3.1 A general pricing formula for a standard asset

In corporate finance, almost all the existing corporate securities can be priced by considering a special form of the standard asset we discussed here. By a standard asset we mean that its cash flow has a linear relationship with the firm’s EBIT, i.e. $aX_t+K$ for some constant $a$ and $K$ when the EBIT falls in an economic state-dependent domain $\mathfrak{D}_{L_t}, L_t\in \{0,1\}$ . The dynamics of the cash flow EBIT $X$ is governed by the Markovian regime-switching GBM (1). We denote by $T_{\mathfrak{D}}$ the first time for the cash flow exiting the current domain, i.e. $T_{\mathfrak{D}}\equiv \inf \{t\geq 0:X_t\not \in \mathfrak{D}_{L_t}\}$ , $L_t\in \{0,1\}$ . At the stopping time $T_{\mathfrak{D}}$ the owner gets nothing but a claim whose value equals to $G(X_{T_{\mathfrak{D}}};\; L_{T_{\mathfrak{D}}})$ . For example, if the standard asset is debt, then $T_{\mathfrak{D}}$ would be the default time and $G(X_{T_{\mathfrak{D}}};\; L_{T_{\mathfrak{D}}})$ represents the liquidation value. If there is no regime-switching, the pricing of the standard asset is well addressed by Tan and Yang (Reference Tan and Yang2017). Model (1) generalizes the models considered by Guo et al. (Reference Guo, Miao and Morellec2005) and Luo and Yang (Reference Luo and Yang2017) $\grave{a}$ la Leland (Reference Leland1994). This generalization makes the pricing challenging.

Our model is time-homogenous and thus we assume the current time is zero without losss of generality. According to the risk-neutral pricing theory, the value of the standard asset is given by

\begin{equation*} Q_{l}(x)=\mathbb {E}\left [\int _{0}^{T_{\mathfrak {D}}} e^{-rt}(aX_t+K)dt+e^{-rT_{\mathfrak {D}}}G(X_{T_{\mathfrak {D}}};\; L_{T_{\mathfrak {D}}}) \mid X_0 =x, L_0=l\right ]. \end{equation*}

We consider two cases: $\mathcal{D}_0\subseteq \mathcal{D}_1$ and $\mathcal{D}_1\subseteq \mathcal{D}_0$ respectively, which are naturally determined in advance to compute these prices. Following Guo et al. (Reference Guo, Miao and Morellec2005) and Luo and Yang (Reference Luo and Yang2017), the rang of the EBIT can be partitioned into three regions if $\mathcal{D}_0\subseteq \mathcal{D}_1$ :Footnote ² $\mathcal{D}_0,\mathcal{D}_1-\mathcal{D}_0$ and $\overline{\mathcal{D}_1}$ (the complement of $\mathcal{D}_1$ ), which are called continuation region, transient region and stopping region respectively. In continuation region, the standard asset generates cash flow in both recession and boom states. In stopping region, or in transient region with the recession state, the claimant of the asset gets nothing but a lump-sum payoff, and by contrast, in transient region with the boom state, the asset continue to generate the cash flow. According to Ito’s formula, the function $Q_{0}(x)$ and $Q_{1}(x)$ satisfy the following system of ordinary differential equations (ODEs):

(2) \begin{equation} \left \{\begin{array}{l} r Q_{0}(x)=ax+K+\mu _{0} x Q_{0}^{\prime }(x)+\frac{\sigma _{0}^{2} x^{2}}{2} Q_{0}^{\prime \prime }(x)+\lambda _{0}\left (Q_{1}(x)-Q_{0}(x)\right )\!, \\[5pt] r Q_{1}(x)=ax+K+\mu _{1} x Q_{1}^{\prime }(x)+\frac{\sigma _{1}^{2} x^{2}}{2} Q_{1}^{\prime \prime }(x)+\lambda _{1}\left (Q_{0}(x)-Q_{1}(x)\right )\!, \end{array}\right . \end{equation}

for $x\in \mathcal{D}_0$ ;

(3) \begin{equation} \left \{\begin{array}{l} Q_{0}(x)=G(x, 0), \\[5pt] r Q_{1}(x)=ax+K+\mu _{1} x Q_{1}^{\prime }(x)+\frac{\sigma _{1}^{2} x^{2}}{2} Q_{1}^{\prime \prime }(x)+\lambda _{1}\left (G_0(x,0)-Q_{1}(x)\right )\!, \end{array}\right . \end{equation}

for $x\in \mathcal{D}_1-\mathcal{D}_0$ ; and

\begin{equation*} \left \{\begin {array}{c} Q_{0}(x)=G(x, 0),\\[5pt] Q_1(x)=G(x, 1), \end {array}\right . \end{equation*}

for $x\in \overline{\mathcal{D}_1}$ .

Define the following quadratic function of variable $\beta$ :

(4) \begin{equation} h_{l}(\beta )=r+\lambda _{l}-\frac{1}{2} \sigma _{l}^{2} \beta (\beta -1)-\mu _{l}\beta, \ l\in \{0, 1\}. \end{equation}

In the following, we denote by $\beta _1$ and $\beta _2$ the two roots of the equation $h_1(\beta )=0$ , by $\beta _3$ and $\beta _4$ the two roots of the equation $h_0(\beta )=0$ . The equation $h_0(\gamma )h_1(\gamma )=\lambda _0\lambda _1$ has two negative roots denoted by $\gamma _1$ and $\gamma _2$ and two positive roots denoted by $\gamma _3$ and $\gamma _4$ . We derive the explicit pricing for all standard assets as follows.

Proposition 3.1. Suppose that the standard asset defined by its terminal payoff $G(x,l)=\phi _l x+\kappa _l$ ,Footnote ³ $l\in \{0,1\}$ , and its cash flow $ax+K$ for some constant $a$ and $K$ if the firm’s cash flow does not exit the domain $\mathcal{D}_l$ . Then if $\mathcal{D}_0\subseteq \mathcal{D}_1$ , the value $Q_l(x)$ of the standard asset has the following form:

(5) \begin{equation} Q_{0}(x)=\left \{\begin{array}{l@{\quad}l} \sum _{i=1}^{4}A_ix^{\gamma _i}+aq_{0} x+\frac{K}{r}, & $if$\ x\in \mathcal{D}_0,\\[5pt] G(x, 0), & $if$\ x\in{\overline{\mathcal{D}_0}},\end{array}\right . \end{equation}

and

(6)

and if $\mathcal{D}_1\subseteq \mathcal{D}_0$ ,

(7) \begin{equation} Q_{1}(x)=\left \{\begin{array}{l@{\quad}l} \sum _{i=1}^{4}\hat{A}_ix^{\gamma _i}+aq_{1} x+\frac{K}{r}, & $if$\ x\in \mathcal{D}_1,\\[5pt] G(x, 1), & $if$\ x\in{\overline{\mathcal{D}_1}},\end{array}\right . \end{equation}

and

(8)

where

\begin{equation*} q_{l}=\frac {r+\lambda _{0}+\lambda _{1}-\mu _{0}-\mu _{1}+\mu _{l}}{\left (r+\lambda _{0}-\mu _{0}\right )\left (r+\lambda _{1}-\mu _{1}\right )-\lambda _{0} \lambda _{1}},\end{equation*}

and $B_i=\frac{h_{0}\left (\gamma _{i}\right )}{\lambda _{0}} A_i, \hat{B}_i=\frac{\lambda _{0}}{h_{0}\left (\gamma _{i}\right )} \hat{A}_i$ .

We point out that to the best of our knowledge, the conclusions of Proposition 3.1 are never derived before. In particular, almost all common corporate securities’ price can be derived by the explicit formulas presented in Proposition 3.1, even if the cash flow generated by the firm is driven by both the Brownian motion and the two-state Markov chain. Accordingly, we extend Leland (Reference Leland1994) and Tan and Yang (Reference Tan and Yang2017)’s closed-form results to the Markovian regime-switching GBM cash flow model.

3.2 The pricing of corporate securities

To price corporate securities, we use a backward induction method and the prices can be derived from some standard asset prices we addressed in the last subsection.

First, after the debt default has taken place, the debtholders take over the firm and get nothing but a claim of which the value equals to $(1-\alpha )P_l(x)=(1-\alpha )(1-\tau _f)q_l x$ . Here, $\alpha$ is the bankruptcy loss rate and $P_l(x)=(1-\tau _f)q_l x$ is the unlevered firm’s value. This is because the unlevered firm has a claim whose cash flow equals to $(1-\tau _f)X_t$ forever, i.e. a special form of the above-mentioned standard asset with $a=(1-\tau _f)$ , $K=0$ and $\mathcal{D}_0=\mathcal{D}_1=(0, \infty )$ .

Now we turn to the valuation of debt for given default thresholds. The default threshold in recession, denoted by $x^b_0$ , and that in boom, denoted by $x^b_1$ are endogenously determined by shareholders. We emphasize that a firm would default earlier in recession than in boom, but the opposition would also hold true, depending on model parameters. This is because a higher drift of the cash flow would induce a later default while a less volatility would lead to an earlier default. To be specific, we assume $x^b_1\lt x^b_0$ , i.e. the firm’s default happens earlier in recession than in boom, in the following derivations.

Debt is perpetual and debtholders receive cash flow $(1-\tau _i)c$ until default happens. At default, the debtholders get nothing but a liquidation value equaling to $(1-\alpha )(1-\tau _f)q_l x$ just derived. Therefore, to price the debt, we take it as a special standard asset with $a=0$ , $K=(1-\tau _i)c$ , $\mathcal{D}_0=(x^b_0, \infty )$ , $\mathcal{D}_1=(x^b_1, \infty )$ , $G(x,0)=(1-\alpha )(1-\tau _f)q_0 x$ and $G(x,1)=(1-\alpha )(1-\tau _f)q_1 x$ . Accordingly, we conclude from Proposition 3.1 the following debt value:

(9)

and

(10)

According to value-matching (continuousness) conditions, we have

\begin{equation*} D_0(x^b_0+)=D_0(x^b_0-),\ D_1(x^b_0+)=D_1(x^b_0-),\ D_1(x^b_1+)=D_1(x^b_1-).\end{equation*}

Additionally, from the differentiability of the value functions within the interior of domain $\mathcal{D}_1$ we conclude $D'_1(x^b_0+)=D'_1(x^b_0-)$ . Hence, we derive that the coefficient vector $(A_1,A_2,C_1,C_2)'$ is the solution of the system of linear equations: $\Gamma _1X=\Phi _1$ , where

\begin{equation*} \Gamma _{1}=\left (\begin {array}{c@{\quad}c@{\quad}c@{\quad}c} \left (x_{0}^{b}\right )^{\gamma _{1}} & \left (x_{0}^{b}\right )^{\gamma _{2}} & 0 & 0 \\[5pt] \frac {h_{0}\left (\gamma _{1}\right )}{\lambda _{0}}\left (x_{0}^{b}\right )^{\gamma _{1}} & \frac {h_{0}\left (\gamma _{2}\right )}{\lambda _{0}}\left (x_{0}^{b}\right )^{\gamma _{2}} & -\left (x_{0}^{b}\right )^{\beta _{1}} & -\left (x_{0}^{b}\right )^{\beta _{2}} \\[5pt] \frac {\gamma _{1} h_{0}\left (\gamma _{1}\right )}{\lambda _{0}}\left (x_{0}^{b}\right )^{\gamma _{1}} & \frac {\gamma _{2} h_{0}\left (\gamma _{2}\right )}{\lambda _{0}}\left (x_{0}^{b}\right )^{\gamma _{2}} & -\beta _{1}\left (x_{0}^{b}\right )^{\beta _{1}} & -\beta _{2}\left (x_{0}^{b}\right )^{\beta _{2}} \\[5pt] 0 & 0 & \left (x_{1}^{b}\right )^{\beta _{1}} & \left (x_{1}^{b}\right )^{\beta _{2}} \end {array}\right ), \end{equation*}

and

\begin{equation*} \Phi _1=\left (\begin {array}{c} (1-\alpha ) (1-\tau _f)q_{0} x_{0}^{b}-\frac {(1-\tau _i)c}{r} \\[5pt] \frac {\lambda _1(1-\alpha )(1-\tau _f)q_0x_0^b}{r+\lambda _1-\mu _1}-\frac {\lambda _1(1-\tau _i)c}{r(r+\lambda _{1})} \\[5pt] \frac {\lambda _1(1-\alpha )(1-\tau _f)q_0x_0^b}{r+\lambda _1-\mu _1}\\[5pt] (1-\alpha )(1-\tau _{f})( q_{1}-\frac {\lambda _1 q_0}{r+\lambda _1-\mu _1}) x_{1}^{b}-\frac {(1-\tau _i)c}{r+\lambda _{1}} \end {array}\right ). \end{equation*}

In a similar way, thanks to Proposition 3.1, we get the following equity value:

(11)

and

(12)

According to the continuousness and differentiability conditions, the coefficient vector $(A_3,A_4,C_3,C_4)'$ satisfies the system of linear equations: $\Gamma _1X=\Phi _2$ , where

\begin{equation*} \Phi _{2}=\left (\begin {array}{c} -\left (1-\tau _{f}\right )\left (q_{0} x_{0}^{b}-\frac {c}{r}\right ) \\[5pt] \left (1-\tau _{f}\right )\left ((\frac {1}{r+\lambda _{1}-\mu _{1}}-q_{1}) x_{0}^{b}+\frac {\lambda _{1} c}{r\left (r+\lambda _{1}\right )}\right ) \\[5pt] \left (1-\tau _{f}\right )\left (\frac {1}{r+\lambda _{1}-\mu _{1}}-q_{1}\right ) x_{0}^{b} \\[5pt] -\left (1-\tau _{f}\right )\left (\frac {x_{1}^{b}}{r+\lambda _{1}-\mu _{1}}-\frac {c}{r+\lambda _{1}}\right ) \end {array}\right ). \end{equation*}

As said before, the bankruptcy boundaries are determined endogenously by shareholders and thus according to smooth-pasting condition $E'_0(x_0^b+)=E'_0(x_0^b-)$ and $E'_1(x_1^b+)=E'_1(x_1^b-)$ , the optimal default threshold $(x_0^b, x_1^b)$ is a solution of the following system of equations:

(13) \begin{equation} \left \{\begin{array}{c} \gamma _{1}A_{3}\left (x_{0}^{b}\right )^{\gamma _{1}}+\gamma _{2}A_{4}\left (x_{0}^{b}\right )^{\gamma _{2}}+(1-\tau _f)q_{0}x_0^b=0, \\[5pt] \beta _{1}C_{3}\left (x_{1}^{b}\right )^{\beta _{1}}+\beta _{2}C_{4}\left (x_{1}^{b}\right )^{\beta _{2}}+\frac{(1-\tau _{f})x_1^b}{r+\lambda _{1}-\mu _{1}}=0. \end{array}\right . \end{equation}

The solution is not explicit but easy to get a numerical solution for it. Therefore, after the project is launched, the total firm value can be derived immediately by $V_l(x)=E_l(x)+D_l(x), l\in \{0,1\}$ .

5.1 The pricing and timing of the option to invest in a project

Government subsides are worthwhile and necessary to motivate a firm to invest in a project that has sufficiently positive externalities when the project itself is not profitable. Government subsidies are generally a fraction ( $\theta$ ) of tax revenues, i.e. $0\le \theta \le 1$ . Actually, the tax subsidy coefficient $\theta$ can be negative ( $\theta \lt 0$ ) for a negative externality project; it also can be greater than 1 ( $\theta \gt 1$ ) for a sufficiently positive externality project.

For a given tax subsidy coefficient $\theta$ , we first fix the optimal investment threshold $x^e_l, l\in \{0,1\}$ under the government subsidy specified in the last section.

Due to the time-homogenous model, we assume that the current time is zero without loss of generality as before. If the current cash flow level (EBIT)Footnote ⁴ is $X_0=x$ and the economy state is $l\in \{0, 1\}$ , the value of the investment option is given by

(18) \begin{equation} R_{l}(x)=\max _{\tau } \mathbb{E}\left [e^{-r\tau }\left (V_{l}\left (X_{\tau }\right )-I+S_l(X_\tau )\right ) \mid X_0=x,L_0=l\right ], \end{equation}

where $\tau$ represent the investment time, a stopping time with respect to $\{\mathcal{F}_t\}_{t\ge 0}$ , and $S_l(X_\tau )=\theta TX_l(X_\tau )$ represents the government subsidy. This is because at the investment time, the entrepreneur’s value is

\begin{equation*} E_l(X_\tau )-I+\frac {(1-\tau _i)c}{r}(1-g_l)=V_{l}(X_{\tau })-I+S_l(X_\tau ) \end{equation*}

according to (16) and (17).

Similar to the preceding discussion on default threshold, we are not sure which of two investment thresholds $x^e_0$ and $x^e_1$ is higher. To be specific, we assume $x^e_1\lt x^e_0$ in the following derivations, i.e. the investment in boom is earlier than in recession, while the discussions under $x^e_1\ge x^e_0$ are similar and thus omitted. Hence, the domain $\mathcal{D}^e_0=(0,x^e_0)$ and $\mathcal{D}^e_1=(0,x^e_1)$ satisfy $\mathcal{D}^e_1 \subseteq \mathcal{D}^e_0$ . Following Guo et al. (Reference Guo, Miao and Morellec2005), the regime-switching space $(0, \infty )$ of real investment can be partitioned into three regions: $\left (0, x_{1}^{e}\right ), \left [x_{1}^{e}, x_{0}^{e}\right )$ and $ \left [x_{0}^{e}, \infty \right )$ , which are called inaction region, transient region and action region, respectively. In inaction region, no investment occurs. Investment is triggered in transient region if the economy is in boom. In action region, investment is exercised immediately no matter what the economy state is.

We derive the following explicit solutions for the pricing and timing of the option to invest in the project with credit guarantees and government subsidies.

Proposition 5.1. Suppose that the default occurs earlier in recession than in boom, i.e. $x_0^b\geq x_1^b$ .Footnote ⁵ Then if $x_1^e\leq x_0^e$ , the value of the option to invest in the project defined by ( 1 ) with the guarantee and government subsidy defined by ( 17 ) is given by

(19)

and

(20)

where $B_7\equiv \frac{\lambda _{0}}{h_{0}\left (\gamma _{3}\right )} A_7, B_8\equiv \frac{\lambda _{0}}{h_{0}\left (\gamma _{4}\right )}A_8, M_{1}\equiv \frac{\lambda _{0}\left (B_{1}+B_{3}+\theta B_{5}\right )}{r+\lambda _{0}-\mu _{0} \gamma _{1}-\sigma _{0}^{2} \gamma _{1}\left (\gamma _{1}-1\right )/2},\ M_{2}\equiv \frac{\lambda _{0}\left (B_{2}+B_{4}+\theta B_{6}\right )}{r+\lambda _{0}-\mu _{0} \gamma _{2}-\sigma _{0}^{2} \gamma _{2}\left (\gamma _{2}-1\right )/2}$ ; and if $x_0^e\lt x_1^e$ , we have

(21)

and

(22)

where ${\hat{B}_7\equiv \frac{h_{0}\left (\gamma _{3}\right )}{\lambda _{0}} \hat{A_7}, \hat{B}_8\equiv \frac{h_{0}\left (\gamma _{4}\right )}{\lambda _{0}}\hat{A}_8},{\hat{M}_{1}\equiv \frac{\lambda _{1}\left (A_{1}+A_{3}+\theta A_{5}\right )}{r+\lambda _{1}-\mu _{1} \gamma _{1}-\sigma _{1}^{2} \gamma _{1}\left (\gamma _{1}-1\right )/2},\ \hat{M}_{2}\equiv \frac{\lambda _{1}\left (A_{2}+A_{4}+\theta A_{6}\right )}{r+\lambda _{1}-\mu _{1} \gamma _{2}-\sigma _{1}^{2} \gamma _{2}\left (\gamma _{2}-1\right )/2}}$ , and function $h_l(\cdot ), \ l\in \{0, 1\}$ , is defined by ( 4 ).

Taking $x_1^e\leq x_0^e$ Footnote ⁶ in particular, we conclude from the value-matching condition and smooth-pasting condition that the column vector $(A_7,A_8,C_7,C_8)'$ is the solution of the system of linear equations: $\Gamma _2X=\Phi _4$ , where

\begin{equation*} \Gamma _{2}=\left (\begin {array}{c@{\quad}c@{\quad}c@{\quad}c} \left (x_{1}^{e}\right )^{\gamma _{3}} & \left (x_{1}^{e}\right )^{\gamma _{4}} & 0 & 0 \\[5pt] \frac {\lambda _{0}}{h_{0}\left (\gamma _{3}\right )}\left (x_{1}^{e}\right )^{\gamma _{3}} & \frac {\lambda _{0}}{h_{0}\left (\gamma _{4}\right )}\left (x_{1}^{e}\right )^{\gamma _{4}} & -\left (x_{1}^{e}\right )^{\beta _{3}} & -\left (x_{1}^{e}\right )^{\beta _{4}} \\[5pt] \frac {\gamma _{3}\lambda _{0} }{h_{0}\left (\gamma _{3}\right )}\left (x_{1}^{e}\right )^{\gamma _3} & \frac {\gamma _{4}\lambda _{0} }{h_{0}\left (\gamma _{4}\right )}\left (x_{1}^{e}\right )^{\gamma _{4}} & -\beta _{3}\left (x_{1}^{e}\right )^{\beta _{3}} & -\beta _{4}\left (x_{1}^{e}\right )^{\beta _{4}} \\[5pt] 0 & 0 & \left (x_{0}^{e}\right )^{\beta _{3}} & \left (x_{0}^{e}\right )^{\beta _{4}} \end {array}\right ), \end{equation*}

and

\begin{equation*} \Phi _{4}=\left (\begin {array}{l} \left (B_{1}+B_{3}+\theta B_{5}\right )\left (x_{1}^{e}\right )^{\gamma _{1}}+\left (B_{2}+B_{4}+\theta B_{6}\right )\left (x_{1}^{e}\right )^{\gamma _{2}}+\left (1-\tau _{f}+\theta \tau _{f}\right ) q_{1} x_{1}^{e}\\[5pt] +(1-\theta ) \frac {\left (\tau _{f}-\tau _{i}\right ) c}{r}-I \\[5pt] M_{1}\left (x_{1}^{e}\right )^{\gamma _{1}}+M_{2}\left (x_{1}^{e}\right )^{\gamma _{2}}+\frac {\lambda _{0}\left (1-\tau _{f}+\theta \tau _{f}\right ) q_{1} x_{1}^{e}}{r+\lambda _{0}-\mu _{0}}+\frac {\lambda _{0}}{r+\lambda _{0}}(\frac {(1-\theta )\left (\tau _{f}-\tau _{i}\right ) c}{r}-I) \\[5pt] \gamma _{1} M_{1}\left (x_{1}^{e}\right )^{\gamma _{1}}+\gamma _{2} M_{2}\left (x_{1}^{e}\right )^{\gamma _{2}}+\frac {\lambda _{0}\left (1-\tau _{f}+\theta \tau _{f}\right ) q_{1} x_{1}^{e}}{r+\lambda _{0}-\mu _{0}} \\[5pt] \left (A_{1}+A_{3}+\theta A_{5}-M_{1}\right )\left (x_{0}^{e}\right )^{\gamma _{1}}+\left (A_{2}+A_{4}+\theta A_{6}-M_{2}\right )\left (x_{0}^{e}\right )^{\gamma _{2}}\\[5pt] +\left (1-\tau _{f}+\theta \tau _{f}\right )(q_{0}-\frac {\lambda _{0} q_{1}}{r+\lambda _{0}-\mu _{0}}) x_{0}^{e}+\frac {r}{r+\lambda _{0}}(\frac {(1-\theta )\left (\tau _{f}-\tau _{i}\right ) c}{r}-I) \end {array}\right ). \end{equation*}

The optimal investment thresholds $x_0^e$ and $x_1^e$ , which are endogenously determined by the entrepreneur, satisfy the following system of equations:

(23) \begin{equation}\! \left \{\!\begin{array}{l} \gamma _{3} A_{7}\left (x_{1}^{e}\right )^{\gamma _{3}}+\gamma _{4} A_{8}\left (x_{1}^{e}\right )^{\gamma _{4}}=\gamma _{1}\left (B_{1}+B_{3}+\theta B_{5}\right )\left (x_{1}^{e}\right )^{\gamma _{1}}+\gamma _{2}\left (B_{2}+B_{4}+\theta B_{6}\right )\left (x_{1}^{e}\right )^{\gamma _{2}}\\[5pt] +\left (1-\tau _{f}+\theta \tau _{f}\right ) q_{1} x_{1}^{e}, \\[5pt] \beta _{3} C_{7}\left (x_{0}^{e}\right )^{\beta _{3}}+\beta _{4} C_{8}\left (x_{0}^{e}\right )^{\beta _{4}}=\gamma _{1}\left (A_{1}+A_{3}+\theta A_{5}-M_{1}\right )\!\left (x_{0}^{e}\right )^{\gamma _{1}}+\gamma _{2}\left (A_{2}+A_{4}+\theta A_{6}-M_{2}\right )\!\left (x_{0}^{e}\right )^{\gamma _{2}}\\[5pt] +(1-\tau _{f}+\theta \tau _{f})(q_{0}-\frac{\lambda _{0} q_{1}}{r+\lambda _{0}-\mu _{0}}) x_{0}^{e}. \end{array}\right . \end{equation}

Proof. To be specific, we focus on the case of $x_0^b\geq x_1^b$ and $x_1^e\leq x_0^e$ in the following since the proofs for other cases are almost the same. The proof is similar to that of the proposition 3.1. We take the investment option as a special standard asset defined in Subsection 3.1. The standard asset has no regular cash flow, and its terminal payoff $G(x,l)=V_l(x)-I+\theta TX_l(x)$ , where $V_l(x)=E_l(x)+D_l(x), l\in \{0,1\}$ , with $D_l(x)$ being given by (9) and (10) and $E_l(x)$ being given by (11) and (12), and $TX_l(x)$ is given by (14) and (15). We stress that the conclusions of Proposition 3.1 do not apply directly because its linear terminal payoff assumption does not hold here.

First, we conclude from the Bellman’s principle (without control) that the value $R_{l}(x)$ defined by (18) of the option to invest satisfies the following ODE:

(24) \begin{equation} \left \{\begin{array}{l} r R_{1}(x)=\mu _{1} x R_{1}^{\prime }(x)+\frac{\sigma _{1}^{2} x^{2}}{2} R_{1}^{\prime \prime }(x)+\lambda _{1}\left (R_{0}(x)-R_{1}(x)\right )\!,\\[5pt] r R_{0}(x)=\mu _{0} x R_{0}^{\prime }(x)+\frac{\sigma _{0}^{2} x^{2}}{2} R_{0}^{\prime \prime }(x)+\lambda _{0}\left (R_{1}(x)-R_{0}(x)\right )\!, \end{array}\right . \end{equation}

for $x \in \left (0, x_{1}^{e}\right )$ ;

(25) \begin{equation} \left \{\begin{array}{l} R_{1}(x)=V_{1}(x)-I+\theta TX_1(x),\\[5pt] r R_{0}(x)=\mu _{0} x R_{0}^{\prime }(x)+\frac{\sigma _{0}^{2} x^{2}}{2} R_{0}^{\prime \prime }(x)+\lambda _{0}\left (V_{1}(x)-I+\theta TX_1(x)-R_{0}(x)\right )\!, \end{array}\right . \end{equation}

for $x \in \left [x_{1}^{e}, x_{0}^{e}\right )$ ; and

(26) \begin{equation} \left \{\begin{array}{l} R_{1}(x)=V_{1}(x)-I+\theta TX_1(x), \\[5pt] R_{0}(x)=V_{0}(x)-I+\theta TX_0(x), \end{array}\right . \end{equation}

for $x \in \left [x_{0}^{e}, \infty \right )$ . After that, we use the guess-and-verify method to solve (24)–(26). Specifically, motivated by (7) and (8) in Proposition 3.1, we guess that (24) (in the continuation region) has the following general solution:

\begin{equation*} R_{1}(x)= A_{7} x^{\gamma _{3}}+A_{8} x^{\gamma _{4}}+\sum _{i=1}^{2}\hat {A}_ix^{\gamma _i}\hbox { and }R_{0}(x)= B_{7} x^{\gamma _{3}}+B_{8} x^{\gamma _{4}}+\sum _{i=1}^{2}\hat {B}_ix^{\gamma _i}. \end{equation*}

Noting that $\lim _{x\rightarrow 0}Q_{1}(x)=\lim _{x\rightarrow 0}Q_{0}(x)=0$ (the option is virtually worthless if the EBIT approaches zero), we get $\hat{A}_1=\hat{A}_2=\hat{B}_1=\hat{B}_2=0$ since $\gamma _1$ and $\gamma _2$ are negative. Substituting the general solution into (24) leads to that $\gamma _3$ and $\gamma _4$ are a root of the equation $h_0(\gamma )h_1(\gamma )=\lambda _0\lambda _1$ . Similarly, if $x \in \left [x_{1}^{e}, x_{0}^{e}\right )$ (transient region), we guess that $R_0(x)=C_{7} x^{\beta _{3}}+C_{8} x^{\beta _{4}}+M_{1} x^{\gamma _{1}}+M_{2} x^{\gamma _{2}}+\zeta _1 x+\zeta _2$ . Substituting it into (25) and using the continuity and differentiability of the value functions, we get (19) and (20) after tedious derivations. Last, we use the well-known smooth-pasting conditions to derive the optimal investment thresholds $x_0^e$ and $x_1^e$ satisfying (23).

Proposition 5.1 explains that in the continuation region ( $x\lt x_{1}^{e}$ ), the real option value is subject to the possible adjustments arsing from the regime-switching in addition to the common investment option value. In the transient region ( $x_{1}^{e}\leq x \lt x_{0}^{e}$ ) with a recession state, the real option value is the common value of waiting plus the increased value due to the possible economy switching into boom when the option is instantly exercised. These conclusions look complicated but fortunately are closed-form. Numerical tests are provided in Section 6.

5.2 Optimal leverage

For a given government subsidy policy, the entrepreneur would sufficiently make use of the policy benefit to maximize the value of the option to invest. Naturally, we wonder what the optimal capital structure or optimal leverage should be if the amount of money borrowed is unrestrained thanks to the guarantee and subsides. We now answer this question.

Clearly, this question comes down to finding optimal coupon rate, denoted by $c_l^*$ , of debt at the investment threshold $x_l^e,\ l\in \{0,1\}$ . To do so, we emphasize that the prices of corporate securities derived before are a function of the coupon rate $c$ of debt, which is exogenously given but must satisfy $(1-\tau _i)c*(1-g)/r\ge I$ , i.e. the loan after paying the guarantee cost must cover the sunk cost $I$ . To highlight their dependence on the coupon rate $c$ , we denote the above-derived option value by $R_l(x,c)$ instead of $R_l(x)$ when the current EBIT is equal to $x$ . In the same way, the above-derived debt value at the investment threshold (time) $x_l^e$ is denoted by $D_l(x_l^e, c)$ instead of $D_l(x_l^e)$ .

Therefore, the optimal coupon rate $c_l^*$ at the investment threshold $x_l^e$ solves the following optimization problem:

(27) \begin{equation} {c_l^*\equiv \arg }\sup _{c} R_l(x_l^e, c), \ l\in \{0,1\}. \end{equation}

And then, we get the following optimal leverage at the investment threshold $x_l^e$ :

(28) \begin{equation} \mathcal{L}^*(x_l^e)=\frac{D_l(x_l^e, c_l^*)}{{V_l(x_l^e,c_l^*)}}, \ l\in \{0,1\}. \end{equation}

5.3 The pricing of the tax revenues

To compute the present value of the future random tax revenues, we note that the tax revenues are like a real option but with the known investment threshold $x_l^e, l\in \{0,1\}$ . Therefore, similar to Proposition 5.1, we provide the following results without a proof.

The tax value prior to investment is given by

\begin{equation*} T_l(x)\equiv \mathbb {E}\left [e^{-r(\tau -t)} (1-\theta )T X_l\left (X_\tau \right ) \mid X_0=x, L_0=l\right ], \end{equation*}

where $\tau$ represents the investment time given by (18).

Suppose that the default occurs earlier in recession than in boom, i.e. $x_0^b\geq x_1^b$ , and that the investment occurs later in recession than in boom, i.e. $x_1^e\leq x_0^e$ . Similar to the real option pricing method, the pre-investment tax value is given by

(29)

and

(30)

where $\!B_9\equiv \frac{\lambda _{0}}{h_{0}\left (\gamma _{3}\right )} A_9,\! B_{10}\equiv \frac{\lambda _{0}}{h_{0}\left (\gamma _{4}\right )}A_{10},\! M_3\equiv \frac{(1-\theta )\lambda _0 B_5}{r+\lambda _0-\mu _0 \gamma _1-\sigma _0^2\gamma _1\left (\gamma _1-1\right )/2},$ $M_4\equiv \frac{(1-\theta )\lambda _0 B_6}{r+\lambda _0-\mu _0 \gamma _2-\sigma _0^2 \gamma _2\left (\gamma _2-1\right )/2}$ . From the value-matching condition, we conclude that the column vector $(A_9,A_{10},C_9,C_{10})'$ is the solution of the system of linear equations $\Gamma _2X=\Phi _5$ , where

\begin{equation*} \Phi _5=\left (\begin {array}{l} (1-\theta )(B_5\left (x_1^e\right )^{\gamma _1}+B_6\left (x_1^e\right )^{\gamma _2}+\tau _f\left (q_1 x_1^e-\frac {c}{r}\right )+\frac {\tau _i c}{r}) \\[5pt] M_3\left (x_1^e\right )^{\gamma _1}+M_4\left (x_1^e\right )^{\gamma _2}+\frac {(1-\theta )\lambda _0 \tau _f q_1 x_1^e}{r+\lambda _0-\mu _0}-\frac {(1-\theta )\lambda _0\left (\tau _f-\tau _i\right ) c}{\left (r+\lambda _0\right ) r} \\[5pt] \gamma _1 M_3\left (x_1^e\right )^{\gamma _1}+\gamma _2 M_4\left (x_1^e\right )^{\gamma _2}+\frac {(1-\theta )\lambda _0 \tau _f q_1 x_1^e}{r+\lambda _0-\mu _0} \\[5pt] \left ((1-\theta )A_5-M_3\right )\left (x_0^e\right )^{\gamma _1}+\left ((1-\theta )A_6-M_4\right )\left (x_0^e\right )^{\gamma _2}\\[5pt] +(q_0-\frac {\lambda _0 q_1}{r+\lambda _0-\mu _0})(1-\theta )\tau _f x_0^e -\frac {(1-\theta )\left (\tau _f-\tau _i\right ) c}{r+\lambda _0} \end {array}\right ). \end{equation*}

In short, we conclude from (13) the optimal default threshold $x_l^b$ for a given coupon rate $c$ , from (23) the optimal investment threshold $x_l^e$ for some given coupon rate $c$ and default threshold $x_l^b$ , and from (27) the optimal coupon rate $c_l^*$ of debt for some given investment threshold $x_l^e$ and default threshold $x_l^b$ . Solving them simultaneously, we derive the optimal default threshold $x_l^b$ , optimal capital structure $c_l^*$ , and optima investment threshold $x_l^e$ . Once decision variables $\{{x_l^b}, x_l^e, c_l^*\}, l\in \{0, 1\}$ , are determined, the values of all other contingent claims including the firm’s leverage can be derived accordingly. More specifically, we produce the following algorithm:

Algorithm 1. The algorithm for the pricing and timing of the real option

6.1 The optimal investment threshold and the guarantee cost

We first consider how the jump intensity from recession to boom impacts on the investment time and guarantee cost. Intuitively, the higher the intensity, the more profitable the project and thus the earlier the investment and due to the less the default risk, the less the guarantee cost. As we expected, Figure 1 tells the same story. Specifically, Figure 1(a) shows that the optimal investment threshold decreases with the jump intensity from recession to boom. In addition, the investment is much earlier in boom than in recession. Figure 1(b) demonstrates that the increase of jump intensity reduces the guarantee cost and the guarantee cost in boom is smaller than that in recession.

Figure 2. The effect of volatility in recession on (a) optimal investment threshold and (b) fair guarantee fee rate.

Figure 2(a) explains that as volatility $\sigma _0$ in recession increases, the investment is postponed both in recession and in boom. This is in agreement with the well-known assertion that a higher volatility increases the value of waiting and delays investment. Moreover, an increase in volatility changes the bankruptcy trigger, postponing the default in recession, such that if volatility $\sigma _0$ surpasses a threshold, the default in recession happens later than in boom as shown in the figure. In addition, as we expected, the investment in boom occurs much earlier than in recession. Figure 2(b) shows that the guarantee fee rate in boom increases with the volatility, while that in recession decreases with the volatility. The former is intuitive since a high volatility has a negative impact on the value of debt, increases the guarantee liability, and naturally induces a high guarantee fee rate in return for the guarantee. However, the latter is counter-intuitive. It comes to light that a higher volatility induces higher investment threshold and government subsidies, which thus leads to a less default risk and a less guarantee liability. The final result is determined by which of the two opposite factors dominates the other. For the same reason, Figure 2(b) displays that when the volatility is high, the guarantee cost in boom is strangely higher than that in recession.

Figure 3. The effect of the tax subsidy coefficient $\theta$ on (a) optimal investment threshold and (b) fair guarantee fee rate.

Figure 3(a) presents the impact of government subsidies through guarantee fee reduction or cash subsidy. It says that the higher the tax subsidy coefficient, the less the optimal investment threshold. This is because the greater the tax subsidy coefficient, equivalently the less the investment cost. Figure 3(b) plots the guarantee fee rate against the tax subsidy coefficient. It states that the increase in the tax subsidy coefficient reduces the guarantee fee rate as we expected. In particular, the guarantee fee rate would be negative, meaning that not only a waiver of the guarantee premium but also a cash subsidy of $-g_l(1-\tau _i)c/r$ is provided by the government. This seems unreasonable at first sight; but it would be really worthwhile for a project having a sufficiently positive externality but lacking in the private value. Figure 3(b) further says that the guarantee fee rate in boom is less than that in recession. Intuitively, this is because the default risk in boom is lower than that in recession.

Figure 4. The effect of corporate effective tax rate $\tau _f$ on optimal investment threshold and fair guarantee fee rate.

Figure 4 illustrates the impact of corporate effective tax rates on investment thresholds and guarantee costs. Figure 4(a) says that an increase in the corporate effective tax rate leads to a steady rise in the optimal investment threshold. This is because the higher the tax rate, the less the firm gets from the investment; thus, the investment is postponed. In addition, economic boom accelerates investment relative to economic recession; this is clear. Figure 4(b) demonstrates that an increase in the tax rate reduces the guarantee cost. The reason behind this is that the higher the tax rate, the higher the investment threshold (i.e. the lower the default risk) and the higher the tax subsidy; both of them lead to a less fair guarantee cost. Last, as we expected, Figure 4(b) states that the guarantee cost is lower in boom than in recession for an arbitrary given tax rate.

Table 1. Optimal coupon rate $c^*_l$ vs. the jump intensity from recession to boom

Table 2. Optimal coupon rate $c_l^*$ vs. the volatility in recession

6.2 The optimal capital structure and option value

More often than not, we do not consider the optimal capital structure problem since the amount of money borrowed is determined by the funding gap for investing in a project. On account of that the government provides subsidies depending on the level of debt in our model, we address the optimal structure and fix the optimal coupon level to maximize the project value. Obviously, due to that the government subsidy is a fraction of the total tax revenue generated by the project, the benefit of issuing debt is decreased. In particular, if tax subsidy coefficient $\theta =1$ , then the merit of tax shields from debt disappears and thus the entrepreneur should not borrow if s/he has enough money to invest in the project. In this subsection, as assumed before, we suppose that the entrepreneur must borrow to start the project and s/he can borrow as much as s/he wants. We discuss how much s/he should borrow to maximize the project value and how the optimal coupon rate is related with the jump intensity $\lambda _0$ , volatility $\sigma _0$ and externality of the project.

Table 1 reports the effect of the jump intensity from recession to boom on the optimal capital structure (i.e. optimal coupon rate $c_l^*$ or optimal leverage $\mathcal{L}_l^*$ , $\ l\in \{0, 1\}$ ) and the real option value. It says that an increased jump intensity increases the level of debt and the leverage both in boom and in recession. The former (the increased debt level) results from the fact that the higher the intensity, the less the default risk, and thus the more the optimal level of debt issued. The latter (the increased leverage) happens because of the increased debt level. However, the leverage may decrease with the jump intensity since the higher the jump intensity, the less the investment threshold, as predicted by Figure 1(a); thus, a lower leverage would be conversely preferred to prevent the firm from default. Moreover, as expected, the optimal level of debt and optimal leverage in boom are higher than that in recession; the higher the jump intensity, the higher the real option value.

Table 2 displays the effect of the volatility on the optimal capital structure and the real option value. It states that the optimal debt coupon both in recession and in boom increases with the volatility $\sigma _0$ . This happens because the higher the volatility, the higher the investment threshold, i.e. the safer the debt, and thus, the higher the optimal debt coupon. By contrast, the change of the leverage is ambiguous since both the debt value and the firm value increases with the volatility. Moreover, the optimal debt coupon in boom is higher than that in recession. The reason behind is that the debt is exposed to less default risk in boom than in recession, which implies that increasing debt scale in boom is profitable relative to the debt scale in recession. Last, the table predicts that a higher volatility increases the value of the option. This is a well-known result.

Table 3 shows the relationship between government subsidies and optimal capital structure. Government subsidies depend on tax revenues; thus, government subsidies would make the tax shields effect insignificant. Therefore, the optimal coupon rate and optimal leverage decrease with the tax subsidy coefficient as argued in the first paragraph of the subsection and predicted by the table. The more the government subsidizes, the less the profit from tax shields and thus the less the optimal level of debt. The table displays further that the increase of the investment externalities induces a higher value of the investment option. This is because the higher the tax subsidy coefficient, the more the subsidies and all the subsidies are harvested by the entrepreneur.

Table 3. Optimal coupon rate $c_l^*$ vs. tax subsidy coefficient

Table 4 illustrates the effect of the corporate effective tax rate on the optimal coupon rate and the value of the real option. It displays that the optimal coupon rate and the leverage increase with the corporate effective tax rate. This is because the higher the tax rate, the tax shield effect becomes more valuable, leading to an increase in both the optimal coupon rate and the leverage. Noting that the tax subsidy coefficient $\theta =0.4$ , we conclude that the value of the real option decreases with the tax rate as shown in the table. Finally, Table 4 explains that the optimal coupon rate, optimal leverage, and the value of the real option in recession are less than their corresponding items in boom. Clearly, these are what we expect.

Table 4. Optimal coupon rate $c_l^*$ vs. corporate effective tax rate

6.3 The subsidy required for an immediate investment

All real investment projects can be divided into three types according to their profitability: One is unprofitable (the project value is negative) and thus should be given up forever; The second type is profitable from the point of view of investors and should be therefore started sometimes; The third type has a positive total social value but it is not profitable for private investors. For the third type of projects governments should provide subsides. For example, when a country is faced with a major industrial or strategic demand and requires a special project to be started immediately. The government should use government funds to support the entrepreneur with the help of credit guarantees and government subsidies. We wonder how much subsides should be provided at least to let the entrepreneur start such a project immediately. Naturally, the larger the government subsidy, the lower the optimal investment threshold. In particular, without government subsidies, it would not lead to an immediate investment if the current cash flow level is too low. More specifically, we can consider the investment threshold $x_l^e$ as a decreasing function of the amount $S_l$ of government subsidies, denoted by $x_l^e(S_l)$ , if Keeping all the other things the same. Clearly, to motivate the entrepreneur to invest instantly, the required minimum subsidy $S_l^*$ is given by $S_l^*\equiv \inf \{S_l|x_l^e(S_l)\leq x\},\ l\in \{0,1\}$ , where $x$ is the current cash flow level. Figure 5 shows that the amount $ S_0$ or $ S_1$ of government subsidies at least is required for motivating the entrepreneur to conduct the immediate investment. Figure 5(a) shows the government subsidies $ S_0$ required to take an immediate investment in recession. If the firm’s cash flow is 14.9, the entrepreneur does not exercise the investment unless s/he gets the amount of government subsidies being $ S_0\approx 11$ . The higher the current level of cash flow, the less the government subsidy required for the immediate investment. In a similar way, Figure 5(b) presents the corresponding results in boom. Last, if the externality of the investment is negative, e.g., an investment inducing environmental deterioration, the tax subsidy coefficient $\theta$ can be less than zero, meaning that the investment is punished in addition to getting no subsidies from a government.

Figure 5. The figure demonstrates the government subsidies required for motivating an immediate investment under the two different economy state and with different cash flow levels.

6.4 The effect of tax rates on the pricing and timing of an alternative real option

Strictly speaking, an investment in a project not only creates the project value but also creates the value of tax revenues in addition to the value derived from its externalities. For an obvious reason, we take the sum of the created project value and the value of the created tax revenues, i.e. $R_l(x)+T_l(x)$ , as the value of the investment option, while the value derived from its externalities is not easy to measure.Footnote ⁷ We call such investment option an alternative real option. We stress that its investment threshold is still determined by the entrepreneur as before; thus, only if $\theta =1$ , the investment threshold chosen by the entrepreneur is also optimal for the alternative real option.

Figure 6(a) says that if the tax subsidy coefficient $\theta$ is greater (less) than 1, the entrepreneur would overinvest (underinvest). Figure 6(b) states that both the overinvestment and underinvestment induce a value loss of the alternative real option. The higher the tax rate, the more significant the effect. All these results are easy to follow thanks to our previous analysis. Naturally, if a project has sufficiently positive externalities, we should let $\theta \gt 1$ to motivate the firm to accelerate investment, even if it would lead to the value loss of the alternative real option. For this reason, the tax subsidy coefficient ( $\theta$ ) is a powerful policy instrument to motivate an entrepreneur to invest in a positive externality project.

Figure 6. The effect of the corporate effective tax rate on the pricing and timing of the alternative real option.