An Analysis of Structural Credit Risk Models and their

Performance in Predicting Credit Spreads of Corporate Bonds

MASTER THESIS

Submitted in Partial Fulfillment of the Requirements for the Degree of

MASTER OF SCIENCE

in Banking and Finance

Univ.-Prof. Dr. Jochen LAWRENZ

Department of Banking and Finance

The University of Innsbruck School of Management

Submitted by

Philipp HADLER, BSc.

Innsbruck, June 2020

Abstract

The financial literature on credit risk models to model default probabilities and corporate

bond spreads has its origins within the structural framework originally established by Mer-

ton (1974). Subsequent generations of structural models which address the shortcomings

of the Merton model have been subject of various studies examining the relative success

and failures of theses models empirically. This master’s thesis empirically tests the perfor-

mance of three structural credit risk models in predicting corporate bond spreads observed

in the market. By using real corporate bond data from Thomson Reuters Datastream

between 2010-2019 and drawing on previous studies covering the validity of the structural

model approach in practice, we test the selected models based on a portfolio-of-zero-

coupon bonds approach to price corporate bonds. We find evidence that the selected

structural models are incapable to generate credit spreads as those witnessed in our bond

sample. While the model performance increases with its complexity, this does come at

the cost of a higher inaccuracy in spread prediction errors. In particular, examining the

model performance based on their spread predictions indicates that the models are unable

to capture risk factors which are being priced in the market.

Acknowledgements

I would like to take this opportunity to thank everybody who has helped and supported

me over the past five years during my studies in Innsbruck and Bangkok. First and

foremost, I would like to thank my thesis advisor, Univ.Prof. Dr. Jochen Lawrenz for

suggesting a potential topic and agreeing to supervise my research topic.

A big thank you to my friends who for the past four months have listened to me, talking

constantly about how much work I had left to do. Special thanks goes to my fellow stu-

dent Simon Kronbichler for continuous discussions and valuable inputs for this thesis.

Finally, I must express my special gratitude to my parents, Werner and Beate who pro-

vided me with continuous encouragement throughout my years of study and through the

process of researching and writing this thesis. This accomplishment would not have been

possible without them. Thank you.

Contents

1 Introduction 1

2 Credit risk modelling in general 2

2.1 Definition of credit risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.2 Scoring and rating methods . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.3 Market price methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 The credit spread puzzle 6

4 Structural credit risk models 9

4.1 The Merton model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4.1.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4.1.2 Derivation and security pricing . . . . . . . . . . . . . . . . . . . . 11

4.1.3 Credit spread and probability of default . . . . . . . . . . . . . . . 13

4.2 Model extensions to Merton . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.2.1 Capital structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.2.2 First-passage models . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.2.3 Interest rate process . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.2.4 Asset value process . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.3 Performance analysis of structural models . . . . . . . . . . . . . . . . . . 19

4.3.1 Merton model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.3.2 Performance of extensions to Merton . . . . . . . . . . . . . . . . . 20

5 Modelling framework 22

5.1 Selection of structural models . . . . . . . . . . . . . . . . . . . . . . . . . 22

5.1.1 Extended Merton model: . . . . . . . . . . . . . . . . . . . . . . . . 23

5.1.2 Longstaff & Schwartz model . . . . . . . . . . . . . . . . . . . . . . 24

5.1.3 Collin-Dufresne & Goldstein model: . . . . . . . . . . . . . . . . . . 29

5.2 Selection of interest rate models . . . . . . . . . . . . . . . . . . . . . . . . 34

5.2.1 Vasicek model calibration . . . . . . . . . . . . . . . . . . . . . . . 34

5.2.2 Nelson & Siegel model calibration . . . . . . . . . . . . . . . . . . . 36

I

6 Data and methodology 37

6.1 Bond and company data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

6.2 Interest rate data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

6.3 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

6.3.1 Capital structure parameters . . . . . . . . . . . . . . . . . . . . . . 40

6.3.2 Interest rate parameters . . . . . . . . . . . . . . . . . . . . . . . . 42

6.3.3 Parameters related to bond features . . . . . . . . . . . . . . . . . . 42

6.4 Summary of model parameters . . . . . . . . . . . . . . . . . . . . . . . . . 43

7 Results 44

7.1 Fitting the yield curve to current market data . . . . . . . . . . . . . . . . 44

7.1.1 Vasicek model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

7.1.2 Nelson & Siegel model . . . . . . . . . . . . . . . . . . . . . . . . . 45

7.2 Performance of structural models . . . . . . . . . . . . . . . . . . . . . . . 46

7.2.1 Performance at end-of-year . . . . . . . . . . . . . . . . . . . . . . . 47

7.2.2 Discussion of results . . . . . . . . . . . . . . . . . . . . . . . . . . 52

8 Conclusion 57

References 64

Appendices 65

A Yield spread in the Merton model 65

B Probability of default in the Merton model 66

C Conditional moments in the CDG model 66

II

List of Figures

1 One Year Transition Matrix (%), Albert Metz (2007) . . . . . . . . . . . . 5

2 Term structure of yield spreads and probabilities of default calculated

through the Merton approach . . . . . . . . . . . . . . . . . . . . . . . . . 14

3 Credit spread term structure for an 8% bond for different values of X and

w`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4 Credit spread term structure in the CDG model compared to the LS model

for different leverage ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5 Vasicek yield curve fits versus US & Euro area market yields . . . . . . . . 45

6 Neslon & Siegel yield curve fits versus US & Euro area market yields . . . 46

List of Tables

1 Summary statistics on bonds and issuers in the sample . . . . . . . . . . . 39

2 Summary of model input parameters . . . . . . . . . . . . . . . . . . . . . 43

3 Aggregate performance of structural models averaged across all available

end-of-year predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4 Performance of structural models averaged across all available end-of-year

predictions - US bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5 Performance of structural models averaged across all available end-of-year

predictions - Eurobonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

III

1 Introduction

Corporations typically have a number of options available when choosing how to finance

their operations. The issuance of corporate bonds represents one of the most popular

methods to borrow money in exchange for a promise on the part of a corporation to make

one or more payments to an investor according to a specified agreement. For various

reasons, the corporation may end up in a situation where it fails to meet the agreements

in the contract. As noted in (Sironi & Resti 2007, p. 277), the risk that this may happen

is generally referred to as credit risk. In general, methodologies to precisely measure and

control the credit risk of corporate bonds critically depend on the accurate assessment of

bond implied credit spreads. Indicated by Clark & Baccar (2018), these spreads change

over time due to varying market conditions, changes in the credit ratings of issuers or in

the expected recovery rate. An extensive large number of mathematical models have been

proposed to provide consistent valuations of corporate bonds and their implied spreads.

Each model attempts to incorporate a subset of factors which influence the risk of a bond

including the value of the firm’s assets, leverage ratios, interest rates and the time to matu-

rity to predict credit spreads as witnessed in real markets (Duffie & Singleton 2003, p. 43).

According to the literature, the two main groups of credit risk models are namely the

structural and reduced-form approaches. The reduced form models established by Jarrow

& Turnbull (1995) and Duffie & Singleton (1999) do not take into account any economic

cause which triggers default and use a pure probabilistic approach to model the default

of a firm. In contrast, structural credit risk models, initiated by Merton (1974), mainly

rely on the economic fundamentals of the underlying company to explain default risk

and to derive credit spreads. Merton (1974) offers an easy and computationally efficient

method of pricing risky corporate bonds, however assuming several simplifying and re-

strictive assumptions. Referring to the shortcomings, the original Merton model has been

extended by several researchers attempting to reduce the shortcomings as well as making

the models more applicable to real world market conditions.

This thesis focuses on testing the performance of three popular structural bond pricing

models including that of Merton (1974), Longstaff & Schwartz (1995) and Collin-Dufresne

1

& Goldstein (2001). The goal is to implement the three structural models over a sam-

ple of bonds in order to make quantitative statements about the relative performance of

each structural model to predict credit spreads. The results should then offer suggestions

about the key shortcomings in each model. Finally, we conclude whether the structural

modelling approach appears to be suitable for predicting market bond prices and their

implied spreads or if further effort is required in order to make the theoretical appealing

approaches more practical and accurate to predict credit spreads.

The remainder of this thesis is organized as follows.

Section 2 outlines different credit risk modelling approaches in general. Section 3 examines

the determinants of credit spreads which is known as the ’credit spread puzzle’. Section 4

reviews the existing literature on structural models, beginning with a detailed analysis of

the original Merton (1974) model. The section continues with the most important exten-

sions to Merton (1974) and an overview of empirical studies that analyze these models.

Section 5 outlines the selection of three structural models for testing and also discusses

the selection of two interest rate models to describe the dynamics of the risk-free interest

rate. Section 6 includes the collection of all data required to perform our test as well as

the characteristics of our sample. Moreover, also the estimation procedures for the input

parameters for the structural models are described. Section 7 presents the results of our

performance analysis and discusses the results. Finally, section 8 offers a summary of this

thesis and gives recommendations for future research topics.

2 Credit risk modelling in general

This section provides an introduction to credit risk both in general and from a modelling

perspective. After the explanation of credit risk, the section continuous with a review on

various frameworks to model credit risk recognized in the academic literature.

2.1 Definition of credit risk

(Duffie & Singleton 2003, p. 3) define credit risk as the potential that the borrower

will fail to meet its obligations in accordance with agreed terms which is associated with

2

unexpected changes in credit quality. Examples of an borrower could be a company that

has borrowed money from a bank, that issues bonds or a household which borrowed money

from a bank to buy a house. Examples for defaults are that a company goes bankrupt, it

fails to pay the coupon on time for some of its issued bonds or that a household fails to

pay amortization or interest on their loan. Credit risk is not only a difficult risk category

from a lenders perspective, but is also an important part of regulations in the financial

industry (Sironi & Resti 2007, p. 547)).

Especially financial institutions are required to measure the credit risk of their portfolios.

The bank of international settlement, defines the following parameters as the most widely

used to measure credit risk (Basel Committee on Banking Supervision 2000):

• Probability of Default (PD)

• Exposure at Default (EAD): The EAD measures the expected outstanding

amount at the time of the default

• Loss Given Default (LGD): Represents the loss rate in the event of default,

measured as a percentage of the EAD. The recovery rate is then given by 1− LGD

Given the parameters above and assuming that all are independent from each other, the

expected loss (EL) on a contract would be given by (Hartmann-Wendels et al. 2019, p.

415):

EL = EAD · LGD · PD

According to (Sironi & Resti 2007, pp. 282-283) credit risk can be measured either

by methods which use a scoring and rating based system or which use market prices to

measure credit risk. Both approaches are the basis of applications in practice.

2.2 Scoring and rating methods

Using accounting historical information to measure credit risk is subject to the scoring

and rating-based model family. A famous credit-scoring model represents the work by

Altman (1968), namely Altman’s Z-score. Altman performs a discrimination analysis

based on accounting data of firms that ended up in bankruptcy and provides a linear

relationship between different financial ratios. Here, credit risk is viewed as a function of

3

five independent variables:

Z = 1.2X1 + 1.4X2 + 3.3X3 + 0.6X4 + 1.0X5 where,

X1 = Working Capital/Total Assets

X2 = Retained Earning/Total Assets

X3 = EBIT/Total Assets

X4 = Market/Book Value

X5 = Sales/Total Assets

For Z < 1.8 high default risk; 1.8 < Z < 3 indeterminate risk; Z > 3 low default risk

The results of Altman (1968) indicate that the model is able to predict corporate

default events. Although easy to implement, (Sironi & Resti 2007, p. 299) note that pre-

dictions of the Z-score heavily rely on accounting ratios rather than market-based data.

Summarizing credit risk in a credit rating belongs to credit rating models, where the

rating of a firm serves as an opinion of the firm’s future credit risk (Sironi & Resti 2007,

p. 369). According to (Hartmann-Wendels et al. 2019, p. 418) this is the most common

method in practice, where the debtor is classified into a rating class depending on his cred-

itworthiness and where each rating-class corresponds to a given (cumulative) probability

of default. (Sironi & Resti 2007, pp. 370-372) note that ratings are typically provided

by specialized agencies like Standard & Poor’s, Moody’s and Fitch, however, ratings can

also be created internally by a bank itself. Those agencies typically provide a cumulative

probability of default over a given time horizon, indicating the probability that the loan

will default within that period of time. For the risk management of credit risk positions

like corporate bonds, it is not only important to determine the probability of default, but

also to explicitly take into account the probability of down- and upgrades (Hartmann-

Wendels et al. 2019, p. 420). Those are given by the rating transition probabilities, which

are typically provided through transitions matrices and should explain the credit migra-

tion risk of firms (Wang 2017). Figure 1 shows the average one year cumulative default

probabilities for U.S corporate bonds for various credit ratings calculated by Moody’s in

4

August 2007 (Albert Metz 2007):

Figure 1: One Year Transition Matrix (%), Albert Metz (2007)

Observe that the default probabilities differ across rating classes and are approximately

zero for investment grade bonds except for AAA and AA rated corporate bonds. The

diagonal shows the one year cumulative probability of a given bond to remain in its

current rating class. While easy to interpret a credit rating in terms of credit risk, the

information of the same should be interpreted carefully as they only provide an evaluation

of the creditworthiness of the corresponding rating class but not on an individual level

(Sironi & Resti 2007, p. 372).

2.3 Market price methods

According to (Duffie & Singleton 2003, p. 100) there are two basic approaches to model

credit risk and spreads based on the capital market prices of assets, which are namely a

structural and reduced-form approach. In the structural approach, explicit assumptions

about the dynamics of a firm’s assets value are made, which include observable variables

such as balance-sheet ratios or business cycle variables. The firm’s inability to honor its

payment obligations is modelled as the default-triggering event. In detail, the firm is

supposed to default when its value drops below a certain threshold level such as the value

of its debt or some percentage of it (Duffie & Singleton 2003, p. 112). Structural models

5

have been established by Black & Scholes (1973) and Merton (1974). These models were

later extended by Black & Cox (1976), Leland (1994), Longstaff & Schwartz (1995) and

Collin-Dufresne & Goldstein (2001). According to (Duffie & Singleton 2003, p. 330),

the benefits of the structural approach are that the models provide an intuitive economic

interpretation and the inputs and outputs of the models are in terms of understandable

economic variables like the ratio of debt to assets and the volatility of asset returns.

Moreover, structural frameworks are widely used in the financial industry representing

the basis for various programs and software to measure credit risk. A discussion of the

basis theoretical framework proposed by Merton and some of the most important exten-

sions of the basic framework will be addressed in section 4.

An alternative approach is the so-called reduced-form approach. Indicated by (Lando

2004, p. 109), the main difference between the two is the definition of the event of de-

fault. Reduced-form models treat default as an unexpected event whose likelihood is

governed by a default-intensity process, most often a Poisson process (Lando 2004, p.

112). This approach was first adopted by Duffie & Singleton (1999), Jarrow & Turnbull

(1995) and others. Apart form their mathematical tractability, default intensity models

only provide little economic interpretation regarding the default event as no balance sheet

data of the firm gets used for measuring credit risk.

3 The credit spread puzzle

The empirical literature on corporate bond spreads finds that the level of credit spreads is

not only explained by credit risk factors. Following Tsuji (2005), this observations refers

to the term ’credit spread puzzle’.

According to (Bomfim 2016, p. 153), the bond yield spreads are named like this be-

cause the assumption is that the difference in yields is primarily due to the corporate

bond’s exposure to credit risk. However, Collin-Dufresne et al. (2001), Campbell & Tak-

sler (2003), Ericsson et al. (2009), Zhang et al. (2009) and Coro et al. (2013) identified

other factors driving the level of bond and credit-default-swap spreads apart from the

theoretical determinants proposed by structural models like default probabilities, interest

6

rates, business cycle fluctuations and leverage ratios. The most notable study was done by

Collin-Dufresne et al. (2001). In their work, they examine whether credit spread changes

are determined by changes in structural model variables by running several regressions

and employing a principal component approach on the residuals. Using monthly corpo-

rate bond data from 1988 to 1997 they made the following assumption on the explanatory

variables:

• Risk-free rate: A higher risk-free rate is expected to increase the of the firm value

process, meaning it reduces the probability of default and therefore the credit spread

• Yield curve slope: As pointed out by Litterman & Scheinkman (1991), the term

structure of interest rates is driven by both the level and slope of the term structure.

As noted in Merton (1974), an increase in the slope of the yield curve is expected

to increase the expected future risk-free rate, which should also lead to a decrease

in credit spreads using the same argument as for the risk-free rate above.

• Leverage: The authors assume that credit spreads are an increasing function of

the firm’s leverage

• Volatility: The structural model approach assumes debt holders to be short in a

put option with the firm value as underlying (Merton 1974). Since option values

increase with volatility, also credit spreads are assumed to increase with volatility.

• Probability of a downward jump in firm value: Indicated by Merton (1974),

implied volatility smiles in option prices (higher implied volatility for options which

are far out of the money) suggest that markets take into account large negative

jumps in the firm value. Therefore, the authors expect credit spreads to increase,

given an increase in the probability of a negative jump.

• Business climate: Spreads are assumed to decrease given a better business climate.

This is measured through the expected recovery rate which should be a function of

the business climate.

The estimated signs (either positive or negative) for the explanatory variables in the re-

gressions are generally in line with the theoretical considerations above. Nevertheless,

they find relatively low explanatory power of the investigated variables above, explaining

7

only about 25% of the variation in market spreads. The only variable which is statistical

and economic significant over all maturities and rating categories is the jump variable.

The variable is measured through changes in the slope of the ”smirk”1 of implied volatili-

ties of deep out-of-the money put options on the S&P 500 futures. An increase in the slope

of the smirk means that there is a higher demand of deep out-of-the-money puts which

is reflected in market participants expectation of a sudden market crash, implying that

credit spreads should increase. Collin-Dufresne et al. (2001) further develop a principal

component approach on the residuals. The results indicate that the residuals are highly

cross-correlated, with the first principal component capturing about 76% of the remain-

ing variation in spread changes. In their final remark, they suggest that the dominant

component of credit spread changes is driven by local supply and demand shocks.

Another linear regression analysis on the relationship between spreads and key variables

suggested by economic theory was conducted by Campbell & Taksler (2003). Their find-

ings provide some evidence that idiosyncratic risk, in the form of equity volatility explains

as much variation in yields as credit ratings.

Ericsson et al. (2009) provided a similar research as Collin-Dufresne et al. (2001) on

credit-default-swap spreads. They are running regressions for both, changes in credit-

default-swap spreads and for the spread level. The authors regress the spread (changes)

on the leverage of the underlying firm, the volatility of its assets and the risk-free rate.

Credit-default-swap data is used from 1999 - 2002. Their results confirm the findings of

Collin-Dufresne et al. (2001) regarding the predicted sign of the variables. However, the

explanatory power and economic significance is higher. In detail, the theoretical variables

account for approximately 60% of the variation in spread levels and around 23% for spread

changes.

Zhang et al. (2009) are the first to explain a significant part of the credit-default-swap

premium using volatility and jump risks as independent variables. Their sample data

contains of U.S. credit-default-swap data from 2001 to 2003. The R2 for the regression

of the credit-default-swap spread level on the historical volatility is 45% while jump risk1Noted in Xing et al. (2010), also called reverse skew pattern, is the observation that especially equity

and index put options which are deep out-of-the money are having a higher implied volatility resultingin higher prices. The explanation behind is that there is a higher demand for out-of-the-money puts dueto the probability of sudden market crashes

8

accounts for 19% in spread variation. Furthermore, the authors calibrate a Merton-type

structural model which incorporates stochastic volatility and jumps in the firm value pro-

cess. According to Zhang et al. (2009) this model outperforms other structural models in

fitting observed market spreads in their point of view.

Another research study by Coro et al. (2013) examines the role of liquidity determinants

on credit-default-swap spread movements. Examining credit-default-swap data from 2006

to 2009, they show that especially during periods of market turmoils, liquidity effects

measured through the bid-ask spreads, demand pressure and trading intensity, dominate

the credit-default-swap price variations compared to firm-specific credit risk factors.

From the literature review we find that structural model variables like the risk-free rate,

leverage or the treasury curve slope show a high correlation with corporate bond yield

spreads. This underlines the economic intuition behind the structural model approach.

However, the determinants seem to contribute less to the variation in market bond spreads

than expected. This is probably due to factors which are not related to credit risk. Vari-

ables like jump risks and stochastic volatility seem to provide a higher explanation of

variations in bonds as well as credit-default-swap spreads. We will further analyze the

application of credit spread determinants in structural models in the next chapter.

4 Structural credit risk models

The following section outlines the most common structural models in the literature. Start-

ing with the famous model by Merton (1974), the section continues with the most popular

extension of Merton’s approach. Moreover, we will give an overview of the empirical per-

formance of the models in practice.

4.1 The Merton model

The structural approach of modelling credit risk has its origin in the modern option pricing

theory developed by Black & Scholes (1973). More specifically, the work by Merton (1974)

serves as the cornerstone for all other structural models. Merton (1974) shows that the

common Black & Scholes (1973) option pricing formulas can be used to price equity and

9

bonds of a firm. The specifications of the Merton model also offer insights regarding the

derivation of credit spreads.

4.1.1 Assumptions

In Merton (1974) the following assumptions are made within his approach:

1. There are no transaction costs, bankruptcy costs and taxes and assets are divisible

2. The Modigliani-Miller theorem holds, meaning the firm value is not affected by

changes in the capital structure

3. Trading in assets takes place continuously in time

4. There are no restrictions to short selling of all assets

5. There are a sufficient number of investors with comparable wealth levels so that

each investor can buy and sell as much of an asset at the market price as he wants

6. The term structure is flat and known with certainty, meaning the risk-free interest

rate r is constant over time.

7. The dynamics for the firm value Vt is described by a diffusion-type stochastic process

with the stochastic differential equation given by:

dVtVt

= (µ− δ)dt+ σdWt (1)

where µ is the expected return on the firms’s asset per unit time, δ the total payout

by the firm per unit time, σ is the volatility on the firm’s assets per unit time and

dWt is a Wiener process.

The basic theoretical framework considers the hypothetical company V having a capital

structure of a single zero-coupon bond with a face value of B and maturity T and the

market value of equity E. The total amount that the company pays back at time T to

the debt-holders is D. The value of the firm’s assets Vt at time t is assumed to be the

sum of equity and liabilities and is then obtained through:

Vt = Bt + Et (2)

10

In the case of Vt > D, i.e. if the value of the assets at time T is higher than the debt, then

the equity-holders are assumed to pay D to the debt-holders. Moreover equity-holders pay

D out of their own pockets because after the debt payments occur, the company still has

some value within as Vt −D > 0, which is kept by the equity owners. If at maturity, the

debt exceeds the asset value, i.e. if Vt < D, the equity-holder will declare the company to

be bankrupt and hand it over to the bond-holders. Therefore, the nominal value of debt

can be viewed as a default threshold. Thus, default is defined as VT < D. Remember

that assumption 5 describes the firm’s asset value process by2:

dVt = rdtVt + σVtdWt

saying that the firm value follows a geometric Brownian motion. By using Ito’s lemma,

the solution is represented by:

Vt = V0e(r−σ

22 )t+σWt (3)

where Vt is the value of the company at time t. We will discuss this assumption in section

4.1.3

4.1.2 Derivation and security pricing

According to the assumptions above, the derivation of debt and equity in the Merton

model is straightforward. If at maturity the firm is assumed to default on its obligations,

the shareholders do not pay back the debt and prefer to transfer their rights to the debt-

holders. Therefore the value received by equity-holder is given by the difference between

the asset value and the value of debt that must be repaid. As for the equity-holder they

receive:

ET = max(0, VT −D) (4)

Similarly, the debt-holders either receive D or VT , whichever is lower at maturity. The

payoff at time T is formally given as:

BT = min(D, VT ) = D −max(0, D − VT ) (5)2Since we are in the Black-Scholes setting we recall that there exists a risk neutral probability measure

Q, which implies that µ is equal to the risk-free interest r

11

The structure of equation (4) and (5) applies to the option-theoretic approach by Black

& Scholes (1973). Turning to equation (4), this means that D can be viewed as the strike

price of a call option, indicating that unless the firm value is less than the debt value,

equity-holders will not exercise their option. The payoff for the equity-holders can be

interpreted as being long in a call option where the value of the company is assumed to

be the underlying. For equation (5), the last term on the right-hand side can be viewed

as the payout of a put option written on the value of the firm’s assets, suggesting that

D again corresponds to the strike price of the option. As V is assumed to possibly fall

below D by time T , the last term on the right-hand side of the expression appears to has

a negative sign. This implies, rather than being the holder of the put option, the debt-

holder wrote the put option. Therefore, the debt-holder’s position represents a portfolio

composed of a long position in a risk-free bond with a face value of D and a short position

in the just described put.

Considering f as the price of a derivative contingent on the firm value V , the variable f

needs to be a function of V and t. From Ito’s lemma the so-called Black-Scholes-Merton

differential equation can be derived. This is necessary as the idea underlying the BSM

model is that the price of any derivative dependent on a non-dividend paying stock needs

to satisfy a certain partial differential equation (PDE), which is given by:

∂f

∂t+ rV

∂f

∂V+ 1

2σ2S2 ∂

2f

∂S2 = rf (6)

The PDE has many solutions which correspond to the various derivatives that can be

defined by the underlying variable V . One of them are the BSM formulas for the prices

of European call and put options described in equation (4) and (5).

Recall from equation (5), that debt is interpreted as being long in a risk-free bond and

short in a put with the firm value as its underlying. By applying the Black-Scholes

formula, the debt value is formally given as:

B = De−r(T−t) − Pt(Vt, D, T − t, σ, r)

= De−r(T−t) − [De−r(T−t)N(−d2)− VtN(−d1)]

= VtN(−d1) +De−r(T−t)N(d2) (7)

12

where Pt denotes the put price and e−r(T−t) represents the discount factor where r is the

continuously compounded risk-free rate. Similarly, for the equity-holders and equation

(4), the Black-Scholes formula can also be applied which gives:

E = VtN(d1)−De−r(T−t)N(d2) (8)

where N(d1,2 ) is the cumulative standard normal distribution function and d1 and d2 are

given by:

d1 =ln(Vt

D) + (r + 1

2σ2T )

σ√T

d2 = d1 − σ√T (9)

Summing up equation (7) and (8) leads again to the equation of the firm value, written

as:

Vt = Et +Bt

Vt = CBS(Vt, D, T − t, σ, r) +De−r(T−t) − PBS(Vt, D, T − t, σ, r) (10)

Where CBS and PBS are representing the Black-Scholes option-pricing formulas for the

call and put option. Through basic option pricing theory, equation (10) can be recognized

as the so called put-call-parity (Hull 2006, p. 221-223).

4.1.3 Credit spread and probability of default

In the Merton model, the yield spread is interpreted as the difference over the risk-free

rate. Actually, Merton (1974) considers the yield to be the promised yield, since there is

a possibility that the bondholder will receive less than D at maturity in case of default.

Studying s as the yield spread and as a function of T , hence s(T ), with t = 0 gives3:

s(t, T ) = y(t, T )− r

s(T ) = − 1T

ln(V0

D(1−N(d1)) + e−r(T−t)N(d2)

)− r (11)

3The complete derivation is given in appendix A

13

where y(T ) is the yield to maturity and r the risk-free rate. The formula for the proba-

bility of default is given by4:

P [VT < D] = N

ln DV0− (r − σ2

2 )Tσ√T

which corresponds to d2 in equation (9) in the way that:

PD = N(−d2) (12)

From this equation it can be seen, that the PD depends on the leverage ratio D/V0, the

volatility of the firm’s assets σ, the risk-free rate r and the time to maturity T .

Figure 2 displays the model implied yield spread and probability of default for differ-

ent combinations of the debt value D and given values for the firm value V , volatility σ

and the risk-free rate r calculated in Python.

Figure 2: Term structure of yield spreads and probabilities of default calculated throughthe Merton approach

V0 = 100, σ = 0.2, r = 0.05

As noted in Merton (1974) a firm is assumed to default at maturity. Thus, if the firm

value is below the debt value for some t < T , that is not treated as a default, as long as

the assets exceed the debt at time T . Figure 2 gives an economic interpretation for the

yield spread and probability of default for different times and debt value characteristics.4The complete derivation is given in appendix B

14

Looking on the right panel, for very short times to maturity and V0 < D, it is extremely

unlikely that Vt > D, i.e. that the assets during a short time period t will increase enough

to exceed the debt D. This is due to the fact that the firm value Vt is modelled as a

geometric Brownian motion which does not allow the firm value to jump to the default

threshold. Therefore, if debt-holders hold bonds with very short maturities and know

that V0 < D, they implicitly recognise that it is almost impossible that the asset value

of the firm will exceed the bond value at maturity. As a compensation, debt-holders will

require a extremely high yield on the bond, resulting in a term structure which has the

form of the blue curve in the right panel. On the other side, using similar arguments, if

V0 > D at t, it is extremely unlikely that the firm value will fall below the debt, meaning

it is very likely that the bond will not default at very short time horizons. For a medium

levered firm, like the orange one in the right panel, a hump-shaped term structure can

be observed. From there it can be seen that the spread rises in the beginning which is

the result of an increase in σ. In detail, the standard deviation σ rises with an increase

in time t at the beginning, which means it is more likely that Vt < D at time t. As time

elapses further, the spread decreases as the exponential growth of the drift term µ begins

to increase.

Summing up the credit spreads obtained through the Merton model, it one concludes

that the result obtained critically depends on the assumption that the firm value Vt

is in some sense a function of a Brownian motion which has continuous paths. Also the

assumptions of no default prior to maturity as well as a constant exogenous default barrier

are chosen because of simplicity. However, while these assumptions are making the model

computational fast and lead to nice closed form expressions for debt, equity and other

relevant quantities, they are sometimes unrealistic. The next section will show extensions

to the Merton model which address to these shortcomings.

4.2 Model extensions to Merton

In response to the shortcomings of the Merton model, numerous more complex models

haven be proposed in the literature. This section outlines a number of popular extensions

which attempt to address the shortcomings of the original Merton (1974).

15

4.2.1 Capital structure

Recall that in Merton (1974) a firm’s debt is viewed as a single zero-coupon bond. Geske

(1977) proposed the first structural model which offers an analytical formula to price

coupon bonds. The author models each coupon payment as an option with the firm value

as its underlying. The model allows equity-holders to default at any coupon payment date

and assumes that equity is issued to make the coupon payments so as to limit changes to

the capital structure. Default occurs if the firm is so low that investors are not attracted

to issue new equity. Although dealing with complex capital structures, the firm in the

Geske model can only default at maturity similar to Merton.

In contrast to a constant default threshold, Collin-Dufresne & Goldstein (2001) introduce

a time depend default barrier. In detail, they incorporate the possibility for firms to adjust

their outstanding debt levels given changes in the firm value which results in a stochastic

mean-reverting leverage ratio. The assumption behind is that firms are assumed to alter

their capital structure in response to changes in the firm value to maintain a target

leverage ratio. This should lead to comparable observed market spreads, especially in the

long run.

4.2.2 First-passage models

In section 4.1 we saw that the Merton model assumes that default can only happen at

maturity. In contrast to that, Black & Cox (1976) make the first attempt to model the

possibility of early default and the effects of bond indentures on bond prices. Their con-

cept is called the first passage of asset value. While focusing on the default prior to

maturity, the authors also study bond covenants, subordinated levels of debt and restric-

tions on changes to the capital structure. Collin-Dufresne & Goldstein (2001) also allow

the firm to default prior to maturity additional to the stochastic default barrier. There

are many more studies considering an exogenous first-passage default barrier, like Briys

& de Varenne (1997), Longstaff & Schwartz (1995) and Brockman & Turtle (2003).

Up to here we only gave an overview for first-passage models which define an exogenous

default barrier, which equity-holders are not able to influence. In endogenous structural

frameworks, default is modelled within the model. In detail, shareholders are assumed

to decide whether a firm defaults on its obligations, because equity-holders are assumed

16

to issue equity to service debt to avoid default. Therefore, this reflects incentive based

approaches, with the shareholders incentive to maximize the equity value (Lando 2004,

p. 60-64).

A popular endogenous structural model is given by Leland & Toft (1996) which include

parameters to capture taxation and bankruptcy costs and consider default to be an en-

dogenously given event triggered by the shareholders. Shareholders are assumed to max-

imize the equity value by finding the lowest default barrier given a trade-off between

bankruptcy costs and taxation. For example, one can think about it that equity-holders

have incentives to take on risky projects given a firm is near bankruptcy or issue debt to

enjoy tax benefits. The work by Leland & Toft (1996) tries to balance these competing

factors, hence representing a traditional trade-off theory model.

The first-passage approach provides a significant improvement to the Merton model as a

firm is not assumed to default prior to the maturity time of debt which results in higher

default probabilities and more realistic credit spreads.

4.2.3 Interest rate process

Remember that the Merton model assumes a constant risk-free interest rate. Obviously,

the market interest rate term structure is not constant, which is the reason one might

also be interested in structural modelling using stochastic interest rates. Pioneers in this

field are Longstaff & Schwartz (1995) which develop a structural first-passage model with

interest rates following a stochastic process described by Vasicek (1977). They assume

that the variability of interest rates influences the firm value process and consider the

Brownian motion to correlate with the interest rate. This is in line with research we

have done in section 3, assuming interest rate risk to be a necessary component of credit

spreads.

Similar to Longstaff & Schwartz, Kim et al. (1993) also incorporate stochastic interest

rates in their model, but in contrast to Longstaff & Schwartz they implement the Cox

et al. (1985) interest rate model for the risk-free rate which is closely related to the Vasicek

model. The difference between both is that for the model by Cox et al., the volatility

includes the square-root of the risk-free rate rather than just the standard deviation of

the firm value and Wiener process as in Vasicek (1977). This extension excludes the

possibility for interest rates to become negative as in the Vasicek model. Similar to

17

Longstaff & Schwartz (1995), the authors show that interest rate risk is a more dominant

factor determining the credit spreads than default risk.

4.2.4 Asset value process

So far the Merton model and its extensions considered the firm value to follow a stochastic

diffusion process, typically a geometric Brownian motion. Under this circumstance, the

changes in the firm value are determined by two parameters:

1. the drift, usually estimated as the long term mean

2. the volatility, usually adding a noise term like a standard Brownian motion to

account for random fluctuations

As a consequence, the firm value will follow a random path around its long term mean

without being affected by sudden up- or downward jumps. This implies that the firm will

never default unexpectedly especially for short periods of time, resulting in low short-term

credit spreads. However, in section 3 we pointed out that jump risk can be viewed as a

statistical as well as economic significant determinant of credit spreads. It was Merton

itself who extended the original geometric Brownian motion process with a jump diffusion

process in Merton (1976), which was then used by Zhou (2001). Actually, Zhou was the

first to model credit spreads using Merton’s jump diffusion model. The jump diffusion

model is given by:

dVt = µdtVt + σVtdWt + dJt, where

dJt = Vtd

(N∑i=0

(Yi − 1))

Here a jump process is added to the underlying diffusion process described in equation

(1) explained earlier. Nt is a Poisson process with rate λ and Yi is a random variable

following a log-normal distribution. Through the above equation two types of changes are

affecting the firm value now. These are the diffusion part described by µdtVt + σVtdWt,

which cause marginal changes and the jump part Jt, which causes discontinuity in the

asset value process due to unexpected shocks. In particular, jumps can be viewed as new

important information becoming available to investors which are industry- or firm-specific.

18

It is reasonable to expect important information to arrive at discrete points in time and for

which a Poisson process with rate λ suits perfectly. The benefit of considering jumps as a

source of credit risk is that it gives rise to multiple term structure shapes of credit spreads

which are for example flat, hump-shaped or down- and upward-sloping (Zhou 2001). This

implies that PDs as well as credit spreads for short maturity times can be different from

zero. Thus a structural model approach incorporating a jump-diffusion process is able to

combine the advantages of the reduced form approach regarding unexpected default events

and the economic explanation of credit risk from the structural framework. Nevertheless,

the parameter estimation of the jump components is very challenging, representing a

disadvantage regarding its practical applications.

4.3 Performance analysis of structural models

We are now going to review empirical studies which compare implied bond spreads and

CDS spreads modelled with the Merton model and its extensions with market observations

in order to evaluate structural models ability to predict market spreads.

4.3.1 Merton model

The first to carry out an empirical study of the original Merton framework were Jones

et al. (1984). They observe bond prices of 27 companies between 1975 and 1981 which

have a relatively simple capital structure. Within their study, they compare the prices

from the Merton model to a risk-free valuation, discounting the cash flows of the bonds

with the risk-free interest rate. Their results indicate that the Merton model yields very

low credit spreads for investment grade bonds and that there is little difference between

the two models. For speculative rated bonds, the Merton framework yields higher credit

spreads while the risk-free valuation has proved to be more suitable for determining credit

spreads.

The study by Eom (2004) also examines the credit spreads obtained through the Merton

model. The authors use the prices of 182 bonds between 1986 and 1997 and compare the

actual credit spreads with spreads calculated through an extended version of the Merton

model using a portfolio-of-zero-coupon bonds approach. The results indicate that the

Merton model calculates credit spreads which are on average 50.42% too low compared

19

to market observations. That holds for investment grade bonds while the spreads on

speculative rated bonds seem to be overestimated by the Merton model.

4.3.2 Performance of extensions to Merton

Additional to the Merton model the study by Eom (2004) further analyzes the perfor-

mance of the models developed by Geske (1977), Leland & Toft (1996), Longstaff &

Schwartz (1995) and Collin-Dufresne & Goldstein (2001). The authors find all models to

have substantial prediction errors for corporate bond spreads. Especially credit spreads

for short maturities are found to be too low. In detail, similar to the results for the Mer-

ton model, model implied spreads of the Geske model are found to be around 30% too

low compared to market spreads. For the stochastic interest rate model by Longstaff &

Schwartz (1995), they show an increase in the model implied spreads on average compared

to Geske and Merton with the percentage spread prediction error being highly positive on

average with 96%. However, the results seem to be very sensitive to volatility estimates

of the Vasicek process. Also the Collin-Dufresne & Goldstein (2001) approach is found

to over-predict spreads by an average percentage spread prediction error of 319%. They

find that the Collin-Dufresne & Goldstein (2001) model over-predicts spreads for firms

with low leverage ratios but underestimates them for high levered firms. For the Leland

& Toft (1996) model they predict higher spreads compared to market data with an aver-

age percentage error of 124% indicating the high dispersion in spread predictions for all

models under investigation.

Huang & Huang (2003) conducted a very extensive performance analysis of structural

models to explain bond spreads. They analyse the same models as Eom (2004) and

also derive a double-exponential jump-diffusion approach in the end. They calibrate the

models, by fitting them to historical bond default data from 1987 to 1997. This should

generate data which more realistically describes the historical intensity of expected losses.

According to their results the Leland & Toft (1996) model explains between 30% to 40%

of market spreads for investment grade bonds and approximately 50% of non-investment

grade bonds. In comparison, the results indicate a weak performance of the Longstaff &

Schwartz (1995) and Collin-Dufresne & Goldstein model, explaining no more than 20%.

Their jump-diffusion model yields better results, where the model spreads account for al-

most 80% in the market spreads for the extreme parameter case. Overall, model implied

20

spreads only account for a small fraction in market spreads for investment grade bonds.

However, the predicted spreads for junk bonds and higher maturity bonds better explain

market observations.

There also exists a number of studies which test structural models ability to forecast

CDS spreads by calculating the survival probability from a given structural model. Er-

icsson et al. (2015) test the accuracy of three endogenous default barrier models among

which is the Leland & Toft (1996) model. They use a sample of CDS data during 1997-

2003 and calculate the CDS premium for a maturity of 5 years. They find that the Leland

& Toft (1996) model provides the highest accuracy with a mean CDS spread being only 2

bps. below market CDS spreads. The authors identify illiquidity to significantly influence

CDS spreads, having a higher impact on credit spreads than on CDS spreads.

Huang & Zhou (2008) conducted a performance analysis of the Merton (1974), Black &

Cox (1976), Longstaff & Schwartz (1995), Collin-Dufresne & Goldstein (2001) and the

jump diffusion model by Huang & Huang (2003). Focusing on senior unsecured CDS con-

tracts on U.S. corporations from January 2002 to December 2004 they provide evidence

that the Merton (1974), Black & Cox (1976) and Longstaff & Schwartz (1995) model

are unable to predict CDS spreads. In detail, the Huang & Huang (2003) and especially

the Collin-Dufresne & Goldstein (2001) model significantly outperform the other three

models. These results are in contrast of those of Eom (2004) and Huang & Huang (2003)

which imply that model extensions do not improve prediction errors. In detail, Huang &

Zhou (2008) found that the jump model performs better for higher rated firms, while the

Collin-Dufresne & Goldstein (2001) model does better for lower rated firms. Schweikhard

& Tsesmelidakis (2012) and Yeh (2010) underline this observation for the CDG model in

their work.

To sum up the literature review on structural models performance, substantial discrep-

ancies between modelled spreads and market observations are identified by the empirical

literature. From a theoretical perspective, the first passage approaches of Black & Cox

(1976), Longstaff & Schwartz (1995) and Collin-Dufresne & Goldstein (2001) provide an

improvement to the default-at-maturity assumption in the Merton model. Capturing the

variability of interest rate in the Longstaff & Schwartz (1995) through stochastic interest

21

rates is also an interesting alternative. Despite increasing in complexity, the model of

Collin-Dufresne & Goldstein (2001) also refers to real market conditions by incorporating

a dynamic capital structure as well as a stochastic interest process.

5 Modelling framework

Based on the structural model analysis in the section before as well as the overview of

the main spread determinants in section 3 we conclude that there is a need of a more

advanced structural model than the one proposed by Merton (1974) to predict market

credit spreads. The approach by Black & Cox (1976) incorporates the possibility of early

default which is more in line with market observations. Notable extension by Longstaff &

Schwartz (1995) and Collin-Dufresne & Goldstein (2001) further address to the constant

interest rate assumption and simple capital structure hypothesis providing an even better

economic interpretation of the structural approach. Hence, starting from the basic Merton

model we will test three structural models which incorporate these specifications, resulting

in three structural models to predict corporate bond prices and their implied spreads. All

three models have closed-form formulas of coupon bond prices which are straightforward

to implement. A detailed explanation of the implied bond pricing formulas for each model

are given in the next subsections.

5.1 Selection of structural models

We follow the work by Eom (2004) where the authors test five structural models including

that of Merton (1974), Longstaff & Schwartz (1995), Collin-Dufresne & Goldstein (2001),

Leland (1994) and Geske (1977). Our study will focus on the models proposed by Merton,

Longstaff & Schwartz and Collin-Dufresne & Goldstein. For simplicity, we will refer to

these models as the LS and CDG model. In order to make the models comparable

to each other we follow the approach by Eom (2004) of modelling coupon bonds as a

simple portfolio of zero-coupon bonds of face value and maturities matching the coupon

payments. Within this approach each coupon is priced as a unique bond with face value

based on the coupon rate c, where the sum of the coupon payments approximates the price

of an equivalent coupon bond. Assuming semiannual payments the approach is written

22

as:

Bcoupon(T ) =t=T∑ti=1/2

c

2Bzero(ti) +Bzero(T ) (13)

The idea of equation (13) offers as an easy to implement and tractable solution to price

risky coupon bonds. The corresponding predicted yields of the prices are then viewed as

bond equivalent yields. The spread is then calculated as the difference between this yield

and a risk-free yield with same maturity.

5.1.1 Extended Merton model:

Following the work by Eom (2004) we price coupon bonds using an extended version of the

original Merton (1974) model which combines equation (13) and (7). As in Eom (2004),

we assume a defaultable bond with maturity T and unit face value that pays semiannual

coupon at an annual rate of c. The price of a coupon bond is then written as:

BM(0, T ) =2T−1∑i=1

B(0, Ti)[(c/2)I{VTi≥K} + min(wc/2, VTi)I{VTi<K}

]+B(0, T )EQ

[(1 + c/2)I{VT≥K} + min(w(1 + c/2), VT )I{VT<K}

](14)

where B(0, T ) represents the value of a risk-free zero-coupon bond at time 0, maturing at

Ti and Vt the value of its assets. I{·} is the indicator function and EQ is the expectation

at time 0 under the risk neutral measure Q and w is the recovery rate. From equation

(12) we know that the risk-neutral survival probability in the Merton model is given by:

EQ[I{VT≥K}

]= N(d2(K, t))

Furthermore,

EQ[I{Vt<K}min(ψ, Vt)

]= V0

B(0, t)e−δtN(−d1(ψ, t)) + ψ [N(d2(ψ, t))−N(d2(K, t))]

d1(x, t) =ln(

V0xD(0,t)

)+ (−δ + σ2

V /2)tσV√t

d2(x, t) = d1(x, t)− σV√t

23

where K are firm’s total liabilities, ψ ∈ [0, K] and N(·) represents the cumulative stan-

dard normal distribution function. Given a default free zero-coupon bond price B(0, T ),

equation (14) can then be used to calculate the price of a defaultable coupon bond un-

der Merton’s assumptions described in section 4.1.1 (Eom 2004). Notice that Merton

(1974) assumes a constant interest rate but offers only little advice to come up with an

appropriate value or a particular interest rate model. We will show in section 5.2 how we

implement an interest rate model in the above bond pricing formula.

5.1.2 Longstaff & Schwartz model

Longstaff & Schwartz (1995) propose a model that delivers an important extension to

the basic framework developed by Merton. In detail, the authors include a first-passage

approach and incorporate interest rate risk which is then used to derive closed-form valu-

ation expression for risky corporate securities and credit spreads. LS model these factors

based on the assumption that a constant interest rate as in Merton (1974) is inappropriate

given real market conditions, in which interest rates may vary widely depending on the

current economic situation and have an impact on a company’s value. In establishing

their model, LS keep several of Merton (1974) assumptions. They also assume that the

firm value follows a geometric Brownian motion as described in equation (1) as well as a

constant volatility of the firm value (σ) and a constant payout ratio (δ). Also the assump-

tions about perfect markets, continuous time trading and a simple debt structure are kept.

Despite the fact that constant interest rates for all maturities are making derivations of

bond pricing formulas even simpler and less computationally-intensive, this assumption

is not observed for risk-free yield curves in the market (see Section 7.1). To capture the

variability of interest rates, LS incorporate the interest rate model proposed by Vasicek

(1977) into their bond pricing model. Hence, the LS model represents a two-factor model,

which depends on both the asset and interest rate process dynamics to influence default

probabilities.

The Vasicek (1977) interest rate model describes the evolution of interest rates as a

mean-reverting stochastic process to model the instantaneous risk-free interest rate r.

24

The interest rate is assumed to have the following dynamics:

drt = κ(θ − rt)dt+ σrdWt (15)

or equivalently

drt = (α− βrt)dt+ σrdWt (16)

where α = κ · θ,β = κ and κ, θ, σr are the rate of mean reversion, long term mean and

volatility of the interest rate respectively. The equation represents an Ornstein-Uhlenbeck

process which is a modification of the arithmetic Brownian motion (Uhlenbeck & Ornstein

1930). This is revealed in the diffusion term, κ(θ−rt), which is the same for the arithmetic

Brownian motion but the drift term includes more elements. The stochastic differential

equation implies, that the direction and magnitude of the drift is not constant but changes

depending on the difference between the value of the process rt and its long term mean θ

at any given point T . For instance, if r happened to be lower than θ, then the drift would

be positive while its magnitude would be proportional to the mean reversion speed as well

as the amount of the difference (θ− rt). Similarly, if r happened to be higher than θ then

the drift would be negative. Based on the model for the interest rate dynamics, Vasicek

(1977) is able to price a zero-coupon risk-free bond at time t with maturity T according

to:

BV (t, T ) = E[exp

(−∫ T

trsds

)|Ft]

(17)

where Ft is a filtration for the interest rate process. That is, Ft is an increasing series

defining all measurable events as the Vasicek process in equation (16) evolves through

time. LS restate this equation in a more understandable form:

BV (r, T ) = e(A(T )−B(T )r) (18)

25

where

A(T ) =(σ2r

2β2 −α

β

)T +

(σ2r

β3 −α

β2

)(e−βT − 1

)−(σ2r

4β3

)(e−2βT − 1

)(19)

B(T ) = 1− e−βTβ

(20)

In contrast to the Merton (1974) assumption, viewing default as a costless event, LS sug-

gest that corporate restructuring costs occur. Instead of modelling a potentially complex

bankruptcy bargaining process between equity and debt holders, they characterize it by

a single parameter w` which represents the percentage write down on the bond in case

of bankruptcy and equals 1− w. Therefore, debt-holderss receive a payout 1 − w` times

the face value of the debt at default. The LS model then calculates the expected payout

based on costly bankruptcy as well as the probability of default under the risk-neutral

measure Q. They propose a solution based on a one-factor Markov process, presented

by Fortet (1947). The LS zero-coupon bond pricing model is defined through a recursive

equation:

BLS(0, T ) = BV (0, T )(1− w`Q(0, T )) (21)

Using the portfolio-of-zeros approach of equation (13) as in Eom (2004) with the above

formula given by LS, the formula to price defaultable coupon bonds is given by:

BLS(0, T ) = c

2

2T−1∑i=1

BV (0, Ti) [1− w`Q(0, Ti)] +(

1 + c

2

)BV (0, T ) [1− w`Q(0, T )] (22)

As indicated by Eom (2004), BV (0, T ) denotes the time 0 value of a default free zero-

coupon bond with maturity Ti given by the Vasicek model and Q(0, Ti) is the time 0

default probability over [0, Ti]. LS propose a numerical solution to the probability of

default estimation which is given by:

26

Q(0, T ) =n∑i=1

qi (23)

q1 = N(ai)

qi = N(ai)−i−1∑j=1

qjN(bi,j), i = 2, 3, ...n

ai =−ln(X)−M( iT

n, T )√

S( iTn

)

bi,j =M( jT

n, T )−M( iT

n, T )√

S( iTn

)− S( jTn

)

M(t, T ) =(α− ρσV σr

β− σ2

r

β2 −σ2V

2

)t

+(ρσV σrβ2 + σ2

r

2β3

)e−βT

(eβT − 1

)+(r

β− α

β2 + σ2r

β3

)(1− e−βT

)−(σ2r

2β3

)e−βT

(1− e−βT

)S(t) =

(ρσV σRβ

+ σ2r

βσ2V

)t

−(ρσV σRβ2 + 2σ2

R

β3

)(1− e−βT

)+(σ2R

2β3

)(1− e−2βT

)

Equation (23) represents the probability under the risk-neutral measure Q that default

occurs, where X = V0/K has to be above 1, meaning if the firm value is already below

the default barrier, the company defaults with a probability of 1 as per construction of

equation (23). The parameter n divides the maturity T into n equal intervals. LS note

that Q(X, r, T, n) → Q(X, r, T ) as n → ∞. Numerical simulations show that setting

n = 200 results in values for Q(X, r, T ) and Q(X, r, T, n) which are virtually indistin-

guishable. The value ρ represents the correlation between the Brownian motion of the

asset value process as in equation (1) and the Brownian motion of the interest rate process

in equation (16).

27

By examining industrial, utility and railroad corporate bond yields for investment grade

companies over a 15 year period, they compare the model implied credit spreads to the

market data. Their findings suggest that corporate bond spreads are heavily influenced by

interest rate volatility, underlining their idea that incorporating stochastic interest rates

in a structural model increases its accuracy to predict market credit spreads. Moreover,

they suggest that the interest rate dynamics better explain credit spreads of risky bonds

than changes to the company asset value. The LS model offers an easy to implement

and tractable solution to price coupon paying bonds and also credit spreads, using equa-

tion (22) and (23). The pricing of the risk-free bond through the Vasicek model through

equation (18) is straightforward as well. Figure 3 represents the term structure of credit

spreads for an eight percent coupon bond for various inverse leverage ratios X in the left

panel and various write-down values w` in the right panel from the original paper in 1997

of LS generated in Python.

Figure 3: Credit spread term structure for an 8% bond for different values of X and w`.

r = 0.04, σV = 0.2, ρ = −0.25, α = 0.06, β = 1 and σ2r = 0.001.

The left panel shows the term structure of credit spreads for different inverse leverage

ratios (X = V0/K). As expected, the credit spreads take higher values for lower values

of X, i.e. higher debt values. We observe a hump shaped term structure for an inverse

leverage ratio of 1.5 similar as in figure 2. For increasing values of the write down param-

eter w`, meaning a lower recovery rate, the credit spread increases. Since the write-down

value w` is related to the priority of debt, the differences in credit spreads shown in the

right panel can be viewed as term structure of priority. However, we observe that short

term spreads are zero, which is mainly because of the firm value following a geometric

28

Brownian motion which does not allow the value to drop unexpectedly in a short period

of time. For the right panel we see that the term structure of credit spreads is generally

downward sloping for long maturities, similar to figure 2.

5.1.3 Collin-Dufresne & Goldstein model:

The model by Collin-Dufresne & Goldstein (2001) constitutes another notable extension

of the basic Merton framework as well as the work of Longstaff & Schwartz (1995). Sim-

ilar to LS, they assume that the firm value evolves according to the geometric Brownian

motion in equation (1), stochastic interest rates evolve according to the Vasicek process

of equation (15) and they allow for bankruptcy costs in the case of default. However,

instead of setting a constant default barrier value K, they implement a dynamic default

boundary and leverage ratio which are mean-reverting. As already described in section

4.2.4, this concept of altering the capital structure is contrary to the earlier structural

models which assume a constant debt value. Hence, the CDG model suggest that firms

continuously adjust their level of debt to maintain a target leverage ratio in response to

changes in the firm value. Based on the work by Opler & Titman (1994), CDG note

that firms generally issue more debt when the value of their assets rises and reduce their

debt values when the value of their assets fall below a certain level. This implies that

leverage ratios would be stationary (mean-reverting) in time. In the setup of their model,

the valuation framework of CDG incorporates a stochastic leverage ratio as well as the

assumption of default prior to the maturity of debt similar to Black & Cox (1976), given

some exogenous specified threshold.

Technically, CDG start by assuming that the firm value follows a geometric Brownian

motion under the risk-neutral measure, similar to Merton (1974), Black & Cox (1976)

and Longstaff & Schwartz (1995). In contrast to these studies, CDG take the logarithm

of the GBM. From equation (1) this implies:

dln(Vt) =(rt − δ −

σ2V

2

)dt+ σV dWt (24)

Second, they assume the risk-free rate rt to follow the dynamics proposed by the Vasicek

(1977) model, similar to equation (16) in the LS model. Thus, the firm value is then

29

assumed to be influenced by two sources of randomness, which are the random values of

its assets ln(Vt) and the random values of the rates rt. CDG, then derive the following

equation for the dynamics of the debt value:

dln(Kt) = λ(ln(Vt)− v − φ(rt − θ)− ln(Kt))dt (25)

where λ > 0 can be interpreted as the speed of mean-reversion of the debt level, φ > 0

as the sensitivity of the debt level to interest rates and v > 0 as a constant, representing

the target leverage ratio. If (ln(Kt) < (ln(Vt) − v − φ(rt − θ)), then dln(Kt) would

be positive, meaning the firm is supposed to increase the debt value. Conversely, if

(ln(Kt) > (ln(Vt) − v − φ(rt − θ)), the firm is assumed to reduce its debt to keep a

stationary leverage ratio.

After describing the dynamics of the asset and debt values through equations (24) and

(25), CDG combine these two, creating a dynamic firm leverage. Defining the logarithm

of the leverage ratio as ln(Lt) = ln(Kt)− ln(Vt) and replacing ln(Kt) and ln(Vt) by their

equations leads to the following dynamics for the log leverage ratio5:

dln(Lt) = λ(ln(Vt)− v − φ(rt − θ)− ln(Kt))dt−((

rt − δ −σ2V

2

)dt+ σV dWt

)

dln(Lt) = λ(L(rt)− ln(Lt))dt− σV dWt (26)

where L(rt) is the risk-neutral target leverage ratio given by:

L(rt) = 1λ

(δ + σ2

V

2

)− v + φθ − rt

(1λ

+ φ)

Equation (26) shows that the log leverage ratio is mean reverting to the target leverage

ratio L(rt), which is a function of the stochastic interest rate given by the Vasicek model.

Default is defined as the first time τ when the log leverage ratio dln(Lt) is zero or equiv-

alently ln(Vt) = ln(Kt).

Similar to Longstaff & Schwartz (1995), CDG derive the price of the bond and credit

spread by assuming that a risky zero-coupon bond with maturity T pays one dollar at5Recall that the leverage ratio is defined as Lt = Kt/Vt, rewriting using the logarithm yields to the

above ln(Lt) = ln(Kt)− ln(Vt)

30

T if τ > T or 1 − w` at time T if τ ≤ T , where w` can be interpreted as the loss given

default. Indicated by Eom (2004), the price of the risky discount and the coupon paying

bond therefore is the same as for the LS model in equations (21) and (22) where only

the calculations of the risk neutral probability of default changes. In particular, the for-

mula presented by Fortet (1947) and used by LS to calculate the probability of default

in equation (23) only correctly calculates the first-passage probability for a one-factor

system. However the LS model is a two-factor model affected by interest rates as well as

asset dynamics, where equation (23) only yields an approximation using Fortet’s formula.

CDG derive an exact solution for the probability of default in their two-factor first-passage

model, offering an improvement to the Fortet version. The derivation of the probability

of default for the CDG model is rather laborious and out of the scope of this thesis, we

will only present the final solution below.

As for the default probability in the LS model, CDG define the default probability as

the sum of discretized probabilities under the T-forward measure, which converges to-

wards the exact solution:

QT (r0, l0, T ) =nT∑j=1

nr∑i=1

q(ri, tj) (27)

q(ri, t1) = ∆rΨ(ri, t1)

q(ri, tj) = ∆rΨ(ri, tj)−

j−1∑v=1

nr∑u=1

q(ru, tv)ψ(ri, tj|ru, tv)

∆r = r − rnr

where:

• nT is the number of equal intervals for which the maturity T is divided

• nr is the number of equal intervals for which the r-space is divided between some

chosen minimum r and maximum r.

r and r represent lower and upper boundaries for the stochastic interest rate. CDG pro-

pose to set these values three standard deviations below and above the risk-free rate’s long

term mean θ which is arbitrarily chosen to optimize for the limits which are statistically

31

unlikely to be reached. Hence:

r = θ − 3√σ2r

2β

r = θ + 3√σ2r

2β

The following equations lead to the final solution of equation (27) and the price for the

risky bond.

Ψ(rt, t) = π(rt, t|r0, 0)N ET

0 (Lt|rt, L0, r0)√V arT0 (Lt|rt, L0, r0)

ψ(rt, t|rs, s) = π(rt, t|rs, s)N

ETs (Lt|rt, Ls = 0, rs)√

V ars(Lt|rt, Ls = 0, rs)

Es(Lt|rt, Ls, rs) = Es(Lt) + CovTs (Lt, rt)2

V arTs (rt)(rt − Es(rt))

V ars(Lt|rt, Ls, rs) = V arTs (Lt)−CovTs (Lt, rt)2

V arTs (rt)

where π is the transition density function for interest rates given by the Vasicek model.

From Aıt-Sahalia (1999), the interest rate in rt of equation (16) has the Gaussian transition

density given by:

π(rt, t|rs, s) = 1√2πγ

e− (rt−θ−(rs−θ)e−β(t−s)2

2γ2 with,

γ2 = V ar(rt|rr) = σr2β

(1− e−2β(t−s)

)

The final solutions for the expected value and variance of Lt and rt can be found in ap-

pendix C. The solution to the PD converges towards the exact PD as the number of sub

intervals {nr, nT} increases. However, its suffice to say that the added features increase

the model complexity and its computation time considerably. For the propose of this

thesis, we set nr = nT = 50, for which it is possible that the estimated PD is outside the

expected interval [0, 1]. For this reason, we implement a check in Python to assure that

the limits are not exceeded.

Figure 4 compares the estimated term structures of credit spreads of the CDG model

32

to the LS model estimates. The left panel shows the credit spreads for parameter values

of the original paper of CDG, while right panel shows the estimation with higher leverage

ratio.

Figure 4: Credit spread term structure in the CDG model compared to the LS model fordifferent leverage ratios

r = 0.06, w = 0.56, σV = 0.2, ρ = −0.2, α = 0.006, β = 0.1, σr = 0.015, φ = 2.8,λ = 0.18, v = 0.6

Figure 4 shows the impact on the credit spread term structure for firms which are

assumed to alter their capital structure continuously. First, the left panel shows that

the CDG model still generates very low credit spreads for short maturities similar to

what we observed in figure 2 and 3 for the Merton (1974) and LS model respectively.

The reason for that is still because of the diffusion property of the Brownian motion.

However, allowing the firm to adjust its debt generates consistently higher credit spreads

for longer maturities, resulting in an upward sloping term structure of credit spreads

which is observed in both panels. One reason is that in the CDG setup, the leverage ratio

is a diffusion type process which makes it an increasing function of time similar to the

firm value, which accordingly yields to higher probabilities of default for longer maturities

and therefore higher credit spreads. The upward sloping curve is therefore more in line

with term structure shapes observed in the market in contrast to the results of the LS

model.

33

5.2 Selection of interest rate models

An integral part of this study is the selection of an appropriate interest rate model as

all previously described models heavily rely on estimations of the risk-free interest rate

as well as our spread calculations. The three selected bond pricing models have different

assumptions regarding the behaviour of the instantaneous risk-free interest rate r. For the

LS and CDG model, the interest rate is assumed to be a stochastic process described by

the Vasicek (1977) mean-reverting model. In the Merton framework a constant interest

rate is assumed which is a rather poor assumption for our spread calculations. Hence,

for the purpose of these thesis two different interest rate models are implemented which

are the previously described Vasicek model and the well known model by Nelson & Siegel

(1987) where the latter is incorporated in the Merton (1974) model.

5.2.1 Vasicek model calibration

The models of Longstaff & Schwartz (1995) and Collin-Dufresne & Goldstein (2001) as-

sume that interest rate dynamics are described by the Vasicek model. Given that we

are implementing both models, using the Vasicek estimates should ensure their internal

consistency as in Eom (2004).

Eom (2004) note that even though the Vasicek model is very popular among academics,

the model contains some theoretically drawbacks. For instance, the variables κ, θ and

σr in equation (15) are assumed to be constant throughout time and the model is not

able to fit all observed term structure shapes in the market. More complex interest rate

models may more accurately capture market interest rate dynamics, but the increasing

complexity for the analytical solutions when incorporated into bond pricing models offsets

this benefit. Following the work by Eom (2004) we implement a least-squares approach

in order to fit the Vasicek parameters κ, θ, r and σr to the market yields. The estimations

are then used as inputs for the LS and CDG model.

In general, the Vasicek model parameters can be either calibrated using historical time

series data or current market yields. Following the studies of Eom (2004) we will apply

the latter one, which uses the current yield curve to calibrate the parameters.

From the Vasicek bond price in equation (18), the theoretical yield-to-maturity is given

34

by(Vasicek 1977):

y(t, T ) = − 1T − t

ln(BV (t, T )

)= −A(t, T ) +B(t, T )rt (28)

where A(t, T ) and B(t, T ) are given in equation (19) and (20) respectively. Consequently

the yield is given by:

y(t, T ) =(σ2r

2β2 −α

β

)(T − t) +

(σ2r

β2 −α

β

)e−β(T−t) − 1

β− σ2

r

2β2e−2β(T−t) − 1

2β (29)

Within the least-squares approach the purpose is to choose the parameter set α, β and σrwhich minimize the sum of squared deviations between the market and Vasicek yield for

all maturities in the data-set. As indicated in Christa Cuchiero (2006), the approach is

to minimize the following optimization problem:

minα,β,σr

(yM(0, T )− yV (0, T )

)′ (yM(0, T )− yV (0, T )

)

where yM(0, T ) denotes the vector of the risk-free market yields and yV (0, T ) the Vasicek

model yields. Following Christa Cuchiero (2006), for today’s short rate r0 the one- or

three-month spot rate is often taken to be the best approximation. Now the parameters

α, β, σr have to be chosen so that yV (0, T ) best matches yM(0, T ). Up to now there was

no optimization function in Python which provides a reliable solution of the optimization

problem above. Hence were using the package ”optimx” in R and call for the function

”nlminb” which uses a quasi-Newton method and allows to set boundary conditions.

In detail, the volatility σr and intensity parameter β have to be larger or equal to 0 by

definition. Following the study by Christa Cuchiero (2006) we implement some additional

boundary constraints as the low number of Vasicek parameters affects the flexibility of the

model to fit various yield curve shapes such as a local minimum. For the speed of mean

reversion β we assume that it does not exceed 4 and for the long-term mean θ we set a

boundary value of 10% as those two values are extreme values. Moreover the volatility

variable is limited between some upper and lower boundary. In particular, 12sr ≤ σr ≤ 3

2sr

where sr denotes that historical standard deviation over an individually chosen past time

horizon of the one year corresponding risk-free rate. Section 7.1 discusses the ability of

our estimation technique to fit the observed term structure.

35

5.2.2 Nelson & Siegel model calibration

The model by Nelson & Siegel (1987) represents one of the most popular among prac-

titioners to fit observed yield curves. Based on the assumption that the current yield

curve only reflects expectations of future interest rates they propose a functional form

for fitting the continuously compounded spot rate curve. Previous tests of the Merton

model in Eom (2004) and Li & Wong (2008) implement the model of Nelson & Siegel

(1987) for their spread predictions. We implement both, the Vasicek and Nelson & Siegel

approach for the Merton model calculation in order to compare both interest rate models.

The Nelson & Siegel approach is simple and very easy to implement in Python using the

common least-squares approach.

Generally, the approach estimates the zero-coupon rate function from the yields observed

on T-bills under an assumed function for forward rates. Utilizing the following function

gives the Nelson & Siegel curve which implies that the implied forward-rate yield curve

is modelled along the entire term structure:

y(t, T ) = α1 + (α2 + α3)[β1

(1− e−(T−t)/β1

T − t

)]− α3e

−(T−t)/β (30)

where α1, α2, α3 and β are parameters which have to be estimated. α1 controls for the

short-term rate, α2 controls for the slope of the yield curve, α3 controls for the hump-

shaped nature of the curve and β controls for the time to decay. Section 7.1.2 will show

this calibration technique applied to real market data.

36

6 Data and methodology

According to Eom (2004) it requires three kinds of data in order to solve for bond prices

and credit spreads for structural models. Those are firm value and capital structure,

interest rate and bond price data. This chapter discusses the various data sources and

selection criteria applied to the data set. The procedures to select company financial

statements, market bond prices, equity valuation and interest rates are outlined in later

subsections. Moreover, a detailed analysis of methods and techniques to estimate the input

parameters required for the three structural models are included. Results from previous

studies will also serve as relevant parameter values for which estimation techniques are

scarce.

6.1 Bond and company data

To test structural model performances, previous researchers like Jones et al. (1984), Eom

(2004) and Li & Wong (2008) select firms with a simple capital structure which for

example incorporate only single levels of debt seniority and a few types of debt. We

restrict our sample to bonds issued by non-financial firms to observe comparable leverage

ratios. In detail, the bonds under considerations need to have standard cash flows with

fixed-rate coupons and principal at maturity and must be issued in either the US or

European market. Moreover, the bonds must be non-callable, non-convertible, have finite

maturity of greater than one year and are not subordinated. For each bond we require

a full history of its daily closing prices starting at the date of issue which is downloaded

from Datastream between January 2010 and December 2019. However, only the prices

on the last trading day starting in the year of the bond issue of each December between

2010 and 2019 are used as we focus the analysis on the year-end observations to match

the price observations with year-end financial data. The time horizon for the date of

issue of the bond sample lasts from January 2010 to October 2016. The financial data

for the corresponding firms is retrieved from Bloomberg & Datastream and is manually

matched with the bond price observations of Datastream. This includes daily observations

of the current market capitalization and end-of-year balance sheet information such as

the total liabilities and dividend yields. Our final sample consists of 30 bonds which is

approximately 20% of those of Eom (2004) and 20% more than that of Jones et al. (1984).

37

Table 1 provides a summary of the bonds and issuer in the sample as well as the industry

in which they are operating. The top two parts show a summary of the issuer divided

into US and European firms. The table indicates that the firms of the US and European

market are fairly large with an average market value of bio. $208.56 and bio. e 106.29

for the US and European market respectively. The leverage ratios of both sub-samples

are approximately equal and around 40% on average. The bond maturities range from

7 to 12 years, all of them representing fixed-coupon bonds. The average spread over the

corresponding risk-free rate is higher for the US than for the European bond sample data

with an average of 101.72 bps. and 78.84 bps. respectively. This indicates that most of the

firms have an investment grade rating with only two of them having a Standard & Poor’s

rating below BBB. The average correlation between the firm value and the corresponding

3 month risk-free rate is positive on average but rather small for both sub-samples. From

the lower panel we observe that most of the firms are operating in the manufacturing or

healthcare business.

6.2 Interest rate data

The ability to accurately model the interest dynamics in the Vasicek and Nelson & Siegel

approach is a key element in the spread predictions of this study. As a part of the portfolio-

of-zeros approach for pricing coupon bonds, it is necessary to download yield curve values

for years prior to bond price observations for each available business day from January

2006 and December 2019. Because our bond sample data contains issuers from the United

States and the Eurozone we retrieve daily observations of the United States risk-free zero

yield curve and euro area yield curve from Thomson Reuters Datastream. For the US

yield curve, Datastream provides observations of the zero yield curve term structure for

each business day modelled as returns from the risk-free financial instrument issued by the

US Treasury. For each daily observation we collect yield curve data for monthly maturity

intervals of 3,6 and 9 months and yearly maturity data from 1 to 30 years. For the

Euro area, daily risk-free yield curve data is obtained from AAA-rated euro area central

government bond daily spot rates. Daily observations for monthly maturities similar to

the US maturities is retrieved as well as yearly maturity data from 1 to 15 years for every

year.

38

Table 1: Summary statistics on bonds and issuers in the sample

Mean Std. Dev. Minimum Maximum

US bonds

Years to maturity 9.87 0.48 8.26 10.05

Coupon (%) 3.38 0.99 2.50 5.75

Yield spread over risk-free rate (bp.) 101.72 45.69 42.47 182.55

Asset market value ($ bio.) 208.56 66.13 41.14 520.35

Market leverage ratio 38.73% 4.97% 16.13% 79.52%

Asset volatility (over 260 days) 14.86% 3.60% 5.81% 24.92%

Correlation between firm valueand 3-month Treasury bills

0.014 0.022 -0.028 0.047

Payout (%) 3.40% 1.94% 0.00% 8.00%

Eurobonds

Years to maturity 8.74 1.67 7.00 12.01

Coupon (%) 2.40 1.12 0.75 4.65

Yield spread over risk-free rate (bp.) 78.84 43.84 23.71 250.70

Asset market value (e bio.) 106.29 17.64 32.05 282.48

Market leverage ratio 42.17% 7.92% 21.32% 80.61%

Asset volatility (over 260 days) 14.62% 3.84% 8.73% 22.82%

Corr between firm value and 3-monthAAA central government bond yield

0.016 0.016 -0.056 0.052

Payout (%) 5.75% 2.35% 2.01% 8.05%

Industry Number of bonds % of sample

Manufacturing 8 26.66%

Healthcare 5 16.66%

Retail 3 10%

Consumer cyclical 3 10%

Consumer defensive 2 6.66%

Telecommunication 2 6.66%

Mines 2 6.66%

Chemicals 2 6.66%

Energy 2 6.66%

Technology 1 3.33%

39

6.3 Implementation

In this section we will outline the necessary steps to implement the selected structural

models in section 5.1 to calculated credit spreads using the parameter estimation tech-

niques of Eom (2004). This includes a description of the values which we estimate as well

as those parameters whose values are assumed based on prior studies.

From 5.1 one can see that the Merton, LS and CDG models offer straightforward formu-

las to price coupon bonds ((Eom 2004)). Each model implied approach is able to price

coupon bonds assuming semiannual coupon payments. For the set of the predicted bond

prices we can calculate their implied spreads from the corresponding yields. The results of

this calculations should then provide insights on each model’s performance in predicting

spreads over a risk-free benchmark given by the interest rate models described in section

5.2. Each of the selected structural models has a set of parameters which have to be es-

timated. Those include firm value and capital structure parameters, payout parameters,

return and volatility estimates, the speed of the mean-reverting leverage process and the

target leverage ratio. Moreover, calibration techniques for the interest rate models as well

as parameters related to bond features are required as well. We will discuss the necessary

estimation techniques in the following chapters which are mainly based on the studies by

Eom (2004) and Huang & Huang (2003).

6.3.1 Capital structure parameters

An essential parameter in all models is the calculation of the default boundary. Following

Eom (2004) we proxy the face value of bonds by the book value of total liabilities.

In order to calculate the leverage ratio we need the asset value of the corresponding firm

which is not observable in the market. A pure proxy approach would estimate it as the

sum of the current market capitalization and the market value of total debt, where the

latter is proxied by its book value. However, this represents a rather poor estimation

technique. For this reason, we implement the most common estimation technique which

has been developed by the KMV corporation in Crosbie & Bohn (2003). The model is most

commonly used to calculated the distance-to-default given by d2 in the original Merton

framework. From Merton (1974), d2 is defined as the number of standard deviations

by which the firm’s asset value must change in order that default occurs some T years

40

from now. Calculations for the distance-to-default are not relevant for this thesis but

the implied estimates for the asset value and volatility are quite attractive for our spread

analysis. In particular, the KMV model estimates the asset value and asset volatility

from the market value and volatility of equity and the book value of liabilities. Crosbie

& Bohn (2003) are using an option pricing based approach recognizing equity as a call

option on the underlying assets of the firm similar to equation (8). From Merton (1974) it

is straightforward to show that by assuming that the equity value E0 follows a geometric

Brownian motion and applying Ito’s Lemma, the equity and asset volatility are related

by the following expression:

σEE0 = ∂E

∂VσV V0 substituting, we obtain

σEE0 = N(d1)σV V0 (31)

From option-pricing theory we know that the partial derivative of the asset value with re-

spect to the equity value equals the Black & Scholes (1973) call option delta(∂E∂V

= N(d1)).

The interpretation behind equation (31) is that the change in the equity value equals the

change in asset value, adjusted for the probability of the firm to survive represented

by N(d1) in equation (9) adjusted for the payout ratio. We now have a system of two

non-linear equations represented by the call price and volatility in equation (8) and (31)

respectively. Solving them simultaneously yields the asset value and volatility implied

by the equity value, equity volatility and liabilities. The equity value and liabilities are

obtained from Bloomberg or Datastream. For the equity volatility we follow Eom (2004)

and estimate the standard deviation of daily historical market capitalization log-returns

over the past 260 trading days and scale it up by a factor of√

260 for 260 trading days

per year. Similar to Eom (2004), we also used a GARCH(1,1) model to calculate equity

volatility but did not come up with significantly different results regarding the spread

calculations.

The parameter δ measures the yearly payout ratio and is calculated as the weighted

average of the bond’s coupon and the firm’s equity payout ratio measured through the

dividend yield in the year of the bond observation. For the target leverage ratio and speed

of mean-reversion in the CDG model we did not come up with a reasonable estimation

technique and therefore follow a constant parameter approach similar to Huang & Huang

41

(2003) and Eisenthal-Berkovitz et al. (2020). We assume identical values to the ones

proposed in the original paper of CDG and set λ = 0.18, v = 0.6, φ = 2.8.

6.3.2 Interest rate parameters

Pricing the bond on a given day requires the estimation of interest parameters for the

structural models. Similar to Eom (2004), parameters are either estimated by the Nelson

& Siegel (1987) yield curve model or the Vasicek (1977) model. In the portfolios-of-zeros

approach in equation (14) of the Merton model, each coupon is priced with the spot rate

of the Nelson & Siegel model. For the LS and CDG model we use the Vasicek risk-free

rate estimates due to internal consistency issues as both models heavily rely on the esti-

mations of the same. The results of fitting the Nelson & Siegel (1987) and Vasicek (1977)

models through the calibration techniques in section 5.2.1 and 5.2.2 to the market yield

curve are given in the next two sections.

In the LS and CDG model the correlation coefficient ρ between the asset returns and in-

terest rates has to be calculated. We follow Eom (2004) by approximating this parameter

through the correlation between equity returns and changes in interest rates. More specif-

ically, the correlation between the 3-month US-Treasury bill rates or Euro area AAA-rate

central government bond yields and stock price data over the last five years prior to the

bond price observation date is calculated.

6.3.3 Parameters related to bond features

Finally, the last input parameter in our structural model approaches is the recovery rate

w. Eom (2004) cite numerous studies related to recovery rates in case of bankruptcy. Even

though they identify that the recovery rate will depend on the nature of the firm’s business

operation and the industry, they assume a universal recovery rate for the face value of the

defaulting bond of w = 51.31%. For the outstanding coupon payments, Collin-Dufresne

& Goldstein (2001) note that those are rarely recovered in the bankruptcy event, therefore

setting the recovery rate of coupons wc = 0%. We follow Eom (2004) and Collin-Dufresne

& Goldstein (2001) and assume the same recovery rates for the bond face value and

coupon payments respectively.

42

6.4 Summary of model parameters

Table 2: Summary of model input parameters

Parameters Description Estimated as Data Source

Bond features

c Coupon rate (%) Given Datastream

T Maturity Given Datastream

K Face value Total liabilities Bloomberg &Datastream

w Face value recovery rate Eom (2004) n/a

wc Coupon recovery rate Collin-Dufresne andGoldstein (2001)

n/a

Firm characteristics

V0 Firm Value KMV model Bloomberg &Datastream

µV Asset returns Average monthlychange in V

Bloomberg &Datastream

σV Asset volatility KMV model Bloomberg &Datastream

δ Payout ratio Weighted average of cand dividend yield

Bloomberg &Datastream

v Target leverage ratio Collin-Dufresne andGoldstein (2001)

n/a

κl Speed of adjustmentto target leverage

Collin-Dufresne andGoldstein (2001)

n/a

φ Sensitivity of targetleverage to interest rates

Collin-Dufresne andGoldstein (2001)

n/a

Interest rates

r Risk-free rate Nelson & Siegel orVasicek model

Datastream

ρ Correlation betweenV and r

Correlation betweenequity returns and r

Bloomberg &Datastream

κ Speed of interestrate mean-reversion

Vasicek model Datastream

θ Long term meanof interest rate

Vasicek model Datastream

σr Interest rate volatility Vasicek model Datastream

43

7 Results

This section is divided into two topics. In section 7.1 we discuss the results of the Vasicek

(1977) and Nelson & Siegel (1987) model to fit observed market yields. Those results

are further implemented in the corresponding firm value models. Section 7.2 discusses in

detail the performance of the Merton (1974), Longstaff & Schwartz (1995) and Collin-

Dufresne & Goldstein (2001) structural firm value models to predict credit spreads.

7.1 Fitting the yield curve to current market data

In this section we will discuss the result of the calibration technique described in section

5.2.1 when fitting the Vasicek (1977) interest rate model to the observe term structure as

well as the results of the fitted yield curve from the Nelson & Siegel (1987) model from

section 5.2.2.

7.1.1 Vasicek model

The Vasicek model appears to perform very different depending on the current term

structure shapes in the market. Figure 5 shows the fitted Vasicek yield curve versus

the US-treasury term structure as well as the AAA euro area central government bond

yields to maturity obtained from Datastream. It is obvious that the Vasicek model does

not fit the market data in an optimal way because it lacks a large range of possible

term structure shapes. For instance it struggles when the current yield curve is sharply

inverted at short maturities like in the lower left panel. Same is true for the US treasury

yield curve at the end of year 2018, where the shape of the term structure for shorter

maturities makes it almost impossible to give a reasonable fit within our least-squares

approach. This observation was already pointed out by Keller-Ressel (2018), underlining

that an inverted curve typically indicates an economic slowdown or the expectation of the

same. Nevertheless, observe that the model performs quite well for a typical smooth term

structure as for the upper-left panel and the Euro area yield curve at the end of year 2018

in the lower-right.

To sum up, the Vasicek model appears to have too few parameters in order to capture

multiple changes of the sign in the slope of observed term structures in the market. One

reason could be that the model does not allow for time-dependent parameters as in the

44

model of Hull & White (1993) but assumes them to be constant over time.

Figure 5: Vasicek yield curve fits versus US & Euro area market yields

7.1.2 Nelson & Siegel model

The Nelson & Siegel curve is very popular among practitioners because of its straight-

forward calculation using an ordinary least-squares approach and its accuracy to fit the

parameters to an observed market yield curve. Figure 6 shows the Nelson & Siegel curve

fitted to US and European market yields to maturity calibrated in Python. In contrast

to the Vasicek estimates, the model is capable of capturing many of the typical shapes of

the yield curve observed in the market.

45

Figure 6: Neslon & Siegel yield curve fits versus US & Euro area market yields

For the US-treasury yield curve we identify some significant deviations from the ob-

served yields for short time to maturities as in the upper-right panel similar to the Vasicek

fit in figure 5. There also some slightly deviations for longer time to maturities as in the

upper-left panel. However, observe that the Nelson & Siegel approach perfectly fits the

market data for the Euro area yield curve in both panels. From BIS (2005) we found

that nine out of the thirteen central banks which report their curve estimations in the

Euro area use the Nelson & Siegel approach to construct their yield curves. Hence, not

surprisingly the fit for the Euro area yield curve is more accurate compared to the US

yield curve fit.

7.2 Performance of structural models

In this section we summarize the performance of the Merton (1974), Longstaff & Schwartz

(1995) and Collin-Dufresne & Goldstein (2001) structural bond pricing models using the

portfolio-of-zeros approach described in section 5.1 and applying the various calibration

46

techniques described in table (2) for the model input parameters. The percentage pricing

errors, the percentage errors in yields, the percentage errors in yield spreads and absolute

percentage errors in spreads are presented to measure the performance of each model.

When calculating spreads, we subtract the predicted bond yield from the yield on a risk-

free bond with the same maturity generated by either the Vasicek (1977) or Nelson &

Siegel (1987) model. The results in the tables below show the average performance of all

prediction error types across the whole sample data of bond prices, yields and spreads.

The performance of each model is measured on an aggregate level in table (3), using

the entire collection for both US and Eurobonds predictions to calculated the average

performance. In addition, the results of table (4) and (5) are included to offer insights

into the performance of each model for US and Eurobonds respectively. All tables refer to

predictions performed on the collection of end-of-year predictions to calculate the average

performance, starting at the first end-of-year observation in the year of the issue date of

each bond.

7.2.1 Performance at end-of-year

Table (3) summarizes the prediction errors of the three models for the whole bond sample

for the entire collection of end-of-year observations including average spread prediction

errors, standard deviations of the model errors as well as medians, minima and maxima

of the average prediction errors. Table (4) and (5) show the same results for US and

Eurobonds respectively. Special attention shall be paid to the average percentage pre-

diction error in spreads as well as the standard deviation of spread prediction error and

the average absolute percentage spread prediction errors as those are the most important

measures regarding the performance of the models.

47

Table 3: Aggregate performance of structural models averaged across all available end-of-year predictions

Model % error inprices

% error inyields

% error inspreads

Absolute % errorin spreads (bps.)

Merton

Mean 2.26% -54.50% -44.76% 68.18%

Std. Dev. 3.23% 92.82% 43.91% 31.77%

Median 2.29% -24.16% -55.05% 66.21%

Min -7.59% -274.21% -92.83% 43.71%

Max 7.78% 40.61% 83.07% 94.14%

LS

Mean 2.19% -38.09% -32.51% 67.83%

Std. Dev. 3.66% 66.01% 56.09% 37.05%

Median 2.01% -22.18% -39.31% 61.84%

Min -4.68% -193.87% -84.54% 33.28%

Max 6.98% 69.25% 108.24% 147.54%

CDG

Mean 0.48% -28.34% -1.13% 74.31%

Std. Dev. 4.07% 92.48% 78.86% 55.57%

Median 0.45% -9.99% -13.03% 66.99%

Min -5.98% -340.82% -76.89% 42.73%

Max 7.15% 54.71% 125.43% 152.78%

Aggregate average empirical performance of each model for end-of-year observations. Each predictedbond price/yield/spread is estimated on the last trading day of each year and compared to the observedprice/yield/spread. The percentage error in prices is predicted price/yield/spread - actual price/yield/spread

actual price/yield/spread .

48

Table 4: Performance of structural models averaged across all available end-of-year pre-dictions - US bonds

Model % error inprices

% error inyields

% error inspreads

Absolute % errorin spreads (bps.)

Merton

Mean 2.98% -14.12% -47.93% 71.39%

Std. Dev. 3.56% 15.09% 41.20% 29.71%

Median 3.31% -17.13% -55.66% 68.94%

Min -7.59% -34.94% -92.83% 45.90%

Max 7.27% 40.61% 83.07% 94.14%

LS

Mean 3.57% -16.38% -42.23% 69.83%

Std. Dev. 4.05% 17.43% 51.20% 31.75%

Median 2.89% -15.83% -42.23% 70.27%

Min 1.65% -35.11% -84.54% 45.43%

Max 6.95% -4.42% 2.85% 106.18%

CDG

Mean 1.54% -7.43% -17.24% 67.09%

Std. Dev. 4.45% 22.08% 65.08% 37.45%

Median 1.55% -5.59% -18.01% 66.45%

Min -2.12% -26.75% -73.61% 42.73%

Max 5.45% 12.34% 41.70% 89.33%

Average empirical performance of each model for end-of-year US bonds observations.

49

Table 5: Performance of structural models averaged across all available end-of-year pre-dictions - Eurobonds

Model % error inprices

% error inyields

% error inspreads

Absolute % errorin spreads (bps.)

Merton

Mean 1.55% -94.87% -41.59% 64.96%

Std. Dev. 2.91% 170.55% 46.62% 33.83%

Median 1.41% -95.42% -43.71% 66.06%

Min -1.58% -274.21% -84.33% 43.71%

Max 7.78% 4.93% 18.49% 87.40%

LS

Mean 0.81% -59.81% -22.78% 65.83%

Std. Dev. 3.27% 114.58% 60.99% 42.35%

Median 0.44% -50.68% -17.95% 61.18%

Min -4.68% -193.87% -78.32% 33.28%

Max 6.98% 69.25% 108.24% 147.54%

CDG

Mean -0.58% -49.25% 14.98% 81.52%

Std. Dev. 3.68% 162.87% 92.65% 73.69%

Median -0.20% -47.63% 18.54% 70.81%

Min -5.98% -340.82% -76.89% 43.15%

Max 7.15% 54.71% 125.43% 152.78%

Average empirical performance of each model for end-of-year Eurobonds observations.

The results for the average prediction errors in bond prices on the aggregate level

for the Merton model in table (3) show that the model overprices the bonds on aver-

age with 2.26% for the end-of-year observations. Conversely, this results in an average

negative prediction error for the yields. Observe that the average percentage error in

yields as well as its standard deviation are very large on the aggregate level which is

mainly due to the results for the Eurobond sub-sample. This is because the sample con-

tains mostly investment grade bonds and therefore very low yields as prices are higher.

More specifically, the predicted yield by the Merton model almost exactly matches that

of an otherwise risk-free bond priced off the Euro area yield curve using the Nelson &

Siegel (1987) model which increasingly turns negative for maturities in the bond sample

starting in 2014. However, for the US bond sample the average errors in yields and the

50

corresponding standard deviations for all models significantly improves. When measuring

the Merton model’s performance through the spread prediction errors, the results suggest

that the model extremely underestimates the market spreads for our bond sample with an

average percentage error of -44.76% indicating the model’s modest predictive power. The

spread error is -0.1493% at best and -100% at worst on a particular day. However, the

Merton model also over-predicts the spreads for four bonds out of the sample but with

rather large standard deviations and average absolute percentage errors indicating a lack

of accuracy. The underestimations in the Merton model remain for the US and Eurobond

sample as well. For the sample data of US bonds in table (4) the average spread error

is -47.93% with nearly the same standard deviation compared to the aggregate results.

Same is true for the Eurobonds. The standard deviation of spread errors and the absolute

spread error suggest that the Merton model more accurately under-predicts observed US

and Euro market spreads.

Similar to the Merton results, the stochastic interest rate model of Longstaff & Schwartz

(1995) also overprices bonds for the end-of-year observations and under-predicts implied

yields as well. The same can be observed for the average spread prediction error on the

aggregate as well as for the subdivided sample level but with a higher standard devi-

ation and quite similar absolute percentage spread error statistics. In contrast to the

Merton model, the LS model is indeed able to raise the average relative spread errors for

all samples. The spread error on the aggregate level is -32.51%. The highest prediction

error is found to be -85% which is still high but is lower compared to the lowest average

spread prediction error of the basic Merton model which is -92.83%. While the estimates

are better, the LS model still consistently under-predicts spreads, with five out of the 30

average spread errors being higher than the average market observations and one out of

the five being a speculative grade bond with BB rating. The LS model performs better

for the Eurobond sample with a low average percentage error for prices and yields and

with the average predicted spread being approximately 20% lower in the sample. However

this comes at the cost of a higher standard deviation of 60.99%, meaning the LS model

performs very erratically for investment grade bonds of the European bond market. The

highest positive spread error is found to be 121.66% for a BB bond and a standard devi-

ation of 162.7% while the lowest average spread prediction error is 0.93% for a AA bond

51

with a standard deviation of 40.45%. For the US bond sample we found that the LS

model only slightly improves the problem of under-prediction in the Merton model. This

indicated by an average mean prediction error of -42.23% compared to -47.93% as in the

Merton model but with a higher dispersion suggested by the standard deviation of spread

errors of 51.20% compared to 41.20% in the Merton results.

For the results of the CDG model we find an even stronger tendency to predict higher

spreads than either the Merton or LS model which is in line with the observations in figure

4. The low average spread error of -1.13% on the aggregate level as well as a positive

error for the Eurobond sub-sample of 14.98% appear to be a major improvement over the

Merton and LS model. The CDG model also does not suffer from serious under-prediction

as in the Merton model indicated by a median of -13.03% compared to the median spread

error of -55.05% in the Merton results. However, examining only the average performances

obscures the fact that the CDG model manages to both greatly under and over-predict

market spreads. This is indicated by a higher average absolute percentage spread error

of 74.40% and a standard deviation of 55.61%. The lowest spread prediction error is still

very low with -76.89% while the highest error is 125.43% both found in the Eurobond

sample for the same BBB rated firm. Nevertheless, the CDG model generates positive

prediction errors for 14 bonds which is nearly 50% of the whole sample. For the positive

prediction errors we found a very low average error in spreads of 0.99% but at the expense

of a large standard deviation of 81.57% underlining the wildly spikes in the estimates. The

lowest average spread prediction error is found to be 0.71% for a BB rated issuer and the

highest for a BBB rated firm with a relative error in spreads of 125.43%. We also can

observe that the dispersion in spread errors is much higher for the Eurobond sample than

for the US bond sample similar to the LS estimates.

7.2.2 Discussion of results

Based on the findings in table (3), (4) and (5) we find little evidence to suggest that the

Merton (1974), Longstaff & Schwartz (1995) and Collin-Dufresne & Goldstein (2001) are

suitable to predict credit spreads for risky corporate bonds in practice. Nevertheless, av-

erage percentage prediction errors in spread decrease as the model complexity increases.

We saw that the average spread prediction errors for all models are either negative or

52

positive and all different from zero. The consistent under-prediction errors in the Merton

estimates in all samples do not justify the use of the Merton model in practical appli-

cations and are in line with the results of Huang & Huang (2003) and Eom (2004). As

with the Merton framework, the average predicted spreads for the model of Longstaff &

Schwartz (1995) are also lower than the average market spreads but somewhat improve

in all samples. The Collin-Dufresne & Goldstein (2001) model predicts spreads which are

noticeably higher than the equivalent estimates of the LS and Merton model in all cases.

Assuming that the risk-free rate follows a stochastic process and thus that the Treasury

or Euroarea risk-free rate impacts the probability of default in the LS model and in-

creases the predictive power of our spread estimations on average. From this perspective

the results indicate that the role of a stochastic interest rates is important for a better

prediction of credit spreads which is not due to the variation of the risk-free rate over dif-

ferent time horizons but because it allows for an additional source of volatility of the firm

value. This leads to higher probabilities of default compared to the Merton estimates

and consequently to higher predicted credit spreads in our portfolio-of-zeros approach.

Furthermore, the first-passage approach, allowing the firm to default prior to maturity in

the original LS model could also be a reason for higher predicted spreads.

The model by CDG differs from the LS model because it assumes a target leverage ratio,

which the firm moves towards. The model requires the estimation of several parameters

which are not estimated in the other two models and is therefore subject to more measure-

ment problems. However, the calculations in table (3) show that the calculated spreads

perform best based on the results for the average percentage error in spreads. One reason

that the CDG model under our calibration approach generates higher credit spreads could

be due to the assumption that firms with higher credit quality may increase their debt

outstanding. From the first perspective the CDG model seems to solve the problem of

structural models generating too low credit spreads for investment grade firms and pre-

dicts higher long-term credit spreads as in figure 4.

Even though the LS and CDG model are able to predict much higher predicted spreads

than the Merton model, we conclude that this does come at the cost of a loss in accuracy

represented by large standard deviations and high absolute percentage errors in line with

the results of Eom (2004). For example, the range of predicted spreads are extreme, there

are 6 observation dates for which the LS model predicts spreads which are close to 1 basis

53

point or lower resulting in a prediction error of nearly -100%. While the CDG model

provides improvements in predicting higher spreads, we still find 4 observation dates for

which the reported error is nearly -100%. On the other side we find that the highest

predicted spread for the LS model is 993 bps. and 775 bps. for the same bond predicted

by the CDG model which are both substantially higher than the corresponding actual

market spreads.

We can identify a number of factors which likely cause the high dispersion of the pre-

dicted spreads as well as the extreme underestimation in some cases:

• Interest rate modelling. One possible source of errors could be the Vasicek model

which describes the dynamics of the risk-free rate in the LS and CDG model and

partly yields poor results as observed in figure 5. For this reason we calculate the

Merton calculations again based on the risk-free estimates from the Vasicek model

and compare them with the results of the Nelson & Siegel approach. We found that

the prediction error decreases by around 7% and the standard deviation increases by

7% compared to the Merton result using the Nelson & Siegel risk-free rate estimates.

This indicates that the Vasicek estimates for the interest rate volatility are probably

a source of the errors in the LS and CDG model. However for all other measures,

the Merton results with the Vasicek risk-free rate estimates change scarcely.

• Portfolio-of-zeros approach. One possible factor which leads to the dispersion

in the prediction errors in most of the samples for all models could be due to the

portfolio-of-zeros approach. While simple to implement, this approach suggests

that there is no default correlation between each coupon payment. Indicated by

Eom (2004), this implies that the approach does not incorporate conditional default

probabilities, meaning that each coupon calculation ignores the possibility that a

previous coupon payment has not been made due to financial distress and results

in an increased likelihood that future coupon payments will also default. Thus, the

portfolio-of-zeros approach, likely understates the risk of coupon bonds. In addition,

this leads to a higher variance of the estimated spreads because firms with higher

probabilities of default are affected by the underestimation of the probability of

default to a greater extent.

54

• Recovery Rates. The problem of the portfolio-of-zeros approach, could be offset

by higher recovery values. However, the recovery rate is assumed to be constant

and the same for all models as the LS and CDG model do not allow an alternative

rule (Eom 2004). However, we saw in figure 3 that the LS model spreads are very

sensitive to changes in recovery rates. Moreover, Eom (2004) note that a constant

recovery rate of the face value assumption decreases the average prediction error

for spreads for the bonds whose predicted spreads are already low implied by the

portfolio-of-zeros approach.

• Parameter estimation methodology. One reason for the consistent underesti-

mation of market spreads in the Merton model could be due to the fact that the

firms in our sample typically have an investment grade with mostly low leverage

ratios across the sample horizon. Remember from table (2) that we use an option

theoretic approach to parameterize leverage following the optimization problem in

equation (31). From option pricing theory we know that call options which are

far in-the-money, i.e. when the asset value is already well above the options strike

price (in our case the total liabilities), have little risk of being out of the money at

expiration date. Note that our bond sample characteristics suggests a rather small

risk, meaning that the equity-as-an option is well in the money on average. Eom

(2004) note that the KMV approach could lead to lower predicted spreads, however

the benefits of estimating the unobserved asset value through an option-theoretic

approach rather than using a pure pure proxy approach by summing up the value

of total liabilities and current market capitalization offsets the disadvantages.

Another cause for the high dispersion in the estimates of the CDG model could be

due to our assumptions about v, κl and φ, the target leverage ratio, the speed of

adjustment toward the target leverage ratio and the sensitivity of the target lever-

age to the risk-free rate respectively which we assume to be constant for the whole

sample. However, a notable study by Huang & Huang (2003) also assumes them to

be constant.

• Non-credit risk factors. From section 3 we found that the academic literature

identifies several non-credit risk factors which affect corporate bond spreads apart

from the credit risk factors of the structural models analysed in our studies. How-

55

ever, the proposed structural models of our study only take into account credit-risk

factors like the risk-free interest, leverage ratio or asset volatility. Hence, our results

represent a more purer measure of credit risk which could also be a reason for the

strong bias in the Merton and LS model.

• Bond sample data. Another factor which may causes the higher dispersion ob-

served in the results is the selection of our bond sample data. While we put an

emphasize that the selection processes is similar to those in Eom (2004) we must

recognize that we have significantly fewer bonds in our sample than in Eom (2004)

(30 vs. 182).

56

8 Conclusion

An extensive analysis of the structural model approach with respect to their applications

in credit risk showed that the mathematics of this type of models translates well to the

relationship between the capital structure of the firm and the default event. More specif-

ically, the models under investigation allow for an economic interpretation of the cause

of default and thus the corporate bond implied credit spreads. We showed that the hy-

potheses and complexities of the structural framework has improved considerably since

the original Merton (1974) model. Interesting examples include the model of Longstaff &

Schwartz (1995), which incorporate a stochastic interest rate process to take into account

interest rate risks to give a better explanation of the default process. Another notable

extension is the work by Collin-Dufresne & Goldstein (2001) which incorporate a station-

ary default barrier in their approach.

From the literature review we selected the above mentioned models for which we analyzed

the underlying concepts and discussed their implications regarding their model implied

spreads. Through a portfolio-of-zero-coupon bonds approach we then performed an em-

pirical test to quantitatively establish whether the model of Merton (1974), Longstaff &

Schwartz (1995) and Collin-Dufresne & Goldstein (2001) are able to predict credit spreads

of a sample of risky coupon bonds observed in the US and European bond market. We

find that the Merton (1974) fails to predict the bond implied spreads for all samples and

significantly under-predicts spreads observed in the market, concluding that the Merton

model under-prices risk factors in the market. The Longstaff & Schwartz (1995) yields

better results in all samples compared to the Merton results, meaning it reduces the spread

prediction errors for all samples. However, the model fails to accurately predict market

spreads indicated by a higher standard deviation as well as a higher absolute percentage

error in spreads. Same is true for the results of the Collin-Dufresne & Goldstein (2001)

model. While the average spread prediction error improves in all samples compared to

the LS and Merton estimates, the predictions of the spreads are wildly inaccurate in all

samples.

We therefore conclude that despite their theoretically appealing nature, the Merton

(1974), Longstaff & Schwartz (1995) and Collin-Dufresne & Goldstein (2001) structural

models remain unsuitable to predict corporate bond credit spreads which is in line with

57

the existing empirical literature. Based on our findings there are several factors which

could improve the model performances and may provide interesting material for further

studies. One promising future topic includes the use of a more sophisticated estimation

technique like the maximum-likelihood estimation technique for the input parameters of

the structural model. Clearly, also the portfolio-of-zeros treatment of coupon bonds used

in the models to price coupon bonds may need to be modified as well as the method

by which recovery rates are specified as this affects the variance of spread prediction er-

rors greatly. Introducing the volatility of interest rates as another source of risk helps

to improve the prediction errors but may require a more advanced model than the Va-

sicek model to do it accurately. Finally, a larger bond sample would likely decrease the

dispersion of the predicted spreads and would perhaps lead to more transparent results.

58

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Appendices

A Yield spread in the Merton model

To derive the model-implied yield spreads, it is necessary to derive the yield to maturity.

Before that, reviewing the put-call parity in (10) makes it much easier. From the formula

V = E +B and equation (7), B is implicitly given as:

Vt = E +B

Bt = Vt − E

Bt = Vt − CBS, that is

Bt = Vt − (VtN(d1)−De−r(T−t)N(d2))

Bt = Vt(1−N(d1)) +De−r(T−t)N(d2)) (32)

The yield y(t, T )) for the bond with price Bt at time t with maturity T and face value

D is defined as:

y(t, T ) = 1T − t

ln(D

Bt

)(33)

That is the rate y(t, T )) which satisfies Btey(t,T )(T−t) = D.

Rewriting (33) as 1T−t ln

(DBt

)= − 1

T−t ln(BtD

)and replacing Bt with the formula in (32),

yields:

y(t, T ) = − 1T − t

ln(VtD

(1−N(d1)) + e−r(T−t)N(d2))

(34)

Finally, this equation can then be used to define the yield spread as:

s(t, T ) = y(t, T )− r (35)

interpreting it as the difference over the risk-free interest rate r. Setting t = 0 gives:

s(T ) = − 1T

ln(V0

D(1−N(d1)) + e−r(T−t)N(d2)

)− r (36)

65

B Probability of default in the Merton model

We are deriving the default probability in the Merton model under the real probability

measure P, which is the same for the risk-neutral probability measure Q except for the

drift term.

Under P it holds that Vt = V0e(r−σ

22 )T+σWT and since ln(x) is an strictly increasing func-

tion, the event VT < D is therefore equivalent with the event lnVT < lnD, that is:

ln(V0e

(µ−σ2

2 )T+σWT

)≤ lnD

lnV0 + (µ− σ2

2 )T + σWT ≤ lnD

Solving for WT gives:

WT ≤ln DV0− (µ− σ2

2 )Tσ

Next, recall that WT ∼ N(0, T ), i.e. is normally distributed with zero mean and variance

T . Therefore, if a random variable X is normally distributed with X ∼ N(0, 1), WT has

the same distribution as√TX, thus:

P[Wt ≤ x] = P[√TX ≤ x] = P

[X ≤ x√

T

]= N

(x√T

)(37)

This observation together with the equation before finally gives the PD:

P[VT < D] = N

ln DV0− (µ− σ2

2 )Tσ√T

(38)

C Conditional moments in the CDG model

To determine the price of the risky bond price in equation (22) of the CDG model, Collin-

Dufresne & Goldstein (2001) derive exact solutions of the expected value and variance of

the log leverage ratio dLt and the stochastic interest rate drt respectively. The dynamics

66

of the variables {Lt, rt} are the following:

dLt = λ

(LQ − 1 + λφ

λrt − Lt + σrρσV

λB(T−t)κ

)dt− σV dW T

t (39)

drt = κ

[θ − rt −

σ2r

κB(T−t)κ

]dt+ σrdW

Tr,t (40)

where,

B(s, β) = 1β

(1− e−βs

)LQ = δ + σ2

V /2λ

− v + φθ

The expected value and variance of the leverage process Lt are given by:

ETs (Lt) = Lse

−λ(t−s) − (1 + λφ)(ru + σ2

r

β2 − θ)e−β(t−s)B(t− s, λ− β)

−(ρσV σrβ

+ (1 + λφ) σ2r

2β2

)e−β(t−s)B(t− s, λ+ β)

+ (1 + λφ) σ2r

2β2 e−β(T−t)e−2β(t−s)B(t− s, λ− β)

+(ρσV σrβ

+ λLQ − (1 + λφ)

(θ − σ2

r

β2

))B(t− s, λ)

and

V arTs (Lt) =(

(1 + λφ)σrλ− β

)2

B(t− s, 2β)

+σ2

V +(

(1 + λφ)σrλ− β

)2

− 2ρσV (1 + λφ)σrλ− β

B(t− s, 2λ)

+ 2ρσV (1 + λφ)σr

λ− β−(

(1 + λφ)σrλ− β

)2B(t− s, λ+ β)

The risk-free rate’s mean and variance are given by:

ETs (rt) = rse

−β(t−s) +(θβ − σ2

r

β

)B(t− s, β) + σ2

r

βe−β(T−t)B(t− s, 2β)

V arTs (rt) = σ2B(t− s, 2β) (41)

67

Finally the covariance between log leverage ratio and stochastic interest rate is equals:

CovTs (Lt, rt) = −(1 + λφ)σ2r

λ− βB(t− s, 2β)

−(ρσV σr −

(1 + λφ)σ2r

λ− β

)B(t− s, λ+ β)

68