Exhibit 4 is based on exactly the same data as Exhibit 3, but displays the data differently. On the vertical axis we show the ratio of effective corporate bond duration to effective synthetic Treasury bond duration, as a function of effective synthetic Treasury bond duration and yield spread. This chart shows that to compute the effective durations of default-prone bonds, one merely needs to compute the effective durations of matched synthetic Treasury bonds and then reduce them by the factors displayed on the vertical axis. (Recall again that the ratios do not reflect any call options that might be present. These would have to be taken into account separately.) In the case of bonds manifesting yield spreads as high as 600 b.p., the reduction in effective duration below that of the matched Treasury appears to approach 20 percent. Most of this reduction is due to credit risk, as will be shown later in Figure 7.

Exhibit 5 is constructed similarly to Exhibit 3, except that the duration measures are taken with respect to a different reference rate of interest. Here we show effective durations based on movements in the instantaneous riskless rate of interest. The term structure of interest volatility is based on the CIR model. The pattern shown is very similar to that in Exhibit 3, with the lowest effective corporate bond durations associated with the highest yield spreads. However, the effective duration measures are much lower, owing to the low volatility of long-term yields relative to that of the short rate. Exhibit 6 is similar to Exhibit 4, but is drawn with respect to movements in the short rate.

Finally, in Exhibit 7 we demonstrate for a 10-year bond how effective duration differs from a simpler derivative-based measure of modified duration. The top curve shows the reduction in duration purely from using increasing discount rates. The middle curve reflects the decline in modified duration that would stem from both increasing discount rates and rising coupon rates. Default risk is not included in the analysis of these top two curves apart from its impact on the yields and coupons. The lowest curve shows the effective duration measure on par corporate bonds, with the reference interest rates being the yields on matched synthetic Treasury bonds. The substantial difference between the lowest curve and the middle curve shows how biased upward the traditional measure of modified duration is for bonds subject to default. We suspect that had this difference been taken into account in the Blume, Keim and Patel [1991] study, their assessment of the relative riskiness of junk bonds would have been increased.

# **VI. Epilogue**

The consensus of all work in this area is that credit risk shortens effective duration of corporate bonds. In this paper we have provided estimates of how much durations shorten due to credit risk. Our estimates are based on observable data and easily estimated bond pricing parameters. Because our duration measures are taken with respect to movements in a common reference rate of interest, they can be used with greater confidence when attempting to compute the duration of a portfolio of bonds subject to varying degrees of credit risk.

Over the past three years, several hundred additional noncallable corporate bonds have been issued and traded. This should provide an even richer data set with which to refine and tune our corporate bond pricing model, and will strengthen our estimates for the impact of credit risk on effective duration. We are currently engaged in assembling these data and plan to report our results in a subsequent paper.

- Page 7 -