Computational Management Science

, Volume 9, Issue 4, pp 515–530 | Cite as

Credit spreads, endogenous bankruptcy and liquidity risk

Original Paper
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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).

It is well known that the structural approach to modeling credit risk often starts with an assumption that the assets of the company are tradeable so that we can construct the risk neutral probability measure \(Q\) as in Goldstein et al. (2001). So we assume the value of the firm’s assets, \(V_t\), is driven by the following geometric Brownian motion under \(Q\),
$$\begin{aligned} \frac{\mathrm{d }V_t}{V_{t}}=(r-\delta )\mathrm{d }t+\sigma \mathrm{d }W_t, \end{aligned}$$
(1)
where \(\sigma \) is the volatility of the asset return, the parameter \(\delta \) denotes the cash flow rate, and \(W_t\) is a standard Brownian motion on a complete probability space \((\Omega ,\mathcal{F },Q)\).

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.

Consider a bond with maturity \(t\), which continuously pays a constant coupon flow \(c(t)\) and has principle \(p(t)\). Denote by \(\rho (t)\) the fraction of asset value \(V_B\) which the bondholder can receive in the event of bankruptcy. Using risk-neutral valuation, we have the value of debt with maturity \(t\),
$$\begin{aligned} d^I(V;V_B,t)&=\int \limits _0^te^{-rs}c(t)[1-F(s;V,V_B)]\mathbf{Q }[N_s=0]\mathrm{d }s\nonumber \\&\quad +\,\,e^{-rt}p(t)[1-F(t;V,V_B)]\mathbf{Q }[N_t=0]\nonumber \\&\quad + \int \limits _0^te^{-rs}\rho (t)V_Bf(s;V,V_B)\mathbf{Q }[N_s=0]\mathrm{d }s\nonumber \\&\quad +\,\,\alpha \mathbf{E }_{\mathbf{Q }}\int \limits _0^te^{-rs}d^L(V_s;V_B,t-s)[1-F(s;V,V_B)]\lambda e^{-\lambda s}\mathrm{d }s, \end{aligned}$$
(2)
where \(f(s;V,V_B)\) is the density of the first passage times to \(V_B\) from \(V\) when the drift rate is \(r-\delta \), \(F(s)\) is the cumulative distribution function of the first passage time to bankruptcy, and \(d^L(V;V_B,t)\) is the value of debt with maturity \(t\) in liquid markets.

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.

Before deriving the explicit expressions of (2), we give the value \(d^L(V;V_B,t)\) in liquid markets. Similarly, adopting the risk-neutral valuation, it holds that
$$\begin{aligned} d^L(V;V_B,t)&=\int \limits _0^te^{-rs}c(t)[1-F(s;V,V_B)]\mathrm{d }s +e^{-rt}p(t)[1-F(t;V,V_B)]\nonumber \\&\quad +\int \limits _0^te^{-rs}\rho (t)V_Bf(s;V,V_B)\mathrm{d }s. \end{aligned}$$
(3)
The closed-form results can be found in Leland and Toft (1996) as follows,
$$\begin{aligned} d^L(V;V_B,t)\!=\!\frac{c(t)}{r}\!+\!e^{-rt}\!\left[p(t)\!-\!\frac{c(t)}{r}\right][1\!-\!F(t)]\!+\! \left[\rho (t)V_B\!-\!\frac{c(t)}{r}\right]G(t), \end{aligned}$$
(4)
where
$$\begin{aligned} G(t)=\int \limits _0^te^{-rs}f(s;V,V_B)\mathrm{d }s. \end{aligned}$$
Moreover,
$$\begin{aligned} F(t)&= N[h_1(t)]+\left(\frac{V}{V_B}\right)^{-2a}\!\!\!N[h_2(t)],\end{aligned}$$
(5)
$$\begin{aligned} G(t)&= \left(\frac{V}{V_B}\right)^{-a+z}\!\!\!N[q_1(t)]+\left(\frac{V}{V_B}\right) ^{-a-z}\!\!\!N[q_2(t)], \end{aligned}$$
(6)
where
$$\begin{aligned}&h_1(t)=\frac{-b-a\sigma ^2t}{\sigma \sqrt{t}};\ \ \ \ \ \ h_2(t)=\frac{-b+a\sigma ^2t}{\sigma \sqrt{t}};\nonumber \\&q_1(t)=\frac{-b-z\sigma ^2t}{\sigma \sqrt{t}};\ \ \ \ \ \ q_2(t)=\frac{-b+z\sigma ^2t}{\sigma \sqrt{t}};\nonumber \\&a=\frac{r-\delta -\frac{\sigma ^2}{2}}{\sigma ^2};\ \ \ \ \ b=\ln \left(\frac{V}{V_B}\right);\ \ \ \ \ z=\frac{[(a\sigma ^2)^2+2r\sigma ^2]^{\frac{1}{2}}}{\sigma ^2}, \end{aligned}$$
(7)
and \(N(\cdot )\) is the cumulative standard normal distribution.

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

Propostition 1

Taking into account the possibility of liquidity risk, the bond value satisfies for a given threshold \(V_B\),
$$\begin{aligned} d^I(V;V_B,t)&=\frac{c(t)(\alpha \lambda +r)}{r(r+\lambda )}+K_1(t)I(r,\lambda , t) \nonumber \\&\quad +K_2(t)(1-F(t))+K_3(t)G(t), \end{aligned}$$
(8)
where
$$\begin{aligned} I(r,\lambda ,t)&= e^{\frac{b(R-r)}{\sigma ^2}}\left\{ \left(\frac{V}{V_B}\right) ^{-A+Z}\!\!\!N[Q_1(t)]+\left(\frac{V}{V_B}\right)^{-A-Z}\!\!\!N[Q_2(t)]\right\} ,\\ K_1(t)&= -\frac{c(t)(\alpha \lambda +r)}{r(r+\lambda )}+\rho (t) V_B -\alpha \left(\rho (t) V_B-\frac{c(t)}{r}\right),\nonumber \\ K_2(t)&= \left[-\frac{c(t)(\alpha \lambda +r)}{r(r+\lambda )}+p(t)-\alpha \left(p(t)-\frac{c(t)}{r}\right)\right]e^{-(r+\lambda )t}\nonumber \\&\quad +\alpha \left(p(t)-\frac{c(t)}{r}\right)e^{-rt},\nonumber \\ K_3(t)&= \alpha \left(\rho (t) V_B-\frac{c(t)}{r}\right),\nonumber \\ R&= \delta -\frac{\sigma ^2}{2}+\sqrt{2(r+\lambda )\sigma ^2+\left(r-\delta - \frac{\sigma ^2}{2}\right)^2-2\delta \sigma ^2},\nonumber \\ Q_1(t)&= \frac{-b-Z\sigma ^2t}{\sigma \sqrt{t}},\ \ \ \ Q_2(t)= \frac{-b+Z\sigma ^2t}{\sigma \sqrt{t}},\nonumber \\ A&= \frac{R-\delta -\frac{\sigma ^2}{2}}{\sigma ^2},\ \ \ \ b=\ln \left(\frac{V}{V_B} \right),\ \ \ \ Z=\frac{[(A\sigma ^2)^2+2R\sigma ^2]^{\frac{1}{2}}}{\sigma ^2},\nonumber \end{aligned}$$
(9)
with \(N(\cdot )\) is the cumulative standard normal distribution and \(F\) and \(G\) are defined in (5) and (6).

Proof of Proposition 1 see Appendix.

It is noteworthy that these expressions extend those obtained by Leland and Toft (1996) by incorporating the possibility of liquidity shocks. Indeed, when \(\lambda =0\ \mathrm{and}\ \alpha =0\), no liquidity shock occurs, our result is consistent with that of Leland and Toft (1996). That is,
$$\begin{aligned}&I(r,0,t)=G(t),\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ K_1(t)=-\frac{c(t)}{r}+\rho (t) V_B,\\&K_2(t)=\left(p(t)-\frac{c(t)}{r}\right)e^{-\lambda t},\ \ \ \ \ \ K_3(t)=0. \end{aligned}$$
Next we will determine the default triggering barrier \(V_B\) endogenously by maximizing equity value, assuming a stationary debt structure, which means that the firm continuously sells a constant amount of new debt, which will redeem at par upon maturity (see, Leland and Toft 1996). Assume that new bonds are issued with identical maturity \(T\), principal value \(p\) and coupon rate \(c\). At each moment in time, the firm has a continuum of bonds outstanding with an aggregate principal \(P=p*T\) and an aggregate coupon payment \(C=c*T\). Similarly, \((1-\rho (t))*T\) equals \(\alpha _1\), which is the fraction of firm asset value lost in bankruptcy. Therefore, we get the value of all outstanding bonds,
$$\begin{aligned} D(V;V_B,T)&= \int \limits _0^Td^I(V;V_B,t)\mathrm{d }t\\&= l_0+l_1(V_B)U(T)+ l_2F(T)+l_3G(T)\\&+\,l_4(V_B)J(T)+l_5I(r,\lambda ,T), \end{aligned}$$
with
$$\begin{aligned} l_0&= \frac{C(\alpha \lambda +r)}{r(r+\lambda )}+\left[-\frac{C(\alpha \lambda +r)}{r(r+\lambda )}+P-\alpha \left(P-\frac{C}{r}\right)\right]\frac{1-e^{-(r+\lambda )T}}{(r+\lambda )T}\\&\quad +\alpha \left(P-\frac{C}{r}\right)\frac{1-e^{-rT}}{r T},\\ l_1(V_B)&= -\frac{C(\alpha \lambda +r)}{r(r+\lambda )}+(1-\alpha _1)(1-\alpha )V_B +\frac{\alpha C}{r},\\ l_2&= \left[-\frac{C(\alpha \lambda +r)}{r(r+\lambda )}+P-\alpha \left(P-\frac{C}{r}\right)\right]\frac{e^{-(r+\lambda )T}}{(r+\lambda )T} +\alpha \left(P-\frac{C}{r}\right)\frac{e^{-rT}}{r T},\\ l_3&= -\frac{\alpha }{r T}\left(P-\frac{C}{r}\right),\\ l_4(V_B)&= \alpha (1-\alpha _1) V_B-\frac{C\alpha }{r},\\ l_5&= -\frac{1}{(r+\lambda )T}\left[-\frac{C(\alpha \lambda +r)}{r(r+\lambda )} +P-\alpha \left(P-\frac{C}{r}\right)\right],\\ U(T)&= \frac{1}{T}\int _0^TI(r,\lambda , t)\mathrm{d }t,\ \ \ \ \ J(T)=\frac{1}{T} \int _0^TG(t)\mathrm{d }t, \end{aligned}$$
and \(F\), \(G\) and \(I(r, \lambda , t)\) are defined in (5), (6) and (). The explicit expression of \(J(T)\) can be found in Leland and Toft (1996),
$$\begin{aligned} J(T)=\frac{1}{z\sigma \sqrt{T}}\left[-\left(\frac{V}{V_B}\right)^{-a+z}\!\!\!N[q_1(T)] q_1(T)+\left(\frac{V}{V_B}\right)^{-a-z}\!\!\!N[q_2(T)]q_2(T)\right]. \end{aligned}$$
Using the similar line of deriving \(I(r, \lambda , t)\) in Proposition 1, we get that
$$\begin{aligned} U(T)=\frac{e^{-\frac{b(R-r)}{\sigma ^2}}}{Z\sigma \sqrt{T}}&\left[-\left(\frac{V}{V_B} \right)^{-A+Z}\!\!\!N[Q_1(T)]Q_1(T)\right.\\&\quad +\left.\left(\frac{V}{V_B}\right)^{-A-Z}\!\!\!N[Q_2(T)]Q_2(T)\right], \end{aligned}$$
where the factors \(a, z,A, Z\) are defined in (7) and Proposition 1.
From Leland (1994), the total market value of the firm, \(\nu \), is given by
$$\begin{aligned} \nu (V;V_B,T)=V+\frac{\tau C}{r}\left[1-\left(\frac{V}{V_B}\right)^{-a-z}\right] -\alpha _1V_B\left(\frac{V}{V_B}\right)^{-a-z}, \end{aligned}$$
where \(\tau \) is the corporate tax rate. So the value of equity is given by
$$\begin{aligned} E(V;V_B,T)=\nu (V;V_B,T)-D(V;V_B,T). \end{aligned}$$
In order to determine the default triggering barrier \(V_B\) endogenously, it is necessary that
$$\begin{aligned} \frac{\partial E(V;V_B,T)}{\partial V}\bigg |_{V=V_B}=0. \end{aligned}$$
Solving the equation gets that
$$\begin{aligned} V_B=\frac{\left(\frac{\alpha C}{r}-\frac{C(\alpha \lambda +r)}{r(r+\lambda )}\right)S_U+l_2S_F+l_3S_G-\frac{\alpha C}{r}S_J+l_5S_I-\frac{\tau C}{r}(a+z) }{1+\alpha _1(a+z)-(1-\alpha _1)(1-\alpha )S_U-\alpha (1-\alpha _1)S_J}, \end{aligned}$$
with
$$\begin{aligned} S_U&= -2\left(Z+\frac{1}{Z\sigma ^2T}+\frac{R-r}{\sigma ^2}\right)N(Z\sigma \sqrt{T})- \frac{2}{\sigma \sqrt{T}}n(Z\sigma \sqrt{T})\\&+Z-A+\frac{1}{Z\sigma ^2 T}+\frac{R-r}{\sigma ^2},\\ S_F&= -2a N(a\sigma \sqrt{T})-\frac{2}{\sigma \sqrt{T}}n(a\sigma \sqrt{T}),\\ S_G&= -2z N(z\sigma \sqrt{T})-\frac{2}{\sigma \sqrt{T}}n(z\sigma \sqrt{T})+z-a,\\ S_J&= -2\left(z+\frac{1}{z\sigma ^2 T}\right)N(z\sigma \sqrt{T})- \frac{2}{\sigma \sqrt{T}}n(z \sigma \sqrt{T})\\&+z-a+\frac{1}{z\sigma ^2 T},\\ S_I&= -2Z N(Z\sigma \sqrt{T})-\frac{2}{\sigma \sqrt{T}}n(Z\sigma \sqrt{T})+Z-A+ \frac{R-r}{\sigma ^2}, \end{aligned}$$
and the factors \(a, z, A, Z\) are defined in (7) and Proposition 1. Also \(N(\cdot )\) and \(n(\cdot )\) denote the cumulative standard normal distribution and the standard normal density function, respectively.
Furthermore, Leland and Toft (1996) introduce a tax cutoff level \(V_T:=\frac{C}{\delta }\) so that the tax deductibility of debt is lost when \(V<V_T\). In this situation, from Leland and Toft (1996), the firm value becomes,
$$\begin{aligned} \left\{ \begin{array}{ll} \nu (V;V_B,T)\!=\!V+A_1V+A_2[1-V^{-a-z}]\!-\!\alpha _1V_B(\frac{V}{V_B})^{-a-z},\ \ \ V_B<V\le V_T,\\ \nu (V;V_B,T)\!=\!V+\frac{\tau C}{r}+B_2V^{-a-z}-\alpha _1V_B(\frac{V}{V_B})^{-a-z},\ \ \ V_T<V, \end{array}\right. \end{aligned}$$
(10)
where
$$\begin{aligned}&A_1=\frac{\tau C (a+z)}{r(a+z+1)V_T},\\&A_2=-\frac{\tau C (a+z)V_B^{a+z+1}}{r(a+z+1)V_T},\\&B_2=-\frac{\tau C (a+z)}{r(a+z+1)V_T}\left(V_B^{z+a+1}+\frac{V_T^{a+z+1}}{a+z+1}\right). \end{aligned}$$
Similarly, we can derive the default triggering barrier \(V_B\) endogenously as follows,
$$\begin{aligned} V_B=\frac{\left(\frac{\alpha C}{r}-\frac{C(\alpha \lambda +r)}{r(r+\lambda )}\right)S_U+l_2S_F+l_3S_G-\frac{\alpha C}{r}S_J+l_5S_I }{1+(\alpha _1+\frac{\tau C}{rV_T})(a+z)-(1-\alpha _1)(1-\alpha )S_U-\alpha (1-\alpha _1)S_J}. \end{aligned}$$
Up to now, we have got the explicit expressions for corporate bonds with maturity \(T\). In the following section, we will investigate the impact of liquidity risk on the valuation of the debt with finite maturity.

3 Numerical illustrations

In this section, we investigate the impact of liquidity risk. Here we mainly compare our results with those of Leland and Toft (1996) for alternative maturities. So we also introduce a tax cutoff level \(V_T:=\frac{C}{\delta }\) so that the tax deductibility of debt is lost when \(V<V_T\), and assume that the coupon is set to make sure new debt sells at par value under the stationary debt structure. Preference parameters are listed in Table 1, which are the values used by Leland and Toft (1996).
Table 1

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 1 depicts the plots of debt value as a function of leverage for alternative maturities. From these plots, we can find that the shape of the curves is similar to that in Fig. 2 in Leland and Toft (1996). Compared with the debt value shown in Leland and Toft (1996), the debt value in the proposed model is lower for different maturities.
Fig. 1

Debt value as a function of leverage. The dotted, dashed, dotdashed and solid lines correspond to maturities \(T=0.5\), \(T=5\), \(T=10\) and \(T=20\), respectively

Fig. 2

Firm value as a function of leverage. The dotted, dashed, dotdashed and solid lines correspond to maturities \(T=0.5\),\(T=5\), \(T=10\) and \(T=20\), respectively

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).

Table 2 shows the characteristics of optimally levered firms with alternative maturities. For comparing with Leland and Toft (1996), we also list their results in Table 3. Optimal leverage increases, as the maturity becomes longer. Optimal leverage ratio, coupon and firm value are all larger than those in Leland and Toft (1996) for every maturity. Credit spreads at the optimal leverage rate are negligible for the debt with maturities of 2 years or less without liquidity risk, but those are significantly high in the proposed model, which is consistent with the empirical findings.
Table 2

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

We adopt the parameter values listed in Table 1

Table 3

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

Table 4 investigates the impact of different parameter values on credit spread and bankruptcy trigger. Credit spread for the debt with the maturity \(T=5\) is smallest for all four cases. Indeed, we get a U-shaped term structure for the credit spreads. Due to the existence of liquidity shocks, the firm’s default boundary \(V_B\) becomes large more obviously for firms issuing bonds with short maturity, comparing the fourth columns in Tables 2 and 3. This leads to a higher premium for bonds with short maturity. The U-shaped credit spreads may tell us to choose appropriate maturity when firms issue new bonds. Credit spread increases when the risk of the firm \(\sigma \) rises. The lower fraction of the price in a perfectly liquid market \(\alpha _1\) also has a significant impact, especially for the debt with the maturity \(T=0.5\), where credit spread rises to \(237.96\). Similarly, the lower bankruptcy cost fraction corresponds to higher credit risk. Furthermore, credit spreads are not negligible for the bonds with shorter maturity for all cases above.
Table 4

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.

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Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.School of Mathematical SciencesNankai UniversityTianjinChina
  2. 2.School of BusinessNankai UniversityTianjinChina

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