# Credit spreads, endogenous bankruptcy and liquidity risk

- 338 Downloads

## Abstract

In this paper, we consider a bond valuation model with both credit risk and liquidity risk to show that credit spreads are not negligible for short maturities. We adopt the structural approach to model credit risk, where the default triggering barrier is determined endogenously by maximizing equity value. As for liquidity risk, we assume that bondholders may encounter liquidity shocks during the lifetime of corporate bonds, and have to sell the bond immediately at the price, which is assumed to be a fraction of the price in a perfectly liquid market. Under this framework, we derive explicit expressions for corporate bond, firm value and bankruptcy trigger. Finally, numerical illustrations are presented.

## Keywords

Liquidity risk Credit risk Credit spreads Endogenous bankruptcy## Mathematics Subject Classification (2000)

60H30 91G40 91G50## 1 Introduction

Issuing corporate bonds is an important way to finance for corporations, who commit themselves to make specified payments to bondholders. However, bondholders may suffer a financial loss, since there may exist the occurrence of default, possibly caused by the firm’s bankruptcy. A rapidly growing body of literature has focused on credit risk. One of the most widely studied framework is the structural approach, where a major issue is the modeling of the evolution of the firm’s value and of the firm’s capital structure. The works devoted to the structural approach include Merton (1974), Black and Cox (1976), Leland (1994), Longstaff and Schwartz (1995), Leland and Toft (1996), Anderson and Sundaresan (1996), Anderson and Sundaresan (2000), etc.

Leland (1994) considers optimal capital structure and the pricing of debt with infinite maturity and credit risk, and gives closed-form results for the value of long-term risky debt and yield spreads. Leland and Toft (1996) extend these results to the case where the debt has finite maturity. They examine the optimal capital structure of a firm that can choose the amount of its debt and its bankruptcy level endogenously. One feature of these papers is that the credit spreads tend to zero as the maturity decreases to zero, which is inconsistent with empirical findings. Furthermore, empirical findings show that levels of credit spreads obtained under most structural models are underestimated (see, e.g., Jones et al. 1984; Delianedis and Geske 2001; Huang and Huang 2002; Yu 2002; Agliardi and Agliardi 2009). Delianedis and Geske (2001) analyze the components of corporate credit spreads and they conclude that credit risk and credit spreads are not primarily explained by default and recovery risk, but are mainly attributable to taxes, jumps, liquidity, and market risk factors. Huang and Huang (2002) find that for high-grade debt, only a small part of the total spread can be explained by credit risk, and for lower quality debt, a larger part of the spread can be explained by default risk. Agliardi (2011) incorporates discrete coupons, bankruptcy costs, taxes and the market risk generated by a stochastic risk-free structure into the classical Geske model for defaultable coupon bonds and provides a comprehensive formula in order to properly disentangle the contribution of several risk factors to credit spreads. Moreover, Agliardi (2011) also considers the duration of defaultable bonds.

On the other hand, jump risk is incorporated to eliminate the feature of credit spreads decreasing to zero at the short maturity. Hilberink and Rogers (2002) allow the value of the firm’s assets to make downward jumps and show that the spreads do not go to zero as maturity goes to zero. Another research stream aiming at overcoming most of the drawbacks of the structural approach is based on incomplete information (see, e.g., Duffie and Lando 2001; Jarrow and Protter 2004; Giesecke and Goldberg 2004; Yu 2005; Giesecke 2006; Agliardi and Agliardi 2009). Duffie and Lando (2001) adopt the Leland–Toft endogenous default model and derive the conditional distribution of the assets, given accounting data and survivorship. Credit spreads are then characterized in terms of accounting information. Giesecke (2006) shows that most incomplete information models can be analyzed from a common vantage point. Moreover, the author introduces the trend of a default model and proves that under mild technical conditions, all incomplete information models lead to generalized reduced-form security pricing formulae in terms of their trend.

However, motivated by empirical findings, we here want to incorporate liquidity risk so that we can get the yield spreads that are more consistent with observed spreads in the financial markets. To our knowledge, there are few papers which have considered both liquidity risk and credit risk (see, e.g., Tychon and Vannetelbosch 2005 and Ericsson and Renault 2006). Tychon and Vannetelbosch (2005) develop a corporate bond valuation model that takes into account both the risk of early default and the risk generated by lack of liquidity and marketability. The liquidity and marketability risk is shown to be a function of the heterogeneity of investors’ valuations, the average belief about the cost of bankruptcy and the bargaining power of bondholders. The model also captures the fact that, soon after the issue, a bond is relatively liquid and later becomes relatively illiquid depending on the underlying asset value. Ericsson and Renault (2006) develop a structural bond valuation model to simultaneously capture both liquidity risk and credit risk. They use the least squares Monte Carlo (LSM) simulation technique suggested by Longstaff and Schwartz (2001) to compute security values and show that levels of liquidity spreads are likely to be positively correlated with credit risk.

In the present paper, we also adopt the structural approach to model credit risk and model liquidity risk in a similar framework as in Ericsson and Renault (2006). That is, we assume that bondholders may encounter liquidity shocks during the lifetime of corporate bonds and they are subjected to random liquidity shocks due to a need to maintain unexpected cash shortages. When such liquidity shocks happens and the firm is solvent, the bondholder has to sell the bond immediately at the price which is assumed to be a fraction of the price in a perfectly liquid market. Moreover, we assume that liquidity shocks follow a Poisson process with parameter \(\lambda \). Here we assume that the credit risk and liquidity shocks are independent. This assumption is in general quite restrictive and its relaxation will be topic of further research. However, the assumption of independence allows us to get explicit results. As for credit risk, Ericsson and Renault (2006) are mainly concerned about bankruptcy renegotiation, adopting debt-equity swaps for out-of-court renegotiation. The interaction between a firm’s claimants in financial distress makes it hard to derive closed-form solutions for bond prices. Moreover, they emphasize term structures of liquidity spreads, not credit spreads. Here we adopt Leland–Toft endogenous default model. Taking into account the possibility of liquidity risk, we can get the closed-form expressions for corporate bond, that makes our numerical illustrations more tractable. Based on these results, we show that credit spreads are not negligible for short maturities. More precisely, we get a U-shaped term structures for the credit spreads.

The rest of this paper is organized as follows. Section 2 is devoted to deriving an explicit expression for corporate bond with both liquidity risk and credit risk. In Sect. 3, a numerical illustration is provided. The concluding remarks are contained in Sect. 4. Most proofs are contained in the Appendix.

## 2 The valuation of corporate bonds

In this section, we first describe our framework for the valuation of risky debt, which captures both credit risk and liquidity risk. Then we derive the explicit formula for corporate debt with finite maturity, based on the closed-form results in Leland and Toft (1996).

Suppose that a credit loss occurs if the market value of the firm’s assets, \(V_t\), is less than some amount \(V_B\), which will be endogenously determined later. Mathematically, \(\tau :=\inf \{t>0|V_t\le V_B\}\) can represent the time at which the default occurs. Once a credit loss occurs, the recovery is assumed to be a constant \(\alpha _1\), while \(1-\alpha _1\) times \(V_B\) represents the deadweight costs due to the bankruptcy or reorganization. Also, we assume that the default-free term structure is flat with an instantaneous riskless rate, \(r\), at which investor may lend and borrow freely.

As for liquidity risk, we also describe it directly under the risk neutral measure \(Q\). We also define liquidity as the ability to sell a security promptly and at a price close to its value in frictionless markets as in Ericsson and Renault (2006). It is assumed that liquidity shocks follow a Poisson process \(N_t\) with parameter \(\lambda \), and that \(\alpha \) is the fraction of the price in a perfectly liquid market when the bondholder has to sell the bond.

The first term on the right of (2) represents the present value of the coupon flow when there is neither liquidity shock nor bankruptcy before maturity and the second term can be interpreted as that of repayment of principle. The third term represents the present value of the fraction of the asset if bankruptcy occurs before liquidity shock and maturity. The last term represents the present value that bondholder will receive if liquidity shock occurs before bankruptcy and maturity.

Based on the expression of \(d^L(V;V_B,t)\), integrating yields the following result.

**Propostition 1**

*Proof of Proposition 1* see Appendix.

## 3 Numerical illustrations

Parameter values

Parameter name | Value |
---|---|

Volatility | \(\sigma =0.2\) |

Initial asset value | \(V_0=100\) |

Bankruptcy cost fraction | \(\alpha _1=0.5\) |

Riskless interest rate | \(r=0.075\) |

Total payout rate | \(\delta =0.07\) |

Tax rate | \(\tau =0.35\) |

Fraction of the price in a perfectly liquid market | \(\alpha =0.98\) |

Intensity of liquidity shocks | \(\lambda =1\) |

Time to maturity | \(T=0.5,1,2,5,10,20\) |

Figure 2 plots firm value as a function of leverage for alternative maturities. The plots are truncated below at firm value of \(100\). These plots show that the optimal leverage increases as the maturity becomes longer and that the maximal value of firm value also increases. All these observations are the same as the results in Leland and Toft (1996).

Optimal leverage ratios and the values of key endogenous variables with alternative maturities

Maturity (years) | Coupon (dollars) | Firm value (dollars) | Bankruptcy trigger (dollars) | Optimal leverage (%) | Credit spread (basis points) |
---|---|---|---|---|---|

0.5 | 1.9 | 105.45 | 29.68 | 20 | 158 |

1.0 | 2.16 | 106.21 | 30.80 | 23 | 128 |

2.0 | 2.49 | 107.16 | 32.00 | 28 | 90 |

5.0 | 3.45 | 109.17 | 35.96 | 38 | 78 |

10.0 | 4.19 | 111.19 | 36.31 | 44 | 116 |

20.0 | 4.56 | 112.62 | 34.83 | 47 | 129 |

Table I in Leland and Toft (1996)

Maturity (years) | Coupon (dollars) | Firm value (dollars) | Bankruptcy trigger (dollars) | Optimal leverage (%) | Credit spread (basis points) |
---|---|---|---|---|---|

0.5 | 1.45 | 104.10 | 27.70 | 19 | 0 |

1.0 | 1.70 | 104.85 | 28.80 | 22 | 0 |

2.0 | 2.10 | 106.00 | 30.55 | 26 | 0 |

5.0 | 3.15 | 108.25 | 35.75 | 37 | 31 |

10.0 | 3.95 | 110.45 | 36.60 | 43 | 89 |

20.0 | 4.35 | 111.95 | 35.30 | 46 | 110 |

The impact of different parameter values on credit spread and bankruptcy trigger

Parameters | \(T=0.5\) | \(T=5\) | \(T=10\) | |
---|---|---|---|---|

\(\sigma =0.20, \alpha _1=0.98,\) | Credit spread | 158.30 | 78.24 | 115.94 |

\( \alpha =0.50\) | Bankruptcy trigger | 29.68 | 35.96 | 36.31 |

\(\sigma =0.25, \alpha _1=0.98, \) | Credit spread | 158.30 | 102.16 | 153.70 |

\( \alpha =0.50\) | Bankruptcy trigger | 22.80 | 30.30 | 39.53 |

\(\sigma =0.20, \alpha _1=0.97, \) | Credit spread | 237.96 | 102.21 | 129.71 |

\(\alpha =0.50\) | Bankruptcy trigger | 30.66 | 36.06 | 36.14 |

\(\sigma =0.20, \alpha _1=0.98, \) | Credit spread | 158.30 | 107.93 | 125.14 |

\(\alpha =0.25\) | Bankruptcy trigger | 39.95 | 42.79 | 40.48 |

## 4 Conclusions

This article presents a bond valuation model capturing both credit risk and liquidity risk. For credit risk, it is assumed that the default triggering barrier is determined endogenously by maximizing equity value. As for liquidity risk, we assume that bondholders may encounter liquidity shocks during the lifetime of corporate bonds, and have to sell the bond immediately at the price, which is assumed to be a fraction of the price in a perfectly liquid market. Under this framework, explicit expressions for corporate bond, firm value and bankruptcy trigger are given. Numerical simulations are also presented and discussed.

For the numerical illustrations, we compared our model with that in Leland and Toft (1996). We found that credit spreads are not negligible for short maturities, since there exists liquidity risk. These results are consistent with the empirical findings. Moreover, due to liquidity risk, credit spreads are also higher than those in Leland and Toft (1996) for the debt with long maturities.

## Notes

### Acknowledgments

The authors would like to thank the anonymous referees and the editor for providing a number of valuable comments that led to several important improvements.

## References

- Agliardi E, Agliardi R (2009) Fuzzy defaultable bonds. Fuzzy Set Syst 160:2597–2607CrossRefGoogle Scholar
- Agliardi R (2011) A comprehensive structural model for defaultable fixed-income bonds. Quant Financ 11:749–762CrossRefGoogle Scholar
- Anderson R, Sundaresan S (1996) Design and valuation of debt contracts. Rev Financ Stud 9:37–68CrossRefGoogle Scholar
- Anderson R, Sundaresan S (2000) A comparative study of structural models of corporate bond yields: an exploratory investigation. J Bank Financ 24:255–269CrossRefGoogle Scholar
- Black F, Cox JC (1976) Valuing corporate securities: some effects of bond indenture provisions. J Financ 31:351–367CrossRefGoogle Scholar
- Delianedis G, Geske R (2001) The components of corporate credit spreads: default, recovery, tax, jumps, liquidity and market factors, working paper, University of California at Los AngelesGoogle Scholar
- Duffie D, Lando D (2001) Term structures of credit spreads with incomplete accounting information. Econometrica 69:633–664CrossRefGoogle Scholar
- Ericsson J, Renault O (2006) Liquidity and credit risk. J Financ 61:2219–2250CrossRefGoogle Scholar
- Giesecke K (2006) Default and information. J Econ Dyn Control 30:2281–2303CrossRefGoogle Scholar
- Giesecke K, Goldberg LR (2004) Forecasting default in the face of uncertainty. J Deriv 12:14–25CrossRefGoogle Scholar
- Goldstein R, Ju N, Leland HE (2001) An EBIT-based model of dynamic capital structure. J Bus 74:483–512CrossRefGoogle Scholar
- Harrison J (1986) Brownian motion and stochastic flow systems. Wiley, New YorkGoogle Scholar
- Hilberink B, Rogers LCG (2002) Optimal capital structure and endogenous default. Financ Stoch 6:237–263CrossRefGoogle Scholar
- Huang JZ, Huang M (2002) How much of the corporate-treasury yield spread is due to credit risk? A new calibration approach, working paper, Penn State UniversityGoogle Scholar
- Jarrow R, Protter P (2004) Structural versus reduced form models: a new information based perspective. J Invest Manag 2:1–10Google Scholar
- Jones EP, Mason SP, Rosenfeld E (1984) Contingent claims analysis of corporate capital structures: an empirical investigation. J Financ 39:611–625CrossRefGoogle Scholar
- Leland HE (1994) Corporate debt value, bond covenants, and optimal captital structure. J Financ 49:1213–1252CrossRefGoogle Scholar
- Leland HE, Toft K (1996) Optimal capital structure, endogenous bankruptcy, and the term structure of credit spreads. J Financ 51:987–1019CrossRefGoogle Scholar
- Longstaff FA, Schwartz ES (1995) A simple approach to valuing risky fixed and floating rate debt. J Financ 50:1767–1774CrossRefGoogle Scholar
- Longstaff FA, Schwartz ES (2001) Valuing American options by simulation: a simple least-aquares approch. Rev Financ Stud 14:113–147CrossRefGoogle Scholar
- Merton RC (1974) On the pricing of corporate debt: the risk structure of interest rates. J Financ 29:449–470Google Scholar
- Tychon P, Vannetelbosch V (2005) A model of corporate bond pricing with liquidity and marketability risk. J Credit Risk 1:3–35Google Scholar
- Yu F (2002) Modeling expected return on defaultable bonds. J Fixed Income 2:69–81CrossRefGoogle Scholar
- Yu F (2005) Accounting transparency and the term structure of credit spreads. J Financ Econ 75:53–84CrossRefGoogle Scholar