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Priced in Corporate Bonds?

Joost Driessen

University of Amsterdam

This Version: March, 2002

I thank Frank De Jong, Siem-Jan Koopman, Bertrand Melenberg, Theo Nijman, Kenneth Singleton, and an

anonymous referee for many helpful comments and suggestions. I also thank seminar participants at the 2001

ESSFM meeting in Gerzensee, INSEAD, Tilburg University, the Tinbergen Institute, NIB Capital Management

and ABN-AMRO Bank for their comments.

This is a revision of an earlier paper that was titled The Cross-Firm Behaviour of Credit Spread Term Structures.

Joost Driessen, Finance Group, Faculty of Economics and Econometrics, University of Amsterdam, Roetersstraat

11, 1018 WB, Amsterdam, The Netherlands. Tel: +31-20-5255263. E-mail: [email protected]

Is Default Event Risk

Priced in Corporate Bonds?

Abstract

We identify and estimate the sources of risk that cause corporate bonds to earn an

excess return over default-free bonds. In particular, we estimate the risk premium

associated with a default event. Default is modelled using a jump process with

stochastic intensity. For a large set of firms, we model the default intensity of each

firm as a function of common and firm-specific factors. In the model, corporate bond

excess returns can be due to risk premia on factors driving the intensities and due to

a risk premium on the default jump risk. The model is estimated using data on

corporate bond prices for 104 US firms and historical default rate data. We find

significant risk premia on the factors that drive intensities. However, these risk

premia cannot fully explain the size of corporate bond excess returns. Next, we

estimate the size of the default jump risk premium, correcting for possible tax and

liquidity effects. The estimates show that this event risk premium is a significant and

economically important determinant of excess corporate bond returns.

JEL Codes: E43; G12; G13.

Keywords: Credit Spread; Default Event; Corporate Bond; Credit Derivative; Intensity Models.

1 Introduction

Given the extensive literature on risk premia in equity markets, relatively little is known about

expected returns and risk premia in the corporate bond market. Recent empirical evidence by

Elton et al. (2001) suggests that corporate bonds earn an expected excess return over default-free

government bonds, even after correcting for the likelihood of default and tax differences. As

shown by Elton et al. (2001), part of this expected excess return is due to the fact that changes

in credit spreads (if no default occurs) are systematic, implying that the risk of these changes

should be priced. The current empirical literature has, however, neglected the possibility that the

risk associated with the default event itself is (also) priced. Typically, a default event causes a

jump in bond prices and this jump risk may have a risk premium. Jarrow, Lando, and Yu (JLY,

2001) and Yu (2001) discuss the possible existence of a default jump risk premium, but do not

estimate the size of this premium.

In this paper, we distinguish the risk of credit spread changes, if no default occurs, and the

risk of the default event itself. We use credit spread data of many different firms and historical

default rates to estimate the size of the default jump risk premium, along with the risk prices of

credit spread changes. We show that, in order to fully explain the size of expected excess

corporate bond returns, an economically and statistically significant default jump risk premium

is necessary, on top of the risk premia that are due to the risk of credit spread changes.

By estimating the default jump risk premium, this paper essentially tests the assumptions

underlying the conditional diversification hypothesis of JLY (2001). These authors prove that,

if default jumps are conditionally independent across firms and if the economy contains an

infinite number of bonds, default jump risk cannot be priced. Intuitively, in this case the default

jump risk can be fully diversified. Our results indicate that default jumps are not conditionally

independent across firms and/or that not enough corporate bonds are traded to fully diversify

default jump risk. A particularly appealing explanation for the existence of a default jump risk

premium is that investors take into account the possibility of a multiple defaults scenario (a

contagious defaults scenario).

The model that we use is specified according to the Duffie and Singleton (1999) framework.

In these intensity-based models, firms can default at each instant with some probability. In case

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of a default event, there is a downward jump in the bond price that equals a loss rate times the

bond price just before default. The product of the risk-neutral default intensity and the loss rate

equals the instantaneous credit spread. Like Duffee (1999) and Elton et al. (2001), we assume

a constant loss rate and allow the default intensity to vary stochastically over time. We model

each firms default intensity as a function of a low number of latent common factors and a latent

firm-specific factor. This extends the analysis of Duffee (1999), who estimates a separate model

for each firm. As in Duffee (1999), all factors follow square-root diffusion processes. We use

a latent factor model, since Collin-Dufresne et al. (2001) show that observable financial and

economic variables cannot explain the correlation of credit spread changes across firms. In line

with empirical evidence provided by Longstaff and Schwartz (1995) and Duffee (1998), the

model also allows for correlation between credit spreads and default-free interest rates, which

are modelled by a two-factor affine model used by Duffie, Pedersen, and Singleton (2001).

Finally, we model the relation between risk-neutral and actual default intensities. The ratio of

the risk-neutral default intensity and the actual intensity defines the jump risk premium, which

we assume to be constant over time.

In total, the model can generate expected excess corporate bond returns in four ways. First,

through the dependence of credit spreads (or, equivalently, default intensities) on default-free

term structure factors. Second, because the risk of common or systematic changes in credit

spreads across firms is priced. Third, via a risk premium on firm-specific credit spread changes,

and, fourth, due to a risk premium on the default jump.1 Empirically, we find that all these terms

contribute to the expected excess corporate bond return, except for the risk of firm-specific credit

spread changes.

We use a data set of weekly US corporate bond prices for 592 bonds of 104 firms, from 1991

to 2000. All bonds in the data set are rated investment-grade. The estimation methodology

consists of four steps. First, using data on Treasury bond yields, we estimate the two-factor

model for the default-free term structure using Quasi Maximum Likelihood based on the Kalman

filter. Second, we estimate the common factor processes that influence corporate bond spreads

of all firms, again using Quasi Maximum Likelihood based on the Kalman filter. Third, the

1Yu (2001) also provides a decomposition of corporate bond returns, but does not estimate the size of the

components.

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residual bond pricing errors are used to estimate the firm-specific factor for each firm. In the

final step, we use data on historical default rates to estimate the default jump risk premium.

The empirical results are as follows. We estimate a model with two common factors and a

firm-specific factor for each firm. The common factors are statistically significant and reduce

the corporate bond pricing errors. These factors have economically and statistically significant

risk prices, while the risk associated with the firm-specific factors of our model is not priced.

Thus, our results indicate that the market-wide spread risk, represented by movements in the

common factors, is priced in the corporate bond prices, whereas the firm-specific risk is not. We

also find a negative relation between credit spreads and the default-free term structure.

Next we show that, if we would not include a default jump risk premium in this model, the

model largely overestimates observed default rates, and, therefore, underestimates expected

excess corporate bond returns. Subsequently, we estimate the size of the default jump risk

premium using historical default rate data, and find an economically and statistically large value

for this parameter. For example, the default jump risk premium accounts for about 68% of the

total expected excess return on a 10-year BBB rated corporate bond. If we correct for tax and

liquidity differences between corporate and government bonds, the estimate for the risk premium

remains economically important and, in most cases, statistically significant.

Our results on the default risk premium are somewhat different from the results on the test

of conditional diversification in JLY (2001), who use the estimates of the Duffee (1999) model.

The main reason for these differences is that JLY (2001) do not use historically observed

cumulative default rates to perform their test, but the cumulative default rates implied by a

Markov model for rating migrations. The observed cumulative default rates are, however, much

lower than these model-implied default rates. Using cumulative default probabilities that are

based on the Markov migration model therefore leads to downward biased estimates of the

default jump risk premium.

We end the paper with an application of our model to the pricing of a nth-to-default swap.

This application highlights the importance of a multiple defaults scenario. Incorporating such

a scenario leads to a large change in the price for a credit default swap, relative to a model with

independent default events. Finally, we note that another practical application of our model is

that it allows financial institutions to extract actual default probabilities from corporate bond

prices, which is useful for risk management purposes.

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The remainder of the paper is organized as follows. Section 2 introduces the model. Section

3 describes the corporate bond data set. In Section 4, the estimation methodology for the factor

model is outlined, and the estimation results for the factor model are presented. In Section 5, we

discuss the estimation of the default jump risk premium and present the results, as well as

corrections for tax and liquidity effects. In Section 6 we apply our model to price basket credit

default swaps. Section 7 concludes.

2 A Model for Defaultable Bond Prices

2.1 Model Setup

The first part of the model describes default-free interest rates. We assume that US Treasury

bonds cannot default. This part of the model is identical to the affine model for the default-free

term structure of Duffie, Pedersen, and Singleton (DPS, 1999). The model implies the following

process for the instantaneous default-free short rate r under the true or actual probability