The authors acknowledge the helpful comments of anonymous referees. 50 0747-5535/87/1500-0050$01.00 copyrightUniversityof Nebraska-Lincoln


Yield curves have been useful for more than the testingof termstructure theories. A yield curve offers a convenient way to obtain yields to maturityand prices of nonpricedbonds. The buildingof these curves is an important first step that has facilitated research in such areas as fixed income securitypricing(and volatility),futurescontracts,contingentclaims, and portfoliomanagement. Livingstonand Jain [17] criticize some of this research. They argue that estimates of forwardrates and zero coupon rates derived from yield curves may not be accurate (particularlyfor long-term bonds). The firstsets of yield curves were drawnfreehandthrougha scatter of observations. The drawbackof this methodis its lack of objectivityand the fact that smooth curves are not reproducible. Subsequently, regression models or equations were used to fit smooth curves to a set of observed data. By fittinglower order polynomialsto the data, the result was a plot that correspondedroughlyto the smooth yield curve obtained by the freehand fitting. These regression models represented a major improvement because they could be evaluated statistically. But most of this work was done with governmentbonds. Only Durand [11] and Johnson[16] build and examine corporate bond yield curves (by the freehandmethod). McCulloch [20] includes corporates in his direct estimates of term structure. In general, the end result is that research with corporates has lagged that of governmentbonds.

^The best known procedure to estimate bond price is to discount a bondfscash flow at a specific yield to maturity. But there are a varietyof ways in which bond prices (yields to maturity) can be estimated. As mentionedpreviously,McCulloch [20] and others have used spline functions to make direct estimates of spot rates that then are used to price bonds. The use of spot rates eliminates the durationproblemcaused by the presence of coupon payments. Merrill Lynch uses anotherapproach knownas matrix pricing. Given normal market relationships,the prices of nontradedbonds are derived fromactual trade data. Finally, Brennanand Schwartz [5], in the spirit of the continuous time option pricing model, show that in equilibrium,bond prices can be determined by the endpoints of the yield curve. 51


This research builds before tax2 yield curves for Canadian corporate bonds at the beginningof each quarter over the period 1973 to 1982 for each of three rating classes. In analyzing the corporate bond market,two questions will be answered. First, does an ad hoc model or a model based on term structuretheoryperformbetter? An ad hoc model is one that is not based on economic theoryand is justifiedon the basis that it worksbest with a given data set. Second, can regressionmodels that have achieved success in buildinggovernmentbond yield curves do the same withcorporates? The well-known,regression models that have been used to build governmentbond yield curves have used either maturityor both maturityand coupon as explanatory variables. At issue is whether the existence of default risk makes corporates more difficult to model. Boardman and McEnally [3] use ordinaryleast squares (OLS) regressionsto test corporate bond price as a functionof the probabilityof call, sinkingfundstatus, security status, marketability,exchange listing, industrial classification, and market risk. Over the 16 regressions (one for each year for each of four differentrating classes), generally,the partial regressioncoefficients were not significantwith the rightsign (i.e., the only coefficientthat was significant and with the right sign in more than 25 percent of the cases was industrialclassification). These results implythat once default risk is controlled by consideringbonds only in each separate rating class, coupon and maturityare sufficientto determine corporate yields to maturity; therefore, the use of governmentbond models is appropriate in the buildingof corporate bond yield curves. This is not to deny the importance of the effect on yield to maturityif a specific feature of a corporate issue deviates frommost other bondsin the same ratingclass. II. MODELS Six regression models are used to develop corporate bond yield curves. The firstmodel is naive and is based onlyon the asymptoticproper-

2Schaefer [23] and McCulloch [21] show the influenceof taxes on the yield curve. But the effect of taxes is not clear. Life insurance companies and pension plans are heavy investorsin the corporate bond market; Reilly [22] notes they are "virtuallytax free institutions." Furthermore,there is no agreement on the specific tax rate to use in the creation of after tax yields. 52


ties of upward and downwardsloping curves. The other five models are prominentin the publishedliteratureon governmentbond yield curves. Each of these representsthe best model of a series the various authors were able to test and/ordevelop. A briefdescriptionof each model is presented. Equation 1 is a basic model that considers yield as a functionof the naturallog of maturityand coupon. (1) Y = a + B1lnT + B2C where: Y T C

= yield to maturity4 = numberof years to maturity = annual coupon rate

Under certain conditions,equation 1 can give a good fit if the scatter of actual points is one of the more basic shapes (i.e., ascending or descending). Fisher [14] builds yield curves for Britishgovernmentsecurities using a stepwise regressionprocedure. His independentvariables are coupon and maturity,including several transformationsof each. He argues that the differentformsof term to maturityallow for nonmonotonicshapes. Equation 2 is Fisherfsmodel. (2) Y = a + BXT + B2T2 + B3T3 + B4 (log10T) + B5C + BßC2 + B? (log10O McCallum [19] find equation 3 to give the best fit (average R2 = .80) for Canadian governmentbond data. Incidentally,this model is the one used by FRI InformationServices Ltd. (a well known Canadian data bank) to generate yield curves.

3Other naive models that are a functionof coupon and maturitythat representgeometric, inverse,and exponentialcurves also were run. Because model 1 outperformedthem,only its results will be reported. 4The use of