Toward a Generalization of the Leland-Toft Optimal …math.cts.nthu.edu.tw/Mathematics/2012SPA-slides/12surya.pdfToward a Generalization of the Leland-Toft Optimal Capital Structure

Toward a Generalization of the Leland-Toft Optimal

Capital Structure Model ∗

Budhi Arta Surya

School of Business and ManagementBandung Institute of Technology

Joint work with

Kazutoshi Yamazaki

Center for the Study of Finance and InsuranceOsaka University

March 5, 2012

∗Based on Surya, B. A. and Yamazaki, K. (2011). Toward a Generalization of the Leland-Toft Optimal Capital StructureModel. arXiv:1109.0897. I would like to thank the organizer and the NCTS Institute for their hospitality and making this

visit possible.

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

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

equity value is maximized.

2012 SPA Workshop, 8-9 March 2012, National Tsing-Hua University, Taiwan 1


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 Levy 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).



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

operator 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)t

Vt

)= 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).



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 =∫ ∞

0e−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

0

e−(r+m)t(

Pρ + p)dt

)+ 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

0

e−rt

1{Vt≥VT }τρPdt)− 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.



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 Levy 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.



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, otherwise

or equivalently, the tax = τρP1{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: τρP1{Xt≥log VT } −→ f2(Xt).

• Coupon: from a constant ρ −→ ρ(Xt).

In the sequel below, we define a function

f1(x) = Pρ(x) + p.



Towards Generalization of the Leland-Toft Model: Continued

By doing so,

• the total value of debt becomes

D(x; B) :=Ex

( ∫ τ−B

0

e−(r+m)t

f1(Xt)dt)

+ Ex

(e−(r+m)τ−

B exp(Xτ−B)(1 − η(X

τ−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

0

e−rt

f2(Xt)dt)− Ex

(e−rτ

−B η(X

τ−B

)).

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



Spectrally Negative Levy Processes

Because of the fact that the Levy measure only charges the negative half-line, the

characteristic exponent is well defined and analytic on (Im(θ) ≤ 0). We refer among

others to Kyprianou (2006).

Hence, it is therefore sensible to define a Laplace exponent

κ(θ) = −µθ +1

2σ

2θ

2+

∫

(−∞,0)

(e

θy− 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

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

The class of spectrally negative Levy processes is very rich. Amongst other things it

allows for processes which have paths of both unbounded and bounded variation.



Spectrally Negative Levy Processes: Continued

It has bounded variation if and only if σ = 0 and∫ 0

−∞|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,

∫ ∞

0

e−λx

W(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

0W (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,

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

instance Surya (2008).



Our Main Results

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

G(q)j (B) :=

∫ ∞

0

e−Φ(q)y

fj(y + B)dy, j = 1, 2

H(q)(B) :=

∫ ∞

0

Π(du){ ∫ u

0

dze−Φ(q)z[

η(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)e

B− 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

2η′(B).

We impose the following assumption throughout:

Assumption 1.∫ ∞

0e−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.



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)

+

∫ ∞

0

Π(du)

∫ u

0

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

)e

x

∫ x

0

e−y

W(q)

(y)dy.



Our Main Results: Continued

The corresponding debt and firm’s values are given by the following:

D(x; B) = ex− e

BΓ

(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.



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.



Sufficient Condition and Examples: Continued

We consider the case

η(x) = η0

(e−a(x−b)

∧ 1), f2(x) = τPρ

(e

x−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 Levy 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.



Numerical Results

-5 0 5-4

-2

0

2

4

6

8

10

12

B

K1

-5 0 5-4

-2

0

2

4

6

8

10

12

B

K1

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.



Numerical Results: Continued

30 35 40 45 50 55-2

0

2

4

6

8

10

12

14

V0

equity value

B*-0.2

B*-0.1

B*

B*+0.1

B*+0.2

30 35 40 45 50 55-2

0

2

4

6

8

10

12

14

V0

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.




30 35 40 45 50 5515

20

25

30

35

40

45

50

55

60

V0

debt value

B*-0.2

B*-0.1

B*

B*+0.1

B*+0.2

30 35 40 45 50 5515

20

25

30

35

40

45

50

55

60

V0

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.




0 20 40 60 80 10095

100

105

110

P

firm value

0 10 20 30 40 50 60 70 80 9095

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.



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

0

e−rs

f2(Xs)ds)− Ex

(e−rτ

η(Xτ)),

whereas Dx(τ) is the total debt outstanding of the firm, given by

Dx(τ) :=Ex

( ∫ τ

0

e−(r+m)s

f1(Xs)ds)

+ Ex

(e−(r+m)τ exp(Xτ)

(1 − η(Xτ)

)).



Main ReferencesReferences

[1] Chen, N., and Kou, S. (2009). Credit spreads, optimal capital structure, and implied volatilitywith 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 Levy Models. arXiv:1104.4563.

[3] Egami, M. and Yamazaki, K. (2010). On Scale Functions of Spectrally Negative Levy 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 Levy Processes with

Applications, Springer-Verlag, Berlin.

[6] Kyprianou, A. E., and Surya, B. A. (2007). Principles of smooth and continuous fit in thedetermination 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 withdefault 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 Levy 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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