We propose a reduced-form model where jumps-to-default are priced because they generate a market-wide jump in credit spreads. While this framework is consistent with a counterparty risk interpretation (e.g., Jarrow and Yu (2001)), it is most naturally interpreted as an updating of beliefs due to an unexpected event. Simple analytic solutions are obtained for the prices of risky debt regardless of the number of firms that share in the contagious response. Empirically, we find that credit events of large firms generate a market wide increase in credit spreads and a significant ‘flight-to-quality’ response in the Treasury market. A calibration exercise suggests that the risk premium for contagion-risk may be considerable, whereas it implies that jump-to-default risk for a typical investment grade firm has an upper bound of only a few basis points.
Recent research has highlighted the fact that structural models of corporate bond prices are incapable of generating reasonable yield spreads (See Eom, Helwege and Huang (2004) and Huang and Huang (2003)). The problem is especially severe among investment-grade bonds with short maturities, where models tend to predict very low credit spreads (Duffie and Lando (2001, henceforth DL). While some suggest the main factors driving bond spreads are taxes and liquidity (e.g., Elton, Gruber, Aggarwal and Mann (2001) and Collin Dufresne, Goldstein and Martin (2001)), others focus on the risk of a jump to default (e.g., Driessen (2005), Berndt, Douglas, Duffie, Ferguson and Schranz (2007)). The latter literature uses the so-called reduced-form models, which directly specifies the jump to default intensity for individual firms. Conveniently, under certain restrictions, these reduced-form models allow to value risky cash-flows by simply discounting under the risk-neutral measure using a default-adjusted short rate. Therefore, defaultable bonds can be valued very similarly to risk-free bonds using standard affine or quadratic models for example.1 Further, by construction, reduced form models can fit any observed risky term structure of credit spreads, even at the short end. However, the implication of large credit spread at short maturities, is that the risk-neutral jump to default probability is high relative to observed historical jump-to default rates. The ratio of risk-neutral (i.e., price implied) default intensity to historical default intensity estimated in these studies is in the range [2,6]. In the reduced-form framework, this ratio can be explained entirely by a risk-premium associated with the actual jump to default itself. Here, we study the magnitude of these implied jump to default risk-premia and their economic implications.
While we observe few such jumps to default, the cost when they occur may explain a good deal of bond spreads. However, we note that the one-year default rate on investment-grade debt is very low, averaging only 0.17% over the past eighty years (and considerably smaller if the Depression era is excluded).2 To the extent that investment-grade firms default, they more often ‘limp’ to default, experiencing multiple spread increases over several years before finally defaulting (e.g., Western Union). In such situations, we would not expect a market-wide response at the default event, because it was not so unexpected. Not only is the one-year default rate quite low, but since 1937 only four firms have defaulted while carrying an investment-grade rating. And, three of the four firms’ bonds traded in the market prior to default as if they were no longer considered safe, which hardly constitutes what we think of as a jump to default.
Not only is it true jump to default extremely rare, but for jump risk to be priced, it must not be conditionally diversifiable (see Jarrow, Lando and Yu (2005)). One reason why the credit event of an individual firm may not be diversifiable is counterparty-risk. This occurs when the default of one firm causes the financial distress of entities with which it has close businessties. This distress in turn is transmitted to a second layer of firms through a domino effect. Jarrow and Yu (2001, henceforth JY) model the pricing of risky debt in the presence of counterparty risk with two parties. However, with only two firms, their framework says little about such a domino effect or spillovers to other industries where business ties are weak.
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Is Credit Event Risk Priced? Modeling Contagion via the Updating of Beliefs
