Lecture 5A: Leland-type Models

Transcription

1 Lecture 5A: Leland-type Models Zhiguo He University of Chicago Booth School of Business September, 2017, Gerzensee

2 Leland Models Leland (1994): A workhorse model in modern structural corporate nance f you want to combine model with data, this is the typical setting A dynamic version of traditional trade-o model, but capital structure decision is static Trade-o model: a rm s leverage decision trades o the tax bene t with bankruptcy cost Relative to the previous literature (say Merton s 1974 model), Leland setting emphasizes equity holders can decide default timing ex post So-called "endogenous default," an useful building block for more complicated models Merton 1974 setting: given VT distribution, default if ev T < F T. No default before T and the path of V t does not matter

3 Firm and ts Cash Flows A rm s asset-in-place generates cash ows at a rate of δ t Over interval [t, t + dt] cash ows is δt dt Leland 94, state variable unlevered asset value Vt = δ t r µ (just relabeling) Cash ow rate follows a Geometric Brownian Motion (with drift µ and volatility σ) dδ t = µdt + σdz t δ t fzt g is a standard Brownian motion (Wiener process): Z t N (0, t), Z t Z s is independent of F (fz u<s g) Given δ0, δ t = δ 0 exp µ 0.5σ 2 t + σz t > 0 Arithmetic Brownian Motion: dδt = µdt + σdz t so δ t = δ 0 + µt + σz t Persistent shocks, i.i.d. return. Today s shock dz t a ects future level of δ s for s > t One interpretation: rm produces one unit of good per unit of time, with market price uctuating according to a GBM n this model, everything is observable, i.e. no private information

4 Debt as Perpetual Coupon Firm is servicing its debt holders by paying coupon at the rate of C Debt holders are receiving cash ows Cdt over time interval [t, t + dt] Debt tax shield, with tax rate τ Debt is deducted before calculating taxable income implies that debt can create DTS The previous cash ows are after-tax cash ows, so before-tax cash ows are δ t / (1 τ) So-called Earnings Before nterest and Taxes (EBT) By paying coupon C, taxable earning is δ t / (1 τ) C, so equity holders cash ows are δt C (1 1 τ τ) = δ t (1 τ) C The rm investors in total get (Modigliani-Miller idea) δ t (1 {z τ) C } Equity + {z} C = δ {z} t Debt Firm s Asset + τc {z} DTS

5 Endogenous Default Boundary Equity holders receiving δ t which might become really low, but is paying constant (1 τ) C When δ t! 0, holding the rm almost has zero value then why pay those debt holders? Equity holders default at δ B > 0 where equity value at δ B has E (δ B ) = 0 and E 0 (δ B ) = 0 Value matching E (δb ) = 0, just says that at default equity holders recover nothing Smooth pasting E 0 (δ B ) = 0, optimality: equity can decide to wait and default at δ B ɛ, but no bene t of doing so At bankruptcy, some deadweight cost, debt holders recover a fraction 1 α of rst-best rm value (1 α) δ B / (r µ) First-best unlevered rm value δb / (r µ), Gordon growth formula Two steps: 1. Derive debt D (δ) and equity E (δ), given default boundary δ B 2. Using smooth pasting condition to solve for δ B

6 General Solution for GBM process with Linear Flow Payoffs rv (y) = f (y) +V 0 (y) µ (y) + 1 {z } {z } 2 σ2 (y) V 00 (y) {z } required return ow (dividend) payo local change of value function Leland setting, f (y) = a + by, µ (y) = µy, and σ (y) = σy t is well known that the general solution to V (y) is V (y) = a r + b r µ y + K γy γ + K η y η where the "power" parameters are given by γ = µ 1 2 σ 2 + r 1 2 σ2 µ 2 + 2σ 2 r σ 2 < 0, η = µ 1 2 σ 2 r 12 σ 2 µ 2 + 2σ 2 r σ 2 > 1 The constants K γ and K η are determined by boundary conditions

7 Debt Valuation (1) For debt, ow payo is C so Two boundary conditions D (δ) = C r + K γδ γ + K η δ η When δ =, default never occurs, so D (δ = ) = C r perpetuity. Hence K η = 0 (otherwise, D goes to in nity) When δ = δb, debt value is (1 α)δ B r µ. D (δ B ) = (1 α)δ B r µ implies that C r + K γδ γ B = (1 α) δ B r µ ) K γ = (1 α)δ B r µ δ γ B C r

8 Debt Valuation (2) We obtain the closed-form solution for debt value D (δ) = C δ γ (1 α) r + δb C δ B r µ r δ γ! (1 α) δ = B δ γ + 1 δ B r µ δ B Present value of 1 dollar contingent on default: E e r τ B δ γ = where τ δ B = inf ft : δ t < δ B g B The debt value can also be written in the following intuitive form Z τb D (δ) = E e rs Cds + e r τ (1 α) δ B B 0 r µ Z C τb = E de rs + e r τ (1 α) δ B B r 0 r µ C = E 1 e r τ B + e r τ (1 α) δ B B r r µ C r

9 Equity Valuation (1) For equity, ow payo is δ t (1 τ) C, so E (δ) = δ r µ (1 τ) C r + K γ δ γ + K η δ η When δ =, equity value cannot grow faster than rst-best rm value which is linear in δ. So K η = 0 When δ = δ B, we have E (δ B ) = Thus δ B r µ E (δ) = (1 τ) C r + K γ δ γ B = 0 ) K γ = (1 τ)c r δ γ B δ (1 τ) C + r µ r {z } Equity value if never defaults (pay (1 τ)c forever) (1 τ) C δ B δ γ r r µ δ {z B } Option value of default δ B r µ

10 Equity Valuation (2) Finally, smooth pasting condition 0 = E 0 (δ) δ=δb 1 (1 = r µ + = 1 r µ + ( γ) τ) C r (1 δ B δ ( γ) r µ δ B τ) C 1 rδ B r µ γ 1 1 δ B δ=δb Thus δ B = (1 τ) C r µ r γ 1 + γ

11 What if the firm can decide optimal coupon At t = 0, what is the optimal capital structure (leverage)? Given δ 0 and C, the total levered rm value v (δ 0 ) = E (δ 0 ) + D (δ 0 ) is δ 0 + τc! δ γ 1 r µ r δ {z } B {z } Unlevered value Tax shield αδ B δ γ r µ δ {z B } Bankruptcy cost Realizing that δ B is linear in C, we can nd the optimal C that maximizing the levered rm value to be C = δ 0 r (1 + γ) 1 + γ + r µ (1 τ) γ αγ (1 τ) 1/γ τ mportant observation: optimal C is linear in δ 0! So called scale-invariance t implies that if the rm is reoptimizing, its decision is just some constant scaled by the rm size

12 Trade-off Theory: Economics behind Leland (1994) Bene t: borrowing gives debt tax shield (DTS) Equity holders makes default decision ex post The rm fundamental follows GBM, persistent income shocks After enough negative shocks, equity holders value of keeping the rm alive can be really low Debt obligation is xed, so when δ t is su ciently low, it is optimal to default Debt-overhang Equity holders do not care if default impose losses on debt holders But, at time zero when equity holders issue debt, debt holders price default in D (δ 0 ) And equity holders will receive D (δ0 )! Hence equity holders optimize E (δ 0 ) + D (δ 0 ), realizing that coupon C will a ect DTS (positively) and bankruptcy cost (negatively) f equity holders can commit ex ante about ex post default behavior, what do they want to do?

13 Leland, Goldstein and Ju (2000, Journal of Business) There are two modi cations relative to Leland (1994): First, directly modelling pre-tax cash ows so-called EBT, rather than after-tax cash ows t makes clear that there are three parties to share the cash ows: equity, debt, and government When we take comparative statics w.r.t. tax rate τ, in Leland (1994) you will ironically get that levered rm value " when τ " n Leland, raising τ does not change δt (which is after-tax cash ows) n LGJ, after-tax cash ows are (1 value τ) δ t, so raising τ lowers rm

14 Leland, Goldstein and Ju (2000, Journal of Business) Second, more importantly, allowing for rms to upward adjust their leverage if it is optimal to do so in the future When future fundamental goes up, leverage goes down, optimal to raise more debt Need x cost to do so otherwise tend to do it too often Key assumption for tractability: when adjusting leverage, the rm has to buy back all existing debt Say that this rule is written in debt covenants As a result, there is always one kind of debt at any point of time After buying back, when equity holders decide how much debt to issue, they are solving the same problem again with new rm size But the model is scale invariant, so the solution is the same (except a larger scale) F face value. A rm with (δ, F ) faces the same problem as (kδ, kf )

15 Optimal Policies in LGJ δ B δ0 = ψ: default factor, δ U δ0 = γ: leverage adjustment factor LGJ: can precommit to γ. No precommitment in Fischer-Heinkel-Zechner (1989)

16 How Do We Model Finite Maturity Perpetual debt in Leland (1994). n practice debt has nite maturity Debt maturity is very hard to model in a dynamic model You can do exponentially decaying debt (Leland, 1994b, 1998) Rough idea: what if your debt randomly matures in a Poisson fashion with intensity 1/m? Exponential distribution, the expected maturity is R 0 x 1 m e x /m dx = m t is memoriless if the debt has not expired, looking forward the debt price is always the same! Actually, you do not need random maturing. Exponential decaying coupon payment also works! So, debt value is D (δ), not D (δ, t) where t is remaining maturity f all debt maturity is i.i.d, large law of numbers say that at [t, t + dt], 1 m dt fraction of debt mature

17 Leland (1998) Using exponentially decaying nite maturity debt Equity holders can ex post choose risk σ 2 fσ H, σ L g Research question: how does asset substitution work in this dynamic framework? How does it depend on debt leverage and debt maturity? Typically with default option, asset substitution occurs optimally (default option gets more value if volatility is higher) With asset substitution, the optimal maturity is shorter, consistent with the idea that short-term debt helps curb agency problems (numerical result, not sure robust) Quantitatively, agency cost due to asset substitution is small

18 Leland (1998) (2) Assume threshold strategy that there exists δ S s.t. σ = σ H for δ < δ S and σ = σ L for δ δ S Solve for equity, debt, DTS, BC the same way as before, with one important change Need to piece solutions on [δ B, δ S ) and [δ S, ) together γ H, η H, γ L, η L : solutions to fundamental quadratic equations D H (δ) = C r + K H γ δ γ H + K H η δ η H for [δ B, δ S ) D L (δ) = C r + K L γ δ γ L + K L η δ η L for [δ S, ) Four boundary conditions to get K H γ, K H η, K L γ, K L η Kη L = 0 because D (δ = ) < C r. The other three: D H (δ S ) = D L (δ S ) (value matching), D H 0 (δ S ) = D L0 (δ S ) (smooth pasting), D H (δ B ) = (1 α)δ B r µ (value matching) Here, smooth pasting at δs always holds, because Brownian crosses δ S "super" fast. The process does not stop there (like at δ B )

19 Leland and Toft (1996) Deterministic maturity, but keep uniform distribution of debt maturity structure Say we have debts with a total measure of 1, maturity is uniformly distributed U [0, T ], same principal P, same coupon C Tough: now debt price is D (δ, t), need to solve a PDE Equity promises to keep the same maturity structure in the future Equity holders cash ows are δ t dt (1 τ) Cdt 1 dt (P D (δ, T )) T Cash ows δt dt; Coupon Cdt; and Rollover losses/gains Over [t, t + dt], there is T 1 dt measure of debt matures, equity holders need to pay 1 dt (P D (δ, T )) T as equity holders get D (δ, T ) T 1 dt by issuing new debt

20 Leland and Toft (1996) First step: solve the PDE rd (δ, t) = C + D t (δ, t) + µδd δ (δ, t) σ2 δ 2 D δδ (δ, t) Boundary conditions D (δ =, t) = C r 1 e rt + Pe rt : defaultless bond δ D (δ = δ B, t) = (1 α) B : defaulted bond r µ D (δ, 0) = P for δ δ B : paid back in full when it matures Leland-Toft (1996) get closed-form solutions for debt values; have a look Better know the counterpart of Feyman-Kac formula. The point is to know it admits closed-form solution

21 Leland and Toft (1996) Equity value satis es the ODE re (δ) = δ (1 τ) C + 1 T [D (δ, T ) P] + µδe δ (δ) σ2 δ 2 E δδ (δ) This is also very tough, given the complicated form of D (δ, T )! Leland and Toft have a trick (Modigliani-Miller idea): E (δ) = v (δ) Z 1 T D (δ, t) dt = T 0 δ Z r µ + DTS (δ) BC (δ) 1 T D (δ, t) dt T 0 DTS (δ) and BC (δ) are much easier to price DTS (δ) is the value for constant ow payo τc till default occurs BC (δ) is the value of bankruptcy cost incurred on default We have derived them given δb After getting E (δ; δ B ), δ B is determined by smooth pasting E 0 (δ B ; δ B ) = 0 n He-Xiong (2012), we introduce market trading frictions for corporate bonds Some deadweight loss during trading, the above trick does not work

22 Calculation of Debt Tax Shield Let us price DTS (δ) which is the value for constant ow payo τc till default occurs We can have Z τb DTS (δ) = E e rs τcds 0 τc = E r Or, F (δ) = DTS (δ) 1 e r τ B = τc r rf (δ) = τc + µδf δ (δ) σ2 δ 2 F δ (δ) 1 δ δ B γ! F (δ) = τc r + K γ δ γ + K η δ η plugging F (δ B ) = 0 and F ( ) = τc r (so K η = 0) we have F (δ) = τc! δ γ 1 r δ B

23 MELLA-BARRAL and PERRAUDN (1997) (1) How to model negotiation and strategic debt service? Consider a rm producing one widget per unit of time, random widget price dp t /p t = µdt + σdz t Constant production cost w > 0 so cash ows are p t w f debt holders come in to manage the rm, cash ows are ξ 1 p t ξ 0 w with ξ 1 < 1 and ξ 0 > 1 Even without debt, p t can be so low that shutting down the rm is optimal This is so called operating leverage One explanation for why Leland models predict too high leverage relative to data: Leland model includes operating leverage For debt holders, if they take over, value is X (p) (need to gure out their hypothetical optimal stopping time by using smooth-pasting condition)

24 MELLA-BARRAL and PERRAUDN (1997) (2) Now imagine the original coupon is b > 0 When p t goes down, what if equity holders can make a take-it-or-leave-it o er to debt holders? Denote the equilibrium coupon service s (p), and resulting debt value L (p) n equilibrium there exist two thresholds p c < p s When pt p s, s (p) = b, nothing happens When pt 2 (p c, p s ), we have s (p) < b and L (p) = X (p). As long as debt service is less than the contracted coupon, the value of debt equals that of debtholders outside option X (p) When pt hits p c, liquidating the rm When s (p) < b we have s (p) = ξ 1 p t holders take the rm. ξ 0 w which is as if debt n the paper, there is some complication of γ > 0 which is the rm s scrap value

25 Miao, Hackbarth, Morellec (2006) Firm EBT is y t δ t, y t aggregate business cycle condition dδ t /δ t = µdt + σdz t y t 2 fy G, y B g : Markov Chain Exponentially decaying debt, etc, same as Leland (1998) Default boundary depends on the current macro state: δ G B and δb B. Same smooth-pasting condition δ G B < δb B, default more in B. Help explain credit spread puzzle Bond seems too cheap in the data. f bond payo is lower in recession, then it requires a higher return Lots of papers about credit spread puzzle use this framework where s 2 fg, Bg or more dδ t /δ t = µ s dt + σ s dz t ODE in vector: x = ln (δ), D (x) = h i 0 D G (x), D B (x) rd (x) = c1 21 +µ 22 D 0 (x) Σ 22D 00 (x) see my recent Chen, Cui, He, Milbradt (2014) if you are interested