56 D.K.Spiess,J.Affleck-Graves Journal of Financial Economics 54 (1999)45-73 3.2.Post-offering performance of convertible debt issuers Table 3 also reports the distribution of holding-period returns for the convert- ible debt issuers.Like the straight debt issuers,these firms underperform their matched control firms.The median convertible debt issuer has a five-year holding-period return of only 3.5%,compared with the median matched firm return of 28.2%,and more than 57%of the sample firms underperform their matched counterparts.The -19.8%median difference in holding-period re- turns is significantly different from zero at the 0.01 level,and the fraction of sample firms that underperform their matched counterparts is significantly different from one half.For the convertible debt issuers,the average holding- period return is 23.2%,while the average holding-period return for their size-and-book-to-market-matched control firms is 60.1%;the -37.0%mean difference in holding-period returns is also significant at the 0.01 level but not statistically different from the mean value for the straight debt sample.This mean underperformance is,however,comparable to the -42.4%five-year underperformance that Spiess and Affleck-Graves(1995)report for seasoned equity issuing firms,which is consistent with interpreting convertible debt as an equity substitute. 4.Alternative models for measuring long-term performance Fama(1998)notes that using an inappropriate model to estimate abnormal returns can lead to significant bias in long-term studies,and he argues that prior long-run event studies show evidence of the bad model problem because differ- ent models of abnormal returns may produce different results and reasonable changes in the model specification even cause the abnormal performance to disappear in some cases.Although there is no way to avoid the potential of a bad model problem,we address this criticism by using four additional measures of long-run abnormal performance.The first two-the 'rolling port- folio'approach suggested by Fama(1998)and the Fama and French(1993) three-factor regression approach-are based on calendar-time averages of short-run abnormal returns.The second two the individual matched firm approach using alternative matching criteria and the benchmark portfolio approach of Lyon et al.(1998)-are based on event-time measures of long-run buy-and-hold returns. 4.1.Rolling portfolios of average monthly returns Fama (1998)notes that statistical issues such as extreme skewness of the computed returns(discussed in Barber and Lyon (1997)and Lyon et al.(1998)) and possible correlation of returns across events(discussed in Brav(1997))make3.2. Post-owering performance of convertible debt issuers Table 3 also reports the distribution of holding-period returns for the convert￾ible debt issuers. Like the straight debt issuers, these "rms underperform their matched control "rms. The median convertible debt issuer has a "ve-year holding-period return of only 3.5%, compared with the median matched "rm return of 28.2%, and more than 57% of the sample "rms underperform their matched counterparts. The !19.8% median di!erence in holding-period re￾turns is signi"cantly di!erent from zero at the 0.01 level, and the fraction of sample "rms that underperform their matched counterparts is signi"cantly di!erent from one half. For the convertible debt issuers, the average holding￾period return is 23.2%, while the average holding-period return for their size-and-book-to-market-matched control "rms is 60.1%; the !37.0% mean di!erence in holding-period returns is also signi"cant at the 0.01 level but not statistically di!erent from the mean value for the straight debt sample. This mean underperformance is, however, comparable to the !42.4% "ve-year underperformance that Spiess and A%eck-Graves (1995) report for seasoned equity issuing "rms, which is consistent with interpreting convertible debt as an equity substitute. 4. Alternative models for measuring long-term performance Fama (1998) notes that using an inappropriate model to estimate abnormal returns can lead to signi"cant bias in long-term studies, and he argues that prior long-run event studies show evidence of the bad model problem because di!er￾ent models of abnormal returns may produce di!erent results and reasonable changes in the model speci"cation even cause the abnormal performance to disappear in some cases. Although there is no way to avoid the potential of a bad model problem, we address this criticism by using four additional measures of long-run abnormal performance. The "rst two } the &rolling port￾folio' approach suggested by Fama (1998) and the Fama and French (1993) three-factor regression approach } are based on calendar-time averages of short-run abnormal returns. The second two } the individual matched "rm approach using alternative matching criteria and the benchmark portfolio approach of Lyon et al. (1998) } are based on event-time measures of long-run buy-and-hold returns. 4.1. Rolling portfolios of average monthly returns Fama (1998) notes that statistical issues such as extreme skewness of the computed returns (discussed in Barber and Lyon (1997) and Lyon et al. (1998)) and possible correlation of returns across events (discussed in Brav (1997)) make 56 D.K. Spiess, J. A{eck-Graves / Journal of Financial Economics 54 (1999) 45}73