Aspects of optimal capital structure and default risk [Elektronische Ressource] / Sarp Kaya Acar
155 Pages
English

Aspects of optimal capital structure and default risk [Elektronische Ressource] / Sarp Kaya Acar

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Published 01 January 2007
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˜Technische Universitat Kaiserslautern
Fachbereich Mathematik
Aspects of Optimal Capital Structure
and Default Risk
Sarp Kaya Acar
Vom Fachbereich Mathematik
der Technischen Universit˜at Kaiserslautern
zur Verleihung des akademischen Grades
Doktor der Naturwissenschaften
(Doctor rerum naturalium, Dr.rer.nat)
genehmigte Dissertation
1. Gutachter: Prof. Dr. Ralf Korn
2. Gutachter: Dr. habil. J˜org Wenzel
Vollzug der Promotion: 11.09.2006To my Dear Mom and Dad...Contents
1 Leland’s Optimal Capital Structure Model 15
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.2 The Model Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.3 PDE-Approach to Price Contingent Claims . . . . . . . . . . . . . . . . . . 18
1.4 Pricing Firm Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.4.1 Debt Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.4.2 Total Firm Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.4.3 The Equity Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.5 Endogenous Default. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.6 Two Step Optimisation Problem of Total Firm Value . . . . . . . . . . . . 29
1.6.1 The First Step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.6.2 An Equivalent Step to the First Step . . . . . . . . . . . . . . . . . 31
1.6.3 The Second Step . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1.7 Optimal Leverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1.8 Credit Spread . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
1.9 Comparative statistics for debt and equity value value . . . . . . . . . . . . 37
2 Extended Optimal Capital Structure Models 41
2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.1.1 Choosing the underlying . . . . . . . . . . . . . . . . . . . . . . . . 42
2.1.2 Relaxing Perpetuity . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.1.3 A Realistic Tax Regime . . . . . . . . . . . . . . . . . . . . . . . . 44
2.2 Difiusion case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
56 CONTENTS
2.2.2 Pricing Firm Derivatives . . . . . . . . . . . . . . . . . . . . . . . . 51
2.2.3 Endogenous Default . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.2.4 Optimal Leverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2.2.5 Credit Spreads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2.2.6 Comparative Statistics . . . . . . . . . . . . . . . . . . . . . . . . . 69
2.3 Jump Difiusion Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
2.3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
2.3.2 Pricing Firm Derivatives . . . . . . . . . . . . . . . . . . . . . . . . 83
2.3.3 Endogenous Default . . . . . . . . . . . . . . . . . . . . . . . . . . 87
2.3.4 Optimal Leverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
2.3.5 Default Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . 90
2.3.6 Credit Spreads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
3 An extension of the Libor Market Model with Default Risk 103
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
3.2 Default model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
3.3 Bond prices and basic rates . . . . . . . . . . . . . . . . . . . . . . . . . . 105
3.4 Dynamics of Rates under the Spot Martingale Measure . . . . . . . . . . . 107
3.4.1 The two factor model . . . . . . . . . . . . . . . . . . . . . . . . . 110
3.4.2 The multi factor model . . . . . . . . . . . . . . . . . . . . . . . . 111
3.5 Forward and Survival Measures . . . . . . . . . . . . . . . . . . . . . . . . 113
3.5.1 The Spot Martingale Measure . . . . . . . . . . . . . . . . . . . . . 114
3.5.2 T -Forward Measure . . . . . . . . . . . . . . . . . . . . . . . . . . 115k
3.5.3 T -Survival . . . . . . . . . . . . . . . . . . . . . . . . . . 116k
3.6 Dynamics of the Rates under the Survival Measures . . . . . . . . . . . . . 119
3.7 Basic Credit Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
3.7.1 Credit Default Swaps . . . . . . . . . . . . . . . . . . . . . . . . . . 125
3.7.2 Credit Default Swaption . . . . . . . . . . . . . . . . . . . . . . . . 128
3.7.3 Sch˜onbucher’s CDSwaption Formula . . . . . . . . . . . . . . . . . 129
3.7.4 An approximation for the valuation of CDSwaptions using multiple
factors and inhomogeneous volatilities . . . . . . . . . . . . . . . . 131CONTENTS 7
3.7.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
3.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
A Modigliani-Miller Results 141
A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
A.2 Propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144Preface
The credit risk is the possibility that a counterparty in a flnancial contract will not fulflll
herobligationsstatedinthecontract. Intheliterature,therearetwodifierentapproaches
tomodelthecreditrisk: thestructuralapproach, andtheintensitybasedorreducedform
approach.
Structuralmodelsareconcernedwithmodelingandpricingthecreditriskthatisassigned
to a particular flrm. Default events are due to the movements of the flrm’s value (assets)
relative to some random or non-random default triggering threshold. Consequently, the
major issue within this framework is the modeling of the evolution of the flrm’s value
and flrm’s capital structure by making explicit assumptions. Therefore, the structural
approachisalsoreferredtoastheflrmvalueapproach. Thedefaulteventismodeledasthe
flrst hitting time of the underlying flrm value process to an exogenously or endogenously
specifled level, which is called the bankruptcy level. Then, by considering the corporate
liabilities of the flrm as contingent claims on the flrm’s value, closed form expressions for
theirpricesarederived. Inmostoftheflrmvaluemodels,itisassumedthatinafrictionless
market the flrm’s value is independent from the flnancial decisions of the flrm. This is
postulatedbyModiglianiandMiller[MM58]inawellknowntheorem,(MM-Proposition1,
seeAppendixA),whichtellsusthatwhenbankruptcycosts,taxadvantages,transactions
costs, agency costs and other frictions in the market are omitted, the flrm’s value is not
1afiected from being an all-equity, all-debt or a leveraged flrm. On the contrary, if one
of these frictions are considered in the market, the flrm’s decision on how it flnances its
2businesses will become important . In this case, additionally to the flrm’s value another
3quantity of the flrm is considered, namely the total flrm value . The total flrm value is
equal to the flrm value plus the net efiect of the frictions of the market, depending on
1Finances its businesses both by issuing debt and equity.
2The efiect of the frictions on the flnancial decisions of a flrm are discussed in details in Appendix A.
3In such an environment, one should make a clear difierence between the flrm value and the total
flrm value notions. The latter one is the value of the flrm, after the debt is issued. This quantity is also
known as levered flrm value. Through out this thesis, we interchangeably use both names, but they
refer to the same quantity. The former one, also called as the unlevered flrm value, is the value of an
all-equity flrm. From now on when we write flrm value, we refer to the unlevered flrm value.the flnancial decision of the flrm. Therefore, the total flrm value is dependent on how
the flrm’s business is flnanced, so that the optimal capital structure of the flrm becomes
a relevant issue. The structural approach can be divided into two, according to how the
bankruptcy level is specifled.
The exogenous bankruptcy level refers to the case when the bankruptcy event is due to
some protective covenant. For instance, the bankruptcy is triggered when the asset value
reaches the exogenously specifled principal value of the debt or an exogenously specifled
threshold process (which can be deterministic or stochastic). Merton [Mer74] considered
a case where the default event can only occur at maturity, if the flrm’s assets are lower
than the face value (principal value) of the debt. But recognizing that a flrm may default
well before the maturity of the debt, Black and Cox [BC76] alternatively assumed that
the flrm goes bankrupt, when the value of its assets hits for the flrst time some lower,
time dependent threshold. Further flrst passage time models with exogenous default are
proposed by Longstafi and Schwartz [LS95], Colin-Dufirense and Goldstein [CDG01].
The notion of the endogenous bankruptcy covers the situations, when the bankruptcy is
declared by the equity holders. Therefore, the bankruptcy level is specifled by some addi-
tionalconditionsinthemodel. Theendogenousapproachiswidelyusedinthe
frameworkoftheoptimalcapitalstructure,whichtakesintoaccountmarketfrictions(e.g.
costsofbankruptcy,taxadvantages,agencycosts,etc.) andexaminestheoptimalpropor-
tionofthedebtoverthetotalflrmvalue, i.e. theoptimalleverage. Withintheframework
of the optimal capital structure approach, equity holders choose the bankruptcy level in
such a way that their value is maximised. The endogenous speciflcation of the default
level enables the analysis of the optimal capital structure. Brennan and Schwartz [BS78]
providetheflrstquantitativeexaminationoftheoptimalcapitalstructureofaflrm,byuti-
lizingnumericaltechniquestodeterminetheoptimalleveragewhentheflrmvaluefollowsa
difiusionprocesswithconstantvolatility. Theproblemoftheoptimalcapitalstructureand
its endogenous default barrier has been considered in a series of papers by Leland [Lel94],
Leland [Lel95], Leland and Toft [LT96]. Other important papers of this framework are
Kane, Marcus, and McDonald [KMM84], Mella-Barral and Perraudin [MBP97], Fischer,
10