Structural Models of Corporate Bond Pricing: An Empirical Analysis

This article empirically tests five structural models of corporatebond pricing: those of Merton (1974), Geske (1977), Longstaffand Schwartz (1995), Leland and Toft (1996), and Collin-Dufresneand Goldstein (2001). We implement the models using a sampleof 182 bond prices from firms with simple capital structuresduring the period 1986–1997. The conventional wisdom isthat structural models do not generate spreads as high as thoseseen in the bond market, and true to expectations, we find thatthe predicted spreads in our implementation of the Merton modelare too low. However, most of the other structural models predictspreads that are too high on average. Nevertheless, accuracyis a problem, as the newer models tend to severely overstatethe credit risk of firms with high leverage or volatility andyet suffer from a spread underprediction problem with saferbonds. The Leland and Toft model is an exception in that itoverpredicts spreads on most bonds, particularly those withhigh coupons. More accurate structural models must avoid featuresthat increase the credit risk on the riskier bonds while scarcelyaffecting the spreads of the safest bonds.     

Risk-neutral and actual default probabilities with an endogenous bankruptcy jump-diffusion model

This paper focuses on historical and risk-neutral default probabilities in a structural model, when the firm assets dynamics are modeled by a double exponential jump diffusion process. Relying on the Leland [(1994a) Journal of Finance, 49, 1213–1252; (1994b) Bond prices, yield spreads, and optimal capital structure with default risk. Working paper no. 240, IBER, University of California, Berkeley] or Leland and Toft [(1996) Journal of Finance, 51(3), 987–1019] endogenous structural approaches, as formalized by Hilberink and Rogers [(2002) Finance and Stochastics, 6(2), 237–263], this article gives a coherent construction of historical default probabilities. The risk-neutral world where evolve the firm assets, modeled by a class of geometric Lévy processes, is constructed based on the Esscher measure, yielding useful and new analytical relations between historical and risk-neutral probabilities. We do a complete numerical analysis of the predictions of our framework, and compare these predictions with actual data. In particular, this new framework displays an enhanced predictive power w.r.t. current Gaussian endogenous structural models.      

A bridge from ruin theory to credit risk

The structural model uses the firm-value process and the default threshold to obtain the implied credit spread. Merton’s (J Finance 29:449–470, 1974) credit spread is reported too small compared to the observed market spread. Zhou (J Bank Finance 25:2015–2040, 2001) proposes a jump-diffusion firm-value process and obtains a credit spread that is closer to the observed market spread. Going in a different direction, the reduced-form model uses the observed market credit spread to obtain the probability of default and the mean recovery rate. We use a jump-diffusion firm-value process and the observed credit spread to obtain the implied jump distribution. Therefore, the discrepancy in credit spreads between the structural model and the reduced-form model can be removed. From the market credit spread, we obtain the implied probability of default and the mean recovery rate. When the solvency-ratio process in credit risk and the surplus process in ruin theory both follow jump-diffusion processes, we show a bridge between ruin theory and credit risk so that results developed in ruin theory can be used to develop analogous results in credit risk. Specifically, when the jump is Logexponentially distributed, it results in a Beta distributed recovery rate that is close to market experience. For bonds of multiple seniorities, we obtain closed-form solutions of the mean and variance of the recovery rate. We prove that the defective renewal equation still holds, even if the jumps are possibly negative. Therefore, we can use ruin theory as a methodology for assessing credit ratings.      

Leverage, options liabilities, and corporate bond pricing

The two major problems with typical structural models are the failure to attain a positive credit spread in the very short term, and overestimation of the overall level of the credit spread. We recognize the presence of option liabilities in a firm’s capital structure and the effect they have on the firm’s credit spread. Including option liabilities and employing a regime switching interest rate process to capture the business cycle resolves the above-mentioned drawbacks in explaining credit spreads. We find that the credit spread overestimation problem in one of the structural models, Collin-Dufresne and Goldstein (J Finan 56:1929–1957, 2001), can be resolved by combining option liabilities and the regime-switching interest rate process when dealing with an investment grade bond, whereas with junk bonds, only the regime-switching interest rate process is needed. We also examine vulnerable option values, debt values, and zero-coupon bond values with different model settings and leverage ratios.      

An ODE approach for the expected discounted penalty at ruin in a jump-diffusion model

Under the assumption that the asset value follows a phase-type jump-diffusion, we show that the expected discounted penalty satisfies an ODE and obtain a general form for the expected discounted penalty. In particular, if only downward jumps are allowed, we get an explicit formula in terms of the penalty function and jump distribution. On the other hand, if the downward jump distribution is a mixture of exponential distributions (and upward jumps are determined by a general Lévy measure), we obtain closed-form solutions for the expected discounted penalty. As an application, we work out an example in Leland’s structural model with jumps. For earlier and related results, see Gerber and Landry [Insur. Math. Econ. 22:263–276, 1998], Hilberink and Rogers [Finance Stoch. 6:237–263, 2002], Asmussen et al. [Stoch. Proc. Appl. 109:79–111, 2004], and Kyprianou and Surya [Finance Stoch. 11:131–152, 2007].      

Structural models of corporate bond pricing with personal taxes

The structural approach offers an integrated framework to deal with yield spreads and default probability simultaneously. However, structural models perform poorly in predicting corporate bond spreads. It is unclear whether this poor performance is caused by characteristics of individual models, missing factors, or different calibration procedures. This study evaluates the performance of four structural models by incorporating two important factors, personal taxes and the liquidity factor, and calibrating these models to data. To ensure our results are not contingent on the calibration method, we further apply the maximum likelihood estimation method to a large sample of individual bonds. Results consistently show that the ability of structural models to predict spreads improves considerably when personal taxes and liquidity are taken into account. Our findings suggest that the poor performance of standard structural models is more likely due to missing factors than the characteristics of individual models or the calibration procedure.     

Principles of smooth and continuous fit in the determination of endogenous bankruptcy levels

We revisit the previous work of Leland [J Finance 49:1213–1252, 1994], Leland and Toft [J Finance 51:987–1019, 1996] and Hilberink and Rogers [Finance Stoch 6:237–263, 2002] on optimal capital structure and show that the issue of determining an optimal endogenous bankruptcy level can be dealt with analytically and numerically when the underlying source of randomness is replaced by that of a general spectrally negative Lévy process. By working with the latter class of processes we bring to light a new phenomenon, namely that, depending on the nature of the small jumps, the optimal bankruptcy level may be determined by a principle of continuous fit as opposed to the usual smooth fit. Moreover, we are able to prove the optimality of the bankruptcy level according to the appropriate choice of fit.      

No-arbitrage in Discrete-time Markets with Proportional Transaction Costs and General Information structure

We discuss the no-arbitrage conditions in a general framework for discrete-time models of financial markets with proportional transaction costs and general information structure. We extend the results of Kabanov et al. (Finance Stoch 6(3):371–382, 2002; Finance Stoch 7(3):403–411, 2003) and Schachermayer (Math Finance 14(1):19–48, 2004) to the case where bid-ask spreads are not known with certainty. In the “no-friction” case, we retrieve the result of Kabanov and Stricker (Preprint 2003). Additionally, we propose a new modelization based on simple orders which appears to be powerful whatever the information structure is.     

A Note on Credit Risk of Vertical <Emphasis Type="Italic">Keiretsu</Emphasis> Firms: Preliminary Evidence from the Japanese Automobile Industry

This paper empirically examines the relationship between the credit risk of Toyota, Nissan and Honda keiretsu-affiliated firms and the credit risk of the respective parent company. As credit spread data for keiretsu-affiliated firms were not available we create a keiretsu default index, as a proxy, using expected default probabilities obtained from the KMV and Leland and Toft (J. Finance 51, 987–1019, 1996) option pricing models. We find parent credit spreads do not Granger cause our keiretsu default index and vice versa in a bivariate vector autoregressive (VAR) framework.JEL classification: G3, L62     

Heterogeneous Agents and the Indifference Pricing of Property Index Linked Swaps

Swap spreads predicted by the traditional risk-neutral valuation models are much lower than the quoted market spreads for property index linked swaps (Patel and Pereira, Journal of Real Estate Finance and Economics, 36:5–21, 2008). This paper attempts to develop a utility indifference-based model for evaluating the reservation spreads of swap receivers and payers based on the principle of expected wealth utility equivalence rather than the traditional risk-neutral argument. Under the proposed model framework, this paper addresses the determination of the swap spreads. When the incompleteness of real estate markets and heterogeneity of representative agents are taken into consideration, it is shown that the agents’ risk preferences and heterogeneous beliefs about expected future property returns are the remarkable determinants for the swap spreads. Our model also identifies market power and the settlement rules in the event of counterparty default as important factors in determining the swap spreads. Our model provides a possible interpretation for the difference between the spreads predicted by the traditional models and the actual market spreads.     

Can long-term performance plans mitigate the negative effects of stock consideration and high cash for acquirers?

Following Travlos (J Finance 42: 943–963, 1987), Loughran and Vijh (J Finance 52: 1765–1790, 1997), Harford (J Finance 54: 1969–1997, 1999), and Oler (Rev Acc Stud 13: 479–511, 2008), we investigate whether acquisitions involving stock consideration and acquirers with high cash levels are associated with poor performance or not. In addition, we investigate whether including a long-term performance plan in top management’s compensation package can mitigate these negative effects. We find that acquirers with a long-term performance plan are less likely to hold a high cash balance and are less likely to use stock consideration, thus avoiding scenarios that are more likely to be value-destructive. Even if an acquirer with a long-term performance plan carries a high cash balance or uses stock, we find that the plan is associated with improved fundamental performance; however, this relationship does not flow through to improved post-acquisition returns.     

Funds of hedge funds: performance, risk and capital formation 2005 to 2010

Using a comprehensive data set of funds-of-hedge funds, we extend the results of Fung et al. (J. Finance 63:1777–1803, 2008) (FHNR) with an augmented version of the Fung and Hsieh (Financ. Anal. J. 60:65–80, 2004a; J. Empir. Finance 18:547–569, 2004b) model to document performance characteristics from January 2005 to December 2010. We find that our sample period is divided into three distinct subperiods: January 2005 to June 2007 (pre-subprime crisis); July 2007 to March 2009; and April 2009 to December 2010 (post-credit crunch) during which the average fund of hedge funds delivered positive alpha only in the first subperiod. We divide the funds of hedge funds sample into those who have alpha and the rest, which we call beta-only. The empirical results show a dramatic decline in the population of alpha producing funds of hedge funds post 2008 compared to the FHNR findings. When we repeat our analysis with a synthetic hedge fund index replicator, we find qualitatively similar results.     

A counter-example to an option pricing formula under transaction costs

In the paper by Melnikov and Petrachenko (Finance Stoch. 9: 141–149, 2005), a procedure is put forward for pricing and replicating an arbitrary European contingent claim in the binomial model with bid-ask spreads. We present a counter-example to show that the option pricing formula stated in that paper can in fact lead to arbitrage. This is related to the fact that under transaction costs a superreplicating strategy may be less expensive to set up than a strictly replicating one.     

Valuation of default-sensitive claims under imperfect information

We propose a valuation method for financial assets subject to default risk, where investors cannot observe the state variable triggering the default but observe a correlated price process. The model is sufficiently general to encompass a large class of structural models and can be seen as a generalization of the model of Duffie and Lando (Econometrica 69:633–664, [2001]). In this setting we prove that the default time is totally inaccessible in the market’s filtration and derive the conditional default probabilities and the intensity process. Finally, we provide pricing formulas for default-sensitive claims and illustrate in particular examples the shapes of the credit spreads.      

Accounting-based versus market-based cross-sectional models of CDS spreads

Models of financial distress rely primarily on accounting-based information (e.g. [Altman, E., 1968. Financial ratios, discriminant analysis and the prediction of corporate bankruptcy. Journal of Finance 23, 589–609; Ohlson, J., 1980. Financial ratios and the probabilistic prediction of bankruptcy. Journal of Accounting Research 19, 109–131]) or market-based information (e.g. [Merton, R.C., 1974. On the pricing of corporate debt: The risk structure of interest rates. Journal of Finance 29, 449–470]). In this paper, we provide evidence on the relative performance of these two classes of models. Using a sample of 2860 quarterly CDS spreads we find that a model of distress using accounting metrics performs comparably to market-based structural models of default. Moreover, a model using both sources of information performs better than either of the two models. Overall, our results suggest that both sources of information (accounting- and market-based) are complementary in pricing distress.     

To what extent are investment bank-differentiating factors relevant for firms floating moderate-sized IPOs?

One explanation provided for the relatively high and increasingly stable spreads for moderate-sized IPOs ($20–$80 million) documented in Chen and Ritter (J Finance 55:1105–1131, 2000) is that issuing firms focus less on price and more on a combination of investment bank-differentiating factors (such as underwriter prestige, analyst coverage, industry expertise, under-pricing, price stabilization activities, liquidity provision, and so on), and banks use industry-based differentiation as a source of market power. Using a new approach developed in a model of firm location choice due to Ellison and Glaeser (J Politi Econ 105:889–927, 1997), this paper presents some evidence on the combined relevance of such bank-differentiating factors, over and above bank size, for firms choosing investment banks for floating IPOs. For moderate-sized IPOs, there is a little, but not much evidence that such factors are a good explanation for high and increasingly stable spreads. Other than in a few of the largest industries, bank-differentiating factors are not significantly relevant for a large proportion of industries. Moreover, one aggregate measure of differentiation is declining over time. We are grateful to Preston McAfee for suggesting this approach to the problem, to Jay Ritter for providing an updated list of IPOs, and to Robert Anderson and anonymous referees for very helpful comments. We are grateful to seminar audiences at the University of Texas at Austin and Macalester College for helpful comments.     

Optimal capital structure and endogenous default

In a sequence of fascinating papers, Leland and Leland and Toft have investigated various properties of the debt and credit of a firm which keeps a constant profile of debt and chooses its bankruptcy level endogenously, to maximise the value of the equity. One feature of these papers is that the credit spreads tend to zero as the maturity tends to zero, and this is not a feature which is observed in practice. This defect of the modelling is related to the diffusion assumptions made in the papers referred to; in this paper, we take a model for the value of the firm's assets which allows for jumps, and find that the spreads do not go to zero as maturity goes to zero. The modelling is quite delicate, but it just works; analysis takes us a long way, and for the final steps we have to resort to numerical methods.     

Analytical approximations for the critical stock prices of American options: a performance comparison

Many efficient and accurate analytical methods for pricing American options now exist. However, while they can produce accurate option prices, they often do not give accurate critical stock prices. In this paper, we propose two new analytical approximations for American options based on the quadratic approximation. We compare our methods with existing analytical methods including the quadratic approximations in Barone-Adesi and Whaley (J Finance 42:301–320, 1987) and Barone-Adesi and Elliott (Stoch Anal Appl 9(2):115–131, 1991), the lower bound approximation in Broadie and Detemple (Rev Financial Stud 9:1211–1250, 1996), the tangent approximation in Bunch and Johnson (J Finance 55(5):2333–2356, 2000), the Laplace inversion method in Zhu (Int J Theor Appl Finance 9(7):1141–1177, 2006b), and the interpolation method in Li (Working paper, 2008). Both of our methods give much more accurate critical stock prices than all the existing methods above.     

On the equivalence of a class of affine term structure models

In specifying a finite factor model for the term structure of interest rates, one usually begins by modeling the dynamics of the underlying factors. In most cases, this is sufficient to completely determine the term structure model. However, a point that is often overlooked is that seemingly different specifications of the factor dynamics may generate indistinguishable term structure models, in the sense that they produce pathwise identical bond prices. Consequently, it is important to be able to determine, at the level of factor dynamics, the conditions under which the models they generate are indistinguishable. In the case of time-homogeneous affine term structure models (ATSMs), such conditions were first described in Dai and Singleton (J Finance 55:1943–1978, 2000). In this paper, we formalize and extend their results to a class of time-inhomogeneous ATSMs, and obtain a simple method for determining the indistinguishability of these models in terms of the underlying factor dynamics.      

Do Credit Spreads Reflect Stationary Leverage Ratios?

Most structural models of default preclude the firm from altering its capital structure. In practice, firms adjust outstanding debt levels in response to changes in firm value, thus generating mean-reverting leverage ratios. We propose a structural model of default with stochastic interest rates that captures this mean reversion. Our model generates credit spreads that are larger for low-leverage firms, and less sensitive to changes in firm value, both of which are more consistent with empirical findings than predictions of extant models. Further, the term structure of credit spreads can be upward sloping for speculative-grade debt, consistent with recent empirical findings.