# **IV. Empirical Validation**

Model-based prices were compared to actual prices given in a data set of all noncallable corporate bonds issued between January 1977 and July 1991. These bonds are also free of other features such as sinking funds or convertibility that might complicate the analysis.<sup>12</sup> Only those bonds that showed a sufficient trading volume were included in our test, to avoid basing our pricing model on stale prices. The use of the simple bonds in this data set allows the focus to be on default risk and interest rate risk only. In order to estimate the other parameters of the model, default-free bonds have been synthesized. This was accomplished by estimating the term structures using "on-the-run" Treasury bonds. The default-free value of the corporate bond's cash flows was determined by discounting these promised cash flows using the Treasury term structures. They give the default-free price of the corporate cash flows and provide the necessary yield-to-maturity information that serves as a reference point from which to compute effective durations of default-prone bonds. Finally, equity data (which are used in the bond pricing model) for each of the firms in the data set were taken from the CRSP tapes.

In an empirical test by Merrill [1994], the model was shown to fit the corporate bond price data better than previous single-factor models, although it was biased slightly high with the bias increasing at higher prices. Therefore, the model prices were fine tuned to the data so that the model could be used with greater confidence when describing the relationship between risky bond value and interest rates. This results in unbiased model prices, which are depicted in Exhibit 2. (A perfect model fit would have all data points line up along the **450** line.) This corporate bond pricing model, fine-tuned to reflect actual corporate bond prices, was then used to estimate the effective corporate bond durations disclosed in the remainder of this study.

# **V. Computation and Comparison of Duration Measures**

In Exhibit 3, we show the effective durations of default-prone bonds relative to those of matched synthetic Treasury bonds. Note that higher yield spreads over the matched Treasuries are associated with lower effective durations relative to those on the matched Treasuries. These effective duration measures are analogous to the modified duration measure in one way, but different in two key respects. They are similar in that they relate the percent change in bond price to change in yield. They are different in that the percent change in this effective duration measure is estimated numerically based on a price/yield curve that incorporates default risk, rather than based on a simple derivative of price relative to yield. Moreover, the effective duration measures of both the corporate and the synthetic Treasury bonds are derived with respect to a change in the yield of the matched synthetic Treasury bonds, whereas Modified duration is typically related to a bond's own yield. By having the effective durations based on changes in the same reference yields, it is easier to compare the interest rate sensitivities of the two kinds of bonds, to isolate the impact of credit risk, and to compute portfolio duration when both kinds of bonds are present.

I<sup>2</sup> The problem of call options, sinking funds, and convertibility has rendered previous empirical studies of risky bond duration, which have not been based on this kind of data set, unable to isolate the impact of pure default risk on duration. See, for example, Fons [1990]. In his sample of 702 "low-rated" corporate bonds, Fons [ 1987] found that over 95% had call provisions. Another difference between this study and that of Fons is that he uses bond ratings to categorize the riskiness of corporate bonds, whereas we use the yield spread between the corporate bond and the synthetic Treasury bond. This allows our duration measures to reflect the most recent market information available about the credit riskiness of the corporate bonds, and also accounts for changing spreads between the Treasury yields and corporate bond yields of a given credit category.

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