Toward a Generalization of the Leland-Toft Optimal...
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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
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 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).
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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
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).
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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 =∫ ∞
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.
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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 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.
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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, 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.
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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
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).
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Toward a Generalization of the Leland-Toft Optimal Capital Structure Model
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.
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Toward a Generalization of the Leland-Toft Optimal Capital Structure Model
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).
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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) :=
∫ ∞
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.
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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)
+
∫ ∞
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.
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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− 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.
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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) = 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.
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Toward a Generalization of the Leland-Toft Optimal Capital Structure Model
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.
2012 SPA Workshop, 8-9 March 2012, National Tsing-Hua University, Taiwan 15
Toward a Generalization of the Leland-Toft Optimal Capital Structure Model
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.
2012 SPA Workshop, 8-9 March 2012, National Tsing-Hua University, Taiwan 16
Toward a Generalization of the Leland-Toft Optimal Capital Structure Model
Numerical Results: Continued
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.
2012 SPA Workshop, 8-9 March 2012, National Tsing-Hua University, Taiwan 17
Toward a Generalization of the Leland-Toft Optimal Capital Structure Model
Numerical Results: Continued
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.
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Toward a Generalization of the Leland-Toft Optimal Capital Structure Model
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)
)).
2012 SPA Workshop, 8-9 March 2012, National Tsing-Hua University, Taiwan 19
Toward a Generalization of the Leland-Toft Optimal Capital Structure Model
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.
2012 SPA Workshop, 8-9 March 2012, National Tsing-Hua University, Taiwan 20