Online Appendix to Financing Asset Sales and Business Cycles

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1 Online Appendix to Financing Asset Sales usiness Cycles Marc Arnold Dirk Hackbarth Tatjana Xenia Puhan August 31, 2015 University of St. allen, Rosenbergstrasse 52, 9000 St. allen, Switzerl. Telephone: oston University Questrom School of usiness, 595 Commonwealth Avenue, oston, MA 02215, United States. Telephone: Swiss Life Asset Managers University of Mannheim, 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 Suppose the continuous-time analog of Epstein-Zin-Weil preferences of stochastic differential utility type e.g., Duffie Epstein 1992a, Duffie Epstein 1992b. The utility index U t over a consumption process C s solves U t = E P [ t ] ρ Cs 1 δ 1 γ U s 1 δ 1 γ ds F 1 δ 1 γ U s 1 δ t, A.1 1 γ 1 in which ρ determines the rate of time preference, γ is the coefficient of relative risk aversion for a timeless gamble, Ψ := 1 δ is the elasticity of intertemporal substitution for deterministic consumption paths. Incorporating the separability of time state preferences assuming that Ψ > 1, i.e., that agents have a preference for early resolution of uncertainty require expected returns that increase in the uncertainty about future consumption, are necessary to capture the impact of aggregate risk on corporate security values. hamra, Kuehn, Strebulaev 2010 Chen 2010 show that solving the ellman equation associated with the consumption problem of the representative agent yields 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.2 in which M t determines the compensated process associated with the Markov chain. r i are the regime-dependent risk-free interest rates. The parameters η i denote the risk prices for systematic rownian shocks affecting aggregate output. The market prices of consumption risk η i increase in the agents risk aversion consumption volatility. κ i are the relative jump sizes of the discount factor when the Markov chain leaves state i, i.e., they are the market prices of discount factor jump risk. 1

3 Risk-free rates, the market prices of consumption jump risk are defined as [ γ δ r i = r i + λ i w γ 1 γ δ 1 w 1 1 ], γ 1 A.3 η i = γσi C, A.4 hj κ i = δ γ log, A.5 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 γ 0 = ρ h δ γ i + 1 γ θ i 1 1 δ 2 γ 1 γ σi C 2 1 γ ρ h 1 γ i + λ i h 1 γ j 1 δ h 1 γ i A.6 The risk-free rates r i consist of the interest rate if the economy stayed in regime i forever, r i, plus a second term adjusting for possible regime switches. The no-jump part of the interest rates, r i, is given by r i = ρ + δθ i 1 2 γ 1 + δ σ C i 2, A.7 w := e κ = e κ A.8 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 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 regime setting follows Hackbarth, Miao, Morellec We consider the case in which the default boundary in good states is lower than the one in bad states, i.e., ˆD < ˆD. If the firm defaults, debtholders receive a fraction Λ i α i of the unleveraged 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. Therefore, an application of Ito s lemma with regime switches shows that debt satisfies the following system of ODEs. 2

4 For 0 X ˆD : ˆd X = α Λ 1 τxy ˆd X = α Λ 1 τxy. A.9 For ˆD < X ˆD : r ˆd X = c + µ X ˆd X σ2 X2 ˆd X + λ α Λ 1 τxy ˆd X ˆd X = α Λ 1 τxy. A.10 For X > ˆD : r ˆd X = c + µ X ˆd X σ2 X2 ˆd X + λ ˆd X ˆd X r ˆd X = c + µ X ˆd X σ2 X2 ˆd X + λ ˆd X ˆd X. A.11 The boundary conditions read lim X ˆd i X X <, i =,, A.12 lim ˆd X = lim ˆd X, A.13 X ˆD X ˆD lim ˆd X = lim ˆd X, A.14 X ˆD X ˆD lim ˆd X = α Λ 1 τd y, X ˆD A.15 lim ˆd X = α Λ 1 τd y. X ˆD A.16 Condition A.12 expresses the no-bubbles condition. The remaining boundary conditions are the value-matching conditions A.13, A.15, A.16, the smooth-pasting condition at the higher default threshold ˆD for the debt function in the good state ˆd, Eq. A.14. The 3

5 functional form of the solution is α i Λ i 1 τxy i X ˆD i i =, ˆd i X = Ĉ 1 X β 1 + Ĉ 2 X β 2 + C3 X + C 4 ˆD < X ˆD, i =  i1 X γ 1 + Âi2X γ 2 + A i5 X > ˆD, i =,, A.17 in which Â1, Â2, Â1, Â2, A 5, A 5, Ĉ1, Ĉ2, C 3, C 4, γ 1, γ 2, β 1, β 2 are real-valued parameters to be determined. First, consider the region X > ˆD. We start by using the stard approach of plugging the functional form ˆd i X = Âi1X γ 1 + Âi2X γ 2 + A i5 into both equations of A.11. Comparing coefficients solving the resulting two-dimensional system of equations for A i5, we find that A i5 = c r j + λ i + λ j = c r i r j + r j λi + r i λj r p, A.18 i that Âk is always 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.19 Due to the no-bubbles condition for debt stated in Eq. A.12, we take the negative roots. Next, we solve the region ˆD X ˆD. Plugging the functional form d X = Ĉ1X β 1 + Ĉ 2 X β 2 + C3 X + C 4 into the first equation of A.10, we find by comparison of coefficients that β1,2 = 1 2 µ σ 2 ± 1 2 µ 2 σ 2 + 2r + λ σ 2 C 3 = λ α Λ 1 τy r + λ µ C 4 = c r + λ. A.20 We then plug the functional form A.17 into conditions A.13 A.16, obtain a four-dimensional 4

6 linear system in the remaining four unknown parameters Â1, Â2, Ĉ1, Ĉ2 : Â 1 ˆDγ 1 + Â2 ˆD γ 2 + A 5 = Ĉ1 ˆD β 1 + Ĉ2 ˆD β 2 + C 3 ˆD + C 4 Â 1 γ 1 ˆDγ 1 + Â2γ 2 ˆDγ 2 = Ĉ1β 1 ˆD β 1 + Ĉ2β 2 ˆD β 2 + C 3 ˆD α Λ 1 τ ˆD y = Ĉ1 ˆD β 1 + Ĉ2 ˆD β 2 + C 3 ˆD + C 4 l 1 Â 1 ˆDγ 1 + l 2Â2 ˆD γ 2 + A 5 = α Λ 1 τ ˆD y. A.21 Define the matrices ˆD γ 1 ˆD γ 2 ˆD β 1 ˆD β 2 γ 1 1 ˆDγ γ 2 2 ˆDγ β ˆM ˆD β 1 1 β ˆD β 2 2 := ˆDβ ˆD β 2 l 1 1 ˆDγ l 2 2 ˆDγ 0 0 A.22 C 3 ˆD + C 4 A 5 C 3 ˆD ˆb := α Λ 1 τ, A.23 y C 3 ˆD C 4 α Λ 1 τ ˆD y A 5 [ such that ˆM Â 1 Â 2 Ĉ 1 ] T Ĉ 2 = ˆb. The solution for the unknown parameters is given by [ ] T Â 1 Â 2 Ĉ 1 Ĉ 2 = ˆM 1ˆb. A.24 The value of the tax shield can be calculated by the formula for the value of debt, in which c is replaced by τc, α is equal to zero. The value of bankruptcy costs is simply obtained by replacing c by zero, α by 1 α. Default policy. The value of equity corresponds to the firm value minus the value of debt. The firm value is given by the value of assets in place plus the value of the option the tax shield minus default costs. Once debt has been issued, managers select the ex post default policy that maximizes the value of equity. Formally, the default policy is determined by equating the first 5

7 derivative of the equity value to zero at the corresponding default boundary: ê ˆD = 0 ê ˆD = 0. A.25 We solve this problem numerically. For a firm that receives scaled earnings after investment, the value of corporate securities is solved similarly by replacing X with the scaled level of 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 in which X < X : We present the derivation of the value of the growth option for a firm that finances the option exercise by issuing equity in good states selling assets in bad states. The value of the growth option for a firm with an alternative financing strategy can be derived similarly. For each regime 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 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.26 For X X < X : X = 1 τs Xy K 1 + Υ r X = µ X X σ2 X2 X + λ 1 τs Xy K 1 + Υ X. A.27 6

8 For X X : X = 1 τs Xy K 1 + Υ X = 1 τs Xy K /Λ. A.28 Whenever the process X is in the option continuation region, which corresponds to system A.26 the second equation of A.27, the required rate of return r i left-h side must be equal to the realized rate of return right-h side. The realized rate of return is calculated by applying Ito s lemma for regime switches. In this region, the last term captures the possible jump in the value of the growth option due to a regime switch. It can be expressed as the instantaneous probability of a regime shift, λ or λ, times the associated change in the value of the option. The first equation of A.27 the system A.28 state the payoff of the option at exercise. The process is in the option exercise region in these cases. The boundary conditions are given by lim i X = 0, i =,, A.29 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 /Λ, X X A.30 A.31 A.32 lim X = 1 τs X y K 1 + Υ. X X A.33 Condition A.29 ensures that the option value goes to zero as earnings approach zero. Conditions A.30 A.31 are the value-matching smooth-pasting conditions of the value function in bad times at the exercise boundary in good times. The remaining conditions A.32 A.33 are the value-matching conditions at the exercise boundaries in a good state a bad state, respectively. 7

9 The functional form of the solution is given by i X = Ā i3 X γ 3 + Āi4X γ 4 0 X < X, i =, C 1 X β 1 + C2 X β 2 + C3 X + C 4 X X < X, i = 1 τs Xy K /Λ X X i = 1 τs Xy K 1 + Υ X X i = A.34 in which Ā3, Ā4, Ā1, Ā2, C 1, C 2, C 3, C 4, γ 3, γ 4, β 1, β 2 are real-valued parameters to be determined. First, consider the region 0 X < X, plug the functional form i X = Āi3X γ 3 +Āi4X γ 4 into both equations of A.26. Comparison of coefficients shows that Āk is a multiple of Ā k, k = 3, 4, with the 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. Using this result when comparing coefficients, we find that γ 3 γ 4 correspond to the positive roots of the quadratic equation µ γ σ2 γγ 1 λ r µ γ σ2 γγ 1 λ r = λ λ. A.35 The reason for taking the positive roots is given by boundary condition A.29. Next, 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.27, 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, C 4 = λ K /Λ r + λ. A.36 The remaining unknown parameters are Ā3, Ā4, C 1, C 2. Plugging the functional form A.34 8

10 into conditions A.30 A.33 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 X β 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.37 A.38 C 1 X β 1 + C 2 X β 2 + C 3 X + C 4 = 1 τs y X K /Λ, A.39 Ā 3 X γ 3 + Ā4X γ 4 = 1 τs y X K 1 + Υ. A.40 This four-dimensional system is linear in its four unknowns Ā3, Ā4, C 1 C 2. We define the matrices l3 X γ 3 l4 X γ 4 X β 1 X β 2 l3 γ 3 X γ 3 M l 4 γ 4 X γ 4 β1 := Xβ 1 β2 Xβ 2, A X β 1 X β 2 X γ 3 X γ C 3 X + C 4 C 3 X b := 3 X C, A τs y X K /Λ 1 τs y X K 1 + Υ [ such that M Ā 3 Ā 4 C1 ] T C2 = b. The solution to the remaining four unknowns is given by [ ] T Ā 3 Ā 4 C1 C2 = M 1 b. A.43 The unleveraged value of the growth option. The unleveraged value of the growth option is 9

11 calculated by additionally imposing the smooth-pasting boundary conditions at option exercise: lim unlev X X unlev X = 1 τs y A.44 lim unlev X X unlev X = 1 τs y. A.45 The solution method is analog to the one for the leveraged option value up to including Eq. A.36. The system of equations A.37 A.40 is augmented by the two equations corresponding to the additional smooth-pasting boundary conditions: C unlev 1 β 1 X unlev β Cunlev 2 β 2 X unlev β C3 = 1 τs y A.46 Ā unlev 3 γ 3 X unlev γ3 1 + Ā unlev 4 γ 4 X unlev γ4 1 = 1 τs y. A.47 The full system is six-dimensional with the six unknowns Āunlev 3, Āunlev 4, C unlev 1, C unlev 2 X unlev, X unlev, linear in the first four unknowns nonlinear in the last two unknowns. It is solved numerically. The case in which X X : The solution of the case X X can be obtained immediately by renaming regimes in the solution of the presented case for X < X. 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. Proof of Proposition 2. An investor requires an instantaneous return equal to the risk-free rate 10

12 r i for holding corporate debt. The application of Ito s lemma with regime switches shows that debt must, consequently, satisfy the following system of ODEs. For 0 X D : d X = α Λ 1 τxy + unlev X d X = α Λ 1 τxy + unlev X. A.48 For D < X D : r d X = c + µ Xd X σ2 X2 d X + λ α Υ 1 τxy + unlev X d X d X = α Λ 1 τxy + unlev X. A.49 For D < X < 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.50 For X X < X : d X = ˆd s + 1X r d X = c + µ Xd X σ2 X2 d X + λ ˆd s + 1X d X. A.51 For X X : d X = ˆd s + 1X d X = ˆd s + 1 K /Λ 1 τx ty X. A.52 In system A.48, the firm is in the default region in both good states bad times. In this region, debtholders receive α i Λ i 1 τxyi + unlev i X at default. The firm is in the continuation region in good state, in the default region in bad states in system A.49. For the continuation region in good states, the left-h side of the first equation is the rate of return required by investors for holding corporate debt for one unit of time. 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 regime 11

13 switch, that triggers immediate default. Eqs. A.50 describe the case in which the firm is in the continuation region in both good bad states. The next system, A.51, 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, 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 = ˆd s + 1 X, which is the first equation in A.51. The second equation in this case is obtained by the same approach as in A.50. The last term captures the notion that a regime switch from bad states to good states triggers immediate exercise of the expansion option with equity financing. Finally, A.52 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 earnings of the firm are scaled by s + 1 K /Λ 1 τx ty. K /Λ 1 τx ty The system is subject to the following boundary conditions. of the assets in place, such that the lim d X = lim d X, X D X D A.53 lim d X = lim d X, X D X D lim d X = α Λ 1 τd y + unlev D X D lim d X = α Λ 1 τd y + unlev D X D A.54, A.55, A.56 lim d X = lim d X, X X X X lim d X = lim d X, X X X X lim d X = ˆd s + 1X, X X A.57 A.58 A.59 lim d X = ˆd s + 1 K /Λ X. A.60 X X 1 τx ty Eqs. A.53 A.54 are the value-matching smooth-pasting conditions for the debt value in the good state at the default boundary of the bad state. Eqs. A.57 A.58 are the 12

14 corresponding conditions for the debt value in the bad state at the option exercise boundary of the good state. Eqs. A.55 A.56 show the value-matching conditions at the default thresholds, Eqs. A.59 A.60 are the value-matching conditions at the option exercise boundaries. The default thresholds option exercise boundaries are chosen by equityholders. Hence, we do not have 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, β1, β 2, β 1, β 2, γ 1, γ 2, γ 3, γ 4 are real-valued parameters to be determined or to be confirmed. 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 + A i5 into both equations of A.50 comparing coefficients, we find that A i5 = cr j + λ i + λ j = c r i r j + r j λi + r i λj r p. A.61 i As in Appendix A.2, A k is always a multiple of A k, k = 1,..., 4, with the 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.62 According to uo 2001, this quadratic equation always has two negative two positive distinct real roots. The value of debt in both regimes is subject to boundary conditions from below default above exercise of expansion option. To meet all 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 Eq. A.62. The no-bubbles condition is not considered again because it is already implemented in the value function ˆd i of a firm with only invested assets. The unknown parameters for this region are A k, k = 1,..., 4. Next, we examine the region D X D. Plugging the functional form d X = C 1 X β 1 + C 2 X β 2 + C3 X + C 4 + C 5 X γ 3 + C 6 X γ 4 into the second equation of A.49, we find by comparison 13

15 of coefficients that β1,2 = 1 2 µ σ 2 ± 1 2 µ 2 σ 2 + 2r + λ σ 2, A.63 C 3 = λ α Λ 1 τy r + λ, µ A.64 c C 4 = r + λ, A.65 C 5 = α Λ l3 l 3 Ā unlev 3, A.66 C 6 = α Λ l4 l 4 Ā unlev 4. A.67 The unknown parameters remaining in this region are C 1 C 2. Finally, we consider the region X < X X. Plugging the functional form 1 X β 1 +2 X β 2 + Z X + λ + ri P r + λ find that c c r + λ into the second equation of A.51 comparing coefficients, we ZX = λ 5 X γ 1 + λ 6 X γ 2. A.68 A.69 The parameters 5 6 are given by 5 = s + 1 γ 1 Â 1 r µ γ σ2 γ 1 γ λ, A.70 6 = s + 1 γ 2 Â 2 r µ γ σ2 γ 2 γ λ. A.71 The unknown parameters remaining in this region are 1 2. To solve for the unknown parameters A 1, A 2, A 3, A 4, C 1, C 2, 1, 2, we plug the func- 14

16 tional form 16 into the system of boundary conditions A.53 A.60: 4 k=1 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 4 k=1 α Λ 1 + τd y + unlev D 4 k=1 4 k=1 A k γ k D γ k = C 1 β 1 D β 1 + C 2β 2 D β 2 + C 3X + C 5 γ 3 X γ 3 + C 6 γ 4 X γ 4 = C 1 D β 1 + C 2D β 2 + C 3D + C 4 + C 5 D γ 3 + C 6D γ 4 l k A k D γ k + A 5 = α Λ 1 + τd y + unlev D l k A k X γ k + A 5 = 1 X β 1 + 2X β 2 + ZX + 4 A.72 4 k=1 4 k=1 l k A k γ k X γ k = 1 β 1 X β 1 + 2β 2 X β 2 + X Z X A k X γ k + A 5 = ˆd s + 1X 1 X β 1 + 2X β 2 + ZX + 4 = ˆd s + 1 K /Λ X. 1 τx ty Using matrix notation, we can write D γ 1 D γ 2 D γ 3 D γ 4 D β 1 D β γ 1 D γ 1 γ 2 D γ 2 γ 3 D γ 3 γ 4 D γ 4 β1 Dβ 1 β2 Dβ D β 1 D β l 1 D γ 1 l 2 D γ 2 l 3 D γ 3 l 4 D γ M := l 1 X γ 1 l 2 X γ 2 l 3 X γ 3 l 4 X γ 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 β1 Xβ 1 β2 Xβ 2 X γ 1 X γ 2 X γ 3 X γ X β 1 X β 2 A.73 15

17 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 Λ 1 τd y + unlev D A 5 + α Λ 1 τd y + unlev D b :=. A.74 A 5 + Z X + 4 X Z X A 5 + ˆd s + 1X Z X ˆd s + 1 K /Λ 1 τx ty X The solution to the remaining unknowns is now given by [ A 1 A 2 A 3 A 4 C 1 C ] T = M 1 b. A.75 The case in which D < D, ˆD < ˆD, X > X : 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 γ 1 D γ 1 γ 2 D γ 2 γ 3 D γ 3 γ 4 D γ 4 β1 Dβ 1 β2 Dβ D β 1 D β l 1 D γ 1 l 2 D γ 2 l 3 D γ 3 l 4 D γ M := X γ 1 X γ 2 X γ 3 X γ 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 β1 Xβ 1 β2 Xβ 2 l1x γ 1 l2x γ 2 l3x γ 3 l4x γ X β 1 X β 2 A.76 16

18 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 Λ 1 τd y + unlev D A 5 + α Λ 1 τd y + unlev D b :=. A.77 A 5 + Z X + 4 X Z X A 5 + ˆd s + 1 K /Λ 1 τx ty X Z X ˆd s + 1X The solution to the unknowns is again given by [ A 1 A 2 A 3 A 4 C 1 C ] T = M 1 b. A.78 Appendix A.5. ankruptcy costs For the calculation of bankruptcy costs, the ODEs are given by the following system: For 0 X D : b X = 1 α Λ 1 τxy + X α Λ unlev X b X = 1 α Λ 1 τxy + X α Λ unlev X. A.79 For D < X D : r b X = µ Xb X σ2 X2 b X + λ 1 α Λ 1 τxy + X α Λ unlev X b X b X = 1 α Λ 1 τxy + X α Λ unlev X. A.80 17

19 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 X + λ b X b X. A.81 For X X < X : b X = ˆd s + 1X rd X = c + µ Xb X σ2 X2 b X + λ ˆd s + 1X b X. A.82 For X X : b X = ˆb s + 1X b X = ˆb s + 1 K /Λ 1 τx ty X. A.83 The boundary conditions are as follows: lim b X = lim b X, X D X D lim b X = lim b X, X D X D A.84 A.85 lim b X = 1 α Λ 1 τd y + D α Λ unlev D, A.86 X D lim b X = 1 α Λ 1 τd y + D α unlev D, A.87 X D lim b X = lim b X, X X X X lim b X = lim b X, X X X X lim b X = ˆb s + 1X, X X A.88 A.89 A.90 lim b X = ˆb s + 1 K /Λ X. A.91 X X 1 τx ty 18

20 Eqs. A.84 A.85 are the value-matching smooth-pasting conditions for bankruptcy costs in good states at the default boundary in bad states. Similarly, Eqs. A.88 A.89 are the corresponding conditions for bankruptcy costs in bad states at the option exercise boundary in good states. Eqs. A.86 A.87 are the value-matching conditions at the default thresholds. They incorporate the fact that upon default, the value of the leveraged growth option switches to the value of the unleveraged growth option. Eqs. A.90 A.91 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 Λ i 1 τxy i α i Λ i unlev i X + i X X D i, i =, C 1 X β 1 + C2 X β 2 + C5 X γ 3 + C 6 X γ 4 α + λ Λ y 1 τ X + c r µ + λ r + λ D < X D, i = A i1 X γ 1 + A i2 X γ 2 + A i3 X γ 3 + A i4 X γ 4 + c r p i D < X X, i =, 1 X β X β 2 + Z X + c λ ri P r + λ r + λ X < X X, i = ˆb s + 1X X > X, i = ˆb s + 1 K /Λ 1 τx ty X X > X, i = A.92 into the system of boundary conditions A.84-A.91. The solution to the unknowns is given by [ A 1 A 2 A 3 A 4 C 1 C ] T = M 1 b, A.93 where 19

21 D γ 1 D γ 2 D γ 3 D γ 4 D β 1 D β γ 1 D γ 1 γ 2 D γ 2 γ 3 D γ 3 γ 4 D γ 4 β1 Dβ 1 β2 Dβ D β 1 D β l 1 D γ 1 l 2 D γ 2 l 3 D γ 3 l 4 D γ M :=, l 1 X γ 1 l 2 X γ 2 l 3 X γ 3 l 4 X γ 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 β1 Xβ 1 β2 Xβ 2 X γ 1 X γ 2 X γ 3 X γ X β 1 X β 2 A.94 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 Λ 1 τd y α Λ unlev D + D A α Λ 1 τd y α Λ unlev D + D b :=, A 5 + Z X + 4 X Z X A 5 + ˆd s + 1X Z X ˆd s + 1 K /Λ 1 τx ty X A.95 C 5 = l Ālev 3 3 α Λ Ā unlev 3, A.96 l 3 C 6 = l Ālev 4 4 α Λ Ā unlev 4. A.97 l 4 20

22 The case in which D < D, ˆD < ˆD, X > X can be solved analogously. 21

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