Calibrating structural models: a new methodology based on stock and credit default swap data

Abstract

This paper presents a modified version of Leland and Toft's [J. Finance, 1996, 51, 987–1019] structural credit risk model, together with a novel calibration methodology based on stock and CDS data: the firm asset value and volatility are consistently derived from equity prices; the default barrier is calibrated from CDS premia. It empirically shows that as long as the appropriate default barrier is selected, the model generates time series of stock market implied credit spreads that fit the times series of CDS spreads. Moreover, CDS implied default barriers prove to be consistent with stockholders’ rationality, with predictions made by structural models with endogenous default, and with historical recovery rates.

Keywords:

• Structural credit risk models
• Calibration
• Default barrier

Acknowledgements

This paper was partially drafted during my visit to the Department of Finance at Tilburg University. I acknowledge financial support from Banco Sabadell and MEC Grants Refs AP2000-1327 and BEC2002-0279. I thank Juan Ignacio Peña, Philippe Gagnepain, Hao Wang, Bas Werker, Max Bruche, Carmen Ansotegui, Lidija Lovreta, and three anonymous referees for helpful suggestions. I also acknowledge comments from seminar audiences at Universidad Carlos III de Madrid, Banco de España, ESADE Business School, XIII Foro de Finanzas, C.R.E.D.I.T. 2005 Conference, EFMA Conference 2006, and FMA European Conference 2007. I am also grateful to Banco Santander for allowing access to their data on CDS spreads. The usual disclaimers apply.

Notes

†It is clear that identifying all the bankruptcy costs with lawyers’ fees is rather restrictive, but seems to be a useful way of thinking of such costs.

†It seems useful to clarify at this point the extent to which the term ‘modified version of the LT model’ is appropriate for a model that avoids dealing with taxes and endogenous default barriers. As an example of the controversy that this term may generate, Leland (2004) refers to Longstaff and Schwartz (1995) to characterize models of the LT type, but with exogenous default boundary. In this paper we simply use the term ‘modified version of the LT model’ to acknowledge that one of the main building blocks in our model, i.e. expression (2) for the value of an individual bond, has been directly taken from the original LT model. Of course, we should expect other models with similar characteristics (e.g. Longstaff and Schwartz's model with constant interest rates) to produce similar results.

‡At the time of issuing debt, the existence of bankruptcy costs will result in creditors claiming a greater interest rate, implying an indirect loss for the shareholders.

§ We prefer using swap rates instead of government bond yields because several studies (Hull et al. 2004, Blanco et al. 2005, Houweling and Vorst 2005, Longstaff et al. 2005) suggest that they are a better proxy for the reference risk-free rate in the CDS market than government bond yields. It is worth mentioning that swap rates have probably lost their ‘close to risk-free and highly liquid’ profile since August 2007; however, this is far from affecting our sample.

†The empirical analysis in section 4 is based on a standard five-year contract.

†In the case of Ford Motor Credit Co. and General Motors Accept. Corp., we collect parent company data.

‡As a robustness check, the main estimations are repeated assuming accounting data to be constant and equal to those reported at close of year 2000. Results (available on request) are not materially affected by this alternative specification.

§The number of firms is not particularly large. However, it is in line with other published studies relating the time series of CDS and bond spreads: the number of non-financial companies is 18 in Blanco et al. (2005), 40 in Norden and Weber (2009) and 16 in Zhu (2006). Moreover, we impose the additional restriction of working with the same (local) currency in the three markets.

†We should remind the reader, however, that predictions of Leland and Toft's (1996) original model of Eom et al. (2004) are compared with corporate-government bond yield differentials, not with CDS premia.

†CDS for Olivetti were included in the original CDS data base.

†For robustness, we include in the regression only company-rating pairs with at least 50 observations.

‡We now omit the leaps between different half-yearly periods in the estimation of σ. This is to prevent any possible bias due to jumps in the value of β.

§All half-yearly periods contain at least 50 observations, while the typical number of observations is in the order of 120.