Dynamic Capital Structure Modelling under Alternative Stochastic ...

Dynamic Capital Structure Modelling

under Alternative Stochastic Processes

by

Simon Seir Bjerrisgaard

and

Denis Fedoryaev

Master’s Thesis

Presented to the Faculty of

the Department of Finance of Copenhagen Business School

in Partial Fulfilment of the Requirements for the Degree of

Elite MSc in Advanced Economics and Finance

(Cand.Oecon.)

Supervisor: Kristian R. Miltersen

Copenhagen Business School

July 2011

No. of pages (characters): 125 (242,958)

Executive Summary

A coherent line of development of research focusing on modelling optimal

capital structure is presented. We begin by revisiting the pricing model in-

troduced in the seminal work of Merton (1974), which represents the funda-

mental building block for most capital structure models. The general partial

differential equation is rederived, and it is demonstrated how it is applied to

price debt and equity as contingent claims. The first stepping stone to our

analysis is the static model along the lines of Leland (1994). We present com-

parative statics for optimally chosen leverage, coupon, and default threshold

as well as for tax advantage to debt, and discuss the drawbacks of the model

in detail. It is evident that under our own base case parameters—based on

the most recent Danish data and empirical estimates—which differ from un-

reasonably high values applied in Leland, the optimal leverage is too high

compared to that observed in practice. We proceed by introducing dynam-

ics in the model of capital structure; rather than focusing on any particular

model, we outline a generalised framework to describe the overall family of

existing dynamic capital structure models. The state variable is the operat-

ing income, and it is the value of a claim to the entire payout of the firm that

is modelled as a stochastic process, which restores the no-arbitrage condition

violated in the static model. As the leverage can be readjusted, the restruc-

turing boundary is incorporated, stipulating when refinancing should take

place, and is obtained from the smooth-pasting condition to ensure incen-

tive compatibility. The key insight from the comparative statics analysis is

that the optimal leverage drops substantially compared to that in the static

i

ii S. Bjerrisgaard and D. Fedoryaev

model, and the default boundary is also lowered, reflecting the fact that the

refinancing option, ceteris paribus, enhances the firm valuation.

Further, we propose our own dynamic model which alters the cash flow

process by assuming mean reversion. The vast majority of existing capital

structure studies rests on the assumption that the state variable follows a ge-

ometric Brownian motion since this ensures mathematical tractability while

modelling the capital structure readjustment. We argue that this underly-

ing process due to its properties is not consistent with dynamics of the firm

fundamental and that the assumption of mean reversion appears to be more

suitable. Not only the latter describes the development of the real sector

better, but is also reinforced by empirical evidence. Besides, by assuming

that earnings follow the mean-reverting process, we obtain a better control

over the process and can thus draw a much clearer distinction between the

industries in terms of both profitability and stability—through varying the

long-term mean and the speed of mean reversion, respectively. As a matter of

fact, optimal capital structure has never been modelled dynamically outside

of the scope of a geometric Brownian motion, presumably due to the loss of

homogeneity property. We assume the modified mean-reverting process with

volatility being proportional to the current earnings level, and derive the or-

dinary differential equation used for pricing debt and equity. Further, using

contingent claims analysis and state pricing, we prove that the new process

still possesses the homogeneity property which implies that the mechanics

of the model is unaltered in time. After that we describe what we call the

optimal capital structure decision framework and formulate how the tuple

that closes it is obtained, which fully determines the capital structure choice

of the owner-manager.

Extensive numerical simulations are conducted, with explicit focus on

numerical algorithms and their shortcomings. We bring together the key op-

timal capital structure frameworks and carry out a cross-model comparison

to uncover the implications that different assumptions have for the results.

When benchmarking our model against the static mean-reverting model of

iii

Sarkar and Zapatero (2003), we find lower optimal leverage and optimal

coupon. If compared to the conventional dynamic GBM-based model, our

model suggests lower optimal leverage and higher restructuring frequency,

stipulated by the difference in expectations of equity holders regarding the

future development of cash flow under the two processes. This relationship is

even more pronounced if we correct for the finiteness of the stationary vari-

ance of the mean-reverting process. Overall, given reasonable assumptions

for base case parameters, the optimal leverage in our model is found to be

closer to the empirical regularities than that in the existing models. All re-

lationships are studied through the prism of mean reversion, and it is shown

that the speed of earnings convergence and the long-term mean value of earn-

ings are indeed important parameters as their impact on the key variables

could be rather substantial. Finally, we extend the analysis by modifying

the GBM-based model to take into account fixed operating costs, so that the

resulting cash flow is not bounded at zero. We demonstrate that higher op-

erating leverage implies lower financial leverage, and find that when the firm

is allowed to have negative cash inflow, it will be less levered compared to an

otherwise identical firm whose earnings are instead mean-reverting but al-

ways positive. Further, we incorporate a call premium in the dynamic model

with mean reversion and show that assuming debt being callable at par may

lead to somewhat understated leverage and too high restructuring frequency.

Keywords: Dynamic capital structure; Optimal bankruptcy; Optimal restruc-

turing; Contingent claims analysis; Mean-reverting earnings; Homogeneity

property

JEL classification: G12; G13; G32; G33; C60; C61; C63

Acknowledgements

We wish to acknowledge with profound gratitude the counsel and constant

encouragement of Kristian Miltersen, whom we are much obliged for numer-

ous conversations, constructive suggestions and help on the coding. We are

also thankful to Michael Genser for providing the code for results in Am-

mann and Genser (2004), which gave us some insight on specific aspects of

the sphere considered. Finally, we would like to thank Sudipto Sarkar and

Fernando Zapatero for the comments on their model, from which we could

draw additional inspiration for our own analysis. We are solely responsible

for all remaining errors.

iv

Contents

1 Introduction 1

1.1 Basic Framework . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Contingent Claims Analysis 15

2.1 Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Ito’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3 Note on Risk Neutrality in Continuous Time . . . . . . . . . . 23

2.4 The General Partial Differential Equation . . . . . . . . . . . 25

2.5 Isomorphism to Option Pricing . . . . . . . . . . . . . . . . . 29

3 Static Model of Capital Structure 34

3.1 Time Independence of Cash Flows . . . . . . . . . . . . . . . . 35

3.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.2.1 Solution to the ODE . . . . . . . . . . . . . . . . . . . 37

3.2.2 Valuation of Claims . . . . . . . . . . . . . . . . . . . . 39

3.2.3 Optimal Leverage . . . . . . . . . . . . . . . . . . . . . 44

3.3 Comparative Statics . . . . . . . . . . . . . . . . . . . . . . . 45

3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4 Family of Existing GBM-based Dynamic Models 50

4.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.2 Comparative Statics . . . . . . . . . . . . . . . . . . . . . . . 57

v

vi S. Bjerrisgaard and D. Fedoryaev

4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5 Proposed Dynamic Model with Mean Reversion 65

5.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 69

5.2.2 Homogeneity Property . . . . . . . . . . . . . . . . . . 74

5.2.3 Optimal Capital Structure Decision Tuple . . . . . . . 80

5.3 Numerical Results and Model Implications . . . . . . . . . . . 87

5.3.1 Note on Numerical Simulations . . . . . . . . . . . . . 87

5.3.2 Numerical Analysis . . . . . . . . . . . . . . . . . . . . 90

5.4 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.4.1 Fixed Operational Costs . . . . . . . . . . . . . . . . . 103

5.4.2 Call Premium . . . . . . . . . . . . . . . . . . . . . . . 108

6 Conclusion 111

References 116

Chapter 1

Introduction

And make sure that the capital structure we have in place is the right capital

structure. I think that’s the reason that we’ve been successful.

(Henry R. Kravis, co-founder of KKR)

How to finance operations and new investment projects is a problem that

CFOs and managers of corporations frequently encounter. The primary rea-

son why it is beneficial to develop a specific leverage policy is that the optimal

combination of debt and equity creates value for the owners of business by

lessening the tax base. Exploiting the tax benefits of debt financing in the

optimal way could help managers potentially increase the firm value by as

much as 10% (Graham, 2000).

The importance of the capital structure choice for the company valuation

has been one of the central questions in academia since the very seminal

work of Modigliani and Miller (1958). Since then, numerous authors have

contributed to the formation of the optimal capital structure study sphere

by introducing different methods and utilising various assumptions. Some

of the earlier attempts to price corporate securities were based on the firm

value dynamics, while subsequent models instead assumed that the manage-

rial expectations regarding the future development of earnings should be the

primary driver. Initially, the papers, aimed to determine the optimal com-

1

2 S. Bjerrisgaard and D. Fedoryaev

position of debt and equity, ignored the firm’s option to change the financing

policy in time, and a comprehensive dynamic framework was not presented

until Fischer, Heinkel, and Zechner (1989a). Moreover, a model allowing

the company to change its leverage while also assuming earnings to be the

governing factor was not introduced until Goldstein, Ju, and Leland (2001).

What brings the capital structure models existing hitherto together is that

irrespective of the underlying factor choice the process describing its dynam-

ics implicitly assumes that the value of the factor always stays positive and

has only a positive trend. Since such assumption seems rather unfeasible,

especially when considering earnings, this leaves some scope for developing a

dynamic capital structure model based on an alternative stochastic process

to rectify these shortcomings.

The paper proceeds as follows. In the rest of this chapter we dwell on the

terminology used throughout the thesis and state the assumptions implic-

itly incorporated in the analysis that follows, pose our research questions,

and provide a literature review on capital structure modelling. Chapter 2

presents necessary theoretical insights from stochastic calculus and highlights

the importance of contingent claims analysis for the theory of capital struc-

ture choice. In Chapter 3 we lay out a static model of capital structure

and analyse it in detail. Chapter 4 follows with a study of a dynamic cap-

ital structure model which serves as a classic example of the conventional

dynamic framework. In Chapter 5 we introduce an alternative assumption

regarding the dynamics of earnings, and thoroughly develop a dynamic cap-

ital structure model, providing corresponding analysis and numerical tests.

Chapter 6 concludes.

1.1 Basic Framework

As there might exist ambiguity in interpretation of the notions and concepts

of capital structure theory, we will first briefly outline some terminology that

will apply throughout. One should note, however, that since the terminology

Chapter 1. Introduction 3

within this field of academic research is not uniform, the below should merely

be viewed as a guide to this thesis rather than as a universal introduction.

Capital Structure

Capital structure is the consensus term for a company’s composition of debt

and equity, and thus specifies the sources of the capital from which a firm

finances its operations and investments. Debt must be serviced irrespectively

of the success of the firm, while equity only has a residual claim to the cash

generated by the company’s operations. This difference in obligation means

that the cash is spent servicing debt payments before it can be returned to

equity holders, and results in a higher return on equity than on debt.

Equity in accounting terms is the sum of common stock, preferred stock,

any capital surplus, retained earnings, treasury stock, stock options, and

reserves; throughout the thesis we will not distinguish between these com-

ponents and will instead consider equity an aggregate parameter represented

only by common stock. We implicitly assume the following: there is no

subordination among shares, no part of earnings is retained as they are paid

out in full to shareholders as dividends, no additional paid-in capital is raised

from equity investors, the company is not holding shares in itself or any other

company, and no reserves are held. We also ignore the fact that shares with

different cash flow or control rights might exist.

Corporate debt is normally comprised of bonds and loans, with the latter

being either syndicated or bilateral facilities and having either revolving or

set instalment repayment schedule. Bonds have a fixed maturity, and thus

usually cannot be paid down gradually, but both loan and bond contracts can

include call provisions allowing the debtor to pay the principal back early. In

this thesis we will exclude loans from the analysis, as the pricing technique

of debt securities that will be presented later implies that the security must

be traded. Despite the existence of the secondary loan market, due to its

relatively small size we will treat the value of debt as being synonymous with

the value of outstanding bonds.

Based on the above, capital structure can be summarised to exhibit in


what proportion the company financing is split between common shares and

bonds. The ratio of debt to the sum of market values of debt and equity

is termed “leverage”. Optimality of capital structure stems from the max-

imisation of the equity value as the financing policy is at sole discretion of

shareholders. Note that since in the models presented in further chapters all

existing debt is retired before the optimal capital structure is derived, the

optimal leverage in fact maximises the firm value.

State Variable

In the models considered in this paper, the optimal leverage is calculated by

utilising contingent claims pricing. The underlying variable which the value

of these claims is dependent on is termed the state variable, and typically

represents a firm fundamental, e.g. value of unlevered assets, earnings, or

another proxy. In Chapters 2 and 3 the state variable is assumed to be the

asset value of the firm, while models in Chapters 4 and 5 consider EBIT

instead.

Default

Bankruptcy is a legal proceeding involving a company that is unable to re-

pay its outstanding debt. A default occurs when the firm has not met its

legal obligation according to the bond contract, i.e. when it has not met a

scheduled payment, or alternatively, has violated a covenant—the latter be-

ing termed technical default. Furthermore, a default can occur not only when

a firm is unable to pay, but also when it is unwilling to pay—the situation

known as strategic default. The unwillingness to pay is a choice made by

equity holders and might be regarded as their last resort to save some value

to themselves after repaying the principal and possibly incurring bankruptcy

costs.1

1 Interestingly, even though strategic debt service has been conventionally considered toresult in lower debt value and higher yield spreads (Anderson and Sundaresan, 1996; Mella-Barral and Perraudin, 1997), Acharya, Huang, Subrahmanyam and Sundaram (2006)demonstrate that when managers are not forced to pay out excess earnings as dividends,the ability to service debt strategically could reduce the likelihood of liquidity-driven de-faults (the extent of such reduction is increasing in firm’s cost of capital), leading tonegligible effect on debt value and yield spread.


We will make no distinction between default and bankruptcy, despite the

obvious discrepancy in the interpretation of the two terms. It should be

noted, however, that the way bankruptcy is modelled will slightly change

throughout the paper. One of the possible assumptions is that bankruptcy

occurs when EBIT reaches an exogenously given critical value; another ap-

proach is that bankruptcy occurs when the state variable reaches an endoge-

nously determined threshold at which it is no longer in the best interest of

equity holders to keep servicing the debt, and the company is liquidated.

The latter is the assumed bankruptcy procedure under the static model in

Chapter 3. Alternatively, the firm can be taken over post-default as a going

concern, and we consider this option in Chapters 4 and 5.

Let us now turn to formulating the general setup of our analysis, delin-

eated by specific confining assumptions that will be implicit throughout the

thesis.

In practice bond contracts involve a great variety of contractual provi-

sions. Despite that, we will refrain from including the following common

bond indenture provisions and liability characteristics in our framework:

Collateral. Collateral is the borrower’s pledge of specific property that

must be forfeited in case of default, and is often included to serve as

a protection for creditors. By excluding this provision we do not have

to consider additionally the transfer and division of collateral upon

default.

Explicit Maturity. Maturity refers to the final repayment date at

which point the principal must be redeemed in full. As will be shown

later, we assume a sufficiently long maturity, so that coupon payments

are essentially the only debt-related cash outflows.

Subordination. Subordination refers to the order of priorities in

claims for ownership of assets. In our model at no point in time a

firm could have two different debt issues outstanding, and therefore

there is no need to model subordination.


Debt Renegotiation. It has been shown by Gilson, John, and Lang

(1990) in a study of 169 financially distressed US firms that around 53%

of the firms went into bankruptcy, while the rest successfully renegoti-

ated the terms of the debt. In this paper, however, we will abstain from

examining debt renegotiation possibilities since this particular aspect

has already been studied extensively in the literature.2

In addition to the restrictions stated above, we will further limit the scope

of our analysis by making the following assumptions:

No Asset Substitution.3 This implies that the investment deci-

sions are unaffected by capital structure, and particularly that the risk

profile is independent of leverage. Note that we also exclude the un-

derinvestment possibilities of Myers (1977) type from our analysis, i.e.

any value-increasing investment opportunity is being accepted by the

equity holders.4

Full Advantage of Tax Shield. It is assumed that all interest ex-

penses can be deducted from the taxable income, even if the earnings

fall below the coupon level, and thus it is implied that the value of the

tax shield could be obtained as a perpetual stream of savings, condi-

tional on the firm remaining solvent.

Constant Risk-free Interest Rate. The risk-free rate has a bearing

on the discounted value of cash flows. Despite the stochastic nature

observed in practice, we assume that this parameter is constant to

simplify the matters.

2 For a theoretical review of how the possibility of renegotiating either coupon orboth coupon and principal affects debt value, see Anderson and Sundaresan (1996) orChristensen, Flor, Lando, and Miltersen (2002), respectively.

3 Jensen and Meckling (1976) present a theoretical analysis of the asset substitutionproblem in their seminal paper.

4 Various studies have also found weak support for asset substitution and underinvest-ment. See, e.g. Graham and Harvey (2001) or Ju and Ou-Yang (2006).


Furthermore, the reader should pay attention to the fact that the abso-

lute priority rule is taken for granted at all times.5 In case assumptions in

addition to those stated above are required, they will be pointed out where

appropriate.

1.2 Problem Formulation

In our analysis we will generally be focusing on two aspects: how to model

optimal capital structure in a dynamic setting, and what effect alternative

assumptions—for instance, regarding the process describing the state variable

dynamics—will have on the composition of debt and equity.

Since empirical evidence shows that firms do indeed alter their financing

policy (e.g. Leary and Roberts, 2005), we find it necessary to incorporate the

option to readjust the capital structure dynamically. Furthermore, allowing

firms to continuously adjust their capital structure according to the changes

in the state variable is more intuitive compared to the static setup, where

leverage is permanent and thus reflects an assumed knowledge of the firm’s

development far into the future. In addition to making the optimal capital

structure dynamic, we aim at utilising an alternative stochastic process to

describe the evolution of the state variable. This is carried out to rectify some

of the drawbacks of the underlying process applied in the majority of existing

models. Apart from that, to make a progression to our model clearer, we

intend to analyse the mechanisms behind existing frameworks, focusing on

their economic interpretation and pointing out the main shortcomings.

Due to multifaceted aspects of corporate financing and restrictive as-

sumptions that need to be made to enable comprehensible derivations, we do

not intend to propose an entirely universal framework, but rather to build a

model that works under certain restrictions. We recognise that a great vari-

5 Even though there exist rational explanations for violation of the absolute priorityassumption (see, for example, Christensen et al., 2002), we choose to adhere to the as-sumption of absolute priority as the effect of deviation on leverage and yield spread hasbeen shown to be marginal, even when different bankruptcy determinants are considered.See, for instance, Leland (1994, Section VI).


ety of factors can contribute to the explanation of observed leverage ratios,

but we will nevertheless limit the scope of our analysis to a specific number

of explanatory parameters. We also respect that the modelling approach

may depend on the type of the state variable considered, but still find it nec-

essary to investigate what implications alternative assumptions have for the

capital structure decision, thereby either reinforcing or undermining previous

findings.

Thus, the main objective of this thesis is to provide answers to the fol-

lowing pivotal questions:

How can a valuation framework to price any corporate security contin-

gent on a firm fundamental be obtained?

What are the main implications of allowing dynamic refinancing in

the traditional model of optimal capital structure which assumes that

earnings follow a geometric Brownian motion?

How can optimal capital structure be modelled dynamically under the

assumption that earnings do not follow a conventional geometric Brow-

nian motion?

Assuming mean reversion in earnings, what inferences could be drawn

from benchmarking the results of the proposed dynamic model against

those in the existing static model?

How does the optimal capital structure decision change when earnings

are assumed to follow a mean-reverting process in a dynamic setup,

compared to that in existing GBM-based models?

If a dynamic GBM-based model with earnings that are allowed to ob-

tain negative values due to the introduction of fixed operating costs is

considered, how does optimal leverage compare to that under a dynamic

model with earnings that exhibit mean reversion?


There are naturally numerous ways to address the questions stated above,

both empirically and theoretically. However, it should be emphasised that

we will restrict our focus to the theoretical analysis combined with numerical

simulations, thus leaving out empirical tests of the proposed model.

1.3 Literature Review

With the basic setup and problem formulation in place, a brief review of

literature concerning the topic of choice is necessary to demonstrate how the

research sphere has evolved through time and what theoretical advancements

we could base our analysis upon.

The question of whether capital structure matters is most famously ad-

dressed by Modigliani and Miller (1958), who argue that the market value

of a firm is independent of its capital structure in a frictionless world. The

idea behind this argument is that investors can freely trade in the finan-

cial markets to make their personal portfolio in accordance with a preferred

capital structure, thus eliminating the need for companies to alter their cap-

ital structures. However, after relaxing the assumption of the absence of

frictions, Modigliani and Miller (1963) found that when companies are al-

lowed to deduct interest expenses from taxable income, the optimal capital

structure under certain conditions is to be fully levered, i.e. firms should

be financed solely by debt. As such examples are virtually non-existent in

practice, many of the subsequent research studies were dedicated to prove

that this conclusion does not hold under a variety of circumstances, includ-

ing the introduction of agency costs, bankruptcy costs, transaction costs or

other assumptions that could be considered when trying to explain observed

leverage ratios.

The academic research following Modigliani and Miller provided different

approaches to explain the determinants of capital structure, and in general

they could be divided into two groups: asymmetric information models and

the trade-off theory models. Myers and Majluf (1984) based their pecking


order theory on the concept of asymmetric information between managers

and investors. The idea behind their work was that managers might be forced

to give up positive-NPV projects in case outside financing is required, and

that internal financing (retained earnings) is preferred to external funding

sources. If outside financing is the only option, then debt is favoured due to

the incentives of the manager to issue new shares only when she expects to

receive a high price, which from investors’ viewpoint would mean that the

stock is overpriced. The new shareholders, knowing this, will adjust the price

they are willing to pay for the new shares, reducing the proceeds from the

issue and diluting the value of existing shares. The manager, acting on behalf

of existing shareholders, will therefore prefer to finance the new project with

debt, which is reinforced by the result that the value of debt is less sensitive

to the private information of the manager than the value of equity.6

A classic alternative to the asymmetric information approach is Kraus

and Litzenberger (1973), who state that optimal capital structure is deter-

mined by balancing the costs and benefits of debt. The benefit of debt is

that companies can deduct the interest expenses from their taxable income,

thus shielding their earnings, while the costs of debt financing are generally

recognised as being the expected direct and indirect bankruptcy costs. When

balancing the costs and benefits of moving towards a target capital structure,

another important factor to consider is adjustment costs, which represent a

potential reason for a wide variation in observed leverage ratios among simi-

lar firms, and might help explain why a firm would deviate from its optimal

capital structure (Myers, 1984).

A drawback of the theories focusing on asymmetric information is that

despite giving indications of which factors need to be considered, they do

not provide an explicit advice on what the leverage level should be. Models

based on the trade-off theory, on the other hand, have been more successful

in determining optimal values of debt and equity, and it has been confirmed

6 This result is derived from the option pricing theory, and rests on more strict as-sumptions than the model presented in Myers and Majluf (1984). See Galai and Masulis(1976).


that this theoretical foundation works fairly well at predicting typical lever-

age levels (e.g. Ju, Parrino, Poteshman, and Weisbach, 2005).7 Subsequent

continuous-time modelling approaches originating from this strand of liter-

ature can be generally categorised as either structural frameworks or more

recently adopted reduced-form frameworks, with Jarrow, Lando, and Turn-

bull (1997) and Duffie and Singleton (1999) as examples of papers studying

reduced-form models, and Merton (1974), Black and Cox (1976), Leland

(1994) and Goldstein et al. (2001) as those dealing with structural models.

Duffie and Lando (2001) present an example of a structural model that is

consistent with reduced-form representation, and thus provide a link between

the two frameworks.

One of the distinctions between structural and reduced-form models con-

cerns the information available to the modeller. Reduced-form models re-

quire a limited information set similar to what is observed by the market,

compared to structural models which assume availability of more compre-

hensive information, akin to what is observed by the manager. Further-

more, reduced-form frameworks are to a greater extent concerned with the

pricing of corporate debt, rather than with explaining optimal capital struc-

ture or debt structure (maturity, subordination, etc). Finally, but perhaps

most importantly, it should be mentioned that structural models link de-

fault explicitly to the first time that the state variable value falls below a

certain level, while reduced-form frameworks model bankruptcy through the

default probability distribution based on publicly available information, e.g.

accounting information, credit rating or business cycle data. Such feature

of the reduced-form approach makes it particularly appealing for modelling

bonds and credit derivatives. However, since our focus is on modelling the

capital structure choice rather than credit risk, we will restrict our attention

7 It is important to acknowledge that the trade-off theory still cannot explain all theobserved differences in capital structure across firms. For instance, in an internationalcomparison Wald (1999) found that the most profitable firms borrowed the least, theexact opposite of what is predicted by the trade-off theory. For a further comparison ofhow the pecking order and the trade-off theories perform empirically, see Frank and Goyal(2003) or Cotei and Farhat (2009).


to structural models.

Structural models commence with an assumption regarding the dynamics

of the state variable. Given this, it is possible to find the price of debt

and equity as the solution to a partial differential equation; to arrive at

this pricing framework, the pioneering results of Black and Scholes (1973)

and Merton (1974) are applied. From the price of debt- and equity-related

securities it is then possible to derive the coupon level which maximises

the firm value, ultimately resulting in an indication of the optimal capital

structure.

Structural models studied in academic literature could be further split

into two categories—static and dynamic models. Leland (1994) serves as a

classic example of the former one, while various authors have studied the cap-

ital structure in a dynamic setting, e.g. Fischer et al. (1989a) or Goldstein et

al. (2001). The main difference between the two groups is, as the name sug-

gests, that static models are derived assuming that the capital structure is a

one-off decision, and can thus only be applied to a narrow range of companies

with unreasonably stable fundamentals. Therefore, one of the key limitations

of this class of models is that they ignore the firm’s optimal restructuring

policy in response to the state variable fluctuations in time. Conversely,

dynamic models view the capital structure decision as an unceasing process.

Since dynamic capital structure modelling is at the core of this paper,

let us briefly outline the development of academic research studying this

class of models. Kane, Marcus, and McDonald (1984) build one of the first

frameworks which serves as a basis for further analysis. They develop a model

of optimal debt policy which incorporates personal taxes and bankruptcy

costs. The underlying asset in their model is allowed to follow a mixed jump-

diffusion process rather than an ordinary diffusion process, and this extension

precludes the use of Black and Scholes (1973) replication derivation, thus

leading them to employ an equilibrium approach. Besides, they extend the

notion of the rate of return shortfall introduced by McDonald and Siegel

(1984) to model the net tax advantage of debt in terms of rate of return (i.e.


measured as a flow variable rather than the standard stock variable). The

key insight from their model is that when tax advantage of debt is strictly

positive, it is never optimal for a firm to be all-equity financed, however large

its bankruptcy costs.

Fischer et al. (1989a) extend the setting of Kane et al. (1984) to allow

a firm to recapitalise at any point in time, but incorporate refinancing costs

to make continuous adjustment costly. They obtain closed-form solutions for

debt and equity as a function of the firm’s dynamic recapitalisation decisions;

further, the optimal recapitalisation policy is derived, which depends on firm-

specific characteristics. They argue that while the leverage declines, the firm

foregoes an increasing amount of debt-related tax shields, and at some point

it is optimal to recapitalise; in the opposite case—while the leverage ratio

increases—it might be optimal to recapitalise to avoid bankruptcy costs.

An important insight from their work is thus that it is useful to study the

leverage ratio range, which helps to explain why similar firms could have

different capital structures—due to the fact that any ratio within the specified

recapitalisation boundaries is in fact optimal.

The next landmark in development of dynamic capital structure mod-

elling is the paper by Goldstein et al. (2001), which introduces a novel model

accommodating the option to increase leverage. As the authors point out,

the fact that Kane et al. (1984) and Fischer et al. (1989a) obtain relatively

simple solutions for capital structure is largely due to the fact that their as-

sumptions effectively reduce the analysis to the one-period framework, and

lead them to argue that the tax advantage to debt is marginal. Goldstein

et al., on the contrary, find that tax benefit of debt increases significantly

as the option to lever up in the future is taken into account. The crucial

novelty of this paper is also in a new interpretation of the state variable as

earnings before interest and taxes, in contrast to the previous studies which

considered the value of unlevered assets. We revisit this assumption in more

detail in Chapter 4.

Since Goldstein et al. (2001) numerous authors have been trying to ex-


tend and refine earlier dynamic capital structure models, and we will only

briefly touch upon some of the issues brought up by them later in our paper.

Among others, Flor and Lester (2002) examine optimal debt maturity and

call premium; Christensen et al. (2002) focus on different aspects of debt

renegotiation; Dangl and Zechner (2004) study the effect of capital structure

dynamics on credit risk; Ammann and Genser (2004) propose a model which

includes finite-maturity debt and multiple bond issues; Francois and Morellec

(2004) investigate how different assumptions regarding bankruptcy affect the

valuation of debt and equity; Titman and Tsyplakov (2007) provide a model

where the firm value is determined by endogenous investment and financing

choices.

Chapter 2

Contingent Claims Analysis

Before proceeding to the discussion of how the optimal capital structure

of the firm could be determined, we will touch upon some fundamentals of

stochastic calculus and introduce the techniques of contingent claims analysis

which represent an important stepping stone to further analysis.

First, we will describe the Wiener process, state the basic properties of

the two stochastic processes—a geometric Brownian motion and the mean-

reverting process with proportional volatility—which will be utilised in the

models we consider, and sketch the basic result of Ito’s lemma as it is critical

for most of the derivations. After that we will elaborate on the question

of attainability of risk neutrality in continuous-time models. Finally, the

method to determine the value of a claim written on a firm fundamental which

evolves according to a diffusion process is presented, and it is also shown

how the obtained valuation framework could be applied to price corporate

securities.

2.1 Stochastic Processes

Let us first show how the Wiener process can be defined as the limit of a

discrete-time process. Consider the value of a stochastic process observed at

time t, z(t). If ∆t is a discrete time change, then the change in z(t) over that

15


time interval is given by

z(t+ ∆t)− z(t) , ∆z =√

∆tε, (2.1)

where ε is a random variable with zero mean and unit variance, and it holds

that Cov[z(t+ ∆t)− z(t), z(s+ ∆t)− z(s)] = 0 for non-overlapping intervals

(t, t+∆t) and (s, s+∆t). It follows that E[∆z] = 0 and Var[∆z] = ∆t, and

that z(t) has serially uncorrelated increments. z(t) represents an example of

a random walk process.

If a change in z(t) over a fixed time interval from 0 to T , consisting of n

intervals of length ∆t, is considered, then

z(T )− z(0) =n∑i=1

∆zi. (2.2)

Applying (2.1), discretised over n time intervals, to (2.2) yields

z(T )− z(0) =n∑i=1

√∆tεi =

√∆t

n∑i=1

εi. (2.3)

Consequently, the first two moments of z(T )− z(0) are

E[z(T )− z(0)] =√

∆tn∑i=1

E[εi] = 0, (2.4)

Var[z(T )− z(0)] = (√

∆t)2

n∑i=1

Var[εi] = ∆t · n · 1 = T, (2.5)

where E[·] and Var[·] denote the mean and variance operators, respectively,

conditional on the information at date t. It should be noticed that, holding

the length of the time interval fixed, the moments of z(T )−z(0) are indepen-

dent of the number of intervals n. This means that when the continuous-time

limit is obtained by letting n go to infinity—or equally, by letting ∆t go to

zero—the central limit theorem, under the assumption that the εi are inde-

Chapter 2. Contingent Claims Analysis 17

pendent and identically distributed, can be applied to state that

plimn→∞

[z(T )− z(0)] ∼ N(0, T ), (2.6)

i.e. z(T ) − z(0) is normally distributed with mean zero and variance T . It

follows that the distribution of a stochastic process z(t) over any finite time

interval [0, T ] could be interpreted as the distribution of a sum of infinitely

many independent increments ∆zi =√

∆tεi, which are drawn from an arbi-

trary distribution. However, since the CLT stipulates that the sum of these

increments converges in distribution to the Gaussian one, we can, without

loss of generality, assume that ∀i εi ∼ N(0, 1).

The limit of an infinitesimal increment could be seen as

dz(t) = limt→0

∆z = limt→0

√∆tε. (2.7)

Due to the previously stated characteristics of ε it holds that E[dz(t)] = 0

and Var[dz(t)] = dt. Given the above, dz is termed a Wiener process and is

often referred to as a (pure) Brownian motion.

The change in z(t) over [0, T ] can now be expressed as

z(T )− z(0) =

∫ T

0

dz(t) ∼ N(0, T ), (2.8)

with the right-hand side being represented by the Ito integral.8 It should be

emphasised that z(t) has an unbounded variation over any finite time interval

and is thus nowhere differentiable.

8 The Ito (stochastic) integral has a more general form I(t) =∫ t

0ϕ(s)dW (s), where

ϕ(s) is an adapted process and W (s) is the Wiener process. This integral is in fact simplya generalisation of a well-known Riemann–Stieltjes integral in the sense that it considersstochastic processes instead of real-valued functions and the integration is carried out withrespect to a (non-differentiable) stochastic process.


0.2 0.4 0.6 0.8 1.0t

-2

-1

1

2

zHtL

Figure 2.1. Sample paths for z(t) as a discrete-time random walk process andas a Wiener process.

From the evolution of the discrete-time process in Figure 2.1 it is evident

that as n → ∞, so that ∆t → 0, the random walk process converges to

the continuous-time Wiener process. The latter, as we will show now, serves

as a fundamental building block for more general continuous-time stochastic

processes.

Consider a new process z′ defined by dz′(t) = σ(t)dz(t), where σ(t) is

a constant. This process has the following distribution over a discrete time

interval [0, T ]:∫ T

0

dz′(t) = σ

∫ T

0

dz(t) ∼ N(0, σ2T ).

If we now additionally introduce a deterministic constant change of µ per unit

of time to the z′(t) process, so that dz′(t) = µdt + σdz(t), the distribution

would change to∫ T

0

dz′(t) = µT + σ

∫ T

0

dz(t) ∼ N(µT, σ2T ). (2.9)

Further, let us consider the time-inhomogeneous analogue of this process, i.e.


with µ and σ both being time-varying, which yields

z′(T )− z′(0) =

∫ T

0

dz′(t)

=

∫ T

0

µ(t)dt+

∫ T

0

σ(t)dz(t). (2.10)

Thus, we have generalised the standard Wiener process z(t) to a new process

z′(t) whose evolution is given by (2.10). This process is often referred to as

the Ito process.

It is important to emphasise here that the described Ito process is a repre-

sentation of a very broad class of stochastic processes with drift and volatility

being dependent not only on time and the current value of the process, but

also possibly on the past process realisations. Therefore, Ito processes gener-

ally do not possess the Markov property which, loosely speaking, states that

given the present, the future is independent of the past. More specifically,

the process described by the stochastic differential equation

dξ(t) = µ(ξ(t), t)dt+ σ(ξ(t), t)dW (2.11)

is a continuous-time Markov process9 if the instantaneous change in the pro-

cess at date t has a distribution that depends only on t and the contem-

poraneous value ξ(t), and not prior values ξ(s), s < t. Moreover, if path

continuity is further assumed, ξ = ξtt≥0 is called a diffusion process.

Geometric Brownian Motion

Perhaps the most widely applied diffusion process is a geometric Brownian

motion, which could be obtain by letting µ(ξ(t), t) = µξ and σ(ξ(t), t) = σξ

in (2.11), so that

dξ = µξdt+ σξdW, (2.12)

9 Markov chain, used primarily in discrete-time analysis, is also an oft-encounteredterm.


where µ ∈ R and σ ∈ R+ are constants. Both drift rate and volatility rate

are proportional to the current value of ξ, which means that (2.12) can be

easily adjusted to express the dynamics in relative terms. The process is

suitable for state variables which grow exponentially at the average rate of

µ, and it exhibits the following properties:

If ξ hits zero (a zero-probability event), it remains at zero

Given a positive starting value, ξ remains positive

At any time s > t, the value ξ(s) is lognormally distributed with

Et[ξ(s)] = ξ(t)eµ(s−t),

Vart[ξ(s)] = ξ2(t)e2µ(s−t)(

eσ2(s−t) − 1

) Given positive drift, both conditional mean and variance are increasing

in time

Due to the above properties, a geometric Brownian motion is commonly

utilised to model stock prices, which can never become negative because of

the limited liability of investors; other examples include asset prices, wage

rates, etc. In Section 2.4 a geometric Brownian motion is used to describe

the dynamics of the state variable, and the importance of this process as a

fundamental building block for finding the optimal capital structure will be

further highlighted in Chapters 3 and 4.

Mean-reverting Process

Another common diffusion process is the mean-reverting process. It should

be noted that—just as with the Brownian motion—there exist different vari-

ations of this process, depending on whether drift and/or volatility are pro-

portional to the current value of the process, the most popular example being

the Ornstein–Uhlenbeck process. However, we will focus on another modifi-

cation which will be underlying our analysis:

dξ = κ(θ − ξ)dt+ σξdW, (2.13)


where θ ∈ R+ is the long-term mean value, σ ∈ R+ is the constant volatility

rate, and κ ∈ R+ is the speed at which the process converges towards the

long-term equilibrium. It can be seen that if the current level of ξ is below

θ, then the drift term is positive, and the process value tends towards the

steady-state value. The opposite relationship takes place when ξt > θ. The

size of increment changes depends on κ; even though the speed of mean

reversion is positive, the process, due to its stochastic nature, can still move

further away from its long-run value, and for higher volatility the potential

deviation is larger. The process has the following characteristics:

If the initial value of the process is positive, then the future value

remains positive

Conditional mean and variance are as follows:

Et[ξ(s)] = θ + (ξ(t)− θ)e−κ(s−t),

Vart[ξ(s)] =θ2σ2

2κ− σ2− e−2κ(s−t)(ξ(t)− θ)2 +

2θσ2(ξ(t)− θ)κ− σ2

e−κ(s−t)

+2κ2(ξ(t)− θ)2 − κσ2ξ(t)(3ξ(t)− 2θ) + σ4ξ2(t)

(κ− σ2)(2κ− σ2)

· e−(2κ−σ2)(s−t)

In the limit the expected value of the process is equal to the long-term

mean θ, and thus the conditional volatility becomes bounded

Given the above, mean-reverting processes are often used to model com-

modity prices, interest rates, currency exchange rates, etc. Later we will

underline the economic intuition behind the utilisation of the described mean-

reverting process with proportional volatility, as opposed to a geometric

Brownian motion. In Chapter 5 the process is applied to describe the evolu-

tion of the state variable which stipulates the optimal capital structure.


2.2 Ito’s Lemma

Since continuous-time Ito processes are not differentiable, we need to apply

Ito’s lemma—known as the fundamental theorem of stochastic calculus—to

differentiate or integrate functions of such processes. Here we will sketch

the intuition behind this result using a Taylor series expansion, omitting the

formal proof.

Let ξ(t) follow the process given by (2.11) and consider a function F (ξ, t)

which is at least twice differentiable in ξ and once in t. We are interested

in finding the total differential of this function, dF . Usual rules of calculus

define this differential in terms of changes in t and ξ:

dF =∂F

∂tdt+

∂F

∂ξdξ +

1

2

∂2F

∂ξ2(dξ)2 +

1

6

∂3F

∂ξ3(dξ)3 + . . . (2.14)

Using (2.11) and the basic rules of Ito calculus (dt)2 = 0, (dW )2 = dt, and

(dtdW ) = 0, (dξ)2 is determined as

(dξ)2 = µ2(ξ, t)(dt)2 + σ2(ξ, t)(dW )2 + 2µ(ξ, t)dtσ(ξ, t)dW

= σ2(ξ, t)dt. (2.15)

Expanding (dξ)3 in the same fashion will result in the expression every term

of which contains dt raised to a power greater than one, so it will go to zero

faster than dt in the limit does. Since the same is valid for higher-order

terms, the differential dF is given by

dF =∂F

∂tdt+

∂F

∂ξdξ +

1

2

∂2F

∂ξ2(dξ)2,

which could finally be expressed as10

dF =

[∂F

∂t+ µ(ξ, t)

∂F

∂ξ+

1

2σ2(ξ, t)

∂2F

∂ξ2

]dt+ σ(ξ, t)

∂F

∂ξdW. (2.16)

10 Equation (2.16) corresponds to the case when F is a function of one Ito process. Multi-

dimensional Ito’s lemma states that dF = ∂F∂t dt+

∑mi=1

∂F∂ξi

dξi+12

∑mi=1

∑mj=1

∂2F∂ξi∂ξj

dξiξj .


The expression in (2.16) is sometimes also referred to as the law of motion

since it exhibits how F (ξ, t) evolves over time, given that ξ(t) follows an Ito

process. It should further be noted that while dF and dξ have different mean

and variance, they both depend on the same source of uncertainty.

2.3 Note on Risk Neutrality in Continuous

Time

Interestingly, despite the remarkable advancement of the theory of stochastic

calculus, the fact remains that the question of risk neutrality still causes

confusion and some of its nuances are frequently overlooked in the existing

literature. In this subsection we will elaborate on how the transition to

the risk-neutral measure in continuous-time modelling could be made, and

pinpoint the specific shortcoming inherent in many pieces of analysis focusing

on optimal capital structure. We will restrict our attention to the question of

the existence of the risk-neutral probability measure, thus leaving aside the

aspect of its uniqueness. Note that here we will only sketch the necessary

key results and their implications, omitting formal proofs and derivations.11

Let X = Xtt∈[0,∞) be a stochastic discounted price process defined on a

filtered probability space (Ω,F ,P). For now assume that filtration F satisfies

what is commonly referred to as the usual conditions in the literature, later

we will elaborate on the specifics of those. The first fundamental theorem of

asset pricing dictates that the absence of arbitrage opportunities is equivalent

to the existence of an alternative probability measure Q such that any X is

a Q-martingale, i.e. X is Ft-adapted and it holds that

∀s Xs = EQ[Xt | Fs], 0 ≤ s ≤ t <∞. (2.17)

Such Q is termed an equivalent martingale measure. Note that the critical

11 See, e.g. Karatzas and Shreve (1991, Chapter 3) or Revuz and Yor (1999,Chapter VIII) for a rigorous analysis.


condition here is the equivalence of the two probability measures, which is

defined as follows. The two measures P and Q are equivalent if and only if

∀A ∈ F P(A) = 0⇔ Q(A) = 0, (2.18)

which is also frequently interpreted as the two measures being mutually abso-

lutely continuous on F . The intuition behind the definition of equivalence is

straightforward: equivalent probability measures always agree on all certain

(i.e. probability-one) and all impossible events.

Now let us turn to the Girsanov theorem which is the most important

result related to the transformation of the probability measure. It should be

mentioned that there are numerous interpretations of the original theorem,

and here we will only outline the general idea behind the basic result of

Girsanov (1960).

Let Y = Ytt∈[0,∞) be an adapted process and

Zt , exp

(∫ t

0

YsdWs −1

2

∫ t

0

Y 2s ds

)under the P-measure. Suppose Z is a martingale.12 Then there exists an

equivalent probability measure Q13 such that the process W = Wtt∈[0,∞)

defined as

WQt , WP

t −∫ t

0

Ysds (2.19)

is a Wiener process under Q. Thus, loosely speaking, a shifted Wiener process

is again a Wiener process provided that the original probability P is replaced

with an equivalent probability Q.

Note that the shift of the Wiener process could also be expressed as

12 It could be shown by applying Ito’s lemma that Z is a local martingale with Z0 = 1.The oft-employed result to verify that Z is a martingale, is the Novikov condition which

states that if E[exp

(12

∫ T0Y 2s ds

)]<∞, then Z(Y ) is a martingale.

13 After Z is shown to be a martingale, the new probability measure could be strictlydefined in terms of the Radon–Nikodym derivative ZT = dQ

dP .


dWt = Ytdt + dWt, which enables the following simple transformation. If

under the probability measure P the process X has the dynamics dXt =

µtdt+σtdWt, then its Q-dynamics would look like dXt = (µt+σtYt)dt+σtdWt,

which implies that in going from the original process to the new one the

diffusion is unchanged.

It should be mentioned that the usual conditions often imposed on filtra-

tion F in the literature could void the Girsanov theorem and consequently

the transition to the risk-neutral probability measure.14 The discussion of

such conditions and the proof of inapplicability thereof are out of the scope of

this thesis.15 However, there is one much less subtle criterion which must be

fulfilled for the Girsanov transformation to hold—the finiteness of the time

horizon. For any finite time T ∈ [0,∞) there exists a unique probability

measure Q on F∞ that is equivalent to P on FT , and the key nuance related

to the Girsanov theorem is that without very limiting restrictions on the

transform process Y the theorem cannot ensure that there exists an appro-

priate probability measure equivalent to P on F∞. Thus, the infinite-time

setup frequently employed in the literature (e.g. Leland, 1994; Goldstein et

al., 2001) may in fact not contain an equivalent martingale measure.16 To

our knowledge, the only paper that acknowledges the problem is Christensen

et al. (2002).

2.4 The General Partial Differential Equation

The optimality of capital structure entails the maximisation of the total

firm value, and is thus dependent on the market values of debt and equity,

which both could be viewed as claims on the underlying state variable. The

14 For example, it could be shown that under the requirement that F0 contains allnegligible sets of F∞, the Girsanov theorem fails. Bichteler (2002, Section 1.3) introducesthe natural conditions which rectify the shortcomings of the conventional usual conditions.

15 Firoozi (2006) provides a note on the martingale property of processes. For a com-prehensive analysis, see Bichteler (2002, Section 3.9).

16 For the formal proof, see Bichteler (2002, Section 3.9). The discrete-time version ofthis result was demonstrated in Back and Pliska (1991).


specific values of these claims can be obtained as the solution to a certain

partial differential equation, which will be derived in this section, with the

applied methods being based on Merton (1974).

The set of assumptions required for the derivation of the general partial

differential equation is as follows.

Financial markets are frictionless (absent transaction costs, taxes or

asset indivisibility issues), arbitrage-free, and complete17

There exists a money market account with a constant risk-free rate r

at which investors can lend and borrow

Assets can be traded continuously in time

The dynamics of the firm value V is described by a diffusion process—a

geometric Brownian motion, i.e.

dV = (µV − ν)dt+ σV dW, (2.20)

where µ is the (instantaneous) expected rate of return on the firm, ν

is the total payout to claim holders if positive, and the net amount re-

ceived by the firm from new financing if negative. The (instantaneous)

variance of the return on the firm is represented by σ2, and W is the

Wiener process.

Suppose there exists a traded security whose market value Y is a function

of V , Y = F (V, t), and whose instantaneous changes in value are described

by

dY = (µY Y − νY )dt+ σY Y dWY , (2.21)

17 Here we imply the conventional definition of absence of arbitrage and completenessthrough existence and uniqueness of an equivalent martingale measure Q, commonly ac-cepted in the asset-pricing field (see, e.g. the seminal work of Harrison and Pliska, 1981),although we do recognise that alternative views exist. For instance, Battig and Jarrow(1999) introduce a definition of market completeness which is independent of any particu-lar probability measure and demonstrate that under this definition arbitrage opportunitiesmay in fact exist in a complete market.


where µY is the expected rate of return, νY is the monetary payout to the

holder of the security, σY is the return volatility, and WY is the Wiener

process. Although in practice companies are often not allowed to liquidate

assets to meet interest payments due to a corresponding negative covenant

frequently observed in debt contracts, we will assume that assets are sold to

finance payouts, following Merton (1974).

Ito’s lemma can now be applied to obtain the differential of the security

value as

dY =

(∂F

∂t+ (µV − ν)

∂F

∂V+

1

2σ2V 2∂

2F

∂V 2

)dt+ σV

∂F

∂VdW. (2.22)

Since both (2.21) and (2.22) describe the evolution in value of the same

security, drift, volatility, and source of uncertainty should be identical, i.e.18

µY Y = Ft + (µV − ν)FV +1

2σ2V 2FV V + νY , µY F (2.23)

σY Y = σV FV , σY F (2.24)

dWY = dW (2.25)

Continuous trading, existence of a risk-free asset, and market complete-

ness stipulate that a replicating portfolio containing the firm and the security

can be formed. The portfolio is financed with a short position in the riskless

asset, implying zero net investment. After rearranging (2.20) to get

dV + νdt

V= µdt+ σdW,

we could express the total return on the portfolio as

dK = M1

(dV + νdt

V

)+M2

(dY + νY dt

Y

)+M3rdt, (2.26)

where M1 is the amount invested in the firm, M2 is the amount invested in

18 For readability purposes we will from now on use the shorthand notation Ft, FV , and

FV V , denoting derivatives ∂F∂t , ∂F

∂V and ∂2F∂V 2 , respectively.


the security and M3 , −(M1 +M2) is the amount borrowed at the risk-free

rate. Substituting from above, we get

dK = M1(µdt+ σdW ) +M2(µY dt+ σY dWY )−M1rdt−M2rdt

= M1(µ− r)dt+M2(µY − r)dt+M1σdW +M2σY dWY ,

which by using (2.25) could be written as

dK = [M1(µ− r) +M2(µY − r)]dt+ [M1σ +M2σY ]dW. (2.27)

The no-arbitrage condition dictates that when there is no uncertainty related

to the portfolio’s expected return, the risk premium must be equal to zero,

and therefore the following system of conditions takes place:

M1σ +M2σY = 0 (2.28)

M1(µ− r) +M2(µY − r) = 0 (2.29)

where (2.28) implies elimination of the source of uncertainty, and (2.29) stip-

ulates zero expected return. Of course, M∗1,2 = 0 is an obvious solution, but

since such a portfolio would be empty, the only non-trivial solution could be

found from

M1σ +M2σY = 0⇔M1 =−M2σY

σ,

−M2σY (µ− r)σ

+M2(µY − r) = 0⇔ µY − rσY

=µ− rσ

. (2.30)

which means that the excess return per unit of risk must be the same for

both assets in the portfolio.


The no-arbitrage condition in (2.30) could be restated using expressions

for µY and σY from (2.23) and (2.24):

µ− rσ

=

Ft+(µV−ν)FV + 12σ2V 2FV V +νY

F− r

σV FVF

⇔

1

2σ2V 2FV V + (rV − ν)FV + Ft − rF + νY = 0. (2.31)

which is a partial differential equation (PDE) validating the absence of arbi-

trage with respect to the replicating portfolio. The sheer beauty of this result

is in the fact that any security whose value is dependent on the firm value

must satisfy this PDE, and we will heavily rely on this condition in further

analysis. Of course, to differentiate between the securities, one should addi-

tionally specify the payout along with the boundary conditions. This will be

demonstrated in the next section when prices of securities are derived.19

From (2.31) it is easy to see that the pricing function F which solves

the general PDE is dependent not only on the current value of the firm and

time, but also on the risk-free rate, the firm’s payout policy, the payout to

holders of the security, and the volatility of the firm value. Interestingly,

F does not depend on the expected return on the firm given by µ, or risk

preferences of investors. This means that two investors who agree on the firm

valuation and its volatility, irrespective of differences in their utility functions

or expectations regarding the firm’s future performance, will accept the same

pricing function to value contingent claims.

2.5 Isomorphism to Option Pricing

Let us now briefly touch upon how corporate securities can be interpreted as

options and therefore valued using the option pricing techniques pioneered

19 An alternative method to obtain the value of a contingent claim is to discount thesecurity’s cash flows at the risk-free rate, and take expectations. This approach assumesthat a risk-neutral measure Q exists. Despite the difference in methodology to the no-arbitrage approach, the Feynman–Kac theorem states that the two methods yield the sameresult.


by Black and Scholes (1973).

In addition to what is stated in the previous section the following as-

sumptions are applied:

There exist two types of claims written on the firm, viz. interest-free

debt and equity, and no other claims can be issued

The bond issue has principal P and must be paid back at time T

The firm cannot pay cash dividends or do share repurchases prior to

debt maturity

If the debt value is denoted as D, then the general PDE from (2.31) can

be written as

1

2σ2V 2DV V + rV DV −Dτ − rD = 0, (2.32)

where νY = 0 as there are no coupon payments, ν = 0 because the firm is

not allowed to attract additional financing or make payouts, τ , T − t is the

time to maturity and thus Ft = −Fτ .To find the function D which solves the PDE, two boundary conditions

and an initial condition need to be laid out. Since the unlevered firm value is

defined as V , D(V, τ) +E(V, τ), where E is the value of equity, then these

conditions are given by

D(0, τ) = E(0, τ) = 0, (2.33)

D(V, τ) ≤ V, (2.34)

D(V, 0) = minV, P, (2.35)

where (2.33) is explained by the fact that debt and equity values always re-

main non-negative, (2.34) is dictated by the limited liability of equity holders,

(2.35) is the initial condition for debt at τ = 0, i.e. at maturity, and is stip-

ulated by the absolute priority rule.


Even though (2.32) could now be solved directly along with (2.33)–(2.35)

by using e.g. standard Fourier transforms, it is possible to ease the procedure

by reducing the problem to the one already solved in literature. Note that

the PDE for equity value is given by

1

2σ2V 2EV V + rV EV − Eτ − rE = 0 (2.36)

s.t.

E(V, 0) = max0, V − P, (2.37)

in addition to conditions corresponding to (2.33) and (2.34).

It is critical to realise that equations (2.36) and (2.37) are identical to

those for a European call option written on a zero-dividend stock, where

V corresponds to the stock price and P—to the strike price. This insight

allows us to write out the solution to (2.36)–(2.37) immediately, using the

Black–Scholes framework:

E(V, τ) = Call(V, P, σ, r, τ) = V Φ(d1)− P e−rτΦ(d2), (2.38)

where Φ(·) is the cumulative distribution function for the Gaussian distribu-

tion, and

d1 =ln(VP

)+(r + 1

2σ2)τ

σ√τ

,

d2 = d1 − σ√τ .

Therefore, from (2.38) and D = V − E, we get

D(V, τ) = P e−rτ −[P e−rτΦ(−d2)− V Φ(−d1)

], (2.39)

and one could notice that in fact the debt value could be represented as the

value of the portfolio comprising the money market account and a short put


position since Put(V, P, σ, r, τ) = P e−rτΦ(−d2)− V Φ(−d1).

The intuition behind the similarities in payoffs to debt and equity owners

and holders of the corresponding contingent claims is as follows. At maturity

equity holders have the option to either to pay the debt principal back or

default. If the value of the firm is sufficiently higher than the principal, then

shareholders will prefer to retire the debt and become the sole claimants of the

then debtless company. This makes the value of their claim at maturity equal

to the difference between the firm value and the principal. If the company

is worth less than the principal, owners of equity will declare bankruptcy

and let debt holders take over the company. The payoff to equity holders is

therefore equivalent to that of an owner of a European call option written

on the firm, with the strike price equal to the principal. Thus, the value of

equity can be obtained by pricing the call option, utilising the Black–Scholes

framework. The value of debt can be derived in a similar fashion, using that

the payoff to the bondholders is equivalent to that of a short position in a

put option written on the firm, where equity holders own the put and can

exercise it at the strike equal to the principal. Therefore, the total value of

debt is equal to the discounted value of the principal less the value of the

put option.

An important implication of the Merton model is that relationships in the

option pricing framework can be directly applied. A typical example is the

impact of an increase in the business risk—measured by the variance of firm

value—which benefits equity holders at the expense of bond holders because

the former have no downside risk. Equity owners will therefore have an in-

centive to transfer value to themselves by taking on more risky projects after

the debt contract has been settled. As a result, if the financing decision is

not separated from the investment decision, the problem of asset substitution

will occur, as extensively discussed in Jensen and Meckling (1976). However,

using numerical analysis, Leland (1998) finds that agency costs related to

the asset substitution problem are small, which is also confirmed by Graham

and Harvey (2001) in an empirical study.


Despite its fundamental significance, Merton’s valuation framework can

only be seen as a basic structural model, and that is amplified by the fact

that several assumptions the model rests upon are rarely observed in practice.

Firstly, the model ignores taxes and thus cannot be used as an appropriate

indication of the optimal capital structure. The lack of applicability is further

emphasised by the possibility of the firm value to either rise to an arbitrar-

ily high level or drop to almost zero without any sort of reorganisation of

the corporate financing. Dynamic models allowing the firm to increase its

leverage depending on the value of the state variable will be presented in

Chapters 4 and 5. Secondly, the model only considers zero-coupon bonds. In

addition to the fact that interest charges on debt are a commonly observed

phenomenon, this assumption implies that the firm cannot default prior to

maturity. The models presented in the chapters that follow, however, do

incorporate a continuous possibility of bankruptcy as well as explicit coupon

payments. Naturally, we will also have to adhere to certain assumptions that

are difficult to justify empirically, e.g. non-stochastic interest rates and con-

stant volatility, but since the primary object of our analysis is the capital

structure, we do not expect the results to change qualitatively because of

that.

Chapter 3

Static Model of Capital

Structure

One of the drawbacks of the Merton model is that it does not consider taxes.

In reality bond investors pay personal taxes on the received interest income

at a certain tax rate τi. Similarly, equity owners incur tax on dividends at

a rate τd. Besides, as owners of equity are subject to double taxation in

the form of both corporate and personal taxes, they face the effective tax

rate τe, which is calculated using (1 − τe) = (1 − τc)(1 − τd). Furthermore,

as companies are generally allowed to deduct interest expenses from the tax

base, they have an incentive to utilise this option by taking on more debt to

finance operations and investments. If the interest expenses are assumed to

be always tax-deductible and it holds that τe > τi, then there is always a tax

advantage to debt. Note, however, that personal taxes will not be explicitly

included in the model described below, following Leland (1994).

In this chapter a model balancing the tax advantage to debt with the

costs of financial distress is presented. The former is seen as a stream of

uncertain tax savings, thereby considering the possibility that the company

might default and thus cannot perpetually utilise the tax shield. The costs

of financial distress are represented by the fraction of the asset base that

is lost due to bankruptcy costs in case the firm defaults. The model yields

34

Chapter 3. Static Model of Capital Structure 35

closed-form solutions, and thus presents explicit guidance on the choice of

optimal capital structure. The underlying assumptions and derivations are

based on the model of Leland (1994).

The chapter is structured as follows. Firstly, we dwell on the assumption

of time independence which is critical for the model derivations. Secondly,

valuation of different contingent claims is presented, following from the spec-

ification of corresponding boundary conditions. Thirdly, the framework is

extended to consider bankruptcy as an endogenous event, as opposed to

modelling default as triggered by the asset value reaching an exogenously

given level. Besides, it is shown how the optimal leverage can be derived

by letting the coupon level be determined from the firm value maximisation

problem. The chapter concludes by presenting the model performance—using

comparative statics analysis—and outlining important drawbacks.

3.1 Time Independence of Cash Flows

Brennan and Schwartz (1978) were the first to take into account the tax

advantage to debt when explicitly modelling optimal capital structure, but

conducted their analysis only numerically. The model developed in Leland

(1994) serves as a general version of that in Brennan and Schwartz as it

provides closed-form solutions for optimal leverage. The former differs by

examining permanent capital structure changes under the infinite maturity

setup, however, as opposed to Brennan and Schwartz who consider finiteness

of debt maturity.

The infinite maturity is in fact one of the key assumptions in Leland

(1994), implying that securities have time-independent cash flows and thus

their valuation is independent of time as well. Generally, however, the value

of a debt security is time-dependent. Corporate bonds, for instance, have fi-

nite maturities, which means that the final cash flow is not the same as those

on interest payment dates and that the impact of the principal repayment

on the bond valuation varies with maturity. The assumption of time inde-


pendence can nevertheless still be justified. Firstly, for sufficiently long bond

maturities the principal redemption is negligible compared to the total payout

value, and can thus be ignored. Therefore, cash flows can be approximated

by interest payments only, essentially making such bond a time-independent

security. Very long bond maturities are not a purely theoretical concept

though: for example, in 2010 DONG Energy, the leading energy company

of Denmark, issued 30-year bonds, in 2008 an American conglomerate Gen-

eral Electric Company placed a 40-year bond issue, and in 2007 a British

retailer Tesco announced an issue of 50-year bonds, among others. Examples

of corporate bond issues with even more extreme maturities do exist as well,

represented by Ford Motor Company’s issue of a 100-year bond in 1997 and

Coca-Cola’s 100-year bond issue in 1993.

Another argument justifying the assumption in question is that revolv-

ing credit agreements—when debt is allowed to be rolled over at maturity

maintaining the interest rate—are often similar to a time-independent setup.

The assumption of time independence is exactly what allows Leland to derive

closed-form solutions, as opposed to preceding authors.20

3.2 The Model

As in Merton (1974), the model considered in Leland (1994) is based on

the dynamics of a firm fundamental. In both models the fundamental is

represented by the unlevered firm value21, and Leland considers the following

20 Note that debt with explicitly finite maturity could also be incorporated in the staticframework. Leland and Toft (1996) develop a model of optimal leverage for a firm thatcontinuously issues finite-maturity bonds, generalising the findings of Leland (1994). Theyshow that not only debt value depends on maturity, but also does bankruptcy whendetermined endogenously. Further, Flor and Lester (2002) extend the results of Lelandand Toft to a dynamic setting. In our analysis we will, however, following the discussion inSection 2.3, assume that all the issued debt has the maturity which is finite, but sufficientlylong to obtain time-independent cash flows.

21 Naturally, the value of the unlevered firm corresponds to the value of assets. Therewill be no distinction between the two terms in the remainder of the thesis.


process to describe its development:22

dV

V= µ(V, t)dt+ σdW, (3.1)

where V is asset value, µ < r is the drift23, and σ is the constant volatility.

V is naturally independent of the financial structure. As the model does

not include any explicit earnings-generating process, and no payout to claim

holders is financed through the sale of assets—as opposed to Merton—Leland

assumes that all claims are serviced by an additional costless issue of equity.

Consider now a claim valued as Y = F (V, t) written on the unlevered

firm, which continuously pays out νY conditioned on the firm being solvent.

Derivations in Chapter 2 suggest that the value F (V, t) of such a security

must satisfy the partial differential equation

1

2σ2V 2FV V + rV FV + Ft − rF + νY = 0. (3.2)

However, due to the assumed time independence and the payout to holders of

the claim, this equation can be reduced to the following ordinary differential

equation (ODE):

1

2σ2V 2FV V + rV FV − rF + νY = 0. (3.3)

3.2.1 Solution to the ODE

We will first solve the homogeneous differential equation

1

2σ2V 2FV V + rV FV − rF = 0. (3.4)

The procedure becomes apparent immediately after one recognises that (3.4)

is a standard Cauchy–Euler equation, i.e. a linear homogeneous ODE with

22 Henceforth, we will be working directly under the risk-neutral probability measureQ, unless otherwise stated.

23 This condition is dictated by the requirement that the state variable has a finite value,and follows from the derivation of the Gordon growth model (Gordon, 1959).


variable coefficients. We will now transform it to the analogous ODE with

constant coefficients to apply the conventional technique of solving a differ-

ential equation using the corresponding characteristic equation.

To deal with the problem of variable coefficients it is common to apply

the following substitution: V = ex. To obtain the first-order derivative in

(3.4), FV , consider

∂F

∂x=∂F

∂V

∂V

∂x=∂F

∂VV, (3.5)

using ∂V∂x

= V . This means that

V FV =∂F

∂x.

The second-order derivative is found in a similar fashion:

∂2F

∂x2=

∂

∂x

(∂F

∂x

)=

∂

∂x

(∂F

∂VV

)=∂F

∂V

∂V

∂x+ V

∂

∂x

(∂F

∂V

)= V

∂F

∂V+ V 2∂

2F

∂V 2, (3.6)

using the chain rule to get ∂∂x

(∂F∂V

)= ∂V

∂x∂2F∂V 2 . Therefore,

V 2FV V =∂2F

∂x2− ∂F

∂x.

Plugging both terms back into (3.4) yields

1

2σ2∂

2F

∂x2+

(r − 1

2σ2

)∂F

∂x− rF = 0, (3.7)

which is a linear homogeneous ODE with constant coefficients, and therefore,

can be solved explicitly using the corresponding characteristic equation

1

2σ2β2 +

(r − 1

2σ2

)β − r = 0. (3.8)


The roots to the characteristic equation are

β1,2 =

(12σ2 − r

)±√

(r + 12σ2)2

σ2,

with

β1 = 1,

β2 = −2r

σ2.

Since both roots are not complex, and due to the fact that r > 0 they

are distinct, the general (and in fact, complete) solution to (3.4) can be

formulated as

F (V ) = B1eβ1x +B2eβ2x,

or, more specifically:

F (V ) = B1V +B2V− 2rσ2 . (3.9)

Adding one particular solution results in the general solution to the inhomo-

geneous ODE in (3.3):

F (V ) = B0 +B1V +B2V− 2rσ2 . (3.10)

Note that the constants B0, B1, and B2 should be determined by the bound-

ary conditions for a specific claim, with B0 taking into account the case when

the security offers a payout νY .

3.2.2 Valuation of Claims

Now we turn to pricing the securities contingent on the unlevered firm value,

using the solution to the ODE derived above.


The two primary claims considered are debt and equity securities, whose

values are given by D(V ) and E(V ), respectively. Two additional claims

could be introduced: tax benefit of debt, whose value is denoted TB(V ),

and bankruptcy costs, whose value is represented by BC(V ). Therefore, the

total firm value can be calculated as A(V ) = V +TB(V )−BC(V ), and since

the equity claim is residual, it could valued as E(V ) = A(V ) − D(V ). As

will be shown below, by applying (3.10) along with the necessary boundary

conditions, closed-form solutions to the values of all these claims could be

derived.

In Leland (1994), owners of corporate bonds are recipients of a perpetual

coupon C, conditioned on the firm being solvent. If the value of assets hits an

exogenously specified threshold Vd, the company defaults and bondholders

pay a fraction 0 < γ < 1 of the unlevered firm value in bankruptcy costs

before taking over the firm, and become the sole owners of the company

valued at (1− γ)Vd.24 Therefore, the boundary conditions for the debt claim

are given by

At V = Vd, D(V ) = (1− γ)Vd, (3.11)

As V →∞, D(V )→ C

r, (3.12)

where the first one states the firm value at default, and the second one implies

that as the firm value increases exponentially, debt essentially becomes risk-

free.

Using condition (3.12) together with (3.10) means that B1 = 0 and

B0 = Cr

. Inserting these expressions in condition (3.11) applied to (3.10),

yields (1 − γ)Vd = Cr

+ B2V− 2rσ2

d , and therefore, B2 =[(1− γ)Vd − C

r

]V

2rσ2

d .

24 The firm value upon bankruptcy is calculated as if the assumption of absolute priorityapplies, which in this model leaves nothing for the equity holders after default. However,in a study of 30 US firms defaulting during the 1980s, Eberhart, Moore, and Roenfeldt(1990) document that equity owners on average received 7.6% of the total value distributedto all claimants. A possible explanation of the seeming contradiction between their findingand the models we present is that we model default differently, assuming the takeover ofthe company, as opposed to restricting the bondholders’ claim only to the principal.


All constants are now known, and the value of the debt security can be

obtained as

D(V ) =C

r+

[(1− γ)Vd −

C

r

](V

Vd

)− 2rσ2

. (3.13)

Consider a new security pd(V ) that has a payoff of 1 DKK if the firm

defaults, and zero in all other states of nature. This claim can thus be

interpreted as an Arrow–Debreu security. Its present value is of the form

pd(V ) = B1V +B2V− 2rσ2 , (3.14)

because B0 = 0 since the claim offers no intermediate cash flows. When

taking into account the boundary conditions

limV→Vd

pd(V ) = 1,

limV→∞

pd(V ) = 0,

it can be seen that

pd(V ) = V2rσ2

d V −2rσ2 =

(V

Vd

)− 2rσ2

. (3.15)

By using this result, debt value can alternatively be written as

D(V ) =C

r+ pd(V )

((1− γ)Vd −

C

r

). (3.16)

The value of debt can therefore be seen as a perpetual stream of coupon pay-

ments, given that the firm is solvent, plus the net payoff in case bankruptcy

takes place, which is equal to the firm value at default less the lost interest

income.

To obtain the levered firm value A(V ) we need to quantify the two key

effects of debt—through bankruptcy costs and tax deductibility of interest

expenses. Since bankruptcy costs are contingent on the unlevered firm value


and time independence of related cash flows also holds, we could apply (3.10)

in the same manner as above to price this claim. This security could be

viewed as the one offering a lump sum payment γVd upon default and zero

otherwise, thus implying the following boundary conditions:

At V = Vd, BC(V ) = γVd,

As V →∞, BC(V )→ 0,

and hence,

BC(V ) = γVdpd(V ). (3.17)

As before, by using (3.15), the value of this claim can be interpreted as the

present value of receiving γVd given that the state of nature when the com-

pany files for bankruptcy is realised. Naturally, the claim value is increasing

in Vd as both the probability of default and the amount received by the se-

curity holder will be larger the higher is Vd, with V counteracting the former

relationship.

The stream of tax savings due to debt financing resembles a claim with a

perpetual payoff τcC contingent on the company remaining solvent, and the

fact that it can fully benefit from the tax deductibility in every period until

default. That brings us to the corresponding boundary conditions:

At V = Vd, TB(V ) = 0,

As V →∞, TB(V )→ τcC

r,

and thus the claim value is

TB(V ) =τcC

r− τcC

rpd(V ). (3.18)

The value of this claim can be interpreted as the value of receiving the tax

benefit continually minus the present value of the cash flows that are lost


after the unlevered firm value falls to the default level.

Now we are able to obtain the total firm value A(V ) as the unlevered firm

value plus tax benefits of debt minus bankruptcy costs, i.e.

A(V ) = V +τcC

r−(τcC

r+ γVd

)pd(V ). (3.19)

It could be noticed that as soon as V = Vd and the bondholders take over

the company, the firm value will be equal to the value of assets less the

bankruptcy costs, as the tax benefits will be lost due to the inability of

the new owners to relever the company after default.25 A less obvious, and

possibly even counterintuitive, implication of (3.19) is that the firm value

is increasing in the corporate tax rate and coupon payment. This can be

explained by the nature of the underlying variable that the model is based

on: since the asset value, unlike e.g. EBIT, is not taxed, the value of the

firm is only affected by taxes and coupons through the value of the tax

benefits, and therefore, the two parameters have no negative impact on the

firm valuation. This problem will be addressed in the models presented in

Chapters 4 and 5.

The equity claim is valued as a residual:

E(V ) = V +τcC

r−(τcC

r+ γVd

)pd(V )

−[C

r+

[(1− γ)Vd −

C

r

]pd(V )

]= V − (1− τc)

C

r+

[(1− τc)

C

r− Vd

]pd(V ). (3.20)

Thus, from the derived claim values we see that the bankruptcy level and the

coupon payment play an important role in determining the capital structure

choice. However, up until now we considered both as being exogenously

given.

25 The value of the firm at bankruptcy can be modelled alternatively, see e.g. Christensenet al. (2002), who allow the new owners to optimally relever the company post-default.


3.2.3 Optimal Leverage

Although the default threshold for the asset value might be given exoge-

nously, as if imposed by a covenant in the debt contract, it can be reason-

ably argued that shareholders will try to choose the boundary optimally to

maximise the value of their claim, stipulating the following smooth-pasting

condition:

∂E(V )

∂V

∣∣∣∣V=Vd

= 0. (3.21)

After simple algebra, the optimal bankruptcy level V ∗d can be found to

be

V ∗d =(1− τc)Cr + 1

2σ2

. (3.22)

Substituting (3.22) into the values of debt and equity claims as well as in the

total firm value found above, yields

D(V ) =C

r+

[(1− γ)(1− τc)C

r + 12σ2

− C

r

][(r + 1

2σ2)V

(1− τc)C

]− 2rσ2

, (3.23)

E(V ) = V − (1− τc)Cr

+

[(1− τc)C

r− (1− τc)C

r + 12σ2

] [(r + 1

2σ2)V

(1− τc)C

]− 2rσ2

,

(3.24)

A(V ) = V +τcC

r−[τcC

r+ γ

(1− τc)Cr + 1

2σ2

][(r + 1

2σ2)V

(1− τc)C

]− 2rσ2

. (3.25)

Note that (3.23)–(3.25) only suggest a certain capital structure given an

arbitrary level of coupon, which in fact does not mean that financing is

optimal. To determine optimal leverage it is necessary to assume that the

firm management can choose the coupon level which maximises the firm


value. Using (3.25) to solve for the optimal coupon yields

C∗(V ) =(r + 1

2σ2)V s

1− τc, (3.26)

where

s =

(τcσ

2

d

)σ2

2r

,

d = 2τcr + τcσ2 + 2rγ − τc2rγ.

Plugging C∗(V ) into (3.23) and (3.25) gives

D∗(V ) =

(r + 1

2σ2)V s

(1− τc)r+

[(1− γ)V s−

(r + 1

2σ2)V s

(1− τc)r

]s

2rσ2 , (3.27)

A∗(V ) = V +τc(r + 1

2σ2)V s

(1− τc)r−

[τc(r + 1

2σ2)V s

(1− τc)r+ γV s

]s

2rσ2 . (3.28)

Note that since C∗(V ) is only a function of V and constant parameters, it

is possible to determine the optimal capital structure simply by knowing the

current asset value. Furthermore, not only the model provides the manage-

ment with the information on the current capital structure given the dynam-

ically changing firm fundamental, but also on how far the current leverage

is from the optimal one, D∗(V )A∗(V )

, thus providing some degree of flexibility and

possibility of application in different scenarios.

3.3 Comparative Statics

The static model described above will serve as a benchmark for the models

in subsequent chapters, and in this section we will look into its performance

using comparative statics analysis. The applied parameter values are based


Table 3.1. Comparative statics for the static model. Base case parameters are:r = 3.5%, γ = 5%, and σ = 25%. The tax advantage of debt is calculated as T = A∗

V − 1.All monetary values are per unit of asset value.

L∗ C∗ χ∗d T

Base case 75.6% 0.042 0.48 0.16

τc = 20% 73.7% 0.039 0.47 0.12

τc = 30% 77.2% 0.046 0.48 0.21

σ = 20% 77.6% 0.039 0.53 0.18

σ = 30% 74.4% 0.047 0.44 0.15

γ = 0% 78.8% 0.045 0.51 0.17

γ = 10% 72.7% 0.040 0.45 0.15

on the Danish data and empirical estimates.

Table 3.1 presents comparative statics for optimal leverage, coupon, and

default threshold, as well as the tax advantage to debt. As it can be noticed,

a higher corporate tax rate increases optimal leverage. This is due to firms

benefitting more from the tax shield, which is illustrated by a higher optimal

coupon payment and a larger tax advantage to debt. From the comparative

statics given in Table 3.1 it is further evident that equity holders of risky firms

will optimally default at lower asset values. Since the value of equity—akin

to the value of a call option—is increasing in volatility, equity holders of risky

companies will continue operations for longer during downturns, compared

to shareholders of safer companies, as the former face a larger potential gain.

Due to limited liability, the expected value of equity in a risky firm is thus

higher than in a safer but otherwise identical company.

The relative amount of debt is higher for less risky firms because the

state variable is less likely to hit the lower boundary, allowing for a more

aggressive leveraging. Interestingly, it can be noted that the optimal coupon

appears to be increasing in the business risk. However, as also mentioned by

Leland (1994), this is due to the specific parameter value: as confirmed by

Figure 3.1(a) below, both high- and low-risk companies optimally choose to

commit to a higher coupon level, while firms with an intermediate volatility

levels pay smaller coupons. However, even though the optimal coupon is


decreasing for some intervals of volatility, borrowing will always be more

expensive for companies with a greater level of uncertainty attached to the

future asset value, as the value of debt is decreasing in volatility. These two

relationships are illustrated in Figures 3.1(e,f).

For an increase in bankruptcy costs the asset value at which the firm

declares bankruptcy drops, cf. Figure 3.1(b). On the other hand, if the

risk-free interest rate rises, the value of equity declines due to the diminished

discounted value of the tax shield, and therefore an increase in r will have

an opposing effect on the default boundary because equity value equals zero

at a higher level of the state variable. These relations are evident from

Figures 3.1(c,d).

3.4 Discussion

The presented model could be criticised on a number of points. Firstly, and

most importantly, if V represents the price of a traded asset after the initial

debt issue, then the model is not arbitrage-free. The risk-free profit can be

earned by investors who buy the unlevered assets of the firm for V , lever up

the company and sell it for a higher value A(V ) > V . To preclude arbitrage,

it hence must hold that A(V ) = V −TC, where TC is the amount required to

be paid in transactions costs to optimally lever the company. If V does not

represent the price of a traded asset, the general PDE from Chapter 2 cannot

be applied. This problem could potentially be circumvented by introducing a

traded derivative written on the assets of the firm, or, alternatively, by using

risk-neutral valuation to arrive at the same ODE.26 One way to rectify this

significant drawback is to model a firm fundamental which is not traded, e.g.

earnings, and we will implement this approach in the next chapter.

A second negative point is the assumption that the firm is liquidated

upon default. In a study of US firms, Alderson and Betker (1995) find that

26 This alternative approach is used in e.g. Cox, Ingersoll, and Ross (1985) to find theprice of contingent claims, although in a different setup than that in Merton (1974).


0.1 0.2 0.3 0.4Σ

0.035

0.040

0.045

0.050

0.055

0.060

C*

(a) Coupon and asset volatility

0.05 0.10 0.15 0.20Γ

0.42

0.44

0.46

0.48

0.50

d

(b) Default boundary and bankruptcy costs

0.02 0.04 0.06 0.08 0.10r

0.24

0.26

0.28

0.30

0.32

E

(c) Equity value and risk-free interest rate

0.02 0.04 0.06 0.08 0.10r

0.35

0.40

0.45

0.50

0.55

0.60

d

(d) Default boundary and risk-freeinterest rate

0.0 0.1 0.2 0.3 0.4Σ

100

200

300

400S

(e) Credit spread and asset volatility

0.1 0.2 0.3 0.4Σ

0.9

1.0

1.1

1.2

1.3

D

(f) Debt value and asset volatility

Figure 3.1. Performance of the static model. Base case parameters are: r = 3.5%,τc = 25%, and σ = 25%. Credit spread is defined as S = C

D − r.


80% of the firms would have created more value to their owners if opera-

tions were continued, compared to if assets were liquidated. Furthermore,

knowledge-based firms with more intangible assets such as pharmaceuticals

and IT companies may have low asset value, and a liquidation of this type

of company thus yields a considerable loss of shareholder wealth, and will

therefore unlikely be seen in practice. Furthermore, if asset specificity is

very high, liquidation costs will surge due to the illiquidity in the secondary

market, which might result in a fire sale (Shleifer and Vishny, 1992). In this

case the decision to continue operations will generally prevail as well.

Lastly, the model could be criticised for not allowing the firm to change

its capital structure after the initial bond issue. This static nature of the

debt level explains why the model suggests a leverage ratio of 76% in the

base case, while empirical research documents much lower estimates.27 This

may be due to the inability to exploit the tax advantage to debt in case

the asset value increases. It would therefore be desirable to allow the debt

level to change dynamically—so that the capital structure is tailored to the

realised value of the firm fundamental continually—and we incorporate that

option in the model presented in the chapter that follows.

27 Gaud, Hoesli, and Bender (2007) in a European study including 172 Danish firmsobserved in the period from 1988–2000 provide evidence that mean leverage in Danishcorporations is around 33%, which falls in the same estimate range as that in most ofother considered countries. For the presented model to yield similar results, bankruptcycosts and asset volatility have to be simultaneously increased to unreasonably high 70%and 45%, respectively, with other base case parameters unchanged.

Chapter 4

Family of Existing GBM-based

Dynamic Models

Changes in capital structure are commonly observed. Leary and Roberts

(2005) conclude that firms on average issue or repurchase their own securities

once a year. The discontinuity in restructuring activity could be explained,

for instance, by the presence of adjustment costs. Consider a firm which

exhibits high operating performance and has positive growth forecasts. To

better exploit the tax advantage to debt, this company has to continually

increase leverage, i.e. to dynamically adjust capital structure according to the

realised earnings. This option is ignored in the class of static models, a classic

example of which was introduced in Chapter 3. In this chapter, however, a

model which does allow for changes in capital structure will be presented.

Optimal leverage in a dynamic setting has previously been investigated by

Fischer et al. (1989a), Goldstein et al. (2001), Christensen et al. (2002),

and Flor and Lester (2002), among others. A distinguishing feature common

to most of existing research papers is the underlying assumption that the

state variable follows a geometric Brownian motion, which allows a simplified

progression from the static setup to the dynamic one due to the homogeneity

property inherent in this particular stochastic process. Therefore, we will try

to lay out a dynamic capital structure model which will describe the whole

50

Chapter 4. Family of Existing GBM-based Dynamic Models 51

family of dynamic GBM-based models in general, rather than represent any

particular setup.

The structure of the chapter is as follows. Firstly, a general introduction

to the model is provided. Afterwards, the set of equations together with

the maximisation problem needed to obtain the optimal capital structure is

presented. The chapter concludes with the comparative statics analysis and

the discussion of the model.

4.1 The Model

As mentioned in Chapter 3, the question of whether to treat the unlevered

firm value as corresponding to the price of a traded asset or not, is a critical

point. The problem is avoided by considering earnings before interest and

taxes as the state variable. Moreover, this makes the model more realistic in

the sense that coupons paid to bondholders are now financed with earnings

rather than by issuing new equity. Thus, the state variable χ denotes the

value of a claim to the entire payout produced by the firm, and its develop-

ment is given by a geometric Brownian motion:

dχ = µχdt+ σχdW, (4.1)

where µ is an exogenously given constant risk-neutral drift rate, σ is a con-

stant volatility of earnings, and r is a constant interest rate on money market

account.

The management will choose to default when the state variable hits a

lower boundary χd , dχ0. However, in addition to models described in previ-

ous chapters, there will now also be introduced an upper boundary χu , uχ0,

at which the management will decide to call all outstanding debt at par and

relever the firm optimally afterwards.28 It is precisely the latter option that

28 Restructuring could in principle be modelled in different ways. For example, a possibleoption would be to model debt as non-callable, and let the firm keep the existing debtand carry out an extra issue in addition. However, in this case additional assumptions


stipulates the dynamics of capital structure. Note that the boundaries for the

state variable could be determined either exogenously, by the management’s

precommitment in the debt contract, or endogenously, by incentive compati-

bility constraints imposed on the equity holders. The retirement of debt will

briefly make current equity holders the sole claimants of the company. Sub-

sequently, the firm will issue new corporate bonds with principal and coupon

exceeding the previous level. It can be shown that due to the properties of

a geometric Brownian motion optimal values for decision variables in case χ

hits the upper boundary χu could be easily obtained by scaling with a factor

u = χuχ0

. This implies that, for instance, optimal default level after the first

refinancing will be equal to uχd, after the second restructuring—u2χd, and

so on, where χd is the lower boundary in the initial period. The same scaling

factor can be applied to find the optimal coupon and restructuring threshold.

Even though both upward restructurings and downward debt repurchases are

observed in practice, we abstain from including the option to reduce the debt

level when earnings reach the lower boundary to simplify the analysis.

Figure 4.1 below captures the management’s option to increase the debt

level in case EBIT increases sufficiently. It can be noticed that when χ is fluc-

tuating within the range (χd, χu), the management will choose to maintain

the capital structure. A possible explanation of this phenomenon is given in

Fischer et al. (1989a), who document that there is a leverage range in which

capital structure will not be changed due to adjustment costs.

Due to the lognormal dynamics of χ, it is not necessary to scale the

parameters µ and σ when reoptimising the capital structure, and the EBIT

process in the second period will have the same properties as in the first

period. This feature implies that the mechanism of the capital structure

decision is in fact unchanged. It can be shown by backward induction that

scaling by a factor u is indeed optimal.29

To obtain the pricing function F of securities contingent on the state

regarding the seniority and credit risk of several debt issues need to be made, which maycomplicate the analysis without changing results qualitatively.

29 See Goldstein et al. (2001, Appendix B).


t

d× Χ0

u×d× Χ0

u2×d× Χ0

Χ0

u× Χ0

u2× Χ0

u3× Χ0

Χ

Figure 4.1. A simulated sample path of the firm’s earnings with lognormaldynamics. Initial earnings level is χ0, and period zero ends with earnings reaching χu,at which point the company calls the entire existing debt and reoptimises the capitalstructure; in the subsequent periods the procedure is repeated.

variable χ, we apply the framework developed in Chapter 2, which boils

down to the following ODE:

1

2σ2χ2Fχχ + µχFχ − rF + νY = 0, (4.2)

where νY denotes the payout specific to a security Y , and Fχ and Fχχ denote

the first- and second-order partial derivatives of F with respect to χ. The

general solution to the corresponding homogeneous differential equation can

be formulated as

F (χ) = B1χβ1 +B2χ

β2 , (4.3)

where β1 and β2 are the roots to the characteristic equation corresponding

to (4.2).

The system of equations describing the model should embrace the bound-

ary conditions for debt and equity and two smooth-pasting conditions. Two

additional equations for the maximum potential firm value and the value of

the principal will be constructed to supplement the final system.


The two value functions that need to be determined are

D(χ) = d1χβ1 + d2χ

β2 +(1− τi)C

r(4.4)

and

E(χ) = e1χβ1 + e2χ

β2 +(1− τe)χr − µ

− (1− τe)Cr

, (4.5)

where the inclusion of τd ensures that the value of equity is decreasing in the

tax rate, which corrects the inconsistency observed under the static model,

when the relationship was positive.

Let us consider the boundary conditions. When EBIT reaches the lower

boundary χd, the firm is taken over by bondholders, after they pay the

bankruptcy costs, γ. Debt value at bankruptcy can thus be expressed as30

D(χd) = (1− τe)(1− γ)χdr − µ

. (4.6)

As shown in (4.6), after the bankruptcy costs have been paid the old bond-

holders become the new equity holders of an unlevered firm. However, we

assume that a company cannot issue new bonds if it has previously defaulted

on debt payments, and therefore the new shareholders simply own a riskless

perpetuity of earnings, and firm value can thus be described by the Gordon’s

growth model (Gordon, 1959). Naturally, the stream of earnings would not

be riskless if, for example, fixed operating costs were considered in the model.

However, as no EBIT-financed payouts except coupon payments are consid-

ered, the firm cannot default in the future as it has no outstanding debt, and

hence becomes risk-free. It can also be seen from (4.6) that the expected

growth in EBIT is unaltered after the firm is taken over by bondholders, as

opposed to e.g. Goldstein et al. (2001) who assume that a firm is liquidated

at default. The assumption of maintaining the firm operations is supported

30 Note that values of debt and equity claims at the boundaries are dependent on theinitial EBIT level χ0, but we suppress it as an argument for convenience in the currentand subsequent chapters.


by empirical evidence that documents that only 5% of financially distressed

firms are eventually liquidated (Gilson et al., 1990). If the new firm owners

are allowed to optimally relever the company post-default, optimal leverage

is expected to increase (Christensen et al., 2002), because debt holders will

be able to exploit the tax shield in the future, thereby increasing the pre-

default value of their claim. This option is, however, not considered in our

model.

At the lower boundary, equity holders will have given up their claim to

residual earnings, and hence

E(χd) = 0 (4.7)

to comply with the absolute priority rule.

At the upper boundary, management will call all outstanding debt, and

under the assumption that no call premium needs to be paid on bonds retired

prematurely,31 the value of the claim of debt holders is

D(χu) = D(χ0). (4.8)

After equity holders have paid back the old debt D(χ0), they will issue

new bonds D(χu) with higher principal and coupon. The value of equity at

restructuring point is thus

E(χu) = E(χu) + (1− q)D(χu)−D(χ0), (4.9)

where q denotes restructuring expenses paid as a fraction of the new debt,

and can be interpreted as issuance costs. As the repayment of the old prin-

cipal is financed by the equity holders, the company will be instantaneously

unlevered, and at this point in time shareholders will choose the new optimal

debt level.

In addition to the two boundary conditions stated, smooth-pasting con-

31 We do not consider any liquidity issues stemming from the presence of this provision.The assumption of the absence of a call premium is relaxed in Subsection 5.4.2.


ditions need to be invoked at both lower and upper EBIT thresholds. These

conditions ensure that it is always in the best interest of equity holders to

either declare bankruptcy or refinance. Even though one or both thresholds

could be determined exogenously—for instance, Goldstein et al. (2001) as-

sume that management precommits to the restructuring level χu, while the

default level χd is determined by the incentive compatibility condition—we

choose to let both upper and lower boundaries be determined endogenously

for consistency. The two smooth-pasting conditions are

∂E(χ)

∂χ

∣∣∣∣χ=χd

= 0, (4.10)

∂E(χ)

∂χ

∣∣∣∣χ=χu

= E(χ0) + (1− q)D(χ0). (4.11)

The high-contact condition is needed to ensure that the decision to refinance

is in fact optimal for the equity owners at that specific EBIT level. If this

condition were not included in the model, equity holders would not maximise

the potential firm value by restructuring at the upper boundary, resulting

in a discrepancy between E(χu) and the shareholders’ incentive-compatible

equity level. The low-contact condition states that it must be optimal for

owners of equity to default exactly when the lower boundary χd is reached.

The system now consists of six equations (4.6)–(4.11), with d1, d2, e1, e2,

d, u as unknowns. This system cannot be solved analytically, and thus we

have to resort to numerical analysis. To simplify expressions utilised in the

numerical procedure, we construct the following extra variables:

Π , E(χ0) + (1− q)D(χ0), (4.12)

Σ , D(χ0). (4.13)

We will refer to the first variable as the “maximum potential firm value”.

This value can be interpreted as the price the owner-manager, who solely

owns the EBIT-generating machine or production technology, can sell the


company for.32 This price is what equity holders want to maximise. The

second definition restates the debt value at issuance as the principal Σ. This

reflects the assumption that bonds are issued at par, with no discount or

premium. Note that both definitions imply pricing per unit of EBIT.

With the two extra equations introduced, the complete system is now

comprised of eight equations, and must be solved for eight unknowns. It could

be reduced to two equations for a pair d, u containing the known parameters

r, σ, µ, γ, q, τe, τi and the coupon C. Due to the non-linearity of these two

equations, we have to utilise numerical methods to obtain values for d and

u, assuming a certain coupon level, and then insert them in the maximum

potential firm value Π, which is further maximised to find the optimal coupon

C∗, i.e.

C∗ = arg maxC

Π. (4.14)

The optimal coupon is then plugged into the algebraic expressions of debt

and equity, which ultimately determine the optimal capital structure. The

results obtained from this optimisation procedure are presented in the next

section. We dwell on the applied numerical methods in the next chapter.

4.2 Comparative Statics

Before proceeding to the numerical results of the model, let us briefly under-

line some common issues associated with simulations. A widespread problem

is an ambiguous choice of the base case parameters. Of course, one might

argue that the exact set of parameters does not matter for the study of

general relationships or relative comparisons, but it is nevertheless obvious

that model predictions in absolute terms do rely (and possibly heavily) on

32 This interpretation is reminiscent of that of Christensen et al. (2002). However,to emphasise that this value could in fact reflect a different leverage level if the owner-manager has initially taken on a suboptimal amount of debt, we prefer to think of Π as thepotential amount the entrepreneur can sell his company for, which implies the optimalityof capital structure. We dwell on the interpretation of this variable in Subsection 5.2.3.


Table 4.1. Comparative statics for the dynamic GBM-based model. Base caseparameters are: r = 3.5%, γ = 5%, µ = 2%, q = 1%, σ = 25%, τi = 37%, τd = 28%,and τc = 25%. The tax benefit of debt is calculated as T = Π

χ0(1−τe)r−µ

− 1. All monetary

values are per unit of EBIT.

L∗ C∗ χ∗d T

Base case 36.1% 1.04 0.19 0.12

τc = 20% 28.9% 0.77 0.15 0.04

τc = 30% 39.7% 1.23 0.21 0.22

σ = 20% 41.7% 1.11 0.25 0.13

σ = 30% 32.0% 1.02 0.15 0.11

γ = 0% 39.3% 1.17 0.21 0.13

γ = 10% 33.3% 0.93 0.18 0.11

the initial choice, and therefore, extensive sensitivity analysis should be con-

ducted. Another shortcoming oft-encountered in the literature is drawing

inferences from some specific values produced by a model (e.g. in Goldstein

et al., 2001) which does not seem appropriate as certain relationships are

in fact not monotonous, and thus graphical analysis is needed to provide a

complete picture in continuity. Apart from the monotonicity issue, there is

also a problem of ignoring strong influence of a third model parameter on the

nature of the relationship considered. Studying cross-relationships of such

kind could provide important insights and that is why we will introduce a

third dimension when necessary.

To carry out a comparison to the static model we preserve the base case

parameters defined in Section 3.3. Note, however, that personal taxes are

now included in the model. The tabulated results are presented below.

The key insight from the comparative statics is that the initial leverage

is now much lower than that observed in the static model. This is a direct

consequence of the fact that the management, while endowed with an option

to relever the firm, will take on less debt at the beginning and instead adjust

the capital structure dynamically in the future to make it correspond to

the earnings level. In the static setting, the initial debt level will have to

reflect the expected future development of the firm, and thus may in fact


be implicitly suboptimal from a dynamic perspective. Moreover, we see that

the bankruptcy threshold is lowered considerably, reflecting the fact that,

ceteris paribus, the value of a firm possessing a restructuring option will be

higher than the value of the firm without one. Thus, in some cases when the

earnings level is low enough for the equity holders to declare default in the

static framework, it will still not be sufficient to prompt them to do so in the

dynamic setting. While comparing the tax advantage to debt in the dynamic

framework with that in the static model, one should note that the latter does

not consider personal taxes. Besides, since Leland (1994) models asset value

and not income, an increase in the tax rate has only a positive influence

since the deduction of interest expenses becomes more valuable. This is not

the case in the dynamic model, where the introduction of a dividend tax

ensures that the value of equity is decreasing in taxes. Therefore, due to

this difference in the assumed state variable and thus in the tax regime, it is

not completely correct to compare the tax benefit of debt directly. The tax

advantage to debt in the dynamic framework is expected to be higher due

to the ability to exploit the benefits of debt financing better because of the

restructuring option; indeed, we do obtain a tax advantage to debt of 33%

in the dynamic setting when we exclude personal taxes from the model.

The effects of changes in parameters on the selected dependent variables

are similar to those obtained in the static model. However, a few merit

comments. The initial earnings level is lower than the optimal coupon in the

base case, but still higher than the bankruptcy threshold, which implies that

the firm may not default even though earnings are insufficient to cover the

interest payments on outstanding bonds. As asset sale or additional debt

issues are not allowed in our model, the deficit has to be financed by equity

owners as long as EBIT fluctuates above χd.

In the dynamic model the spread in optimal values for varying parameter

values is wider. Hence, not only optimal values per se differ, but so does the

impact of slight changes in input parameters. For instance, if bankruptcy

costs increase by five percentage points from the base case, optimal coupon


is reduced by more than 10% in the dynamic model, while in the static frame-

work it only decreases by 6%. Furthermore, if volatility deviates from the

base case by five percentage points in the dynamic model, optimal leverage is

in the range 32–42%, compared to the interval 74–78% in the static setting.

Thus, as opposed to the much smaller effort required to determine optimal

leverage in a static setting, precision of estimates and calibration of input

parameters are essential if firms wish to fully benefit from the opportunity

to adjust capital structure dynamically.

As the dynamic model allows for debt increase, management, when decid-

ing on capital structure changes, also needs to take readjustment costs q into

consideration, in addition to the parameters in a static setup. If restructur-

ing is more costly, this will result in new bonds being issued less frequently

and will prolong the commitment to the coupons related to the current debt

level. Therefore, the equity holders will prefer lower coupon payments in the

future, and accordingly lower leverage and decreased probability of default.

There is also a counter-effect present, however: as the cost of rebalancing the

capital structure rises, the equity holders may prefer to have a higher initial

leverage and higher coupon payments to better utilise the tax advantage to

debt. These opposing effects help explain why the incremental change in

leverage in Figure 4.2(a) is so narrow—the level drops by a mere 0.5 percent-

age points when restructuring costs increase from 0.5% to 2%. Overall, the

interplay of the described relations is evident from Figures 4.2(a,e).

The advantage of taking on debt to finance operations and investments

is decreasing in the tax rate on interest income. As the inclusion of τi only

affects the debt value, the maximum potential firm value decreases when the

tax on interest income increases. This results in a lower tax benefit of debt,

making equity financing more favourable. This leads to the optimal leverage

being lower for higher personal taxes, as can be seen from Figure 4.2(b).

Figure 4.2(f) demonstrates that as the effective tax rate increases, the

optimal restructuring level falls. This is because the tax benefit of debt

is positively related to τe, and thus for higher taxes there is a wider net


0.005 0.010 0.015q

0.357

0.358

0.359

0.360

0.361

0.362

L*

(a) Leverage and restructuring costs

0.38 0.39 0.40 0.41Τi

0.30

0.31

0.32

0.33

0.34

0.35

0.36

L*

(b) Leverage and tax on interest income

0.25 0.30 0.35 0.40 0.45 0.50 0.55Σ

400

500

600

700

800

900

1000S

(c) Credit spread and earnings volatility

0.25 0.30 0.35 0.40 0.45 0.50 0.55Σ

1.05

1.10

1.15

C*

(d) Coupon and earnings volatility

0.005 0.010 0.015q

1.1

1.2

1.3

1.4

1.5u

(e) Restructuring boundary andrestructuring costs

0.44 0.46 0.48 0.50 0.52Τe

1.28

1.30

1.32

1.34

1.36

1.38

1.40u

(f) Restructuring boundary andeffective tax rate

Figure 4.2. Performance of the the dynamic GBM-based model. Base caseparameters are: r = 3.5%, τc = 25%, τd = 28%, τi = 37%, γ = 5%, q = 1%, and σ = 25%.


refinancing gain which equity owners can realise by adjusting the capital

structure. Therefore, the restructuring level u is lowered to accommodate

more frequent readjustment.

Results in Table 4.1 imply that riskier firms pay a higher coupon. How-

ever, conclusions based on tabulated comparative statics have the obvious

drawback of implying monotonicity. From Figure 4.2(d) it can be seen that

the coupon is increasing in volatility only for higher-risk firms. As in the

static model, both safer and riskier companies will optimally commit to

high coupons, when compared to an otherwise similar firm with a mid-range

volatility level. Safe firms will prefer higher coupons to better exploit the

tax advantage of debt, whereas risky companies will choose high coupon pay-

ments to attract bond investors. Furthermore, the absence of monotonicity in

the relationship between earnings volatility and optimal coupon means that

the change in coupon cannot provide an unambiguous guidance on whether

the price of borrowing has increased. Indeed, we see from Figure 4.2(c) that

firms with higher business risk always have to face a wider credit spread—

because the debt value is strictly decreasing in volatility—even though an

increase in volatility may in fact imply a lower coupon payment.

If there is a greater uncertainty attached to a firm’s future earnings level,

such company will, in addition to the higher credit spread, also have a sub-

stantially lower optimal leverage than an otherwise identical firm, but the

relationship is less pronounced if the effective tax rate faced by equity hold-

ers is increased, as illustrated by flattening curvature of the optimal leverage

plane for higher τe in Figure 4.3(a). This could be explained by the fact that

higher tax benefit of debt counteracts the negative impact of volatility, which

makes debt financing more favourable.

Further, an increase in the effective tax rate prompts the management

to commit to higher optimal coupon. If bankruptcy is expensive, however,

interest payments will be lowered due to a decreased earnings level at which

equity holders declare bankruptcy and therefore lowered default probability.

This balancing of the positive and negative effects of debt exactly repre-


(a) Leverage, earnings volatility andeffective tax rate

(b) Coupon, bankruptcy costs andeffective tax rate

Figure 4.3. Optimal leverage for varying earnings volatility and effective taxrate, and optimal coupon for varying bankruptcy costs and effective tax rate.Base case parameters are: r = 3.5%, τc = 25%, τd = 28%, τi = 37%, γ = 5%, q = 1%,and σ = 25%.

sents the classic trade-off that managers encounter when considering optimal

capital structure, as depicted in Figure 4.3(b).

4.3 Discussion

The dynamic model presented in this chapter allows the firm management

to increase the debt level, enabling a better utilisation of the tax shield for

growing earnings. The company’s option to reduce the outstanding debt is

excluded, which is further emphasised by the inability of firms to accumulate

cash savings. This assumption might be justified by some empirical evidence

which suggests that financially distressed firms will not reduce debt levels due

to transaction costs (Gilson, 1997). Furthermore, strategic debt service mod-

els indicate that equity holders do not have the ability to reduce debt close

to bankruptcy, as earnings are used to bargain with creditors and the level

will be too low for a successful renegotiation of the coupon (Mella-Barral and

Perraudin, 1997). Following the arguments of Jensen and Meckling (1976)

and Jensen (1986), the holding of cash encourages “empire building” and

aggravates agency costs of free cash flow, respectively, and the assumption of

full payout and thus zero retained earnings can be seen as a way to mitigate


these issues. Hence, excluding the downward restructuring option is not a

critical limitation.

One should note that most of the dynamic capital structure models ignore

the dynamics at the lower earnings threshold by modelling default through

liquidation or by restricting the firm to be only all-equity financed post-

bankruptcy, as is the case with the model presented in this chapter. This

might, of course, lead to understated leverage since the debt holders will not

be able to reoptimise the capital structure, which leads to a lower initial

valuation of the debt claim.

As the value of the state variable reaches the upper threshold, the com-

pany increases the debt level by calling all outstanding debt, and issuing new

debt with a larger principal and coupon. In the presented framework, how-

ever, no premium is paid on bonds retired prematurely. This will give equity

holders an incentive to continuously call the debt to exploit the tax benefit

of debt, as long as the tax advantage is sufficient to cover restructuring costs.

Consequently, the model may underestimate the optimal leverage as upward

restructurings will occur too often.

One of the cornerstones of the model is its scalability, which allows opti-

mal future coupon, bankruptcy and restructuring thresholds to be calculated

by simply scaling the current values. This property rests on the assumption

that the state variable follows a geometric Brownian motion. Even though a

closer analysis does reveal some drawbacks of this process, it has neverthe-

less been commonly used for mathematical tractability. To circumvent some

of the known issues, the underlying state variable can instead be modelled

with an alternative process. In the next chapter we propose a novel dynamic

capital structure model which addresses some of the shortcomings inherent

in the family of GBM-based models.

Chapter 5

Proposed Dynamic Model with

Mean Reversion

Having analysed the fundamental models of capital structure in the academic

literature, we now turn to the development of our own model, which extends

existing research in several dimensions described below.

The chapter is organised as follows. Section 5.1 dwells on the motiva-

tion behind the model and emphasises the key points of differentiation from

existing dynamic capital structure models. Section 5.2 focuses on the deriva-

tion of the model and its properties, and lays out the system to be solved to

close the model. Section 5.3 considers tests of the model, elaborates on the

numerical methods employed, and presents the numerical results and their

implications. Section 5.4 extends the analysis by incorporating additional

assumptions into the basic setting and illustrates obtained results in com-

parison with those from the main mean-reverting model and the dynamic

GBM-based model.

5.1 Motivation

It is important to stress from the outset that in this paper we do not aim

to build a completely new framework for analysing optimal leverage; rather,

65


our novelty is in providing a new perspective on the modelling of capital

structure decisions, and thus our model will hopefully serve as an extension

of existing research and give some significant insights on the matter.

The overwhelming majority of existing capital structure studies—as a

matter of fact, almost every one of them—is based on the assumption that

the underlying state variable follows the stochastic process represented by a

geometric Brownian motion. Such a widespread adherence to this assump-

tion is not surprising: the process embraces some very attractive properties33

that ensure mathematical tractability and in some special cases even allow

analytical solutions for the optimal capital structure. Only very few papers

consider the arithmetic Brownian motion as the underlying process, for in-

stance, Ammann and Genser (2004)—Genser (2006) for further extension—

and Bank and Lawrenz (2005), and they only study capital structure in a

static setting. Deviation from the Brownian motion in general is even rarer.

In this thesis we present a dynamic model of capital structure with the op-

erating income as the state variable, which follows a mean-reverting process,

and show the implications for the study of capital structure choice. Admit-

tedly, as all the other authors in the field considering dynamic modelling,

we are also unable to derive analytical solutions, but nevertheless we get re-

markably far. The model is analysed in great detail and extensive numerical

simulations are carried out, with the shortcomings and potential areas of

improvement being highlighted.

One of the key contributions of this paper is the analysis of capital struc-

ture under an alternative process which exhibits mean reversion. The intu-

ition behind our approach is as follows. Firstly, the assumption of either ge-

ometric or arithmetic Brownian motion implies that the state variable could

in time reach infinitely high values, and that due to the commonly assumed

positive drift rate the value constantly increases, with short-run falls being

restricted by the volatility and not having any impact in the medium or long

term. In practice, however, such variable as operating income or, more gen-

33 In particular, the process possesses the homogeneity property, which permits dynamicmodel transformation. We will elaborate on this property later in the chapter.

Chapter 5. Proposed Dynamic Model with Mean Reversion 67

erally, earnings is very often consistent with a certain long-term mean value

which is industry-specific, and ignoring that fact may severely undermine the

application of a model; a potential caveat here is that the long-term mean will

change through time, and we take that into account in our model. Secondly,

by assuming mean reversion we obtain more control over the process as we

are able to directly specify not only the speed of mean reversion which is in

a way similar to the drift rate of a geometric Brownian motion, but also the

long-term mean. That effectively allows the modeller to distinguish between

different industries both in terms of their profitability, through the mean

value of earnings, and their stability, through varying how fast the earnings

will converge. Finally, there exists empirical evidence of mean-reverting pat-

terns in the evolution of earnings, e.g. Bajaj, Denis, and Sarin (2004), Fama

and French (2000), Lipe and Kormendi (1994).

To the best of our knowledge, the only paper that models mean-reverting

earnings while considering capital structure choice in a continuous-time frame-

work is Sarkar and Zapatero (2003).34 However, they only develop a static

model which is akin to that of Leland (1994), and as such it does not allow for

changes in leverage, which is a fairly restrictive setting. In fact, the dynamics

of capital structure has never been considered so far outside of the scope of

the geometric Brownian motion assumption. Here we specifically emphasise

that by the notion “dynamic” we mean the fact that capital structure can

be varied through time at the discretion of management, which acts in the

interests of existing shareholders. Although modelling in a static setting may

still provide interesting insights and serves as a good starting point, it is the

dynamics that really makes the model relevant by bringing more flexibil-

ity to it and making its predictions more consistent with observed financing

practices. Moreover, we further differ from Sarkar and Zapatero (2003) by

employing a more complex setting to our model. For instance, we explicitly

model the complete tax structure by considering corporate and personal tax-

ation imposed on both equity and debt holders, while they only incorporate

34 Raymar (1991) also considers mean reversion, but works in the discrete-time setting,assuming that EBIT follow a first-order autoregressive process.


the net tax advantage of debt financing—an aggregate parameter, supposed

to account for everything at once, which leads to certain ambiguities. For

example, that parameter does not affect the debt claim valuation, which ac-

tually makes the setup identical to simply ignoring tax on interest income.

Thus, the tax advantage to debt is always present in their model, which could

ultimately lead to overstated leverage.

Another important dimension of our analysis is explicit focus on numerical

methods—a part of research that is frequently being ignored in the literature.

In fact, numerical simulations are often presented without even mentioning

which specific algorithm has been applied, and, as will be shown later, since

different methods bear different implications one should take great care to

avoid inherent caveats. The two mathematical problems embedded in the

analysis of dynamic capital structure are the following: solving a system of

nonlinear algebraic equations and running nonlinear local optimisation. The

general algorithmic toolbox for tackling those is not as extensive as one could

initially imagine, and we will try to discuss the main methods used.

The model derived in the next section is extended afterwards to incorpo-

rate a call premium, which relaxes the assumption that equity holders are

able to retire the old debt at par at the restructuring point. Apart from

that, we also extend the dynamic GBM-based model to take into account

fixed operational costs, which allows us to resolve another widespread issue

of capital structure analysis—the cash flow being restricted to take on only

positive values. Since most start-ups do exhibit negative operating income

as do mature firms during economic downturns, this issue is fairly limiting

from a practical angle.


5.2 The Model

5.2.1 Preliminaries

We assume that the operating income of a firm follows the mean-reverting

stochastic process under the risk-neutral probability:

dχ = κ(θ − χ)dt+ σχdW, (5.1)

where κ is the speed of mean reversion, θ is the long-term mean value of

earnings35, σ is volatility of earnings, and W is the Wiener process.36 Note

that, as opposed to the conventional arithmetic Ornstein–Uhlenbeck process,

the volatility term is not constant and is proportional to the current earnings

level. In the special case of θ = 0 the process becomes a geometric Brownian

motion with drift µ = −κ. The non-negativity of earnings is implicitly

assumed due to the nature of the process, but we relax this assumption in

Section 5.4.

We will now derive the Gordon growth model for the assumed mean-

reverting stochastic process, which will be required in the valuation of claims

that follows in Subsection 5.2.3.

Lemma 1. Let ξ be a random variable following (5.1), then ∀s > t

Et

[∫ ∞t

ξse−r(s−t) ds

]=θ

r+ξt − θr + κ

. (5.2)

Proof. Since the corresponding integral is finite, by Fubini’s theorem we have

Et

[∫ ∞t

ξse−r(s−t)ds

]=

∫ ∞t

e−r(s−t)Et [ξs] ds. (5.3)

35 Henceforth we will use the notions “operating income”, “EBIT” and “earnings” in-terchangeably, ignoring possible differences stipulated by accounting practices.

36 The specific functional form of mean reversion with proportional volatility was intro-duced by Bhattacharaya (1978).


It could be shown that the expectation of the mean-reverting process is37

Et [ξs] = θ + (ξt − θ)e−κ(s−t), (5.4)

and thus substituting in the right-hand side of (5.3) yields∫ ∞t

e−r(s−t)(θ + (ξt − θ)e−κ(s−t)) ds

=

∫ ∞t

(θer(t−s) + (ξt − θ)e(t−s)(r+κ)

)ds

= −(θ

rer(t−s) +

(ξt − θr + κ

)e(t−s)(r+κ)

)∣∣∣∣∞t

=θ

r+ξt − θr + κ

.

In fact, the result in Lemma 1 is quite intuitive: if earnings exhibit mean

reversion, the expected value of future cash inflows will be equal to the long-

term mean value discounted in perpetuity, which is represented by the first

term, plus what we call the convergence term—the extra cash flow related

to the fact that during a certain period of time, actual earnings will deviate

from their long-term mean.

The general partial differential equation for pricing contingent claims is

obtained by the absence of arbitrage condition as in previous chapters. We

omit most of the derivations.

Let Υ(χ) be the value of a claim written on the total earnings χ of the

firm. Assume Υ(χ, t) is twice differentiable in χ and once in t, and apply

Ito’s lemma to (5.1) to get

dΥ =

(Υt + κ(θ − χ)Υχ +

1

2σ2χ2Υχχ

)dt+ σχΥχdW. (5.5)

If the claim additionally offers a continuous payoff νY , then the expected

return it yields must equal the risk-free rate to preclude arbitrage opportu-

37 See, e.g. Tsekrekos (2010).


nities, i.e.

rΥdt = EQ [dΥ + νY ]

= EQ

[(Υt + κ(θ − χ)Υχ +

1

2σ2χ2Υχχ

)dt+ σχΥχdW + νY

].

Due to the common properties of the Wiener process described in Chapter 2

and the fact that Υ(χ, t) and its derivatives are known at time t, the above

could be rewritten as the following partial differential equation:

1

2σ2χ2Υχχ + κ(θ − χ)Υχ − rΥ + Υt + νY = 0. (5.6)

Note that in order for an equivalent martingale measure to exist, we have

to assume that the time horizon is finite,38 but sufficiently long to neglect

the relative value of the principal and thus obtain time independence of the

payouts. So, the value of a claim on the operating income will then satisfy

the ordinary differential equation

σ2

2χ2Υχχ + κ(θ − χ)Υχ − rΥ + νY = 0. (5.7)

To derive the general solution to this ODE let us first solve the corre-

sponding homogeneous differential equation

σ2

2χ2Υχχ + κ(θ − χ)Υχ − rΥ = 0 (5.8)

by demonstrating that it could be represented as a special case of a specific

differential equation, whose solution form is commonly known.

Consider the general confluent equation (Abramowitz and Stegun, 1972,

Eq. 13.1.35), which is the generalised form of the well-known Kummer’s

38 See Section 2.3 for more details on the question of existence of an equivalent martingalemeasure.


equation:

ω′′(Z) +

[2A

Z+ 2f ′(Z) +

(bh′(Z)

h(Z)− h′(Z)− h′′(Z)

h′(Z)

)]ω′(Z)

+

[(bh′(Z)

h(Z)− h′(Z)− h′′(Z)

h′(Z)

)(A

Z+ f ′(Z)

)+A(A− 1)

Z2

+2Af ′(Z)

Z+ f ′′(Z) + (f ′(Z))2 − a(h′(Z))2

h(Z)

]ω(Z) = 0. (5.9)

The general solution to this differential equation is given by

ω(Z) = B1Z−Ae−f(Z)Φ(a, b, h(Z)) +B2Z

−Ae−f(Z)Ψ(a, b, h(Z)), (5.10)

where B1 and B2 are arbitrary constants, Φ(a, b, h(Z)) is the Kummer’s con-

fluent hypergeometric function, and Ψ(a, b, h(Z)) is the Tricomi’s confluent

hypergeometric function. The Kummer’s confluent hypergeometric function

could be expanded in a generalised hypergeometric series

Φ(a, b, h(Z)) =∞∑0

(a)n(h(Z))n

(b)nn!,

where (a)n and (b)n are the rising factorials with (a)0 = 1 and (b)0 = 1.

The Tricomi’s confluent hypergeometric function is most frequently defined

in terms of the Kummer’s function:

Ψ(a, b, h(Z)) =π

sin πb

(Φ(a, b, h(Z))

Γ(1 + a− b)Γ(b)

− (h(Z))1−b Φ(1 + a− b, 2− b, h(Z))

Γ(a)Γ(2− b)

),

where Γ(·) is the gamma function.39

Let A = −β, f(Z) = 0, and h(Z) = 2κθσ2Z

. Then h′(Z) = − 2κθσ2Z2 and

39 To be completely strict, one should additionally specify the domains of Φ(·, ·, ·) andΨ(·, ·, ·), which is a somewhat delicate matter, given that the functions are undefined inseveral special cases, e.g. when a and b are non-positive integers.


h′′(Z) = 4κθσ2Z3 . Substituting into (5.9) gives after simple algebra

ω′′(Z) +

[−2β

Z+

(− bZ

+2κθ

σ2Z2+

2

Z

)]ω′(Z)

+

[(− bZ

+2κθ

σ2Z2+

2

Z

)(− βZ

)+β(β + 1)

Z2− 2aκθ

σ2Z3

]ω(Z) = 0.

Multiplying both sides of the above by Z yields

Zω′′(Z) +

[−2β − b+

2κθ

σ2Z+ 2

]ω′(Z)

+

[−2κθ(β + a)

σ2Z2+β2 + β(b− 1)

Z

]ω(Z) = 0. (5.11)

The homogeneous ODE of interest (5.8) could be rewritten as

χΥχχ +

(2κθ

σ2χ− 2κ

σ2

)Υχ −

2r

σ2χΥ = 0, (5.12)

and one could immediately notice that if we let Z = χ and ω(Z) = Υ(χ),

then this differential equation would be identical to (5.11) if and only if the

following system of equations holds:

− 2β − b+2κθ

σ2χ+ 2 =

2κθ

σ2χ− 2κ

σ2

− 2κθ(β + a)

σ2χ2+β2 + β(b− 1)

χ= − 2r

σ2χ

This system is equivalent to the following one:

b = −2β + 2 +2κ

σ2

a = −β

1

2σ2β(β − 1)− κβ − r = 0.


The roots of the quadratic polynomial are

β1,2 =2κ+ σ2 ±

√(2κ+ σ2)2 + 8rσ2

2σ2,

completing the two sets of solutions to the system of equations.

Thus, the general solution to our homogeneous ODE becomes

Υ(χ) = B1χβ1Φ

(a(β1), b(β1),

2κθ

σ2χ

)+B2χ

β2Ψ

(a(β2), b(β2),

2κθ

σ2χ

),

(5.13)

where a(·) and b(·) symbolically denote values of parameters depending on

the root of the polynomial. B1 and B2 will be determined by the corre-

sponding boundary conditions. Note that we will not present the solution

to the inhomogeneous ODE in its general form here, but will formulate the

particular solutions in Subsection 5.2.3, depending on the type of a claim

considered.

5.2.2 Homogeneity Property

Now that the solution to the ODE is derived, we are ready to formulate the

result which is central to the dynamics of capital structure in our model—the

homogeneity property of the assumed mean-reverting process.

Lemma 2. The process given by equation (5.1) is homogeneous of degree one

in the pair (χ, θ).

Proof. We will demonstrate the validity of this statement using contingent

claims analysis. Let χ ∈ [χd, χu] follow (5.1), χd and χu being exogenously

determined levels. Consider a simple claim contingent on χ paying 1 DKK

when χ falls to χd before the upper threshold χu has been reached. Following

the approach from the previous subsection, the price Pd of such claim could


be obtained as a solution to the following differential equation:40

σ2

2χ2∂

2Pd∂χ2

+ κ (θ − χ)∂Pd∂χ− rPd = 0 (5.14)

subject to the two boundary conditions

Pd(χd) = 1 and Pd(χu) = 0.

The solution to this system is

Pd(χ) =

(1

χ

)− 12− κσ2−√

4κ2+4κσ2+8rσ2+σ4

2σ2(

1

χd

) 12

+ κσ2

+

√4κ2+4κσ2+8rσ2+σ4

2σ2

·

( 1

χ

)√4κ2+4κσ2+8rσ2+σ4

σ2

Φ(χu) Ψ(χ)−(

1

χu

)√4κ2+4κσ2+8rσ2+σ4

σ2

· Φ(χ) Ψ(χu)

)/

( 1

χd

)√4κ2+4κσ2+8rσ2+σ4

σ2

Φ(χu) Ψ(χd)−

(1

χu

)√4κ2+4κσ2+8rσ2+σ4

σ2

Φ(χd) Ψ(χu)

where Φ(χ) and Ψ(χ) are the Kummer’s and Tricomi’s confluent hypergeo-

metric functions, respectively, and due to space constraints they are symbol-

ically denoted as

Φ(χ) = Φ

(−1

2− κ

σ2−√

4κ2 + 4κσ2 + 8rσ2 + σ4

2σ2,

1−√

4κ2 + 4κσ2 + 8rσ2 + σ4

σ2,

2κθ

σ2χ

)

40 For simplicity we ignore taxes as the inclusion of them does not impact the line ofderivation.


and

Ψ(χ) = Ψ

(−1

2− κ

σ2+

√4κ2 + 4κσ2 + 8rσ2 + σ4

2σ2,

1 +

√4κ2 + 4κσ2 + 8rσ2 + σ4

σ2,

2κθ

σ2χ

)

Utilising the notation from the previous subsection, the solution could be

rewritten in a more convenient form

Pd(χ) =

(1χ

)−β1 (1χd

)β1 [(1χ

)β1−β2Φ(χu)Ψ(χ)−

(1χu

)β1−β2Φ(χ)Ψ(χu)

](

1χd

)β1−β2Φ(χu)Ψ(χd)−

(1χu

)β1−β2Φ(χd)Ψ(χu)

,

(5.15)

where β1 > 0 and β2 < 0.

Let ρ ,(

1χd

)β1−β2Φ(χu)Ψ(χd) −

(1χu

)β1−β2Φ(χd)Ψ(χu), then the above

becomes

Pd(χ) =1

ρ

[(χ

χd

)β1 (χβ2−β1 Φ(χu) Ψ(χ)− χβ2−β1u Φ(χ)Ψ(χu)

)]. (5.16)

By analogy, we can derive the price Pu of a symmetric claim which pays

1 DKK when χ reaches χu before the lower boundary has been reached:

Pu(χ) =1

ρ

[(χ

χu

)β1 (χβ2−β1 Φ(χd) Ψ(χ)− χβ2−β1u Φ(χ) Ψ(χd)

)]. (5.17)

The two claims priced above could be seen as Arrow–Debreu securities in

the dual state of nature system i ∈ d, u, with either low or high earnings

being realised. Thus, the existence of such claims implies the completeness

of our system, and we are able to price any security in this market.

Define a new claim that continuously pays aχt+b ∀t. Using Lemma 1 it is

easy to see that the price of this claim is given by aχt+δθ+η, with α = ar+κ

,


δ = aκr(r+κ)

, and η = br. Consider now a general claim which is contingent on

χ and pays out a continuous dividend aχt + b ∀t as long as χ remains within

(χd, χu). When χ either falls to χd or reaches χu, the claim yields the closing

one-off payment Υi. The price of this claim could be derived as41

Υ(χ, θ) = aχ+δθ+η+[Υd−(aχd+δθ+η)]Pd(χ)+[Υu−(aχu+δθ+η)]Pu(χ).

(5.18)

The intuition behind this equation is trivial. As long as the earnings oscillate

within the specified interval, at any time t the value of the payoff offered by

the claim is aχt + δθ + η; as soon as the state variable reaches any of the

boundaries, the claim pays out Υi, but the net inflow is lower due to the lost

future value of the dividend stream and is thus reduced to Υi−(aχi+δθ+η).

The discounted value of the net payoff is obtained through the price of the

Arrow–Debreu security corresponding to that state of nature, Pi(χt).42

Inserting expressions for Arrow–Debreu prices from (5.16) and (5.17) into

the price function gives

Υ(χ, θ) = aχ+ δθ + η + [Υd − (aχd + δθ + η)]

· 1

ρ

[(χ

χd

)β1 (χβ2−β1Φ(χu)Ψ(χ)− χβ2−β1u Φ(χ)Ψ(χu)

)]+ [Υu − (aχu + δθ + η)]

· 1

ρ

[(χ

χu

)β1 (χβ2−β1Φ(χd)Ψ(χ)− χβ2−β1u Φ(χ)Ψ(χd)

)].

(5.19)

To formally demonstrate the required homogeneity property it needs to be

41 Note that even though the pricing function depends only on the current level ofearnings χ, wwe also make the long-term mean value of earnings θ an explicit argumentsince this parameter is fundamental in our model.

42 We will omit the time subscript in further derivations to make notation less confusingand reserve the subscript solely for denoting the state of nature.


proven that Υ(λχ, λθ) = λΥ(χ, θ).43 Since the second and third terms in

(5.19) are symmetric, it suffices to show that this relation holds for any one

of them. We will consider the downside case, ignoring the other term.

Υ(λχ, λθ) , Υ(λχ, λθ;α, δ, λη, λχd, λχu, λΥd, λΥu)

= aλχ+ δλθ + λη + [λΥd − (aλχd + δλθ + λη)]

· 1

ρ′

[(λχ

λχd

)β1 ((λχ)β2−β1Φ(λχu)Ψ(λχ)

−(λχu)β2−β1Φ(λχ)Ψ(λχu)

) ],

where ρ′ is equal to ρ with the respective parameters scaled up, i.e.

ρ′ =

(1

λχd

)β1−β2

Φ(λχu)Ψ(λχd)−(

1

λχu

)β1−β2

Φ(λχd)Ψ(λχu)

= λβ2−β1[χβ2−β1d Φ(λχu)Ψ(λχd)− χβ2−β1

u Φ(λχd)Ψ(λχu)].

Plugging that into Υ(λχ, λθ) gives after some simplification

Υ(λχ, λθ) = λ(aχ+ δθ + η) + λ[Υd − (aχd + δθ + η)]

(χ

χd

)β1· χ

β2−β1Φ(λχu)Ψ(λχ)− χβ2−β1u Φ(λχ)Ψ(λχu)

χβ2−β1d Φ(λχu)Ψ(λχd)− χβ2−β1

u Φ(λχd)Ψ(λχu).

After noticing that Φ(λχ, λθ) = Φ(χ, θ) and Ψ(λχ, λθ) = Ψ(χ, θ), the above

43 It should be noted that the parameters expressed in monetary units, e.g. η thatenters the claim price, are also scaled up accordingly by the same factor of λ.


becomes

Υ(λχ, λθ) = λ

(aχ+ δθ + η + [Υd − (aχd + δθ + η)]

·1ρ

[(χ

χd

)β1 (χβ2−β1Φ(χu)Ψ(χ)− χβ2−β1u Φ(χ)Ψ(χu)

)])

= λΥ(χ, θ),

which implies

Υ(λχ, λθ) = λΥ(χ, θ).

The formal statement of Lemma 2 could be interpreted as follows. When-

ever the upper earnings boundary is reached and the capital structure is read-

justed, the management of the firm will face the same decision in the sense

that the mechanics of the optimal capital structure model will be unchanged;

moreover, not only earnings will be scaled as in the case of a geometric Brow-

nian motion, but also the long-term mean earnings will be shifted accordingly.

A straightforward implication behind this result is that the model vari-

ables and parameters are independent of the underlying currency. For in-

stance, if we redefine the units of the model in terms of EUR instead of

DKK, this would not have any bearing on the pricing function as the latter

will just be corrected accordingly. More generally, the homogeneity property,

loosely named a “scaling feature” in Goldstein et al. (2001), allows a simple

transformation in time of the value of any claim through a time-invariant

factor which we will define later. The importance of this result is mainly

expressed in the salient simplicity of the derivations in the dynamic setting,

which is the main reason behind the ubiquitous assumption of a geometric

Brownian motion in the academic literature.


5.2.3 Optimal Capital Structure Decision Tuple

The setup is effectively the same as the one we introduced in the previous

chapter, with all the notation preserved. At any point in time the firm can

have only a single class of callable debt outstanding, with sufficiently long

finite maturity. Bankruptcy and restructuring44 decisions are endogenised

through the lower and upper earnings thresholds, χd and χu, respectively. In

case of bankruptcy, the debt holders take over the firm as a going concern

after incurring the cost of bankruptcy procedure, γ. While restructuring,

equity holders are assumed to call the entire debt at par, and issue new

debt of a larger amount at a cost q; the case of downward restructuring is

omitted. The tax structure is as follows. The corporate earnings are taxed at

the corporate tax rate τc, and debt holders face a tax rate of τi on the received

interest payment. After the coupon is paid out, the rest of the earnings is

distributed to equity holders as a dividend, and thus is taxed at τd. The

effective tax rate borne by the firm owners is denoted τe.

As a note aside, let us dwell on the factors that determine the tax advan-

tage to debt. Choose a moment in time when the firm belongs in its entirety

to the owner-manager, which means that she owns claims on both debt and

equity payouts, and consider the aggregate after-tax inflow that is entitled

to her. The debt claim would bring a coupon payment C, taxed at τi, and

thus the after-tax payoff would be (1− τi)C. The equity claim is subject to

the coupon payout and dual taxation on corporate and personal levels, which

overall yields (1− τd)(1− τc)(χ−C). Thus, the total after-tax payoff of the

owner-manager becomes

(1− τe)χ+ (τe − τi)C. (5.20)

We see that the first term in (5.20) is independent of the capital structure

decision, but the second term is strictly increasing in the factor (τe− τi) and

44 The term “restructuring” sometimes refers to the bankruptcy procedure in the liter-ature; throughout the paper we will, however, utilise it implying debt refinancing, or thereadjustment of capital structure.


represents the tax advantage to debt (the increasing relation to the coupon

is obvious as the manager owns the debt claim). Therefore, the tax benefit

of debt exists if and only if the debt is coupon-bearing and the effective tax

rate on equity holders’ payout is higher than the tax on interest income, with

the benefit increasing in the tax difference. We will assume throughout our

theoretical analysis that τe > τi, and will demonstrate the impact of the tax

rates on optimal capital structure while presenting our numerical results in

Section 5.3.

It is important to note that the expression for the tax advantage to debt

from above has an important implication: it could potentially result into

an overstated leverage. The caveat stems from the fact that the assumed

tax schedule is perfectly symmetric, meaning that coupon payments are de-

ductible even when the firm’s current earnings are insufficient to service the

debt, i.e. χt < C. Such taxation scheme favours debt and thus may lead

to the situation when the firm is so levered that the coupon exceeds even

the initial level of earnings. The key problem is that the tax advantage to

debt is independent of the earnings level, and the conventional method to

resolve this issue is to introduce another parameter ε ∈ [0, 1], which repre-

sents the proportional reduction of the effective tax rate when the earnings

net of interest (EBT) become negative. The tax rate in that case will be ετe,

thereby diminishing the benefit of leverage. However, although Christensen

et al. (2002) confirm that in a model with debt renegotiation the assumption

of full tax deduction (ε = 1) is unreasonable and very frequently results in

negative net operating income, they also show that under the general setting

this effect is somewhat minor, and changes in the main results are negligible.

Therefore, in our analysis we will assume the symmetric tax schedule.

To determine the optimal capital structure we will first derive values of

debt and equity, utilising Lemma 2. In actual fact, both of these claims

are special cases of a general claim considered in the previous subsection,

and thus could be easily priced according to Υ(χ, θ), with the corresponding

model parameters that need to be determined.


Let χ0 be the initial level of earnings defined when the first debt issue

occurred, and d ∈ (0, 1) and u ∈ (1,∞) denote the factors for obtaining the

downside and upside earnings levels, respectively. Note that χ0 will natu-

rally change through time when subsequent restructurings will take place, so

it should generally be denoted as χtj , where tj is the time of the jth read-

justment of leverage; we will nevertheless consider only the case of the very

first change in capital structure, since all the other ones will be identical.

It follows from the complete solution to the ODE derived in Subsection 5.2.1

that the value of the debt claim is given by

D(χ) = d1χβ1Φ

(−β1,−2β1 + 2 +

2κ

σ2,

2κθ

σ2χ

)

+ d2χβ2Ψ

(−β2,−2β2 + 2 +

2κ

σ2,

2κθ

σ2χ

)+

(1− τi)Cr

, (5.21)

where

β1,2 =2κ+ σ2 ±

√(2κ+ σ2)2 + 8rσ2

2σ2, β1 > 0, β2 < 0.

The equity claim is priced in a similar fashion by applying Lemma 1:

E(χ) = e1χβ1Φ

(−β1,−2β1 + 2 +

2κ

σ2,

2κθ

σ2χ

)

+ e2χβ2Ψ

(−β2,−2β2 + 2 +

2κ

σ2,

2κθ

σ2χ

)

+ (1− τe)(θ

r+χ− θr + κ

)− (1− τe)C

r. (5.22)

Now we need to specify the appropriate boundary conditions to find the

constants d1, d2, e1, and e2. It should be emphasised that the derivations

that follow rely heavily on Lemma 2, whereby we are able to scale optimal

coupon as well as restructuring and default boundaries by u = χuχ0

.45 For now

45 Since the downward restructuring option is omitted—as is the case when debt holders


the coupon C is taken as given; we will state the optimisation programme

it solves after specifying the system of equations needed to price the claims.

Without loss of generality, we let χ0 = 1 and consider all the model variables

in monetary terms to be per unit of earnings.

When earnings fall to χd, the equity holders will choose to default and the

firm will be taken over by debt holders, as we rule out any debt renegotiation

possibilities. An important point is that the company is not liquidated at

default, but is rather maintained to continue operations after the transfer of

control. Thus, the debt value at the lower boundary after correcting for the

bankruptcy costs is

D(χd) = (1− γ)(1− τe)(θ

r+dχ0 − θr + κ

). (5.23)

The equity holders will give up all control rights, so the value of their

claim at default is zero:

E(χd) = 0. (5.24)

When earnings rise to χu, the current debt is called in its entirety. Note

that here we assume that the equity holders can always call the debt at par,

but relax this assumption later. The value of the debt claim at the upper

earnings threshold is

D(χu) = D(χ0). (5.25)

Immediately after the debt is paid back the equity holders will face the same

capital structure choice as they did initially, apart from the fact that the

earnings would now amount to χu. They will optimally determine the new

leverage and debt structure (interest payment in our case) so that the equity

are allowed to relever the firm afterwards and effectively become the new equity holders—scaling down by a corresponding factor d = χd

χ0is not considered.


claim will be valued as

E(χu) = [E(χ0) + (1− q)D(χ0)]u−D(χ0), (5.26)

taking into account the restructuring costs q, which are calculated as a per-

centage of the principal of the new debt and could be interpreted as issuance

expenses paid to a broker, e.g. an investment bank arranging the placement.

The interpretation of the equity claim may seem a little ambiguous at this

stage as it appears that it embraces both equity and debt instruments, but

we will elaborate on this below.

To make further analysis more intuitive, we also introduce two additional

variables defined as

Π , E(χ0) + (1− q)D(χ0), (5.27)

Σ , D(χ0). (5.28)

As has been briefly mentioned in Chapter 4, Π denotes the maximum

potential firm value, which is the price the owner-manager who solely owns

the EBIT-generating machine can sell the firm for. We prefer to call this

value “potential” as the entrepreneur due to e.g. high risk aversion or per-

sonal preferences might choose a suboptimally low leverage or even not have

any debt at all, and thus forego the tax benefit of debt entirely. In case an

acquirer with different preferences, e.g. a PE fund, takes over the company,

the offer price will reflect the potential to increase the total firm value by

taking on debt and exploiting the tax shield, and this is the price the current

equity owners would like to maximise. This principle is central to the LBO

valuation technique commonly applied by investment banks nowadays, but

might also be encountered in other deals where debt financing can potentially

play an important role after the target is acquired. Thus, Π is an abstract

variable which explicitly takes into account the full potential of value cre-

ation from the optimal financing viewpoint. Variable Σ in (5.28) denotes the


principal of the current debt, so the definition implies that debt is issued

at par. Finally, it is convenient to express leverage in terms of these two

variables: L = ΣΠ

.

A careful reader would notice that in the boundary conditions d and u

were assumed to be deterministic, but in fact they must be obtained en-

dogenously through the smooth-pasting conditions, implicit in the capital

structure decision of the equity holders. The two smooth-pasting conditions

stipulate the exact points in time when the shareholders should choose to

default or restructure. From a game theory perspective, they can be seen

as the binding incentive compatibility constraints, ensuring that it is always

optimal to declare bankruptcy when the earnings fall to χd and to lever up

the firm when the earnings reach χu. The conditions are as follows:

∂E(χ)

∂χ

∣∣∣∣χ=χd

= 0, (5.29)

∂E(χ)

∂χ

∣∣∣∣χ=χu

= Π. (5.30)

The interpretation of these conditions is fairly straightforward if one recog-

nises that the decision faced by the equity holders represents a classic optimal

stopping problem. The solution to such problem is the set of the smooth-

pasting conditions complemented by the corresponding value-matching con-

ditions (represented by the respective boundary conditions from above). Con-

sider, for example, the determination of the lower earnings threshold, χd. If

the low-contact condition did not hold and the value function were minimised

at some χd < χd, then it would have been optimal for the equity holders to

default after the earnings fall to χd; the reasoning in the opposite case is

analogous, and the two cases together reveal the logic behind (5.29): if the

condition holds, it is optimal for the equity holders to default exactly when

the earnings equal the lower boundary level χd.

Overall, equations (5.23)–(5.30), when combined, form the system of eight

nonlinear algebraic equations in eight unknowns, d, u, d1, d2, e1, e2, Π,


Σ, which could be reduced to two equations in the pair of unknowns d,

u. Due to the nonlinearity of the system as well as numerous confluent

hypergeometric functions being involved in each equation, it is not possible

to obtain analytical solutions, and we will solve the system using numerical

methods in the next section.

Having described the procedure for obtaining prices of the debt and equity

claims, we can state the simple optimisation programme that the owner-

manager will have to solve to find the optimal initial coupon:

C∗ = arg maxC

Π, (5.31)

which means that the owner-manager will choose the debt structure so as to

maximise the potential firm value. (5.31) cannot be solved numerically for

the same reasons as the system of equations described earlier, and we will

apply nonlinear local optimisation methods to solve it in Section 5.3.

Thus, we are now ready to formulate what we call the optimal capital

structure decision framework and the tuple that closes the model and pro-

vides the final solution. The operations in the decision framework are as

follows. Before the initial debt issuance the owner-manager computes the

optimal coupon C∗ in accordance with (5.31); she also finds the optimal

earnings thresholds for restructuring and default as the solution to the sys-

tem of equations (5.23)–(5.30), thus implicitly determining the horizon of

the capital structure decision; finally, she solves for the respective debt and

equity values and determines the optimal leverage L∗. Simultaneous exe-

cution of the decision framework yields the tuple 〈C∗, d∗, u∗,Σ∗,Π∗〉, which

represents the generalised solution to the problem of the owner-manager.46

46 Note that the explicit decision variables are only C∗, d∗, and u∗, while the optimalleverage is directly obtained once they have been determined.


5.3 Numerical Results and Model Implica-

tions

The aim of this section is to compare the performance of the dynamic cap-

ital structure framework presented above to that of existing models, and to

investigate whether the assumption of an alternative underlying stochastic

process brings the optimal leverage closer to observed ratios. We carry out

extensive numerical simulations and show both cross-model comparisons as

well as general relationships under our framework.

5.3.1 Note on Numerical Simulations

Before presenting the results of our model we will consider some of the aspects

underlying the numerical procedure utilised to obtain them. We should stress

that since numerical methods form a vast independent research field, we do

not aim to cover all the nuances of numerous advanced techniques involved,

but rather try to sketch the algorithms applied in our numerical simulations

and point out the main inherent drawbacks. We will focus primarily on the

methods that are realised through known built-in functions in traditional

technical computing programmes, e.g. Mathematica or MATLAB, as these

are the ones most commonly used by researchers.

The two mathematical problems in the numerical procedure we carry out

are solving a system of nonlinear algebraic equations and running nonlinear

local optimisation. We will omit the discussion of the optimisation methods

as there is plenty of systematised literature on the topic47; moreover, our local

optimisation programme nests the system of nonlinear algebraic equations,

and the evaluation procedure revealed that most of the computational issues

resulted from the latter.

Numerical algorithms utilised to solve the above problems could be broadly

split into gradient-based methods and direct search methods. The former

ones utilise gradients (or Jacobians, in vector calculus) or Hessians, while the

47 See, for example, Nocedal and Wright (2006).


latter ones do not. Such general distinction is logical as the use of derivatives

could give rise to significant complications as will be discussed below. Direct

search methods, on the other hand, tend to be computationally more expen-

sive, but are more robust. The most widespread fundamental root-finding

algorithms for dealing with nonlinear systems are: Newton’s method, the

secant method, and Brent’s method, with only the first one belonging to the

family of gradient-based techniques. Interestingly, Newton’s method, despite

having a very long history, is still underlying in its different variations the

majority of built-in functions, and is considered a very powerful technique

with quadratic convergence at a simple root and linear convergence at a

multiple root; yet, it still bears certain limitations.

The method can be loosely described as follows. Assume ∃p ∈ [a, b] | f(p) =

0. If f ′(p) 6= 0, then ∃δ > 0 such that the sequence pn∞n=0 defined by the

recurrence relation

pn+1 = g(pn) = pn −f(pn)

f ′(pn), n = 0, 1, 2, . . .

will converge to p for any initial approximation p0 ∈ [p− δ, p + δ].48 One of

the central problems related to the non-convergence of the algorithm is its

dependence on the initial approximation to the root, p0. It is important to

stress here that since equations in our system involve complicated expressions

with nonlinearity of varying degree and numerous confluent hypergeometric

functions, there is no systematic procedure for finding all solutions, even

numerically. Therefore, if the starting approximation is not sufficiently close

to the desired root, the sequence is likely to converge to some other root.

Moreover, an improper starting point may cause an infinite cycle with the

sequence being repeated or almost repeated, thus preventing convergence

entirely. Extensive numerical tests reveal that the behaviour of the functions

in our system is rather unstable, and as a result even a slight change in initial

approximation may indeed have a tangible impact on the solution. Another

48 Since f(p) = 0 it is easy to see that Newton’s method is executed by computing afixed point of the iteration function g(x), i.e. g(p) = p.


aggravating issue stems from the use of a derivative of a function in the

iteration. The derivative may turn out to be zero at a particular iteration,

yielding the division-by-zero error, and the value obtained may not be an

acceptable approximation to the root; the derivative at the root may not

exist, and the sequence might in fact be diverging; if the derivative at the

root is discontinuous, the convergence may never take place in an acceptable

neighbourhood of the root; further, there are other less obvious examples that

could lead to potentially distorted results, e.g. in the case when |g′(x)| ≥ 1

on an interval containing the root p, there is a chance of divergent oscillation

(Mathews and Fink, 1999, Section 2.4).49

The secant method is effectively a finite difference approximation of New-

ton’s method and thus has the advantage of avoiding the use of derivatives:

pn+1 = g(pn, pn−1) = pn − f(pn)pn − pn−1

f(pn)− f(pn−1), n = 1, 2, . . .

However, being a two-point iteration mechanism, it exhibits a serious draw-

back as well, as both of the initial points should be close to the root. Besides,

the convergence of this method is normally slower, and the tests on our sys-

tem do not discover its superiority to Newton’s method.

Brent’s method combines the bisection method with the secant method,

imposing some extra conditions at each iteration. The validity of this method

in our case is severely undermined, however, due to the absence of function

continuity, and the results are clearly inferior to those produced under New-

ton’s method. We also try to implement other algorithms, viz. Muller’s

method and Steffensen’s method, but irregularities in the model output still

persist. It appears that the key problem lies in the somewhat high sensitivity

of discussed methods to the initial approximation(s) and thus the algorithm

49 Usually most of the built-in functions offer additional options to Newton’s method,supposed to mitigate some of the mentioned pitfalls. Examples include varying the numberof iterations, controlling for tolerance, introducing a damping factor—which should ideallybe chosen to be the multiplicity of the desired root, to speed up the convergence—orincreasing the precision beyond machine precision. However, as the tests show, if the levelof function complexity is too high, applying these techniques would still not guaranteethat all the issues are rectified.


that is less sensitive or insensitive to the starting point would be desired.

A possible example of such algorithm—the modified Newton’s method—is

described in Atluri, Liu, and Kuo (2009), but its implementation is outside

of the scope of this thesis. Thus, in our final programme module we ap-

ply the damped Newton’s method with additional specifications, and even

though we do conduct an extensive search procedure to choose a correct ini-

tial approximation, the obtained results should nevertheless be interpreted

carefully.

Another important issue related to the numerical procedure is the si-

multaneity of the optimal capital structure decision framework, which may

create additional noise in the results. The problem is that the system (5.23)–

(5.30) should be solved for a given coupon value, while the optimal coupon

is obtained from the optimisation programme (5.31) which in turn implies

exogenously given default and restructuring boundaries. Since the entire de-

cision framework has to be solved numerically, the described simultaneity

stipulates a high dependence of results on the initial value of the coupon.

In an attempt to rectify this issue, we iteratively search for an appropriate

starting point which produces a relatively stable output.

5.3.2 Numerical Analysis

We start out by benchmarking our results against those produced under

the mean-reverting process in the static setting, considered in Sarkar and

Zapatero (2003). Note that to carry out a correct comparison we apply

the same base case parameters that are used in their model, even though

the choice is not easily justifiable.50 Utilising the notation from previous

50 Consider, for instance, bankruptcy costs of 50%, which are the same as in Leland(1994). The value is unreasonably high, contradicting most empirical estimates, and isespecially surprising given that, for example, a well-developed bankruptcy procedure inthe US clearly does not reflect the fact that on average half of the firm value is lost upondefault. Further, the risk-free rate of 7% does not seem to match the historical data eitheras the 30-year US Treasury bond yield, which is a conventional proxy for the risklesssecurity, averaged roughly 5% in 1998–2003.


chapters, we set r = 7%, γ = 50%, τc = 15%, τi = 0%51, σ = 40%, κ = 0.1,

and θ = 1.0.

Hypothesis 1. The dynamic mean-reverting model predicts, ceteris paribus,

lower optimal leverage and optimal coupon than the static mean-reverting

model does.

Static MR

Dynamic MR

0.3 0.4 0.5 0.6 0.7 0.8Σ

0.65

0.70

0.75

0.80

0.85

L*

(a) Leverage and earnings volatility

Static MR

Dynamic MR

0.06 0.08 0.10 0.12 0.14 0.16Κ0.2

0.3

0.4

0.5

0.6

0.7

0.8C*

(b) Coupon and speed of mean reversion

Dynamic MR

Static MR

0.04 0.06 0.08 0.10 0.12 0.14 0.16Κ0.2

0.3

0.4

0.5

0.6

0.7

0.8

L*

(c) Leverage and speed of mean reversion

Figure 5.1. Comparison between dynamic and static mean-reverting models.Base case parameters are: r = 7%, γ = 50%, τc = 15%, τi = 0%, σ = 40%, κ = 0.1, andθ = 1.0. Numerical results of the static model in Sarkar and Zapatero (2003) are taken asgiven. Note that (a) assumes a different value for the speed of mean reversion: κ = 0.2(as in Sarkar and Zapatero), which explains a difference in leverage levels.

The juxtaposition in Figure 5.1 demonstrates that the optimal leverage is

indeed lower in the dynamic model—the result reminiscent of the comparison

51 Even though parameter τ in Sarkar and Zapatero (2003) is meant to take into accountpersonal and corporate taxation, it effectively plays the role of the corporate tax in theclaim valuation, and thus in the capital structure decision. Moreover, tax on interestincome does not enter the value of the debt claim, and debt holders receive the couponpayment in full.


between dynamic and static GBM-based models.52 The interpretation is also

similar: in the static model the leverage level is too high as the management

needs to optimise given the infinite expected earnings path and thus chooses

to commit to the capital structure that would yield the largest net gain in

the long term; however, as soon as the option to relever the firm in the future

is granted, a more conservative capital structure is chosen upfront, optimally

balancing bankruptcy costs of debt and the tax shield benefits already in

the short run. Thus, the dynamics of the model enables a more accurate

calibration of the debt level, and the latter is lower as a consequence. Note

that the difference between the leverage levels in dynamic and static models

is not as sharp as in the case when earnings follow a geometric Brownian

motion, which could be explained by the inherent stability of the mean-

reverting process dictated by reversion to mean and bounded conditional

long-term variance. Applying the same reasoning as above it is easy to see

why the optimal coupon is lower under the dynamic setting.

The direction of relationships is analogous between the two models, as

expected. The optimal leverage is decreasing in earnings volatility, which

implies that firms try to neutralise the effect of increased business risk, to

prevent the default probability from soaring. Interestingly, the gap between

the optimal leverage levels widens as volatility grows substantially. This re-

flects the logic described above: the implicit time horizon of the optimisation

problem faced by the management is so long under the static setup that the

volatility elasticity of leverage is very low. On the other hand, in the dynamic

model managers are much more cautious—knowing that the opportunity to

lever up will still be present in the future—which is further amplified by the

fact that in this case there is also an upper earnings threshold present and

52 Here we should stress that since the mathematical representation of the programmewe are solving is extremely computationally-intensive due to the presence of numerousconfluent hypergeometric functions and nonlinearity of varying degree, it is not possible tocreate continuous graphical forms, and therefore we resort to the procedure of discretisationand interpolation. The unstable behaviour of these functions also explains occasionalnarrowness of considered intervals.


restructuring is costly.53 Furthermore, as the speed of mean reversion aug-

ments, the earnings converge to their long-term mean faster, so the cash flow

stream becomes more predictable. This effect shows that the mean reversion

speed counteracts the influence of volatility which explains why the optimal

leverage rises. Hence, direct relationships in Results 2 and 3 from Sarkar and

Zapatero (2003) are confirmed under the dynamic setting.

Model dynamics also has an important implication for the lower earnings

threshold:

Hypothesis 2. The static GBM-based model offers a low default bound-

ary, while the static mean-reverting model suggests a much higher bankruptcy

threshold. The intermediate value is reached when the dynamics is introduced

under the assumption of mean reversion, i.e.

χStaticGBMd < χDynamicMRd < χStaticMR

d

and the threshold is further decreased when the earnings converge slower.

It should firstly be noted that the default boundary in the GBM-based

model is predicted to be lower than that in the static mean-reverting model.

This is due to the fact that a short-term downswing in earnings of the same

magnitude would be perceived differently in the two cases, the former being

generally less sensitive. Equity holders are aware that in the static GBM-

based model the earnings will keep following a positive trend, dictated by

the drift, while in the static mean-reverting model they are bounded by the

occasional fluctuation above the long-run mean, and thus it will be unlikely

to recoup the incurred losses as fast. Besides, the default earnings threshold

in the dynamic mean-reverting model is expected to be lower than that in

the static mean-reverting model. The intuition behind this is simple: the

possibility of altering the capital structure dynamically will, ceteris paribus,

53 It should be pointed out that the comparison of the discrepancy in e.g. optimalleverage across comparative statics for different parameters would not be valid as there isno bijective mapping between them.


Static GBM

Dynamic MR

Static MR

0.2 0.3 0.4 0.5 0.6 0.7Σ0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7d

(a) Default boundary and earningsvolatility

Static GBM

Dynamic MR

Static MR

0.10 0.15 0.20 0.25Κ

0.1

0.2

0.3

0.4

0.5

0.6

0.7d

(b) Default boundary and speed of meanreversion

Figure 5.2. Comparing default boundary in the dynamic mean-revertingmodel, the static mean-reverting model, and the static GBM-based model.Base case parameters are: r = 7%, γ = 50%, τc = 15%, τi = 0%, σ = 40%, κ = 0.1,and θ = 1.0. Results under the static mean-reverting model are simulated by solvingnumerically the system in Sarkar and Zapatero (2003).

enhance the valuation of the firm and the shareholders would try to salvage

such firm even if its earnings fall to the bankruptcy level of the static case.

Consequently, what is left to determine is the relationship between the

threshold values in the case of the dynamic mean-reverting and the static

GBM-based models. This is easy to see if one recognises that even though

the earnings are no longer bounded when the capital structure dynamics is

incorporated in the mean-reverting setup—since the long-term mean will be

shifted in time—the overall growth will still be slower due to the fact that

the drift becomes negative as soon as the current value exceeds the long-term

mean. In fact, the diverging nature of the geometric Brownian motion process

stipulates a much lower boundary value: even with a somewhat moderate

volatility estimate of 25%, equity holders will declare bankruptcy only after

the earnings drop to a quarter of their initial value.

Figure 5.2 confirms the statement of Hypothesis 2 for varying earnings

volatility and speed of mean reversion. From the direct relationship between

the latter and the default threshold it is clear that as the speed of earn-

ings convergence to the long-term mean slows down, our model suggests a

lower boundary value. Moreover, we also find that if the industry outlook is


improved, i.e. if analysts predict higher expected long-term earnings, then

the default boundary is lowered. This effect could be explained by the fact

that given a more positive forecast, equity holders will become more patient

and thus even when earnings fall to the level when default would have previ-

ously been declared, they will still keep the company alive, anticipating the

optimistic prospects.

The subsequent numerical results are based on our set of parameters

which is represented by the latest available Danish data as well as empiri-

cal findings. We set r = 3.5%, which corresponds to the average yield for

2008–11 on the Danish government bonds with longest maturity available,

10 years; Danish corporate tax rate is τc = 25%54, and dividends are taxed at

τd = 28%, yielding an effective tax rate for equity holders of τe = 46%; inter-

est income in Denmark is taxed at τi = 37%. The rest of the parameters are

determined as follows. Recent empirical studies estimate bankruptcy costs to

vary within the interval 2–15% of the firm value at bankruptcy (Strebulaev,

2007; Bris, Welch, and Zhu, 2006; Bris, Schwartz, and Welch, 2005). We only

consider direct bankruptcy costs as the indirect costs of financial distress have

a very high variation; even direct costs of bankruptcy procedure could change

substantially depending on firm-specific characteristics, e.g. size (Grinblatt

and Titman, 2002, p. 560), and since the capital structure optimisation is

most relevant for larger corporations, we choose a more conservative esti-

mate of γ = 5%. Note that the difference in assumed bankruptcy costs is

striking—in the dynamic model we do not have to assume such unreasonably

high values as in Sarkar and Zapatero (2003) or Leland (1994) to obtain re-

sults consistent with practice. We assume readjustment costs q to comprise

1% of the new debt issue, consistent with empirical estimates (Davydenko

and Strebulaev, 2007; Kim, Palia, and Saunders, 2003) and fee structures

proposed by investment banks for larger financings across debt capital mar-

kets. There are almost no papers that document the volatility of operating

income and since it is known to be generally lower than that of net income

54 To preserve the homogeneity property of the model we ignore the variability of taxrates across firms or time.


(Petrovic, Manson, and Coakley, 2009), we apply a conservative estimate of

25%. Finally, the two key parameters describing the mean-reverting nature

of the process—the long-term mean and the convergence speed—are set to

θ = 1.0 and κ = 0.1, respectively. It is important to emphasise that values of

mean reversion parameters play more an indicative role, since they are very

industry-dependent per se and as such are merely meant to facilitate mod-

elling of particular industries. Widely-varying discrepancy of mean reversion

patterns in earnings and profitability between industries has been confirmed

in the literature (Nordal and Næs, 2010; Altunbas, Karagiannis, Liu, and

Tourani-Rad, 2008).

Let us now turn to comparing the output of numerical simulations under

the dynamic model with mean reversion and the dynamic GBM-based model

presented in Chapter 4 to draw some more subtle inferences.

Hypothesis 3. The dynamic mean-reverting model suggests lower optimal

leverage and higher restructuring frequency than the dynamic GBM-based

model does, and the former relationship is more pronounced if the long-term

industry outlook is worsened or the speed of earnings convergence is slowed

down.

Conclusions in Hypothesis 3 are reinforced by Figure 5.3. First, we ob-

serve that assuming the underlying mean-reverting process brings down the

optimal leverage compared to that under a geometric Brownian motion. This

phenomenon has an intuitive explanation: GBM-based models always imply

a positive earnings trend due to the nature of the drift, thus offering a more

favourable setup as opposed to the mean-reverting model, where drift could

take both positive and negative direction depending on the current level of

earnings. Moreover, it is important to point out that assuming the same

volatility of earnings for both models may downplay the difference in opti-

mal leverage. This follows from the fact that our process has an inherently

bounded variance as a result of reversion to mean, by contrast with a geo-

metric Brownian motion whose conditional variance is infinite. Therefore, to

carry out a more accurate comparison the earnings volatility assumed in the


Dynamic GBM

Dynamic MR

0.26 0.27 0.28 0.29 0.30 0.31 0.32Σ

0.24

0.26

0.28

0.30

0.32

0.34

0.36

L*

(a) Leverage and earnings volatility

Dynamic MR, Σ = 40 %

Dynamic MR, Σ = 25 %

Dynamic GBM

0.03 0.04 0.05 0.06 0.07Γ

0.25

0.30

0.35

0.40L*

(b) Leverage and bankruptcy costs

Dynamic MR

0.10 0.11 0.12 0.13 0.14 0.15Κ

0.35

0.40

0.45

0.50

0.55

0.60

0.65L*

(c) Leverage and speed of mean reversion

Dynamic MR

0.98 0.99 1.00 1.01 1.02 1.03Θ

0.315

0.320

0.325

0.330

0.335L*

(d) Leverage and long-term mean

Dynamic GBM

Dynamic MR

0.004 0.006 0.008 0.010 0.012 0.014q1.05

1.10

1.15

1.20

1.25

1.30

1.35

1.40

u

(e) Restructuring boundary andrestructuring costs

Figure 5.3. Comparison between dynamic mean-reverting and dynamic GBM-based models. Base case parameters are: r = 3.5%, τc = 25%, τd = 28%, τi = 37%, γ =5%, q = 1%, σ = 25%, θ = 1.0, κ = 0.1, and µ = 2%. (b) additionally exhibits thedynamic mean-reverting model with an increased volatility level of 40%, while all otherparameters kept unchanged.


base case should be higher for the mean-reverting model. It could be easily

shown that the stationary variance of the process given by (5.1) is

V[χs] =θ2σ2

2κ− σ2, (5.32)

and so we increase the initial volatility level accordingly, to bring the vari-

ance of the process closer to that of a geometric Brownian motion. The

outcome is demonstrated in Figure 5.3(b): we see that the leverage in our

model is lowered, making the gap between the two considered models more

pronounced.

Further, one should note that the above discrepancy was observed for

the base case with implied neutral outlook on earnings and average mean

reversion speed. If the long-term earnings forecast turns negative, i.e. the

predicted mean value of operating income is lower than the current value,

then the optimal leverage would be decreased, thus also widening the gap

between the GBM-based and mean-reverting models. The same impact has

slowing down the earnings convergence speed which reduces the stability of

the earnings stream. Both effects are evident from Figures 5.3(c,d), and

could be observed in practice as well, e.g. during the period of economic

stagnation: at its beginning, sector analysts would publish more negative

forecasts, and after that would maintain them for longer-than-usual as the

economy keeps stagnating and recovery of the real sector is slow. Besides, we

also see that the optimal leverage is much more sensitive to the changes in the

mean reversion speed. This could be explained by the fact that altering the

convergence speed has an immediate bearing on the firm’s activities through

increasing the certainty of cash inflows, as opposed to the long-term forecast

that only has a partial effect at the current point in time and comes into play

more gradually.

Higher restructuring frequency for the mean-reverting model could be ob-

served in Figure 5.3(e).55 If the earnings deviate too much from the long-term

55 Our interpretation of frequency is akin to that in physics—viz. frequency of waves—toemphasise the continuity of the process. Therefore, higher frequency does not necessarily


0.05 0.10 0.15 0.20 0.25 0.30 0.35Σ0.6

0.7

0.8

0.9

1.0

1.1

1.2

d,u

(a) Earnings boundaries andearnings volatility

0.04 0.06 0.08 0.10 0.12 0.14 0.16Κ0.4

0.6

0.8

1.0

1.2d,u

(b) Earnings boundaries and speed ofmean reversion

Figure 5.4. Upper and lower boundaries in the dynamic mean-reverting model.Base case parameters are: r = 3.5%, τc = 25%, τd = 28%, τi = 37%, γ = 5%, q =1%, σ = 25%, θ = 1.0, and κ = 0.1.

mean, they will start reverting back straight away, and thus the restructuring

boundary should be lower compared to that in the GBM-based model, oth-

erwise the threshold may never be reached. It is also worth mentioning that

the difference is quite substantial: in the base case, for example, the expected

earnings growth up to the restructuring point is almost 15 percentage points

higher in the GBM-based model.

Finally, we will point out some general relationships under the dynamic

mean-reverting framework, so henceforth the focus is restricted solely to the

results under our model.

Figure 5.4(a) presents the dependence of the lower and upper earnings

thresholds on the earnings volatility. We see that the gap is diverging, which

reflects the fact that when the business risk is increased, both boundaries

will be altered accordingly as equity holders prefer to keep their strategic

decisions unaffected by short-run fluctuations in operating income. Higher

speed of earnings convergence has the opposite effect, as expected: more

stable businesses will decrease the upper boundary to keep exploiting the tax

advantage to debt; the other relationship is more subtle, however, because it

may not be instantly clear why it is optimal to hike the default boundary. As

imply higher number of times the boundary has been hit.


0.370 0.375 0.380 0.385 0.390Τi

0.319

0.320

0.321

0.322

0.323

0.324

0.325

0.326

L*

(a) Leverage and tax on interest income

0.46 0.47 0.48 0.49 0.50 0.51 0.52Τe

0.33

0.34

0.35

0.36

0.37L*

(b) Leverage and effective tax rate

Figure 5.5. Optimal leverage and tax structure in the dynamic mean-revertingmodel. Base case parameters are: r = 3.5%, τc = 25%, τd = 28%, τi = 37%, γ =5%, q = 1%, σ = 25%, θ = 1.0, and κ = 0.1.

the speed of mean reversion augments, equity holders will expect to finance

the debt service deficit56 from their own pockets more and more often as the

optimal coupon will be rising—cf. Figure 5.1(b)—and the situation when

the earnings are insufficient will be occurring more frequently. Therefore,

the lower boundary will be pushed up, diminishing the spread between the

boundaries, just as would be the case if the earnings volatility went down.57

From Figure 5.5 it is evident that the optimal leverage is decreasing in

the tax on interest payments and increasing in the effective tax rate imposed

on equity holders. These relationships are a direct consequence of the form of

the after-tax payoff of the owner-manager, described in (5.20). When the tax

rate on interest income rises or, alternatively, the effective tax rate declines,

the tax advantage to debt shrinks, thus reducing the incentive to utilise debt

financing which ultimately results into lower optimal leverage. The direct

impact of changes in the tax structure could also be seen from the valuation

of debt and equity claims.

56 Since asset sale or extra debt issues are not allowed in the model, consistent withprotective covenants utilised in practice, every time the situation χt < C occurs, therequired cash is provided by equity holders.

57 Perhaps one could understand this relationship a bit easier if a decrease in the speed ofmean reversion is considered. Slower earnings convergence means that they will stay belowthe long-term mean for longer, and thus, given the same volatility, the default probabilityis increased, prompting equity holders to decrease the bankruptcy threshold.


0.015 0.020 0.025 0.030 0.035r

1.145

1.150

1.155

1.160

1.165

1.170

1.175

u

(a) Restructuring boundary andrisk-free rate

0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.050r0.0

0.2

0.4

0.6

0.8

C*

(b) Coupon and risk-free rate

0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.050r0.5

0.6

0.7

0.8

0.9d

(c) Default boundary and risk-free rate

0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.050r

400

450

500

550

600

650S

(d) Credit spread and risk-free rate

Figure 5.6. Effect of varying risk-free rate on restructuring boundary, optimalcoupon, default boundary, and credit spread (in bps) in the dynamic mean-reverting model. Base case parameters are: r = 3.5%, τc = 25%, τd = 28%, τi =37%, γ = 5%, q = 1%, σ = 25%, θ = 1.0, and κ = 0.1.

A consequence of the risk-free rate hike is that the discounted value of the

tax benefit of debt received in future periods is lowered, and hence the owner-

manager chooses a higher coupon to counterbalance this effect, which in turn

enhances the debt valuation. The overall impact on the debt value—through

both increased risk-free rate and optimal coupon—is more pronounced com-

pared to that on the coupon payment and therefore debt becomes relatively

cheaper when the risk-free interest rate rises, as illustrated by the credit

spread in Figure 5.6(d).

To interpret the relationship between the restructuring boundary and the

risk-free rate first notice that as the latter rises, the optimal coupon is in-

creased, and the effect is somewhat strong. Therefore, a low riskless rate


results in a low optimal coupon, meaning that the utilisation of the tax

shield is quite weak. This induces the equity holders to set the restructuring

threshold low to begin exploiting the tax benefit of debt earlier. On the other

hand, if the risk-free interest rate is high, this implies that the tax shield is

already utilised heavily, and that the firm may already be running a high

risk of bankruptcy because the debt service constitutes a large proportion of

the earnings. This stipulates a high value of the upper earnings boundary.

The monotonicity of the relationship between the two extremes we consid-

ered is also easily shown: whenever the risk-free rate rises, this eventually

leads to increased default probability58, so the equity holders will prefer not

to restructure as often and the upper threshold will be raised. Another ar-

gument in favour of the positive relationship stems from the intertemporal

discrepancy in cash flows: the costs of restructuring are incurred now, while

the present value of the benefits is diminished as they are discounted with a

higher interest rate. This result is interesting as it contrasts with that ob-

tained in the dynamic GBM-based model of Goldstein et al. (2001), whose

tabulated simulations show that the restructuring boundary declines as the

interest rate rises. The authors do not comment on the relationship, and

even though it could be partially justified by the nature of the process—for

low values of the risk-free rate, a slight increase may lead to the decline in

the restructuring threshold as the equity holders anticipate a positive drift

in earnings and a slightly higher default probability might not matter as

much—we still find such relationship quite puzzling generally, and expect

our result to be more consistent with practice.

5.4 Extensions

In this section we will extend the analysis of dynamic models under alterna-

tive cash flow processes. Of course, possible directions of extension are nu-

58 This follows from the fact that when a coupon is higher, the low-contact conditioncombined with the boundary condition for equity at default will stipulate a higher defaultthreshold. This effect is demonstrated in Figure 5.6(c).


merous, and examples include considering endogenous finite maturity, debt

subordination, post-bankruptcy restructuring option, and so forth; technical

extensions such as e.g. stochastic interest rates or non-constant volatility

could also be considered. Nevertheless, we choose to restrict our attention to

the following two: incorporating fixed operating costs in the dynamic GBM-

based model and including a call premium in the dynamic mean-reverting

model. The former one plays a dual role as it enables us to resolve one of the

well-known issues of capital structure modelling—cash flow being bounded

at zero—and to directly compare the effects of the two changes in the nature

of the cash flow process, viz. allowing cash flow to become either negative or

mean-reverting. The second extension shows that the basic dynamic model

with mean reversion introduced in this chapter might understate the optimal

leverage if equity holders possess the option of redeeming the entire debt at

par.

5.4.1 Fixed Operational Costs

Let us first consider the model with the earnings being unrestricted. An

obvious shortcoming of GBM-based models as well as of the mean-reverting

model proposed above is that they always imply a positively-valued state

variable. Negative earnings are fairly often observed in practice, e.g. in

start-ups or in mature firms during the periods of recession, and could also

be attributed to other industry-specific factors, for instance, to the prevalence

of fixed costs in the cost structure combined with demand cyclicality or

intensified competition.

One of the ways to introduce a possibility of non-positive earnings is to

assume that the state variable follows an arithmetic Brownian motion, but

that would lead to the loss of homogeneity property, significantly complicat-

ing the modelling of capital structure dynamics. Therefore, we apply another

solution—redefining the state variable as the firm revenue and incorporating

fixed operating costs in the model. These costs, denoted F , are continuously

paid by equity holders, and thus the resulting net cash flow is χ−F −C, so


that the earnings, due to the stochastic nature of the state variable, could

take on negative values even before the debt is serviced.

In this subsection we explore the implications for optimal capital structure

that incorporating fixed operating costs in the dynamic GBM-based model

described in Chapter 4 has. The obtained results are benchmarked against

those under the dynamic mean-reverting model. Note that the inclusion of

fixed costs would lead to the following changes in the system of equations.

First, the value of the equity claim becomes lower as the costs are entirely

borne by equity holders:

E(χ) = e1χβ1 + e2χ

β2 +(1− τe)χr − µ

− (1− τe)(C + F )

r. (5.33)

Further, the boundary condition for debt at default is also altered: to pre-

clude any strategic considerations, debt holders are assumed to manage the

company just as well as equity holders did and thus have to pay the operating

costs of the same amount after taking over the firm upon bankruptcy, i.e.

D(χd) = (1− τe)(1− γ)dχ0

r − µ− (1− τe)F

r. (5.34)

After implementing the above changes, we solve for the optimal capital

structure decision tuple numerically in the same fashion as before, and apply

the same set of base case parameters, viz. r = 3.5%, τc = 25%, τd =

28%, τi = 37%, γ = 5%, q = 1%, σ = 25%, and µ = 2%. For the fixed

costs we use a moderate estimate of F = 0.2.

First we study the dependence of the key variables on operating fixed

costs in the dynamic GBM-based model, using the framework introduced in

Chapter 4 as a benchmark.

Hypothesis 4. In the dynamic GBM-based model with fixed costs, higher

operating leverage59 implies higher optimal coupon, increased default and re-

59 Strictly speaking, for operating leverage to be defined, the model should embracevariable costs as well. However, for the purposes of comparative statics analysis variablecosts are assumed to be zero, so that varying fixed operating costs would unambiguously


structuring boundaries, higher credit spread, and lower financial leverage.

Figure 5.7(a) demonstrates the reverse impact on the financial leverage.

We see that when fixed costs are assumed to be zero, the two models yield

the same optimal leverage, and when we increase the costs, the financial

leverage gradually drops. Comparison between the two base cases reveals

that even when the fixed costs amount to a moderate value of 0.2, the optimal

leverage drops by almost 10 percentage points. The intuition behind the

relationship is as follows. An increase in fixed costs implies a higher default

probability because the equity value falls to zero at a higher earnings level,

i.e. the bankruptcy threshold is augmented. Bondholders, ceteris paribus,

will demand a higher coupon as well as yield since debt becomes riskier: with

higher fixed costs even a short-term downswing dictated by volatility could

already trigger default.60 As a consequence of the interplay between these

effects, the optimal leverage goes down as well. Further, equity holders will

naturally raise the restructuring boundary to balance out the readjustment

frequency with riskiness of debt.

The cross-model comparison leads to the following hypothesis:

Hypothesis 5. The dynamic GBM-based model that accommodates fixed

operating costs, exhibits lower optimal leverage than the dynamic model with

mean reversion does, i.e.

L∗DynamicGBMFC< L∗DynamicMR < L∗DynamicGBM .

The hypothesis is reinforced by Figure 5.8 which shows the dynamics of

the optimal leverage for varying bankruptcy costs and cash flow volatility.

This result can be explained by the different nature of cash flow process mod-

ifications that we consider. Incorporating fixed operating costs in the model

stipulate the change in operating leverage.60 It may appear odd to the reader that the optimal leverage and the optimal coupon

payment are in fact moving in the opposite directions. However, we emphasise that this isprimarily due to the fact that they are both determined simultaneously from the optimalcapital structure decision framework; moreover, this relationship does not always takeplace—cf. Figures 5.1(a,b).


Dynamic GBM

Dynamic GBM with FC

0.0 0.1 0.2 0.3 0.4 0.5 0.6F

0.20

0.25

0.30

0.35

L*

(a) Leverage and operating costs

Dynamic GBM with FC

Dynamic GBM

0.0 0.1 0.2 0.3 0.4 0.5 0.6F

1.0

1.2

1.4

1.6

1.8

2.0C*

(b) Coupon and operating costs

Dynamic GBM

Dynamic GBM with FC

0.1 0.2 0.3 0.4 0.5 0.6F

0.2

0.3

0.4

0.5

d

(c) Default boundary and operating costs

Dynamic GBM with FC

Dynamic GBM

0.0 0.1 0.2 0.3 0.4 0.5 0.6F

1.2

1.4

1.6

1.8

2.0

u

(d) Restructuring boundary andoperating costs

Dynamic GBM with FC

Dynamic GBM

0.0 0.1 0.2 0.3 0.4 0.5 0.6F300

400

500

600

700S

(e) Credit spread and operating costs

Figure 5.7. Effect of varying operating leverage in the dynamic GBM-basedmodel with fixed costs (FC). The standard dynamic GBM-based model serves as abenchmark. Base case parameters are: r = 3.5%, τc = 25%, τd = 28%, τi = 37%, γ =5%, q = 1%, σ = 25%, µ = 2%, and F = 0.2.


Dynamic GBM with FC

Dynamic MR

Dynamic GBM

0.035 0.040 0.045 0.050 0.055 0.060 0.065 0.070Γ

0.25

0.30

0.35

L*

(a) Leverage and bankruptcy costs

Dynamic GBM

Dynamic MR

Dynamic GBM with FC

0.25 0.26 0.27 0.28 0.29 0.30 0.31 0.32Σ

0.20

0.25

0.30

0.35

L*

(b) Leverage and earnings volatility

Figure 5.8. Comparison between the dynamic GBM-based model with andwithout fixed costs and the dynamic mean-reverting model. Base case parametersare: r = 3.5%, τc = 25%, τd = 28%, τi = 37%, γ = 5%, q = 1%, σ = 25%, θ = 1.0, κ =0.1, µ = 2%, and F = 0.2.

has an immediate negative bearing on the cash flow process and there is no

uncertainty in the effect as the costs are constant and continuous; the effect

of introducing mean reversion in earnings is much more subtle: firstly, it is

not unambiguous—depending on the current earnings value, mean reversion

can have both positive and negative influence on the cash flow development,

and secondly, it is less pronounced if a reasonable assumption regarding the

magnitude of earnings convergence speed is made. Consequently, the opti-

mal capital structure is more sensitive to the introduction of fixed operating

costs in the model. It is worth noting, however, that the optimal leverage

predicted by this model seems to provide estimates that are lower than em-

pirical regularities, even for a moderate value of fixed costs. In the base case,

for example, it yields the leverage of 26%, as opposed to 33% observed un-

der the dynamic mean-reverting model, whereas studies investigating Danish

corporates document estimates in the range 30–35% (among others, Gaud

et al., 2007; Chen and Hammes, 2003). Plus, as shown above, the optimal

leverage drops somewhat fast when fixed costs are increased, bringing the

model forecast even further from the magnitude consistent with practice.


5.4.2 Call Premium

The dynamic mean-reverting model presented in Section 5.2 considered debt

being callable, so that equity holders could change the leverage level before

the outstanding bonds matured. More importantly, it was assumed that this

option could be exercised at no cost, potentially resulting in too frequent

restructuring. Furthermore, empirical evidence suggests that corporations

indeed do pay a premium when bonds are retired early (Mitchell, 1991; Fis-

cher et al., 1989b; Vu, 1986). Therefore, the model will now be adjusted so

that a call premium needs to be paid by equity holders when the old debt

is redeemed prior to maturity. Since the inclusion of a call premium only

affects the value of debt and equity claims at the restructuring point, only

these boundary conditions will be stated—other equations remain unchanged

from the previous section.

When the upper boundary is reached and equity holders will optimally

decide to retire old bonds, the value of debt will be given by

D(χu) = (1 + Λ)D(χ0),

where Λ denotes a call premium, modelled as a fraction of the bond value.61

Following the same logic, the equity value at restructuring is now

E(χu) = [E(χ0) + (1− q)D(χ0)]u− (1 + Λ)D(χ0).

Note the distinction between the new debt level, uD(χ0), and old debt level,

D(χ0), as restructuring costs are a fraction q of the new bond value, while

the call premium is paid only on top of the old principal. Further, as the call

premium is paid before equity holders decide on the optimal capital structure,

61 We acknowledge that empirical research documents call premia being not constant,but rather increasing in time to maturity and dependent on firm characteristics, e.g.volatility (Fischer et al., 1989b). However, for the purposes of tractability we assume a callpremium to be an exogenously given constant. Additionally, we abstain from includingany deferral period and thus allow equity holders to call outstanding bonds at any timeafter issuance.


Π and Σ from (5.27) and (5.28) are unaltered, as is the maximisation problem

in (5.31). The optimal bankruptcy and readjustment thresholds d and u are

still found using numerical procedures, and leverage is calculated through

the coupon which optimises the maximum potential firm value.

It should also be mentioned that even though some studies consider a

possibility of choosing the call premium optimally by equity holders (Hen-

nessy and Tserlukevich, 2008; Flor and Lester, 2002; Fischer et al., 1989b),

we exclude this option and instead assume that this parameter is exogenously

given—following Christensen et al. (2002) and Dangl and Zechner (2004)—

to maintain the focus on those decision variables which are more directly

associated with the capital structure decision.

To see what implications the call premium has for the optimal composi-

tion of debt and equity, let us consider some numerical results.

Hypothesis 6. If retiring the outstanding debt prior to maturity entails pay-

ing a call premium, then equity holders will choose a higher optimal leverage

and will readjust the capital structure less often.

0.00 0.01 0.02 0.03 0.04 0.05L

0.30

0.35

0.40

0.45

0.50L*

(a) Leverage and call premium

0.00 0.01 0.02 0.03 0.04 0.05L1.10

1.15

1.20

1.25

1.30

1.35

1.40u

(b) Restructuring boundary and callpremium

Figure 5.9. Effect of varying call premium on optimal leverage and restructur-ing boundary in the dynamic mean-reverting model. Base case parameters are:r = 3.5%, τc = 25%, τd = 28%, τi = 37%, γ = 5%, q = 1%, σ = 25%, θ = 1.0, andκ = 0.1.

Introducing the call premium has a positive effect on the optimal leverage.

Since the capital structure decision is made at the point where both debt


and equity claims belong to the owner-manager, the latter would favour debt

financing as the value of a premium-bearing debt is higher. Moreover, the

benefit from the enhanced debt valuation comes today, while the downside of

having to pay the premium lies only in the future, when restructuring takes

place, and thus it is less pronounced due to discounting. The mathematics

of this relationship could be easily seen from the expression for leverage, viz.

L = ΣE(χ0)+(1−q)Σ , as the nominator is always increasing faster in the call

premium because of the presence of restructuring costs in the model and the

reverse impact of the call premium on the equity value.

Furthermore, as the call premium rises, it will become more and more

expensive for equity owners to retire the old debt prematurely, and thus

they will have less incentive to restructure. Therefore, when a premium to

calling outstanding bonds before maturity is introduced, the firm will decide

to refinance less frequently, and this effect is illustrated by Figure 5.9(b).

Chapter 6

Conclusion

In this thesis we analyse in detail the development of research focusing on

modelling optimal capital structure, and attempt to extend it by proposing

the dynamic mean-reverting model that rectifies some of the shortcomings

of existing studies. The scope of analysis is further broadened to consider

an alternative modification of the cash flow process—allowing earnings to

obtain negative values. We bring together the key dynamic models and try

to uncover the implications that different frameworks have for the results

through comprehensive numerical simulations.

We start out by revisiting the pricing model introduced in the seminal

work of Merton (1974), which represents the fundamental building block for

most capital structure models. The general partial differential equation is

rederived along the Black–Scholes lines, and it is shown how this valuation

framework could be applied to price debt and equity as contingent claims;

an intuitive analogy with the option pricing model is further provided. We

also demonstrate that the oft-employed assumption of time independence—

used to make a transition from the partial differential equation to the ordi-

nary differential equation—may violate the conditions for the existence of an

equivalent martingale measure.

The first stage of our analysis is considering the basic static model akin

to that in Leland (1994). The valuation of claims is derived using the ODE

111


and corresponding boundary conditions for the case when the bankruptcy

decision of equity holders is endogenised and the optimal coupon is found

from the firm value maximisation problem. We present comparative statics

for optimally chosen leverage, coupon, and default threshold as well as for tax

advantage to debt. It is evident that under our own base case parameters—

based on the most recent Danish data and empirical estimates—which differ

from unreasonably high values applied in Leland, the optimal leverage is

too high compared to that observed in practice. Moreover, the prominent

inconsistency of this model lies in the fact that the state variable is the value

of unlevered assets which represents the price of a traded asset, thus opening

up arbitrage opportunities. Apart from that, the model embraces a very

simple setup, ignoring e.g. personal taxation, and the assumed static nature

of capital structure is very limiting as well.

Therefore, the natural progression of our analysis is introducing dynam-

ics in the model of capital structure. Rather than focusing on any particular

model, we outline a generalised framework to describe the overall family of

existing dynamic capital structure models. The state variable is the operat-

ing income, which restores the absence of arbitrage in the model. The firm

value is reinterpreted as the maximum potential firm value, which reflects the

debt financing potential at the point in time when both claims on debt and

equity belong to the owner-manager. Further, some other previous inconsis-

tencies are rectified due to a more elaborate setup, e.g. equity value is no

longer increasing in the effective tax rate. As the leverage can be readjusted,

the upper earnings boundary is incorporated, stipulating when restructur-

ing should take place; it is obtained from the high-contact condition which

makes the refinancing decision incentive-compatible from the viewpoint of

equity holders. The key insight from the comparative statics analysis of the

model is that the optimal leverage drops substantially compared to that in

the static model, and the default boundary is also lowered, reflecting the fact

that the refinancing option, ceteris paribus, enhances the firm valuation.

In the rest of the thesis we develop our own model which alters the cash

Chapter 6. Conclusion 113

flow process by assuming mean reversion. The vast majority of existing cap-

ital structure studies rests on the assumption that the state variable follows

a geometric Brownian motion since this ensures mathematical tractability

while modelling the capital structure readjustment. However, this underly-

ing process implies that the state variable—represented by earnings—could

in time reach infinitely high values, and moreover, that due to the positive

drift, its value is always expected to increase, with short-term falls being

governed only by volatility. We argue that such behaviour is perhaps more

consistent with the development of the stock price, rather than with dy-

namics of the firm fundamental, and that the assumption of mean reversion

appears to be more suitable. Not only the latter is generally more consistent

with the realities of the business world, but is also confirmed by a number of

empirical studies. Furthermore, by assuming that operating income follows

the mean-reverting process, we are able to draw a much clearer distinction

between the industries through a better control over the process—by varying

the long-term mean and the speed of mean reversion, which correspond to

the industry profitability and stability, respectively.

To our knowledge, optimal capital structure has never been modelled dy-

namically outside of the scope of a geometric Brownian motion, presumably

due to the loss of homogeneity property. We assume the modified mean-

reverting process with volatility being proportional to the current earnings

level, and derive the ordinary differential equation used for pricing debt and

equity. Further, using contingent claims analysis and state pricing, we prove

that the process still possesses the homogeneity property, which implies that

the mechanics of the model is unaltered in time. The importance of this

result is expressed in the salient simplicity of the derivations in the dynamic

setting, which is the main reason behind the ubiquitous assumption of the ge-

ometric Brownian motion in the academic literature. After that we describe

what we call the optimal capital structure decision framework—comprised

of the system of boundary conditions and smooth-pasting conditions—and

formulate how the tuple that closes it is obtained, which fully determines the


capital structure choice of the owner-manager.

We conduct extensive numerical tests, focusing explicitly on the numeri-

cal methods applied in the procedure and pointing out inevitable shortcom-

ings, the main ones being the high dependence on the initial approximation

and the difficulty of dealing with the simultaneity of the system. Numerical

results could be split into three sets: comparing our model to the static mean-

reverting model developed in Sarkar and Zapatero (2003), benchmarking our

model against the conventional dynamic GBM-based model, and examining

general relationships under our own setup. Firstly, we find that the dynamic

mean-reverting model predicts lower optimal leverage and optimal coupon

than the static mean-reverting model does. The interpretation of this result

is as follows. In the static model the leverage is too high as equity holders

are forced to choose the capital structure given the infinite expected earnings

path and thus have to optimise over the long-run horizon, while if the relev-

ering option is granted, a more conservative leverage is chosen a priori, to

optimally balance the costs of debt financing with the tax benefits already in

the short term. Secondly, the dynamic model with mean reversion suggests

lower optimal leverage and higher restructuring frequency than the dynamic

GBM-based model does, which could be explained by the difference in ex-

pectations of equity holders regarding the future development of the cash

flow under the two processes. When earnings follow a geometric Brownian

motion, the trend always remains positive, while in case of mean reversion it

varies depending on the current earnings value. Therefore, equity holders in

the GBM-governed setup are excessively optimistic and thus set the optimal

leverage too high. Moreover, we emphasise that since the stationary variance

of the mean-reverting process is finite, as opposed to that of a geometric

Brownian motion, we need to hike the base case volatility in the former case

to make a comparison more correct. This reveals an even more pronounced

gap in the optimal leverage levels. Overall, the optimal leverage in our model

is found to be in the range 30–35%, given reasonable assumptions for base

case parameters, which brings it closer to the empirical estimates than the


existing models. All relationships are studied through the prism of mean

reversion, and it is shown that the speed of earnings convergence and the

long-term mean value of earnings are indeed important parameters as their

impact on the key variables could be rather substantial.

Besides the above, we consider two additional extensions: first, the GBM-

based model is modified to take into account fixed operating costs, so that

the resulting cash flow can obtain negative values, and second, the mean-

reverting model is changed to incorporate a call premium paid on the debt

that is retired prematurely. We demonstrate that higher operating leverage

implies lower financial leverage, and find that when the firm is allowed to

have its accounts in the red, it will be less levered compared to an otherwise

identical firm whose earnings are instead mean-reverting but always positive.

Finally, it is shown that assuming debt being callable at par may lead to

understated leverage and too high restructuring frequency.

References

Abramowitz, M. and I.A. Stegun (1972). Handbook of Mathematical Func-

tions, Ninth Edition, Dover Publications Inc.

Acharya, V., J. Huang, M. Subrahmanyam, and R.K. Sundaram (2006).

When Does Strategic Debt Service Matter? Economic Theory, Vol. 29 (2),

363–378.

Alderson, M.J. and B.L. Betker (1995). Liquidation Versus Continuation:

Did Reorganized Firms Do the Right Thing? Ohio State University working

paper No. 9512.

Altunbas, Y., A. Karagiannis, M-H. Liu, and A. Tourani-Rad (2008). Mean

Reversion of Profitability: Evidence From the European-Listed Firms, Man-

agerial Finance, Vol. 34 (11), 799–815.

Ammann, M. and M. Genser (2004). A Testable Credit Risk Framework with

Optimal Bankruptcy, Taxes, and a Complex Capital Structure, University of

St. Gallen working paper series in finance No. 1.

Anderson, R.W. and S. Sundaresan (1996). Design and Valuation of Debt

Contracts, The Review of Financial Studies, Vol. 9 (1), 37–68.

Atluri, S.N., C-S. Liu, and C-L. Kuo (2009). A Modified Newton Method

116


for Solving Non-Linear Algebraic Equations, Journal of Marine Science and

Technology, Vol. 17 (3), 238–247.

Back, K. and S.R. Pliska (1991). On the Fundamental Theorem of Asset

Pricing With an Infinite State Space, Journal of Mathematical Economics,

Vol. 20 (1), 1–18.

Bajaj, M., D.J. Denis, and A. Sarin (2004). Mean Reversion in Earnings

and the Use of E/P Multiples in Corporate Valuation, Journal of Applied

Finance, Vol. 14 (1), 4–10.

Bank, M. and J. Lawrenz (2005). Informational Asymmetry Between Man-

agers and Investors in the Optimal Capital Structure Decision, SSRN working

paper series.

Battig, R.J. and R.A. Jarrow (1999). The Second Fundamental Theorem of

Asset Pricing: A New Approach, The Review of Financial Studies, Vol. 12

(5), 1219–1235.

Bhattacharya, S. (1978). Project Valuation with Mean-Reverting Cash Flow

Streams, The Journal of Finance, Vol. 33 (5), 1317–1331.

Bichteler, K. (2002). Stochastic Integration With Jumps, Cambridge Univer-

sity Press.

Black, F. and J.C. Cox (1976). Valuing Corporate Securities: Some Effects

of Bond Indenture Provisions, The Journal of Finance, Vol. 31 (2), 351–367.

Black, F. and M. Scholes (1973). The Pricing of Options and Corporate Li-

abilities, The Journal of Political Economy, Vol. 81 (3), 637–654.


Bradley, M., G.A. Jarrell, and E.H. Kim (1984). On the Existence of an

Optimal Capital Structure: Theory and Evidence, The Journal of Finance,

Vol. 39 (3), 857–878.

Brennan, M.J. and E.S. Schwartz (1978). Corporate Income Taxes, Valua-

tion, and the Problem of Optimal Capital Structure, The Journal of Business,

Vol. 51 (1), 103–114.

Bris, A., A. Schwartz, and I. Welch (2005). Who Should Pay for Bankruptcy

Costs? The Journal of Legal Studies, Vol. 34 (2), 295–341.

Bris, A., I. Welch, and N. Zhu (2006). The Costs of Bankruptcy: Chapter 7

Liquidation versus Chapter 11 Reorganization, The Journal of Finance, Vol.

61 (3), 1253–1303.

Chen, Y. and K. Hammes (2003). Capital Structure Theories and Empirical

Results—A Panel Data Analysis, SSRN working paper series.

Christensen, P.O., C.R. Flor, D. Lando, and K.R. Miltersen (2002). Dynamic

Capital Structure with Callable Debt and Debt Renegotiations, SSRN work-

ing paper series.

Cotei, C. and J. Farhat (2009). The Trade-Off Theory and the Pecking Order

Theory: Are They Mutually Exclusive? North American Journal of Finance

and Banking Research, Vol. 3 (3), 1–16.

Cox, J.C., J.E. Ingersoll, and S.A. Ross (1985). An Intertemporal General

Equilibrium Model of Asset Prices, Econometrica, Vol. 53 (2), 363–384.

Dangl, T. and J. Zechner (2004). Credit Risk and Dynamic Capital Struc-

ture Choice, Journal of Financial Intermediation, Vol. 13 (2), 183–204.


Davydenko, S.A. and I.A. Strebulaev (2007). Strategic Actions and Credit

Spreads: An Empirical Investigation, The Journal of Finance, Vol. 61 (2),

565–608.

Dixit, A.K. and R.S. Pindyck (1994). Investment under uncertainty, Prince-

ton University Press.

Duffie, D. and D. Lando (2001). Term Structures of Credit Spreads with

Incomplete Accounting Information, Econometrica, Vol. 69 (3), 633–664.

Duffie, D. and K.J. Singleton (1999). Modeling Term Structures of Default-

able Bonds, The Review of Financial Studies, Vol. 12 (4), 687–720.

Eberhart, A.C., W.T. Moore, and R.L. Roenfeldt (1990). Security Pricing

and Deviations from the Absolute Priority Rule in Bankruptcy Proceedings,

The Journal of Finance, Vol. 45 (5), 1457–1469.

Fama, E. and K.R. French (2000). Forecasting Profitability and Earnings,

Journal of Business, Vol. 73 (2), 161–175.

Fink, K.D. and J.H. Mathews (1999). Numerical Methods Using MATLAB,

Third Edition, Prentice Hall.

Firoozi, F. (2006). On the Martingale Property of Economic and Financial

Instruments, International Review of Economics and Finance, Vol. 15 (1),

87–96.

Fischer, E.O., R. Heinkel, and J. Zechner (1989a, March). Dynamic Capital

Structure Choice: Theory and Tests, The Journal of Finance, Vol. 44 (1),

19–40.


Fischer, E.O., R. Heinkel, and J. Zechner (1989b, December). Dynamic Re-

capitalization Policies and the Role of Call Premia and Issue Discounts, The

Journal of Financial and Quantitative Analysis, Vol. 24 (4), 427–446.

Flor, C.R. and J. Lester (2002). Debt Maturity, Callability, and Dynamic

Capital Structure, University of Southern Denmark working paper.

Francois, P. and E. Morellec (2004). Capital Structure and Asset Prices:

Some Effects of Bankruptcy Procedures, The Journal of Business, Vol. 77

(2), 387–411.

Frank, M.Z. and V.K. Goyal (2007). Trade-off and Pecking Order Theories

of Debt, SSRN working paper series.

Galai, D. and R.W. Masulis (1976). The Option Pricing Model and the Risk

Factor of Stock, Journal of Financial Economics, Vol. 3 (1-2), 53–81.

Gaud, P., M. Hoesli, and A. Bender (2007). Debt-Equity Choice in Europe,

International Review of Financial Analysis, Vol. 16 (3), 201–222.

Gilson, S.C., K. John, and L.H.P. Lang (1990). Troubled Debt Restructur-

ings: An Empirical Study of Private Reorganization of Firms in Default,

Journal of Financial Economics, Vol. 27 (2), 315–353.

Girsanov, I.V. (1960). On Transforming a Certain Class of Stochastic Pro-

cesses by Absolute Continuous Substitution of Measures, Theory of Proba-

bility and its Applications, Vol. 5 (3), 285–301.

Goldstein, R., N. Ju, and H.E. Leland (2001). An EBIT-Based Model of

Dynamic Capital Structure, The Journal of Business, Vol. 74 (4), 483–512.


Gorbenko, A.S. and I.A. Strebulaev (2010). Temporary versus Permanent

Shocks: Explaining Corporate Financial Policies, The Review of Financial

Studies, Vol. 23 (7), 2591–2647.

Gordon, M.J. (1959). Dividends, Earnings, and Stock Prices, The Review of

Economics and Statistics, Vol. 41 (2), 99–105.

Graham, J.R. and C.R. Harvey (2001). The theory and practice of corporate

finance: evidence from the field, Journal of Financial Economics, Vol. 60

(2-3), 187–243.

Grinblatt, M. and S. Titman (2002). Financial Markets & Corporate Strat-

egy, Second Edition, McGraw–Hill.

Harrison, J.M. and S.R. Pliska (1981). Martingales and Stochastic Integrals

in the Theory of Continuous Trading, Stochastic Processes and their Appli-

cations, Vol. 11 (3).

Hennessy, C.A. and Y. Tserlukevich (2008). Taxation, Agency Conflicts,

and the Choice Between Callable and Convertible Debt, Journal of Eco-

nomic Theory, Vol. 143 (1), 374–404.

Jarrow, R.A., D. Lando, and S.M. Turnbull (1997). A Markov Model for the

Term Structure of Credit Risk Spreads, The review of Financial Studies, Vol.

10 (2), 481–523.

Jensen, M.C. (1986). Agency Costs of Free Cash Flow, Corporate Finance,

and Takeovers, The American Economic Review, Vol. 76 (2), 323–329.

Jensen, M.C. and W.H. Meckling (1976). Theory of the Firm: Managerial


Behavior, Agency Costs and Ownership Structure, Journal of Financial Eco-

nomics, Vol. 3 (4), 305–360.

Ju, N. and H. Ou-Yang (2006). Capital Structure, Debt Maturity, and

Stochastic Interest Rates, The Journal of Business, Vol. 79 (5), 2469–2502.

Ju, N., R. Parrino, A.M. Poteshman, and M.S. Weisbach (2005). Horses and

Rabbits? Trade-off Theory and Optimal Capital Structure, The Journal of

Finance and Quantitative Analysis, Vol. 40 (2), 259–281.

Kane, A., A.J. Marcus, and R.L. McDonald (1984). How Big is the Tax

Advantage to Debt? The Journal of Finance, Vol. 39 (3), 841–853.

Karatzas, I. and S.E. Shreve (1991). Brownian Motion and Stochastic Cal-

culus, Second Edition, Springer–Verlag.

Kim, D., D. Palia and A. Saunders (2003). The Long-Run Behaviour of Debt

and Equity Underwriting Spreads, NYU Stern working paper No. S-DRP-

03-03.

Kraus, A. and R.H. Litzenberger (1973). A State-Preference Model of Opti-

mal Financial Leverage, The Journal of Finance, Vol. 28 (4), 911–922.

Leary, M.T. and M.R. Roberts (2005). Do Firms Rebalance Their Capital

Structures? The Journal of Finance, Vol. 60 (6), 2575–2619.

Leland, H.E. (1994). Corporate Debt Value, Bond Covenants, and Optimal

Capital Structure, The Journal of Finance, Vol. 49 (4), 1213–1252.

Leland, H.E. (1998). Agency Costs, Risk Management, and Capital Struc-

ture, The Journal of Finance, Vol. 53 (4), 1213–1243.


Leland, H.E. and K.B. Toft (1996). Optimal Capital Structure, Endogenous

Bankruptcy, and the Term Structure of Credit Spreads, The Journal of Fi-

nance, Vol. 51 (3), 987–1019.

Lipe, R. and R. Kormendi (1994). Mean Reversion in Annual Earnings and

Its Implications for Security Valuation, Review of Quantitative Finance and

Accounting, Vol. 4 (1), 27–46.

McDonald, R. and D. Siegel (1984). Option Pricing When the Underlying

Asset Earns a Below-Equilibrium Rate of Return: A Note, The Journal of

Finance, Vol. 39 (1), 261–265.

Mella-Barral, P. and W. Perraudin (1997). Strategic Debt Service, The Jour-

nal of Finance, Vol. 52 (2), 531–556.

Merton, R.C. (1974). On the Pricing of Corporate Debt: The Risk Structure

of Interest Rates, The Journal of Finance, Vol. 29 (2), 449–470.

Mitchell, K. (1991). The Call, Sinking Fund, and Term-to-Maturity Features

of Corporate Bonds: An Empirical Investigation, The Journal of Financial

and Quantitative Analysis, Vol. 26 (2), 201–222.

Modigliani, F. and M.H. Miller (1958). The Cost of Capital, Corporation Fi-

nance and the Theory of Investment, The American Economic Review, Vol.

48 (3), 261–297.

Modigliani, F. and M.H. Miller (1963). Corporate Income Taxes and the

Cost of Capital: A Correction, The American Economic Review, Vol. 53 (3),

433–443.


Myers, S.C. (1977). Determinants of Corporate Borrowing, Journal of Fi-

nancial Economics, Vol. 5 (2), 113–269.

Myers, S.C. (1984). The Capital Structure Puzzle, The Journal of Finance,

Vol. 39 (3), 575–592.

Myers, S.C. and N.S. Majluf (1984). Corporate Financing and Investment

Decisions When Firms Have Information That Investors Do Not Have, Jour-

nal of Financial Economics, Vol. 13 (2), 187–221.

Nocedal, J. and S.J. Wright (2006). Numerical Optimization, Second Edi-

tion, Springer.

Nordal, K.B. and R. Næs (2010). Mean Reversion in Profitability of Non-

Listed Firms, European Financial Management, Wiley Online Library, ac-

cessed June 2011.

Petrovic, N., S. Manson and J. Coakley (2009). Does Volatility Improve UK

Earnings Forecasts? Journal of Business Finance & Accounting, Vol. 36 (9),

1148–1179.

Raymar, S. (1991). A Model of Capital Structure when Earnings are Mean-

Reverting, The Journal of Financial and Quantitative Analysis, Vol. 26 (3),

327–344.

Revuz, D. and M. Yor (1999). Continuous Martingales and Brownian Mo-

tion, Third Edition, Springer–Verlag.

Sarkar, S. and F. Zapatero (2003). The Trade-off Model with Mean Revert-

ing Earnings: Theory and Empirical Tests, The Economic Journal, Vol. 113

(490), 834–860.


Shleifer, A. and R.W. Vishny (1992). Liquidation Values and Debt Capac-

ity: A Market Equilibrium Approach, The Journal of Finance, Vol. 47 (4),

1343–1366.

Strebulaev, I.A. (2007). Do Tests of Capital Structure Theory Mean What

They Say? The Journal of Finance, Vol. 62 (4), 1747–1787.

Titman, S. and S. Tsyplakov (2006). A Dynamic Model of Optimal Capital

Structure, McCombs research paper series No. FIN-03-06.

Tsekrekos, A.E. (2010). The Effect of Mean Reversion on Entry and Exit

Decisions under Uncertainty, Journal of Economic Dynamics & Control, Vol.

34 (4), 725–742.

Vu, J.D. (1986). An Empirical Investigation of Calls of Non-Convertible

Bonds, Journal of Financial Economics, Vol. 16 (2), 235–265.

Wald, J.K. (1999). How Firm Characteristics Affect Capital Structure: An

International Comparison, The Journal of Financial Research, Vol. 22 (2),

161–187.

Date post:	02-Feb-2017
Category:	Documents
Upload:	haduong
View:	214 times
Download:	0 times

Dynamic Capital Structure Modelling under Alternative Stochastic ...

Documents