Online Appendices to Financing Asset Sales and Business Cycles

Transcription

1 Online Appendices to Financing Asset Sales usiness Cycles Marc Arnold Dirk Hackbarth Tatjana Xenia Puhan August 22, 2017 University of St. allen, Unterer raben 21, 9000 St. allen, Switzerl. Telephone: oston University, 595 Commonwealth Avenue, oston, MA 02215, United States. Telephone: Swiss Life Asset Managers, eneral-uisan-quai 40, 8002 Zurich, Switzerl. Telephone:

2 Appendix A. Derivations Appendix A.1. The stochastic discount factor, risk-free rates, market prices of risk According to hamra, Kuehn, Strebulaev 2010 Chen 2010, solving the ellman equation associated with the consumption problem of the representative agent implies that the stochastic discount factor m t follows the dynamics dm t m t = r i dt η i dw C t + e κ i 1 dm t, A.1 in which M t is the compensated process associated with the Markov chain. r i are the regimedependent risk-free interest rates, η i the risk prices for systematic rownian shocks affecting aggregate consumption, κ i the market prices of jump risk: [ ω δ r i = r i + λ i w ω 1 ω δ 1 w 1 1 ], A.2 ω 1 η i = ωσi C, A.3 hj κ i = δ ω log, A.4 h i with i, j =,, i = j. The parameters h, h solve the following non-linear system of equations e.g. hamra, Kuehn, Strebulaev 2010: 1 ω 0 = ρ h δ ω 1 δ i + 1 ω θ i 1 2 ω 1 ω σi C 2 1 ω ρ h 1 ω 1 δ i + λ i h 1 ω j h 1 ω i A.5 The parameters κ i denote the relative jump sizes of the discount factor when the Markov chain leaves state i. The risk-free rates r i contain the interest rate if the economy stayed in state i forever, r i, plus a second term that incorporates a possible switch in the state. The no-jump part of the interest rates, r i, are r i = ρ + δθ i 1 2 ω 1 + δ σ C i 2, A.6 w := e κ = e κ A.7 measures the size of the jump in the real-state price density when the economy shifts from bad states to good states see for example Proposition 1 in hamra, Kuehn, Strebulaev

3 Finally, r p i reflects the perpetual risk-free rate given by r p i = r i + r j r i p + r j p f j. A.8 Appendix A.2. Derivation of the values of corporate securities after investment The valuation of corporate debt. Our valuation of corporate debt of a firm that consists of only invested assets in a two state setting follows Hackbarth, Miao, Morellec 2006 Arnold, Wagner, Westermann We illustrate the case in which the default boundary in good states is lower than that in bad states, i.e., ˆD < ˆD. If a firm defaults, debtholders receive a fraction Ξ i α i of the unlevered after tax asset value 1 τxy i. A debt investor requires an instantaneous return equal to the risk-free rate r i. The instantaneous debt return corresponds to the realized rate of return plus the coupon proceeds from debt. Hence, an application of Ito s lemma with possible state switches shows that debt satisfies the following system of ODEs. For 0 X ˆD : { ˆ d X = α Ξ 1 τxy ˆ d X = α Ξ 1 τxy. For ˆD < X ˆD : { rd ˆ X = c + µ Xd ˆ X σ2 X2 ˆ For X > ˆD : ˆ d X = α Ξ 1 τxy. X + λ α Ξ 1 τxy d ˆ X d r d ˆ X = c + µ Xd ˆ X σ2 X2 d ˆ X + λ d ˆ X d ˆ X r d ˆ X = c + µ Xd ˆ X σ2 X2 d ˆ X + λ d ˆ X d ˆ X. A.9 A.10 A.11 The boundary conditions read lim X lim X ˆD lim X ˆD dˆ i X X <, i =,, A.12 ˆ d X = lim X ˆD ˆ d X = lim X ˆD ˆ d X, lim d ˆ X = α Ξ 1 τ ˆD y, X ˆD A.13 ˆ d X, A.14 A.15 2

4 lim d ˆ X = α Ξ 1 τ ˆD y. X ˆD A.16 Condition A.12 captures the no-bubbles condition. The other boundary conditions are the continuity condition A.13, the smoothness condition A.14, the value-matching conditions A.15 A.16 at the default thresholds ˆD ˆD for the debt values in a good state a bad state, dˆ d ˆ. The functional form of the solution is dˆ i X = α i Ξ i 1 τxy i X ˆD i i =, Ĉ 1 X β 1 + Ĉ 2 X β 2 + C 3 X + C 4 ˆD < X ˆD, i =  i1 X γ 1 + Âi2 X γ 2 + A i5 X > ˆD, i =,, A.17 in which  1,  2,  1,  2, A 5, A 5, Ĉ 1, Ĉ 2, C 3, C 4, γ 1, γ 2, β 1, β 2 are real-valued parameters to be determined. First, we consider the region X > ˆD. We start with the stard approach by plugging the functional form ˆ d i X =  i1 X γ 1 +  i2 X γ 2 + A i5 A.18 into both equations of A.11. Comparing coefficients solving the resulting two-dimensional system of equations for A i5, we obtain A i5 = c r j + λ i + λ j = c r i r j + r j λ i + r i λ j r p, A.19 i find that  k is a multiple of  k, k = 1, 2, with the factor l k := 1 λ r + λ µ γ k 1 2 σ2 γ kγ k 1, i.e.,  k = l k  k. Using these results when comparing coefficients again, it can be shown that γ 1 γ 2 are the negative roots of the quadratic equation µ γ σ2 γγ 1 λ r µ γ σ2 γγ 1 λ r = λ λ. A.20 To satisfy the no-bubbles condition for debt in Equation A.12, we take the negative roots. Next, we consider the region ˆD X ˆD. Plugging the functional form d X = Ĉ 1 X β 1 + Ĉ2 X β 2 + C3 X + C 4 A.21 3

5 into the first equation of A.10, we find by comparison of coefficients that β 1,2 = 1 2 µ σ 2 2 ± 1 2 µ σ 2 + 2r + λ σ 2 C 3 = λ α Ξ 1 τy r + λ µ C 4 = c. r + λ A.22 We then plug the functional form A.17 into conditions A.13 A.16, obtain a four-dimensional linear system in the remaining four unknown parameters  1,  2, Ĉ 1, Ĉ 2 :  1 ˆD γ 1 +  2 ˆD γ 2 + A 5 = Ĉ 1 ˆD β 1 + Ĉ 2 ˆD β 2 + C 3 ˆD + C 4  1 γ 1 ˆD γ 1 +  2 γ 2 ˆD γ 2 = Ĉ 1 β ˆD β Ĉ 2 β ˆD β C 3 ˆD α Ξ 1 τ ˆD y = Ĉ 1 ˆD β 1 + Ĉ 2 ˆD β 2 + C 3 ˆD + C 4 l 1  1 ˆD γ 1 + l 2Â2 ˆD γ 2 + A 5 = α Ξ 1 τ ˆD y. A.23 We define the matrices ˆD γ 1 ˆD γ 2 ˆD β 1 ˆD β 2 γ ˆM := 1 ˆD γ 1 γ 2 ˆD γ 2 β ˆD β 1 1 β ˆD β 2 2 ˆD β 1 ˆD β 2 l 1 ˆD γ 1 l 2 ˆD γ 2 C 3 ˆD + C 4 A 5 ˆb := C 3 ˆD α Ξ 1 τ ˆD y C 3 ˆD C 4, α Ξ 1 τ ˆD y A 5 A.24 A.25 such that ˆM [Â1  2 Ĉ 1 Ĉ 2 ] T = ˆb. The solution for the unknown parameters is [Â1  2 Ĉ 1 Ĉ 2 ] T = ˆM 1ˆb. A.26 The value of the tax shield is calculated by using the formula for the value of debt, in which c is replaced by τc, α is equal to zero. Similarly, we obtain the value of bankruptcy costs by simply replacing c with zero, α with 1 α. 4

6 Default policy. The value of equity equals firm value minus the value of debt. Firm value is given by the value of assets in place plus the value of the growth option the tax shield minus default costs. After debt has been issued, firms choose the ex post default policy that maximizes the value of equity. Formally, the default policy is determined by equating the first derivative of the equity value to zero at the corresponding default boundary level: We solve this problem numerically. { ê ˆD = 0 ê ˆD = 0. A.27 For a firm that receives scaled earnings after investment, the value of corporate securities is solved analogously by replacing X with the scaled earnings. For example, if the firm exercises the option in the good state finances the exercise cost by issuing equity, the scaled earnings correspond to s + 1X. The default boundaries ˆD ˆD are then expressed in terms of the scaled earnings levels. Appendix A.3. Derivation of the value of the growth option The case with X < X : We present the derivation of the value of corporate securities for a firm that issues equity in good states sells assets in bad states. For each state i, the option is exercised immediately whenever X X i option exercise region; otherwise, it is optimal to wait option continuation region. This structure results in the following system of ODEs for the option s value function. For 0 X < X : { r X = µ X X σ2 X2 X + λ X X r X = µ X X σ2 X2 X λ X X. A.28 For X X < X : { X = 1 τs Xy k 1 + φ r X = µ X X σ2 X2 X λ 1 τs Xy k 1 + φ X. A.29 For X X : { X = 1 τs Xy k 1 + φ X = 1 τs Xy k /Ξ. A.30 When X is in the option continuation region, which corresponds to system A.28 the second 5

7 equation of A.29, the required rate of return r i left-h side must be equal to the realized rate of return right-h side. We calculate the realized rate of return by using Ito s lemma for state switches. In this region, the last term captures the possible jump in the value of the growth option due to a state switch. It can be expressed as the instantaneous probability of a shift in the state, that is, λ or λ, times the corresponding change in the value of the option. The first equation of A.29 the system A.30 describe the payoff of the option at exercise. The process is in the option exercise region in these cases. The boundary conditions are lim i X = 0, i =,, A.31 X 0 lim X = lim X, X X X X lim X = lim X, X X X X lim X = 1 τs X y k /Ξ, X X A.32 A.33 A.34 lim X = 1 τs X y k 1 + φ. X X A.35 Equation A.31 ensures that the option value goes to zero as earnings approach zero. Equations A.32 A.33 are the continuity smoothness conditions of the value function in bad states at the exercise boundary in good states. Equations A.34 A.35 are the value-matching conditions at the exercise boundaries in a good state a bad state, respectively. The functional form of the solution is Ā i3 X γ 3 + Ā i4 X γ 4 0 X < X, i =, C 1 X β 1 + C 2 X β 2 + C 3 X + C 4 X i X = X < X, i = 1 τs Xy k /Ξ X X i = 1 τs Xy k 1 + φ X X i =, A.36 in which Ā 3, Ā 4, Ā 1, Ā 2, C 1, C 2, C 3, C 4, γ 3, γ 4, β 1, β 2 are real-valued parameters that need to be determined. First, we consider the region 0 X < X plug the functional form i X = Ā i3 X γ 3 + Ā i4 X γ 4 into both equations of A.28. A comparison of coefficients implies that Ā k is a multiple of Ā k, k = 3, 4, with the multiple factor l k := 1 λ r + λ µ γ k 1 2 σ2 γ kγ k 1, i.e., Ā k = l k Ā k. 6

8 Using this result when comparing coefficients, we find that γ 3 γ 4 are the positive roots of the quadratic equation µ γ σ2 γγ 1 λ r µ γ σ2 γγ 1 λ r = λ λ. A.37 oundary condition A.31 implies to use the positive roots. Next, we consider the region X X < X. Plugging the functional form X = C 1 X β 1 + C 2 X β 2 + C 3 X + C 4 into the second equation of A.29, we find by comparison of coefficients that β 1,2 = 1 2 µ σ 2 ± 1 2 µ σ r + λ, σ 2 C 3 = λ 1 τs y r µ + λ, A.38 C 4 = λ k /Ξ r + λ. The remaining unknown parameters are Ā 3, Ā 4, C 1, C 2. Plugging the functional form A.36 into conditions A.32 A.35 yields C 1 X β 1 + C 2 X β 2 + C 3 X + C 4 = l 3 Ā 3 X γ 3 + l 4 Ā 4 X γ 4, C 1 β 1 Xβ 1 + C 2 β 2 X β 2 + C 3 X = l 3 Ā 3 γ 3 X γ 3 + l 4 γ 4 Ā 4 X γ 4, A.39 A.40 C 1 X β 1 + C 2 X β 2 + C 3 X + C 4 = 1 τs y X k /Ξ, A.41 Ā 3 X γ 3 + Ā 4 X γ 4 = 1 τs y X k 1 + φ. A.42 This four-dimensional system is linear in its four unknowns Ā 3, Ā 4, C 1 C 2. We define the matrices M := l 3 X γ 3 l 3 γ 3 X γ 3 l 4 X γ 4 X β 1 X β 2 l 4 γ 4 X γ 4 β1 Xβ 1 β2 Xβ 2, A.43 X β 1 X β 2 X γ 3 X γ 4 7

9 C 3 X + C 4 b := C 3 X C 3 X C τs y X k /Ξ, 1 τs y X k 1 + φ A.44 such that M [Ā3 Ā 4 C 1 C 2 ] T = b. The solution to the remaining four unknowns is [Ā3 Ā 4 C 1 C 2 ] T = M 1 b. A.45 The unlevered value of the growth option. The unlevered value of the growth option corresponds to the value of an option in a firm without debt all equity firm. It is calculated by additionally imposing the smooth-pasting boundary conditions at option exercise: lim unlev X X unlev X = 1 τs y A.46 lim unlev X X unlev X = 1 τs y. A.47 The solution method is analog to that for the levered growth option value up to including Equation A.38. The system A.39 A.42 needs to be augmented by the two equations corresponding to the additional smooth-pasting boundary conditions: C 1 unlev β1 X unlev β C 2 unlev β2 X unlev β C 3 = 1 τs y A.48 Ā unlev 3 γ 3 X unlev γ3 1 + Ā unlev 4 γ 4 X unlev The full system is six-dimensional with the six unknowns Ā unlev 3 γ4 1 = 1 τs y. A.49, Ā unlev 4, C 1 unlev, C unlev 2 X unlev, X unlev, linear in the first four unknowns nonlinear in the last two unknowns. We solve this system numerically. 8

10 The case with X X : The solution of the case X X is obtained immediately by renaming states in the solution of the presented case for X < X. Alternative financing strategies: Values of growth options for any alternative financing strategy can be derived analogously by replacing the friction adjusted option exercise cost terms in Equations A.29, A.30, A.34, A.35. Appendix A.4. Firms with invested assets an expansion option We first present a proof for the value of corporate debt in the case in which D < D, ˆD < ˆD, X > X. The argument of the proof is adapted from Arnold, Wagner, Westermann Proof of Proposition 2. A debt investor requires an instantaneous return equal to the risk-free rate r i. The application of Ito s lemma with state switches shows that debt must satisfy the following system of ODEs. For 0 X D : { d X = α Ξ 1 τxy + unlev X d X = α Ξ 1 τxy + unlev X. A.50 For D < X D : For D < X < X : { r d X = c + µ Xd X σ2 X2 d X + λ α φ 1 τxy + unlev X d X d X = α Ξ 1 τxy + unlev X. r d X = c + µ Xd X σ2 X2 d X + λ d X d X r d X = c + µ Xd X σ2 X2 d X + λ d X d X. A.51 A.52 For X X < X : { d X = d ˆ s + 1X r d X = c + µ Xd X σ2 X2 d X + λ d ˆ s + 1X d X. A.53 9

11 For X X : d X = ˆ d s + 1X d X = d ˆ s + 1 k /Ξ X 1 τx ty. A.54 In system A.50, the firm is in the default region in both good bad states. In this region, debtholders receive at default α i Ξ i 1 τxy i + unlev i X. The firm is in the continuation region in good states in the default region in bad states in system A.51. For the continuation region in good states, the left-h side of the first equation is the instantaneous rate of return required by investors for holding corporate debt. The right-h side is the realized rate of return, computed by Ito s lemma as the expected change in the value of debt plus the coupon payment c. The last term expresses the possible jump in the value of debt in case of a state switch that triggers immediate default. Equations A.52 describe the case in which the firm is in the continuation region in both good bad states. The next system, A.53, treats the case in which the firm is in the exercise region in good states in the continuation region in bad states. After exercising the option, the firm owns total assets in place with value 1 τxy i + 1 τs ī Xy i, A.55 reflecting the notion that the exercise cost of the growth option can be financed by issuing equity in good states. The value of debt must then be equal to the value of debt of a firm with only invested assets, i.e., d X = ˆ d s + 1 X, A.56 which is the first equation in A.53. We obtain the second equation in this case by using the same approach as in A.52. The last term captures the notion that a switch from bad states to good states triggers immediate exercise of the expansion option with equity financing. Finally, A.54 describes the case in which the firm is in the exercise region in both good bad states. In good states, the earnings of the firm are scaled by s + 1. In bad states, the exercise cost k is financed by selling k /Ξ 1 τx ty A.57 10

12 of the assets in place, such that the earnings of the firm are scaled by The system is subject to the following boundary conditions. s + 1 k /Ξ 1 τx ty. A.58 lim d X = lim d X, X D X D lim d X = lim d X, X D X D lim d X = α Ξ 1 τd y + unlev D X D lim d X = α Ξ 1 τd y + unlev D X D lim d X = lim d X, X X X X lim d X = lim d X, X X X X lim d X = d ˆ s + 1X, X X A.59 A.60, A.61, A.62 A.63 A.64 A.65 lim d X = d ˆ s + 1 k /Ξ X. A.66 X X 1 τx ty Equations A.59 A.60 are the continuity smoothness conditions for the debt value in the good state at the default boundary of the bad state. Equations A.61 A.62 show the value-matching conditions at the default thresholds, Equations A.63 A.64 reflect the continuity smoothness conditions for the debt value in the bad state at the option exercise boundary of the good state. Equations A.65 A.66 are the value-matching conditions at the option exercise boundaries in the good state the bad state. The default thresholds option exercise boundaries are chosen by equityholders. Hence, we do not need the corresponding smooth-pasting conditions for debt. To solve this system, we start with the functional form of the solution, in which A 1, A 2, A 1, A 2, C 1, C 2, C 3, C 4, C 5, C 6, 1, 2, 4, β 1, β 2, β 1, β 2, γ 1, γ 2, γ 3, γ 4 are real-valued parameters to be determined. We first consider the region D < X X. Plugging the functional form d i X = A i1 X γ 1 + A i2 X γ 2 + A i3 X γ 3 + A i4 X γ 4 + Ai5 into both equations of A.52 comparing coefficients, we ob- 11

13 tain A i5 = cr j + λ i + λ j = c r i r j + r j λ i + r i λ j r p. A.67 i As in Appendix A.2, A k is a multiple of A k, k = 1,..., 4, with the multiple factor l k := 1 λ r + λ µ γ k 1 2 σ2 γ kγ k 1, i.e., A k = l k A k. Using this relation comparing coefficients, it can be shown that γ 1, γ 2, γ 3, γ 4 correspond to the roots of the quadratic equation µ γ σ2 γγ 1 λ r µ γ σ2 γγ 1 λ r = λ λ. A.68 According to uo 2001, this quadratic equation always has two negative two positive distinct real roots. The value of debt in both states is subject to boundary conditions from below default above exercise of expansion option. To satisfy these boundary conditions, we use four terms with the corresponding factors A ik as well as the exponents γ k, which requires the usage of all four roots of Equation A.68. We do not incorporate the no-bubbles condition again because it is already implemented in the value function ˆ d i of a firm with only invested assets. The unknown parameters for this region are A k, k = 1,..., 4. C 2 X β 2 Next, we examine the region D X D. Plugging the functional form d X = C 1 X β C 3 X + C 4 + C 5 X γ 3 + C 6 X γ 4 into the second equation of A.51, we find by comparison of coefficients that β 1,2 = 1 2 µ σ 2 α C 3 = Ξ 1 τy λ, r + λ µ c C 4 =, r + λ 2 ± 1 2 µ σ 2 + 2r + λ σ 2, A.69 A.70 A.71 C 5 = α Ξ l3 l 3 Ā unlev 3, A.72 C 6 = α Ξ l4 l 4 Ā unlev 4. A.73 The unknown parameters left in this region are C 1 C 2. 12

14 Finally, we consider the region X < X X. Plugging the functional form 1 X β X β 2 + Z X + λ r P i c c + r + λ r + λ A.74 into the second equation of A.53 comparing coefficients, we find that ZX = λ 5 X γ 1 + λ 6 X γ 2. A.75 The parameters 5 6 are given by 5 = s + 1 γ 1 Â 1 r µ γ σ2 γ 1 γ λ, A.76 6 = s + 1 γ 2 Â 2 r µ γ σ2 γ 2 γ λ. A.77 The unknown parameters in this region are 1 2. To obtain the unknown parameters A 1, A 2, A 3, A 4, C 1, C 2, 1, 2, we plug the functional form into the system of boundary conditions A.59 A.66: α Ξ 1 + τd y + unlev D 4 A k D γ k + A 5 = C 1 D β 1 + C 2D β 2 + C 3X + C 4 + C 5 X γ 3 + C 6 X γ 4 k=1 4 A k γ k D γ k = C 1 β 1 Dβ 1 + C 2β 2 Dβ 2 + C 3X + C 5 γ 3 X γ 3 + C 6 γ 4 X γ 4 k=1 = C 1 D β 1 + C 2D β 2 + C 3D + C 4 + C 5 D γ 3 + C 6D γ 4 4 l k A k D γ k + A 5 = α Ξ 1 + τd y + unlev D k=1 4 l k A k X γ k + A 5 = 1 X β 1 + 2X β 2 + ZX + 4 A.78 k=1 4 l k A k γ k X γ k = 1 β1 Xβ 1 + 2β2 X β 2 + X Z X k=1 4 A k X γ k + A 5 = d ˆ s + 1X k=1 1 X β 1 + 2X β 2 + ZX + 4 = d ˆ s + 1 k /Ξ X 1 τx ty. 13

15 Using matrix notation, we can write D γ 1 D γ 2 D γ 3 D γ 4 D β 1 D β 2 γ 1 D γ 1 γ 2 D γ 2 γ 3 D γ 3 γ 4 D γ 4 β 1 Dβ 1 β 2 Dβ 2 D β 1 D β 2 M := l 1 D γ 1 l 2 D γ 2 l 3 D γ 3 l 4 D γ 4 l 1 X γ 1 l 2 X γ 2 l 3 X γ 3 l 4 X γ 4 X β 1 X β 2 l 1 γ 1 X γ 1 l 2 γ 2 X γ 2 l 3 γ 3 X γ 3 l 4 γ 4 X γ 4 β1 Xβ 1 β2 Xβ 2 X γ 1 X γ 2 X γ 3 X γ 4 X β 1 X β 2 A.79 A 5 + C 3 D + C 4 + C 5 D γ 1 + C 6D γ 2 C 3 D + γ 1 C 5 D γ 1 + γ 2C 6 D γ 2 C 3 D C 4 C 5 D γ 3 C 6D γ 4 + α Ξ 1 τd y + unlev D A 5 + α Ξ 1 τd y + unlev D b := A 5 + Z X + 4. A.80 X Z X A 5 + d ˆ s + 1X Z X d ˆ s + 1 k /Ξ X 1 τx ty The solution to the remaining unknowns is [ A 1 A 2 A 3 A 4 C 1 C ] T = M 1 b. A.81 The case with D < D, ˆD < ˆD, X > X : 14

16 oing through the same steps as in the previous case gives us D γ 1 D γ 2 D γ 3 D γ 4 D β 1 D β 2 γ 1 D γ 1 γ 2 D γ 2 γ 3 D γ 3 γ 4 D γ 4 β 1 Dβ 1 β 2 Dβ 2 D β 1 D β 2 M := l 1 D γ 1 l 2 D γ 2 l 3 D γ 3 l 4 D γ 4 X γ 1 X γ 2 X γ 3 X γ 4 X β 1 X β 2 γ 1 X γ 1 γ 2 X γ 2 γ 3 X γ 3 γ 4 X γ 4 β 1 Xβ 1 β 2 Xβ 2 l1x γ 1 l2x γ 2 l3x γ 3 l4x γ 4 X β 1 X β 2 A.82 A 5 + C 3 D + C 4 + C 5 D γ 1 + C 6D γ 2 C 3 D + γ 1 C 5 D γ 1 + γ 2C 6 D γ 2 C 3 D C 4 C 5 D γ 3 C 6D γ 4 + α Ξ 1 τd y + unlev D A 5 + α Ξ 1 τd y + unlev D b := A 5 + Z X + 4. A.83 X Z X A 5 + d ˆ s + 1 k /Ξ X 1 τx ty Z X d ˆ s + 1X The solution to the unknowns is again determined by [ A 1 A 2 A 3 A 4 C 1 C ] T = M 1 b. A.84 The case with X X : The solution of the case X X is obtained immediately by renaming states in the solution of the presented case for X < X. Alternative financing strategies: Debt values for any alternative financing strategy can be derived analogously by replacing earnings levels at option exercise in Equations A.53, A.54, A.65, A

17 Appendix A.5. Derivation of the value of bankruptcy costs For the calculation of bankruptcy costs, the ODEs are given by the following system: For 0 X D : { For D < X D : b X = 1 α Ξ 1 τxy + X α Ξ unlev X b X = 1 α Ξ 1 τxy + X α Ξ unlev X. r b X = µ Xb X σ2 X2 b X + λ 1 α Ξ 1 τxy + X α Ξ unlev X b X b X = 1 α Ξ 1 τxy + X α Ξ unlev X. For D < X < X : { r d X = c + µ Xb X σ2 X2 b X + λ b X b X r d X = c + µ Xd X σ2 X2 b λ b X b X. A.85 A.86 A.87 For X X < X : { b X = ˆ d s + 1X For X X : rd X = c + µ Xb X σ2 X2 b X + λ ˆ d s + 1X b X The boundary conditions are as follows: b X = ˆb s + 1X b X = ˆb s + 1 k /Ξ X. 1 τx ty. A.88 A.89 lim b X = lim b X, X D X D lim b X = lim b X, X D X D A.90 A.91 lim b X = 1 α Ξ 1 τd y + D α Ξ unlev D, A.92 X D lim b X = 1 α Ξ 1 τd y + D α unlev D, A.93 X D lim b X = lim b X, X X X X lim X X b X = lim X X b X, lim b X = ˆb s + 1X, X X A.94 A.95 A.96 16

18 lim b X = ˆb s + 1 k /Ξ X. A.97 X X 1 τx ty Equations A.90 A.91 are the continuity smoothness conditions for bankruptcy costs in good states at the default boundary in bad states. Similarly, Equations A.94 A.95 are the corresponding conditions for bankruptcy costs in bad states at the option exercise boundary in good states. Equations A.92 A.93 are the value-matching conditions at the default thresholds. They incorporate the fact that upon default, the value of the levered growth option switches to the value of the unlevered growth option. Equations A.96 A.97 are the value-matching conditions at the option exercise boundaries. To solve for the unknown parameters, we plug the functional form b i X = 1 α i Ξ i 1 τxy i α i Ξ i i unlev X + i X X D i, i =, C 1 X β 1 + C 2 X β 2 + C5 X γ 3 + C 6 X γ 4 α + Ξ y 1 τ c D < X D, i = λ X + r µ + λ r + λ A i1 X γ 1 + Ai2 X γ 2 + A i3 X γ 3 + A i4 X γ 4 + c r p D < X X, i =, i c 1 X β X β 2 + Z X + λ ri Pr + λ + c X < X X, i = r + λ ˆb s + 1X X > X, i = ˆb s + 1 k /Ξ 1 τx ty X X > X, i = A.98 into the system of boundary conditions A.90 A.97. The solution to the unknown parameters is given by [ A 1 A 2 A 3 A 4 C 1 C ] T = M 1 b, A.99 in which 17

19 D γ 1 D γ 2 D γ 3 D γ 4 D β 1 D β 2 γ 1 D γ 1 γ 2 D γ 2 γ 3 D γ 3 γ 4 D γ 4 β 1 Dβ 1 β 2 Dβ 2 D β 1 D β 2 M := l 1 D γ 1 l 2 D γ 2 l 3 D γ 3 l 4 D γ 4 l 1 X γ 1 l 2 X γ 2 l 3 X γ 3 l 4 X γ 4 X β 1 X β, 2 l 1 γ 1 X γ 1 l 2 γ 2 X γ 2 l 3 γ 3 X γ 3 l 4 γ 4 X γ 4 β1 Xβ 1 β2 Xβ 2 X γ 1 X γ 2 X γ 3 X γ 4 X β 1 X β 2 A.100 A 5 + C 3 D + C 4 + C 5 D γ 1 + C 6D γ 2 C 3 D + γ 1 C 5 D γ 1 + γ 2C 6 D γ 2 C 3 D C 4 C 5 D γ 3 C 6D γ α Ξ 1 τd y α Ξ unlev D + D A α Ξ 1 τd y α Ξ unlev D + D b := A 5 + Z X + 4, X Z X A 5 + d ˆ s + 1X Z X d ˆ s + 1 k /Ξ X 1 τx ty A.101 The case with X X : C 5 = l Ālev 3 3 l α Ξ Ā unlev 3, A C 6 = l Ālev 4 4 l α Ξ Ā unlev 4. A The solution of the case X X is obtained immediately by renaming states in the solution of the presented case for X < X. Alternative financing strategies: ankruptcy costs for any alternative financing strategy can be derived analogously by replacing the earnings levels at option exercise in Equations A.88, A.89, A.96, A

20 References Arnold, Marc, F. Wagner, Alexer, Ramona Westermann, 2013, rowth options, macroeconomic conditions the cross-section of credit risk, Journal of Financial Economics 107, hamra, Harjoat S., Lars-Alexer Kuehn, Ilya A. Strebulaev, 2010, The levered equity risk premium credit spreads: A unified framework, Review of Financial Studies 23, Chen, Hui, 2010, Macroeconomic conditions the puzzles of credit spreads capital structure, Journal of Finance 65, uo, Xin, 2001, An explicit solution to an optimal stopping problem with regime switching, Jounal of Applied Probability 38, Hackbarth, Dirk, Jianjun Miao, Erwan Morellec, 2006, Capital structure, credit risk, macroeconomic conditions, Journal of Financial Economics 82,