[PPT] - Toward a Generalization of the Leland-Toft Optimal Capital Structure PowerPoint Presentation

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Toward a Generalization of the Leland-Toft Optimal Capital Structure Model ∗

Budhi Arta Surya School of Business and Management Bandung Institute of Technology

Joint work with

Kazutoshi Yamazaki Center for the Study of Finance and Insurance Osaka University

March 5, 2012

∗Based on Surya, B. A. and Yamazaki, K. (2011). Toward a Generalization of the Leland-Toft Optimal Capital Structure

Model. arXiv:1109.0897. I would like to thank the organizer and the NCTS Institute for their hospitality and making this

visit possible.

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Toward a Generalization of the Leland-Toft Optimal Capital Structure Model

The Central Issue of the Model

According to the works of

Leland (Journal of Finance, 1994),
Leland and Toft (Journal of Finance, 1996)

the optimal capital structure problem can be formulated as follows. It is concerned with capital raise of a firm by issuing a debt within a given time

interval. The debt will pay in exchange to the investor streams of payments paid

continuously prior to and at default. A portion of each debt payment made is applied towards reducing the debt principal and another portion of the payment is applied towards paying the interest (coupon) on the debt; similar to amortizing bond. In case of default, part of the firm’s asset will be liquidated to pay the default settlement and the remaining of which will go to the debt holder. Should default

ccur, the firm’s manager tries to find an optimal default level in which the firm’s

equity value is maximized.

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Toward a Generalization of the Leland-Toft Optimal Capital Structure Model

Literature Review and Contribution

As the source of randomness in the firm’s asset, Leland (Journal of Finance, 1994) and Leland and Toft (Journal of Finance, 1996) employed diffusion process. The model has been extended to those allowing jumps in the firm’s asset. Extensions to the Jump-Diffusion process with jumps of exponential type:

Hilberink and Rogers (Finance & Stochastics, 2002) - one-sided jumps.
Chen and Kou (2009) (Mathematical Finance, 2009) - two-sided jumps.

Extensions to the Spectrally Negative L´ evy process with general structure of jumps:

Kyprianou and Surya (Finance & Stochastics, 2007)

Our contribution: In Surya and Yamazaki (2011) we extend the above works by allowing bankruptcy costs, coupon rates and tax rebate to be dependent on the asset

value. In the calculation, we use a few results from Egami and Yamazaki (2011).

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Toward a Generalization of the Leland-Toft Optimal Capital Structure Model

The Leland-Toft Optimal Capital Structure Model

Let X = (Xt : t ≥ 0) be source of randomness1 in the firm’s underlying asset. We denote by Px the law of X under which the process Xt started at x ∈ R. For convenience we write P = P0 and we shall write Ex (resp., E) the expectation

perator associated with Px (resp., P). The firm’s asset value Vt evolves as Vt = eXt.

We assume the existence of a default-free asset that pays a continuous interest rate r > 0. Furthermore, assume that under P, the discounted value e−(r−δ)tVt of the firm’s asset is P−martingale, i.e., E

e−(r−δ)tVt
= 1

where δ > 0 is the total payout rate to the firm’s investors (bond and equity holders). Default happens at the first time τ −

B the underlying falls to some level B or lower;

τ −

B := inf{t ≥ 0 : Xt < B},

B ∈ R.

1DP, Leland-Toft (1994,1996); JDP, Hilberink-Rogers (2002), Chen-Kou (2009); SNLP, Kyprianou-Surya (2007). 2012 SPA Workshop, 8-9 March 2012, National Tsing-Hua University, Taiwan 3

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Toward a Generalization of the Leland-Toft Optimal Capital Structure Model

The Leland-Toft Model: Continued As the firm may declare default prior to debt maturity, the debt may not be paid back. Hence, the debt holder will charge a higher interest m on the debt than a default-free

asset. Suppose that the up-front payment of the outstanding loan is P with principal

p to be repaid in a periodical basis. Since the debt loan is an amortizing loan, the credit spread m can be determined as such that P = ∞ e−mtpdt, i.e., m = p

P .

Following the aforementioned literature, the total value of debt can be written as

D(x; B) :=Ex

τ− B

e−(r+m)tP ρ + pdt

+ Ex

e−(r+m)τ−

B Vτ− B

1 − η 1{τ−

B <∞}

, where ρ is the coupon rate and η is the fraction of firm’s asset value lost in default.

The firm value is given by

V(x; B) := ex+Ex

τ− B

e−rt1{Vt≥VT }τρP dt

− Ex
e−rτ−

B ηVτ− B

.

Here, it is assumed that there is a corporate tax rate τ and its (full) rebate on coupon payments is gained if and only if Vt ≥ VT for some cut off level VT > 0.

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Toward a Generalization of the Leland-Toft Optimal Capital Structure Model

The Leland-Toft Model: Continued The optimal default level B ∈ R is found by solving the problem: Listing 1: Finding the optimal default boundary max{x≥B} E(x; B) := V(x; B) − D(x; B), subject to the limitedliability constraint E(x; B) ≥ 0. (1)

The diffusion model admits analytical solutions (e.g., Leland and Toft, 1996).
The spectrally negative model admits semi-analytical solutions in terms of the scale

function (e.g., Kyprianou and Surya, 2007). Depending on the path regularity of the underlying L´ evy process X, the optimal boundary is found by employing

Smooth-pasting condition when X has paths of unbounded variation.
Continuous-pasting condition when X has paths of bounded variation.

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Toward a Generalization of the Leland-Toft Optimal Capital Structure Model

Towards Generalization of the Leland-Toft Model

The original model model of Leland-Toft assumes that
Default costs is a constant fraction η of the asset value Vt = eXt.
The tax is a stepwise function of the asset value, as

the tax =

τρP,

when Xt ≥ log VT 0,

therwise
r equivalently, the tax = τρP 1{Xt≥log VT }.
Coupon is a constant fraction ρ of up-front payment P of total loan.
In our work, we attempt to generalize the above assumptions in the following sense:
Default costs: ηeXt −

→ η(Xt).

The tax: τρP 1{Xt≥log VT } −

→ f2(Xt).

Coupon: from a constant ρ −

→ ρ(Xt). In the sequel below, we define a function f1(x) = P ρ(x) + p.

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Toward a Generalization of the Leland-Toft Optimal Capital Structure Model

Towards Generalization of the Leland-Toft Model: Continued By doing so,

the total value of debt becomes

D(x; B) :=Ex

τ− B

e−(r+m)tf1(Xt)dt

+ Ex
e−(r+m)τ−

B exp(Xτ− B )

1 −

η(Xτ−

B )

1{τ−

B <∞}

,

where η(x) := e−xη(x) is the ratio of default costs relative to the asset value.

The firm value is given by

V(x; B) := ex+Ex

τ− B

e−rtf2(Xt)dt

− Ex
e−rτ−

B η(Xτ− B )

.

The optimal default boundary B ∈ R is found by solving the problem (1).

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Toward a Generalization of the Leland-Toft Optimal Capital Structure Model

Spectrally Negative L´ evy Processes

Because of the fact that the L´ evy measure only charges the negative half-line, the characteristic exponent is well defined and analytic on (Im(θ) ≤ 0). We refer among

thers to Kyprianou (2006).

Hence, it is therefore sensible to define a Laplace exponent κ(θ) = −µθ + 1 2σ2θ2 +

(−∞,0)
eθy − 1 − θy1{y>−1}
Π(dy),

and, hence, we see that the identity E

eθXt

= etκ(θ) holds whenever Re(θ) ≥ 0. We denote by Φ : [0, ∞) → [0, ∞) the right continuous inverse of κ(λ), so that κ(Φ(λ)) = λ for all λ ≥ 0

r, i.e., Φ(α) is the largest positive root of Φ(α) = sup{p > 0 : κ(p) = α}.

The class of spectrally negative L´ evy processes is very rich. Amongst other things it allows for processes which have paths of both unbounded and bounded variation.

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Toward a Generalization of the Leland-Toft Optimal Capital Structure Model

Spectrally Negative L´ evy Processes: Continued It has bounded variation if and only if σ = 0 and

−∞ |x|Π(dx) < ∞.

In that case one may rearrange the Laplace exponent into the form κ(λ) = dλ −

(−∞,0)

(1 − eλx)Π(dx) for some d > 0. Definition 1. [Scale function] For a given SNLP X with Laplace exponent κ there exists for every q ≥ 0 a increasing function W (q) : R → [0, ∞) such that W (q)(x) = 0 for all x < 0 and otherwise is differentiable on [0, ∞) satisfying,

∞

e−λxW (q)(x)dx = 1 κ(λ) − q, for λ > Φ(q). In the sequel, we will use the function Z(q)(x) := 1 + q x

0 W (q)(y)dy. The scale

function W (q)(x) and Z(q)(x) appear in Laplace transform of exit times of SNLP. In some cases, there are explicit expressions of the scale function W (q)(x). In general,

ne can apply numerical inversion of Laplace transform to compute W (q)(x). See for

instance Surya (2008).

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Toward a Generalization of the Leland-Toft Optimal Capital Structure Model

Our Main Results

Before we state our main results, let us define for all B ∈ R the following: G(q)

j (B)

:=

∞

e−Φ(q)yfj(y + B)dy, j = 1, 2 H(q)(B) :=

∞

Π(du)

u

dze−Φ(q)zη(B) − η(B − (u − z)) J(r,m)(B) :=

r + m

Φ(r + m) − r Φ(r)

η(B) −
H(r)(B) − H(r+m)(B)
K(r,m)

1

(B) := κ(1) − (r + m) 1 − Φ(r + m) eB − G(r+m)

1

(B) + G(r)

2 (B) − J(r,m)(B)

K(r)

2 (B)

:= G(r)

2 (B) +

r Φ(r)η(B) + H(r)(B) + σ2 2 η′(B). We impose the following assumption throughout: Assumption 1. ∞ e−Φ(q)xfj(x)dx < ∞, j = 1, 2, for some q > 0. Assumption 2. η ∈ C2(R) is bounded on (−∞, B) for a fixed B ∈ R.

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Toward a Generalization of the Leland-Toft Optimal Capital Structure Model

Remarks 1. The Assumption 2 is required just to show monotonicity of the function B → E(x; B), but not necessary to later prove the optimality of default boundary. This makes the equity value E(x; B) is well defined for any x > B. Remarks 2. If η(B) is increasing in B, then K(r)

2 (B) is uniformly positive.

Remarks 3. As Φ(q) is increasing in q and η(.) > 0, J(r,m)(B) ≥ 0 ∀B ∈ R. Also we define functions M(q)

j (x; B), j = 1, 2, Λ(q)(x; B) and Γ(q)(x) as follows

Λ(q)(x; B) = η(B)

Z(q)(x − B) −

q Φ(q)W (q)(x − B)

−W (q)(x − B)H(q)(B)

+

∞

Π(du)

u

dzW (q)(x − z − B)

η(B) − η(z + B − u)
M(q)

j (x; B)

= W (q)(x − B)G(q)

j (B) −

x

B

W (q)(x − y)fj(y)dy Γ(q)(x) = κ(1) − q 1 − Φ(q)W (q)(x) +

κ(1) − q
ex
x

e−yW (q)(y)dy.

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Toward a Generalization of the Leland-Toft Optimal Capital Structure Model

Our Main Results: Continued The corresponding debt and firm’s values are given by the following: D(x; B) = ex − eBΓ(r+m)(x − B) + M(r+m)

1

(x; B) − Λ(r+m)(x; B) V(x; B) = ex + M(r)

2 (x; B) − Λ(r)(x; B),

Proposition 1. It can be shown after some algebra that for a fixed x ∈ R ∂ ∂BE(x; B) = −

Θ(r+m)(x − B)K(r,m)

1

(B) +

Θ(r)(x − B) − Θ(r+m)(x − B)
K(r)

2 (B)

∀B < x.

where Θ(q)(x) := W (q)′(x) − Φ(q)W (q)(x) is the resolvent measure of the ascending ladder height process of X. For a fixed x > 0, Θ(q)(x) ց q. Listing 2: Optimality If there exists B⋆ such that K(r,m)

1

(B) ≥ 0 ⇔ B ≥ B⋆ and K(r)

2 (B) ≥ 0

forevery B ≥ B⋆, then B⋆, if it exists, is theoptimal default boundary.

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Toward a Generalization of the Leland-Toft Optimal Capital Structure Model

Sufficient Condition and Examples

Listing 3: Example Suppose that (1) η is increasing, (2) η isdecreasing, (3) ρ is decreasing, (4) f2 is increasing, (5) 0 ≤ η(.) ≤ 1. Then B⋆ is theoptimal default boundary. In other words, the optimality holds when, monotonically in the asset value,

the loss amount at default is increasing,
its proportion relative to the asset value is decreasing,
the coupon rate is decreasing,
and the value of tax benefits is increasing.

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Toward a Generalization of the Leland-Toft Optimal Capital Structure Model

Sufficient Condition and Examples: Continued We consider the case

η(x) = η0
e−a(x−b) ∧ 1
,

f2(x) = τP ρ

ex−c ∧ 1
.

and a constant coupon rate ρ. Regarding the source of randomness, we consider B as a jump diffusion process with σ > 0 and jumps of exponential type with L´ evy measure: Π(dx) = λβe−βxdx. See, e.g., [5] and [3] for an explicit expression of the Scale function W (q). The model parameters used in the computation are given by r = 7.5%, δ = 7%, τ = 35%, σ = 0.2, λ = 0.5, β = 9. We set V0 = 100 We look at two cases:

Case 1: η0 = 0.9, a = 0.5, b = 0 and c = 5,
Case 2: η0 = 0.5, a = 0.01, b = 5 and c = 0.

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Toward a Generalization of the Leland-Toft Optimal Capital Structure Model

Numerical Results

5

5

4
2

2 4 6 8 10 12 B K 1

5

5

4
2

2 4 6 8 10 12 B K 1

case 1 case 2 Figure 1: The plots of K(r,m)

1

(B). Applying the bisection method to K(r,m)

1

(B) = 0, we obtain B⋆ = 3.61 and B⋆ = 3.64 for cases 1 and 2, respectively.

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Toward a Generalization of the Leland-Toft Optimal Capital Structure Model

Numerical Results: Continued

30 35 40 45 50 55

2

2 4 6 8 10 12 14 V 0 equity value B *-0.2 B *-0.1 B * B *+0.1 B *+0.2 30 35 40 45 50 55

2

2 4 6 8 10 12 14 V 0 equity value B *-0.2 B *-0.1 B * B *+0.1 B *+0.2

equity value (case 1) equity value (case 2) Figure 2: The equity/debt/firm values as a function of V0 for various values of B.

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Toward a Generalization of the Leland-Toft Optimal Capital Structure Model

Numerical Results: Continued

30 35 40 45 50 55 15 20 25 30 35 40 45 50 55 60 V 0 debt value B *-0.2 B *-0.1 B * B *+0.1 B *+0.2 30 35 40 45 50 55 15 20 25 30 35 40 45 50 55 60 V 0 debt value B *-0.2 B *-0.1 B * B *+0.1 B *+0.2

debt value (case 1) debt value (case 2) Figure 3: The equity/debt/firm values as a function of V0 for various values of B.

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Numerical Results: Continued

20 40 60 80 100 95 100 105 110 P firm value 10 20 30 40 50 60 70 80 90 95 100 105 110 P firm value

case 1 case 2 Figure 4: The firm value as a function of P for the two-stage problem. The optimal face values of debt are given by P ⋆ = 73.7 and P ⋆ = 39, respectively.

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Some Ideas for Future Works

These are some ideas which can be pursued to extend the current work.

Allowing the jumps of X to be two-sided, having an explicit Wiener-Hopf

factorization, such as jump-diffusion process with exponential jumps or phase-type.

Another consideration would be the following finite-time model. In case of the firm

is in financial distress, the firm’s manager may be looking for a stopping time τ in a finite time interval [0, t], such that the equity value of the firm is maximized. That is to say that he/she is trying to solve the optimal stopping problem: sup

{0≤τ≤t}

Ex(τ) := Vx(τ) − Dx(τ), where Vx(τ) is the firm’s value defined as Vx(τ) := ex+Ex

τ

e−rsf2(Xs)ds

− Ex

e−rτη(Xτ), whereas Dx(τ) is the total debt outstanding of the firm, given by Dx(τ) :=Ex

τ

e−(r+m)sf1(Xs)ds

+ Ex
e−(r+m)τ exp(Xτ)1 −

η(Xτ) .

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Main References References

[1] Chen, N., and Kou, S. (2009). Credit spreads, optimal capital structure, and implied volatility with endogenous default and jump risk. Math. Finance, Vol. 19, 343378 [2] Egami, M. and Yamazaki, K. (2011). On the Continuous and Smooth Fit Principle for Optimal Stopping Problems in Spectrally Negative L´ evy Models. arXiv:1104.4563. [3] Egami, M. and Yamazaki, K. (2010). On Scale Functions of Spectrally Negative L´ evy Processes with Phase-type Jumps. arXiv:1105:0064. [4] Hilberink, B., and Rogers, L. C. G. (2002). Optimal capital structure and endogenous default. Finance Stoch., Vol. 6, No. 2, 237-263. [5] Kyprianou, A. E. (2006).Introductory Lectures on Fluctuations of L´ evy Processes with Applications, Springer-Verlag, Berlin. [6] Kyprianou, A. E., and Surya, B. A. (2007). Principles of smooth and continuous fit in the determination of endogenous bankruptcy levels. Finance Stoch., Vol. 11 No. 1, p. 131-152. [7] Leland, H. E. (1994). Corporate debt value, bond covenants, and optimal capital structure with default risk, J. Finance, Vol. 49, p. 1213-1252. [8] Leland, H. E., and Toft, K. B. (1996). Optimal capital structure, endogenous bankruptcy, and the term structure of credit spreads. J. Finance, Vol. 51, 987-1019. [9] Surya, B. A. (2008). Evaluating scale functions of spectrally negative L´ evy processes. J. Appl. Probab., Vol. 45 No. 1, 135-149. [10] Surya, B. A. and Yamazaki, K. (2011). Toward a Generalization of the Leland-Toft Optimal Capital Structure Model. arXiv:1109.0897.

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