Essays in Finance:
Corporate Hedging,
Mutual Fund Managers’ Behavior, and
Cryptocurrency Markets
Dissertation zur Erlangung des akademischen Grades
Doctor rerum politicarum
der Fakultät für Wirtschaftswissenschaften
der Technischen Universität Dortmund
Gerrit Köchling
February 1, 2021

Contents
List of Figures V
List of Tables VII
1 Introduction 1
1.1 Publication Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Price Delay and Market Frictions in Cryptocurrency Markets 11
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Data and Delay Measures . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Empirical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3 Does the Introduction of Futures Improve the Efficiency of Bitcoin? 17
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Data and Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.3 Empirical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4 Volatility Forecasting Accuracy for Bitcoin 23
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.3 Data, Empirical Analysis, and Results . . . . . . . . . . . . . . . . . . 25
4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5 Managerial Behavior in Fund Tournaments - the Impact of TrueSkill 33
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
5.2 Fund Tournaments & Skill . . . . . . . . . . . . . . . . . . . . . . . . . 34
5.2.1 The Economics of Tournaments . . . . . . . . . . . . . . . . . . 34
III
Contents
5.2.2 The Impact of Prior Performance through TrueSkill . . . . . . . 35
5.2.3 Identifying Skill Based Tendencies in Risk-Shifting . . . . . . . 38
5.3 Empirical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.3.1 Measuring Performance with TrueSkill . . . . . . . . . . . . . . 41
5.3.2 Skill Driven Risk-Shifting . . . . . . . . . . . . . . . . . . . . . 41
5.3.3 Robustness Tests . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
6 Do Firms Hedge in Order to Avoid Financial Distress Costs? 53
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
6.2 Modeling Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
6.2.1 Estimation of the FDC Function . . . . . . . . . . . . . . . . . 57
6.2.2 Implications of an FDC-induced Hedging Equilibrium . . . . . . 59
6.3 Empirical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
6.3.1 Data Description and Methodology . . . . . . . . . . . . . . . . 62
6.3.2 Absolute FDC and Hedging . . . . . . . . . . . . . . . . . . . . 67
6.3.3 Marginal FDC Benefits and Hedging . . . . . . . . . . . . . . . 70
6.3.4 Robustness Checks . . . . . . . . . . . . . . . . . . . . . . . . . 72
6.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
Appendices 81
A Appendix for Chapter 5 81
A.1 General Tournament Behavior . . . . . . . . . . . . . . . . . . . . . . . 81
A.2 Tables – Robustness Tests . . . . . . . . . . . . . . . . . . . . . . . . . 83
B Appendix for Chapter 6 89
B.1 Estimating the Financial Distress Cost Parameter η from a Capital
Structure Equilibrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
B.2 Construction of the FDC Variables . . . . . . . . . . . . . . . . . . . . 91
B.3 Correlation Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
B.4 Industry Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
Bibliography 97
IV
List of Figures
2.1 Evolvement of the Average Price Delay over Time . . . . . . . . . . . . 13
4.1 Standard GARCH(1,1) Volatility Forecasts, Volatility Proxies, and Losses 28
4.2 Visualization of one-day-ahead Volatility Forecasts . . . . . . . . . . . . 30
5.1 Schematic Work of TrueSkill as a Factor Graph . . . . . . . . . . . . . 37
5.2 Managers per Tournament and Benchmark Related Performance . . . . 40
5.3 Correlation of TrueSkill and the Underlying Information Ratio . . . . . 42
5.4 Comparison of Skill Developments by TrueSkill and ELO . . . . . . . . 51
B.1 Visualization of the First Optimization Problem . . . . . . . . . . . . . 92
B.2 Visualization of the Second Optimization Problem . . . . . . . . . . . . 93
B.3 Contour Plot of the Theoretical Hedge Ratio . . . . . . . . . . . . . . . 93
B.4 3D Plot of the Theoretical Hedge Ratio . . . . . . . . . . . . . . . . . . 94
V

List of Tables
2.1 Correlations and Descriptive Statistics . . . . . . . . . . . . . . . . . . 14
2.2 Regressions of Price Delay on Market Liquidity . . . . . . . . . . . . . 15
3.1 Descriptive Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2 Results for the Pre- and Post-Futures Periods . . . . . . . . . . . . . . 21
4.1 Descriptive Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.2 Number of Superior Models . . . . . . . . . . . . . . . . . . . . . . . . 27
4.3 Results of the MCS Procedure . . . . . . . . . . . . . . . . . . . . . . . 29
5.1 Risk Transitions for Managers of Different Skill Levels . . . . . . . . . . 43
5.2 Risk Transitions Based on Extreme Risk-Shifting . . . . . . . . . . . . 45
5.3 Regressions of Risk-Shifting on Different Skill-Levels . . . . . . . . . . . 50
6.1 Variable Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
6.2 Descriptive Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
6.3 Regressions of Hedging Extent on Hedging Incentives including the
Absolute Amount of Expected FDC . . . . . . . . . . . . . . . . . . . . 71
6.4 Regressions of Hedging Extent on Hedging Incentives including the
Marginal Benefits of FDC Reduction . . . . . . . . . . . . . . . . . . . 73
6.5 Regressions of Hedging Extent on Hedging Incentives for Different Sub-
samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6.6 Regressions of Alternative Proxies of Hedging Extent on Hedging Incentives 77
A.1 Contingency Tables of Annual Tournaments following Brown et al. (1996) 82
A.2 Risk Transitions for Managers of Different Skill Levels - Active Return 83
A.3 Rank Correlations Based on Different Performance Measures . . . . . . 84
A.4 Risk Transitions for Different Hyperparameters: σ = 12.5 . . . . . . . . 85
A.5 Risk Transitions for Different Hyperparameters: σ = 6.25 . . . . . . . . 86
VII
List of Tables
A.6 Risk Transitions for Managers of Different Skill Levels - Secondary
Benchmarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
B.1 Correlation Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
B.2 Industry Distribution of the Dataset . . . . . . . . . . . . . . . . . . . 96
VIII
Acknowledgment
I would like to thank Professor Peter N. Posch, my mentor. He gave me the opportu-
nity to write this thesis, gave me academic freedom, and provided me with constant
encouragement and guidance. I am also profoundly indebted to my co-authors, Philipp
Schmidtke, Janis Müller, Lutz Hahnenstein, and Alexander Swade who have contributed
greatly to my personal and professional growth. I am also grateful to the finance research
group at TU Dortmund University, with whom I spent an amazing time not only as
colleagues but also as friends. Special thanks go to Tobi and Yannick who have been
my companions since the beginning of my bachelor’s studies.
Finally and with upmost importance, I am sincerely grateful for the unconditional
support and love of my parents and my brother. Thank you mum and dad, Klaus,
Beate, and Nils.
IX

1 Introduction
A central concept of financial economics is the informational efficiency of financial mar-
kets. It describes a market in which the prices of financial assets represent all available
information and, consequently, a situation in which investors cannot systematically beat
the market on a risk-adjusted basis. The theoretical roots of this concept go back a
long way (Bachelier, 1900; Mandelbrot, 1963). Fama (1970) operationalized the efficient
market hypothesis forming the basis for several theories, such as asset pricing models
that provide a relationship between an asset’s required return and its risk.
This dissertation covers research from the fields of corporate finance, asset man-
agement, and cryptocurrency markets whose underlying questions can be broadly
summarized under the concepts of market efficiency and risk management. In two
separate contexts, Chapters 2 and 3 empirically investigate the efficiency of the relatively
new market for cryptocurrencies. Assessing the market efficiency is a crucial challenge
due to the novelty, concerns about the intrinsic value of cryptocurrencies (cf. Quiggin
(2013) among others), and massive public interest in the market. Equally important,
Chapter 4 examines the risk, or more precisely, the volatility forecasting capabilities of a
broad collection of generalized autoregressive conditional heteroscedasticity (GARCH)-
type models (Bollerslev, 1986) for Bitcoin returns. The fifth chapter explores how
mutual fund managers deal with risk in a tournament environment.1 Introducing a
new theoretical approach, Chapter 6 uses a proprietary dataset of corporate derivatives
usage to provide a fresh perspective on whether firms hedge (bankruptcy) risk.
Market efficiency refers to the state of a market in which prices at any time “fully
reflect” all available information (Fama, 1970, 1991). Fama (1970) divides price ad-
justments into three information subsets: historical prices, public information, and
monopolistic access to any information relevant to price formation. Empirical testing
of the inability of taking advantage of such information to systematically profit from
trading the underlying assets indicates weak-form, semi-strong, and strong market
efficiency, respectively.
1
1 Introduction
Starting with Urquhart (2016), the weak-form efficiency has almost exclusively been
studied for Bitcoin, whereas only a few papers have focused on cryptocurrencies in
general (Brauneis and Mestel, 2018; Wei, 2018). The present dissertation expands
this field by concentrating on the average price delay of the cryptocurrency market
concerning new information. Using the three price delay measures initially proposed
by Hou and Moskowitz (2005), Chapter 2 demonstrates that the price delay of the
cryptocurrency market has decreased significantly over the past three years. As the
delay measures parsimoniously capture the severity of market frictions that cause a
delay of information incorporation into cryptocurrency prices, Chapter 2 concludes that
the cryptocurrency market becomes more informational efficient over time. Further
tests reveal that this improvement is substantially linked to an increase in market
liquidity.
Exemplary for market frictions are short-sale constraints (Saffi and Sigurdsson, 2011).
Due to the novelty of crypto assets, they are less regulated than the established
equity markets, which likely imposes barriers for institutional investors to access the
market due to a lack of infrastructural and regulatory frameworks. Institutional market
engagement, in turn, fosters efficiency (Boehmer and Kelley, 2009). Chapter 3 addresses
the effect of the launch of Bitcoin futures in December 2017 on Bitcoin’s spot-price
efficiency. Carrying out multiple empirical tests to quantify various aspects of the
return predictability of Bitcoin prices from historical prices, Chapter 3 finds that the
return predictability decreases after introducing futures, suggesting an increase in the
weak-form informational efficiency of Bitcoin.
The unprecedented development of Bitcoin2 combined with its association with
financing illegal activity (Foley et al., 2019), its highly speculative components (Cheah
and Fry, 2015), and its unregulated nature illustrates the importance of understanding
the risks associated with a Bitcoin investment. Chapter 4 focuses on out-of-sample one-
day-ahead Bitcoin return volatility forecasts of GARCH-type models and builds model
confidence sets (MCS) (Hansen et al., 2011) for a statistically sound distinction between
models of equivalent performance and underperformance. The findings indicate that
smaller MCS (i.e., greater distinction between equal- and underperforming models), are
facilitated by a jump-robust volatility proxy based on intra-day returns and asymmetric
loss functions. However, the relatively large final sets confront the modeler with
numerous statistically equivalent models in terms of forecasting accuracy, suggesting
2
caution when making model recommendations for Bitcoin volatility.
The remainder of the thesis moves away from cryptocurrency markets but maintains
a focus on risk. Chapter 5 analyzes the risk-shifting tendencies of mutual fund managers
in a tournament environment, first introduced by Brown et al. (1996). In this context,
losing managers (i.e., those ranked below the median manager at the midpoint of the
year) are incentivized to proactively raise their portfolio risk in the second half of the
year to increase their chances of being among the winning managers at year-end, which
can result in higher fund inflows and compensation (Sirri and Tufano, 1998; Kempf and
Ruenzi, 2008b). Within this framework, Chapter 5 focuses on how prior performance
affects this behavior. For this, the Microsoft’s TrueSkill algorithm (Microsoft Research,
2005) is employed, which is a skill-based ranking system based on the Bayesian network
theory, originally designed for the gaming industry. Our findings suggest a positive
correlation of prior performance and risk-shifting in line with the literature emphasizing
the “overconfidence" of successful managers (Ammann and Verhofen, 2007; Puetz and
Ruenzi, 2011). Furthermore, skilled fund managers seem to hold or increase their risk
in the second half of the year, regardless of their relative success in the first half.
The last chapter of this dissertation covers the use of derivatives by corporations to
hedge their risk. In a perfect and frictionless market, the use of derivatives to hedge
risk should not affect the firm value because shareholders can replicate the firm’s chosen
capital structure (Modigliani and Miller, 1958). Following this, incentives to hedge risk
arise from market imperfections such as tax benefits and financial distress costs (FDC)
(Smith and Stulz, 1985), among others. In a Merton (1974) setting, Chapter 6 proposes
a new theoretical approach by which ex-ante expected FDC are estimated based on the
value of an asset-or-nothing put option on the firm’s assets. Chapter 6 further proposes
optimal hedge ratios that capture the marginal benefits of a further FDC reduction,
trading off the ex-ante expected FDC and expected transaction costs for hedging in an
equilibrium setting. For a proprietary dataset of the use of derivatives of 189 German
middle-market companies, our approach explains a significant share of the empirically
observed cross-sectional variations in hedge ratios.
While the present dissertation adds to a diverse set of literature strands, the following
three contributions are most important. First, the work adds to the growing literature
on cryptocurrencies. The expansive growth of the cryptocurrency market marks a
(possibly) new era of decentralization. Cryptocurrencies impose severe challenges to
3
1 Introduction
regulators worldwide, from being absolutely or implicitly banned to being completely
legal without any rigorous rules. Hence, a complete understanding of the market is of
utmost importance. By examining market efficiency and risk, this dissertation adds to
this understanding, which is of interest to cryptocurrency regulators, developers, users,
risk managers, and others involved in this emerging asset class. Second, the dissertation
adds to the literature studying the influence of behavioral factors on individuals’ decision-
making, particularly the literature studying the relationship between past performance
and future choices of shifting risk. The dissertation extends the empirical work in the
area of fund tournaments by highlighting the self-confident behavior of skilled managers
in increasing their portfolio risk in the second half of the calendar year. Third, this
thesis adds to the literature on corporate hedging by introducing a novel estimation
approach of ex-ante expected FDC, closely tied to corporate finance theory. The thesis
provides a fresh perspective on whether firms hedge to escape the expense of distress,
and it enriches the literature from the perspective of German middle-market companies
and a financial intermediary.
4
1.1 Publication Details
1.1 Publication Details
Paper I (Chapter 2):
Price delay and market frictions in cryptocurrency markets
Authors:
Gerrit Köchling, Janis Müller, Peter N. Posch
Abstract:
We study the efficiency of cryptocurrencies by measuring the price’s reaction time
to unexpected relevant information. We find the average price delay to significantly
decrease during the last three years. For the cross-section of 75 cryptocurrencies we
find delays to be highly correlated with liquidity.
Publication Details:
Economics Letters 174 (2019), 39–41
https://doi.org/10.1016/j.econlet.2018.10.025
5
1 Introduction
Paper II (Chapter 3):
Does the introduction of futures improve the efficiency of Bitcoin?
Authors:
Gerrit Köchling, Janis Müller, Peter N. Posch
Abstract:
The introduction of futures on Bitcoin eases the access of institutional investors to
the market and offers an efficient way to short the cryptocurrency. We investigate
the effect of this event on the market’s price efficiency and find the Bitcoin market to
turn efficient. We conduct commonly used tests for market efficiency and check the
robustness of our results by investigating Bitcoin Cash, a hard fork of Bitcoin, where
we do not find a change in market’s efficiency.
Publication Details:
Finance Research Letters 30 (2019), 367-370
https://doi.org/10.1016/j.frl.2018.11.006
6
1.1 Publication Details
Paper III (Chapter 4):
Volatility forecasting accuracy for Bitcoin
Authors:
Gerrit Köchling, Philipp Schmidtke, Peter N. Posch
Abstract:
We analyze the quality of Bitcoin volatility forecasting of GARCH-type models applying
different volatility proxies and loss functions. We construct model confidence sets and
find them to be systematically smaller for asymmetric loss functions and a jump robust
proxy.
Publication Details:
Economics Letters 191 (2020): 108836
https://doi.org/10.1016/j.econlet.2019.108836
7
1 Introduction
Paper IV (Chapter 5):
Managerial behavior in fund tournaments – the impact of TrueSkill
Authors:
Alexander Swade, Gerrit Köchling, Peter N. Posch
Abstract:
Measuring mutual fund managers’ skills by Microsoft’s TrueSkill algorithm, we find
highly skilled managers to behave self-confident resulting in higher risk-taking in the
second half of the year compared to less skilled managers. Introducing the TrueSkill
algorithm, which is widely used in the e-sports community, to this branch of literature,
we can replicate previous findings and theories suggesting overconfidence for mid-years
winners.
Publication Details:
Journal of Asset Management (2021)
https://doi.org/10.1057/s41260-020-00198-7
8
1.1 Publication Details
Paper V (Chapter 6):
Do firms hedge in order to avoid financial distress costs?
New empirical evidence using bank data
Authors:
Luth Hahnenstein, Gerrit Köchling, Peter N. Posch
Abstract:
We present a new approach to test the financial distress costs theory of corporate
hedging empirically. We estimate the ex ante expected financial distress costs, which
serve as a starting point to construct further explanatory variables in an equilibrium
setting, as a fraction of the value of an asset-or-nothing put option on the firm’s
assets. Using single-contract data of the derivatives’ use of 189 German middle-market
companies that stems from a major bank as well as Basel II default probabilities and
historical accounting information, we are able to explain a significant share of the
observed cross-sectional differences in hedge ratios. Hence, our analysis adds further
support for the financial distress costs theory of corporate hedging from the perspective
of a financial intermediary.
Publication Details:
Journal of Business Finance & Accounting (2020)
https://doi.org/10.1111/jbfa.12489
9

2 Price Delay and Market Frictions in
Cryptocurrency Markets
The following is based on Köchling et al. (2019b):
Köchling, G., J. Müller, and P. N. Posch (2019b). Price delay and market frictions in
cryptocurrency markets. Economics Letters 174, 39–41.
https://doi.org/10.1016/j.econlet.2018.10.025
11
2 Price Delay and Market Frictions in Cryptocurrency Markets
12
13
2 Price Delay and Market Frictions in Cryptocurrency Markets
14
15

3 Does the Introduction of Futures Improve
the Efficiency of Bitcoin?
The following is based on Köchling et al. (2019a):
Köchling, G., J. Müller, and P. N. Posch (2019a). Does the introduction of futures
improve the efficiency of Bitcoin? Finance Research Letters 30, 367–370.
https://doi.org/10.1016/j.frl.2018.11.006
17
3 Does the Introduction of Futures Improve the Efficiency of Bitcoin?
18
19
3 Does the Introduction of Futures Improve the Efficiency of Bitcoin?
20
21

4 Volatility Forecasting Accuracy for Bitcoin
The following is based on Köchling et al. (2020):
Köchling, G., P. Schmidtke, and P. N. Posch (2020). Volatility forecasting accuracy for
Bitcoin. Economics Letters 191, 108836.
https://doi.org/10.1016/j.econlet.2019.108836
23
4 Volatility Forecasting Accuracy for Bitcoin
24
25
4 Volatility Forecasting Accuracy for Bitcoin
26
27
4 Volatility Forecasting Accuracy for Bitcoin
28
29
4 Volatility Forecasting Accuracy for Bitcoin
30
31

5 Managerial Behavior in Fund Tournaments -
the Impact of TrueSkill
The following is based on Swade et al. (2021).
5.1 Introduction
Fund managers compete for investor’s money by signaling their ability to generate risk-
adjusted returns (or alpha) to the market. Using Microsoft’s TrueSkill to estimate each
manager’s skill, we study the impact on the portfolio’s risk level. We find highly skilled
managers to take systematically more risk within one year’s tournament compared to less
skilled managers. These results are robust regarding different market phases, different
years with pronounced risk-shifting incentives, and different empirical approaches.
Our work contributes to the existing literature by introducing Microsoft’s TrueSkill
algorithm as a new measure and thus regarding the tournament nature of the fund
managers as a "game". Building upon Bayesian network theory, TrueSkill identifies
and tracks the skills of managers in a competitive setting in which the belief about a
manager’s skill is estimated on the basis of a manager’s past performance relative to
all other active managers. Despite broad evidence for the long-term underperformance
of active managers against a benchmark (Fama, 1965), individual managers seem
to outperform the market in the short-term, resulting in higher fund inflows and
compensation (Sirri and Tufano, 1998; Kempf and Ruenzi, 2008b), hence promoting a
competitive environment among fund managers.
Second, we extend the empirical work in the area of fund tournaments, which was
first introduced by Brown et al. (1996). They analyze the behavior of mutual fund
managers within one year and detect a risk-seeking investment style for mid-term losers.
Replicating their findings, our results indicate winners increasing their risk suggesting
a different trend of individual behavior in tournaments in recent years. We then follow
33
5 Managerial Behavior in Fund Tournaments - the Impact of TrueSkill
Kempf et al. (2009) and highlight risk-shifting differences in years driven by incentives
(winners are rewarded for their outperformance) and years driven by unemployment risk
(losers are facing high chances of having their funds closed due to underperformance).
We extend this area of research by detecting certain investment patterns based on the
individual skill level of the managers and highlight the correlation between skill and
risk-seeking.
The remainder is structured as follows: In Section 5.2, we introduce the fund
tournaments’ setup and Microsoft’s TrueSkill, Section 5.3 contains the empirical analysis,
while the final section concludes.
5.2 Fund Tournaments & Skill
5.2.1 The Economics of Tournaments
Research in the fields of managerial tournaments is considered as a subset of the
agency theoretic contracting theory, which deals with the disparity between principals’
and agents’ interests and risk aversions. Bolton et al. (2005) summarize the basic
assumptions and implications for multiple scenarios in different areas of economics.
The underlying premise of our analysis is to view the market for portfolio management
service as a multi-period decision making. This implies that investors decide in a cyclical
pattern which fund service to invest in. One significant aspect of this investment process
is the established compensation structure within the fund industry. Fund managers are
often compensated based on their funds’ assets under management which implies large
incentives to generate high fund inflows. Empirical evidence for the positive correlation
in a multi-period context of the past performance of the individual fund and new fund
inflows has been provided for example by Sirri and Tufano (1998).
This correlation leads to the plain risk adjustment hypothesis in literature of losing
managers increasing their risk at mid-term in order to catch up on the leading managers
within their peer (cf. Brown et al. (1996)):
σ2L σ2W
> (5.2.1)
σ1L σ1W
where σpL indicates the risk level of a loser’s portfolio in period p ∈ {1, 2} of a
34
5.2 Fund Tournaments & Skill
two-period annual tournament and σpW the risk level of a winner’s portfolio, respec-
tively. Multiple researchers followed this hypothesis and analyzed various aspects and
implications such as different time periods, competition within fund families, the impact
of the selected fund segment, among others. Important ideas and results can be found
in the works of Chevalier and Ellison (1997), Busse (2001), Deli (2002), Kempf and
Ruenzi (2008a), Kempf and Ruenzi (2008b), and Bär et al. (2010). Despite the findings
of all these researchers, there are still contrary opinions about the existence of such
tournament behavior between managers and especially the exact behavioral aspects for
winners and losers, respectively.
5.2.2 The Impact of Prior Performance through TrueSkill
New fund inflows are positively correlated with the standings of the individual fund at
the end of the tournament, i.e. the end of the year. Most investors tend to trust in
the past performance of a fund and expect it to result in positive returns at or around
the benchmark level once the fund claims a top-level within a certain year. Hence,
investors update their beliefs about the strength of an individual manager based on
past, observed returns, and prior beliefs. In empirical research, this behavior has been
modeled for example by Berk and Green (2004), who use a model that includes two key
aspects: First, the performance of fund managers is not persistent and, second, investors
behave as Bayesians. The first aspect can be interpreted as fund managers are not
outperforming a passive benchmark continuously. Second, investors update their belief
about the strength of an individual manager based on past, observed returns, and prior
beliefs. This leads to the concept of conditional probabilities also known as Bayesian
probability where the probability is interpreted as some reasonable expectation based
on prior beliefs and knowledge.
The TrueSkill algorithm has been developed by a team from Microsoft Research in
2005 and is used for match-making in various online games ever since. The purpose of
this ranking system is to detect and track the skill of individual players despite playing
in teams, derive public rankings, and implement a match-making system that allows
players of the same skill to play against each other. The general idea behind TrueSkill
is to update the presumption about a player’s skill based on the observed outcome of a
given game. This technique is called Bayesian inference as explained for example by
35
5 Managerial Behavior in Fund Tournaments - the Impact of TrueSkill
Box and Tiao (2011). TrueSkill characterizes the belief of a manager’s skill as Gaussian
uniquely described by its mean µ and standard deviation σ (cf. Microsoft Research
(2005)). The parameter µ can be interpreted as the average manager’s skill belief whilst
σ describes the uncertainty about that skill level. The more games a participant plays,
the smaller becomes his σ and therefore, the knowledge about a player’s skill becomes
more precise. Furthermore, his average skill level µ is updated based on the match
outcome.
One of the most important advantages of TrueSkill is its adaptivity to any underlying
setup of ranking match outcomes. It only needs a clear ranking for each match - whether
teams are compared with each other or individuals. We will give a brief overview of
the underlying process of TrueSkill in order to derive a basic understanding of its
functionality. However, we will not explain every mathematical step and its technical
realization within the algorithm but refer to the paper of Herbrich et al. (2006).
Let k managers with a total of n funds {1, ..., n} compete in a match. Each fund is
uniquely assigned to a single manager resulting in k disjunct subsets Aj ⊂ {1, ..., n}. For
each match, the outcome r := (r1, ..., rk) ∈ {1, ..., k} indicates the match specific ranks
rj for each manager j in an ascending order; i.e. rj = 1 is the winner and possible draws
are given as ri = rj . Making use of Bayes’ rule, the conditional probability P (r|s, A) of
the game outcome r given the individual skills s := (s1, ..., sn) of all participating funds
in their manager assignments A := (A1, ..., Ak) leads to the posterior distribution of
P (r|s, A)p(s)
p(s|r, A) =
P (r|A) . (5.2.2)
The prior distribution of the funds’ skill (s) := ∏f n 2i=1N (µi, σi ) is assumed to be
a factorizing Gaussian whilst each fund i has a performance fi ∼ N (si, β2i ) in the
match, centered around their individual skill si with fixed variance β2i . With TrueSkill,
the performance mj of a manager∑j is defined as the sum of its individual funds’performances indicated by mj := i∈A fi (cf. Herbrich et al. (2006)). Figure 5.1j
shows the exemplary process of TrueSkill as a factor graph. This methodology is used
in information technologies to describe complicated ’global’ functions consisting of
many variables which are most likely derived themselves from various ’local’ functions.
Those global functions factor as a product of the local functions and can therefore be
described in a bipartite graph called factor graph. Further information can be found in
36
5.2 Fund Tournaments & Skill
Kschischang et al. (2001).
Figure 5.1: Schematic Work of TrueSkill as a Factor Graph
Notes: Schematic work of TrueSkill illustrated as a factor graph (based on Herbrich et al.
(2006)) for the resulting joint distribution p(s, f, ,|r, A) of three managers with a total of four
funds and manager 1 winning whilst manager 2 and manager 3 draw (k = 3, A1 = {1},
A2 = {2, 3}, A3 = {4} and the ranking r := (1, 2, 2)). The black boxes represent the factor
functions which are used to calculate the local variables - visualized by the light grey circles.
The grey arrows indicate the initial calculation of the skill level for all three managers followed
by the ’inner iteration circle’. This circle is used to approximate the new skill level of all
managers whilst after that, the black arrows indicate the updates of the skill beliefs for each
individual fund.
37
5 Managerial Behavior in Fund Tournaments - the Impact of TrueSkill
5.2.3 Identifying Skill Based Tendencies in Risk-Shifting
In a first step, we calculate the six-month rolling information ratio as a performance
measure of each fund. We use these ratios to create a rating of the funds on a monthly
base to feed-forward to the TrueSkill algorithm. At this stage, funds with less than
one year of tracking record prior to the start of the tournament year are also included
due to the initial calculation of skill levels. Second, the funds included in the annual
tournaments compete against each other on a monthly base whereas their skill level -
and therefore the skill level of each manager - is calculated by TrueSkill based on the
performance rankings. To compare the skill level of different fund managers, we use only
each manager’s expected average skill level µ once the skill development is calculated.
To overcome biases for new managers who have not reached their intentional skill level
yet, we only consider managers and therefore funds with at least one year of tracking
record. This leads to at least 18 matches between all managers and their funds before
they are categorized at the end of a tournament’s interim period for the first time.
To analyze the skill dependent risk-shifting, we use conditional transition matrices
for the best 20% (high skill), the next 60% (medium skill), and the least 20% (low skill)
of each year’s managers. We follow the work of Ammann and Verhofen (2009) and
adapt this transition approach, commonly known from credit default analyses. The
transitions are based on the historical volatility of each manager’s portfolio, whereas
each manager is assigned to a risk tercile:
(e 2i1, ei2) ∈ {1, . . . , 3} , i = 1, . . . , 3 (5.2.3)
with ei1 characterizing the risk tercile of manager i in the interim period and ei2 the
risk tercile in the second half of the year’s tournament. Here, 1 indicates the highest
risk tercile and 3 the lowest, respectively. These migration events of the same kind are
now aggregated in a 3x3 matrix of migration frequencies where the generic element
∑3
cjk = 1{(ei1, ei2) = (j, k)} (5.2.4)
i=1
is the number of migration events from j to k and 1{. . . } the indicator function.
Furthermore, we assume that the observations ei2 are the realization of the random
variables ẽi2 with conditional probability distribution
38
5.3 Empirical Results
∑3
pjk = P (ẽi2 = k | ẽi1 = j) , pjk = 1 (5.2.5)
k=1
with the probability pjk of the risk level of a manager’s portfolio to change from the
j-th to the k-th tercile. Therefore, we use the migration rates as observed:
= cjkp̂jk (5.2.6)
nj
with ∑n = 3j k=1 cjk. To identify any differences between the differently skilled
managers, we use a chi-squared test to check for pairwise homogeneity of the transition
matrices. The test statistic
∑∑ ( )23 22 cjk(s)− nj(s)p̂+= jkχ ( ) + (5.2.7)k=1 s=1 nj s p̂jk
is asymptotically χ2-distributed with two degrees of freedom. The variable p̂+jk models
the estimated probability based on the aggregated data of the two transition matrices
and s is the index for the respective sample, e.g. high- and medium-skilled manager.
By the nature of this approach our analyses put emphasis on the whole dynamics of
the risk-shifting tendencies of differently skilled fund managers. Transition matrices as
employed in this study are, inter alia, widely used in the literature on credit risks (cf.
Höse et al. (2009) for details) and in previous studies focusing on prior performance
and risk-taking of mutual fund managers (Ammann and Verhofen, 2007, 2009).
5.3 Empirical Results
Our empirical analysis builds on the two databases Morningstar and Bloomberg. Follow-
ing Brown et al. (1996) and further researchers in their choice of taking growth-oriented
US equity funds due to the high interest of financial press and direct investor involve-
ment, we include all funds classified by the Morningstar categories US OE Large Growth,
US OE Small Growth and US OE Mid-Cap Growth. We use monthly closing prices by
Bloomberg of the categorized funds for the period of 1991 to 2017. This long period
allows analyzing the behavior of the managers in various market situations since the
selected period combines multiple different aspects such as financial crisis and market
39
5 Managerial Behavior in Fund Tournaments - the Impact of TrueSkill
phases with a positive long-term trend, e.g. 2009-2017. All funds are listed in US dollar
and we clean them for survivorship bias.
Figure 5.2: Managers per Tournament and Benchmark Related Performance
Notes: Black (white) bars indicate a positive (negative) average active return in the respective
year.
Furthermore, we tackle the fact that various funds are team-managed and multiple
managers handle more than one fund by using a string matching algorithm to identify
funds managed by the same managers. We exclude all team-managed funds and match
the remaining funds clearly to a single manager. This results in 559 individual managers
who hold at least one fund on their own within the given time period.
We include all funds in each year’s tournament which have at least one year of
tracking record and do not miss any data point in the given period. Also, we use two
periods of six months to analyze the risk-shifting, which leads to June being the end of
the interim period. Those managers above the average at that point are classified as
winners and those below as losers. Managers with two or more funds fulfilling these
requirements are considered to hold an equally weighted portfolio of their funds to reduce
the impact of pro-active risk-shifting across multiple funds. To calculate benchmark
related performance measures, we use the data of the MSCI North America for the
40
5.3 Empirical Results
same period. An overview of the annual tournaments and the average performance of
its participating manager against the benchmark is given in Figure 5.2.
There are several options to measure risk-levels of mutual funds. Examples are the
return standard deviation, the tracking-error standard deviation which is the standard
deviation of the excess returns of the fund over a benchmark, or the systematic risk a
fund takes which is commonly estimated via a market model. However, the latter two
are rather uncommon in mutual fund tournament studies. We follow previous studies
and measure risk by the annualized standard deviation of the monthly fund returns
(Brown et al., 1996; Kempf and Ruenzi, 2008b).
5.3.1 Measuring Performance with TrueSkill
We start our empirical analysis by demonstrating TrueSkill’s capability to take prior
performance into account. Figure 5.3 shows the development of the Pearson correlation
coefficients between the TrueSkill based rankings of all participating funds within the
tournament of five, two, and one years and their information ratio rankings. The left
panel shows the correlation with TrueSkill levels being calculated for 4 years prior to
2015, the middle one with 1 year prior, and the right one with TrueSkill establishment
just starting in 2015. Hence, Figure 5.3 underlines the time dependence of TrueSkill
and its adaptation of prior performance while establishing skill levels. Since investors’
decisions are often based on behavioral aspects such as prior performance or performance
of fund family members (e.g. Sirri and Tufano (1998), Nanda et al. (2004)), TrueSkill
is an adequate skill measure due to its capability of incorporating these aspects.
5.3.2 Skill Driven Risk-Shifting
Table 5.1 shows the aggregated risk-shifting tendencies for the whole sample period.
It is structured into four panels - the first one is showing the unconditional transition
rates based on the risk terciles in the first and second half of the year and the other
three panels are showing the transition rates for the different skill levels. Thus, the
three skill-based transition matrices are sub-samples of the unconditional case. The
χ2-values are representing the H0-hypotheses of conditional transitions being equal to
the unconditional. Panel D shows significant differences to the unconditional case at
the 5% and 1% level for winners and losers, respectively. Indeed, the tendencies in
41
5 Managerial Behavior in Fund Tournaments - the Impact of TrueSkill
Figure 5.3: Correlation of TrueSkill and the Underlying Information Ratio
Notes: This figure shows the evolution of the Pearson rank correlations between TrueSkill and
its underlying performance measure over different time spans for an exemplary year (2015).
More precisely, the left (middle, right) figure shows the rank correlation between the TrueSkill
rankings estimated over the trailing five (two, one) years and the rankings based solely on the
information ratio over the trailing six months to the corresponding month in 2015.
42
5.3 Empirical Results
increasing the risk levels are much lower for managers with less skill than for those with
high skill.
Table 5.1: Risk Transitions for Managers of Different Skill Levels
Values in % Winner Loser
Risk tercile high medium low χ2 high medium low χ2
Panel A: Unconditional
high 62.4 25.7 11.8 61.3 25.0 13.7
medium 30.0 44.8 25.2 26.6 40.9 32.5
low 11.1 32.9 56.0 11.2 28.1 60.7
Panel B: High Skill
high 62.5 27.0 10.5 64.8 23.6 11.5
medium 29.0 47.1 24.0 25.8 43.0 31.1
low 12.8 31.1 56.2 0.93 12.3 30.4 57.3 2.94
Panel C: Medium Skill
high 63.0 24.6 12.3 61.0 24.8 14.3
medium 31.7 42.6 25.6 27.1 42.7 30.2
low 11.3 35.0 53.7 2.74 12.2 29.1 58.7 2.08
Panel D: Low Skill
high 60.2 28.0 11.8 59.0 27.1 13.8
medium 25.4 49.2 25.4 25.8 34.3 40.0
low 8.1 29.1 62.8 7.43** 7.9 24.4 67.8 13.1***
Notes: This table shows the risk-shifting tendencies from the year’s first-half tercile to the
tercile in the second half of the year for the full aggregated dataset. Panel A represents the
whole sample, whilst Panel B to D show the transitions for different skill levels. Each manager
is classified as being a winner (loser) if his performance measured by the information ratio lies
above (below) the median at the end of the interim period. χ2-values testing H0-hypotheses
of conditional transitions being equal to the unconditional. ***, **, and * indicate significance
at the 1%, 5%, and 10% level (two-tailed tests), respectively.
The first observable pattern is the difference in general risk-seeking between winners
and losers in general. Whilst winners tend to stay at their initial level or even increase
the risk in the second period of the year, losers act the other way around. With
transition rates of 30.0, 11.1, and 32.9 for winner compared to rates of 26.6, 11.2,
and 28.1 for losers, Panel A demonstrates the risk-seeking behavior of winners. Vice
43
5 Managerial Behavior in Fund Tournaments - the Impact of TrueSkill
versa rates of risk decreasing by 25.7, 11.8, and 25.2 compared to 25.0, 13.7, and 32.5
complement this pattern. The subsamples given by Panel B to D indicate similar
patterns across the different skill levels. Still, a lot of managers stay within their first
half risk tercile with transitions up to 62.4. The transitions for remaining managers are
highest for the extreme risk terciles of high and low risk.
Looking at the impact of different skill levels for either tournament standing, we find
clear tendencies of high-skilled managers increasing their risk more often than those
with less skill regardless of their first-half performance. The comparison of Panel B and
Panel D shows higher risk increasing rates for skilled managers in both positions. Hence,
the risk decreasing rates are always higher for managers with less skill. Additionally,
the risk remaining transitions are bigger for high-skilled managers in the highest risk
tercile and low-skilled managers in the lowest risk tercile, respectively.
The subsample for high- and medium-skilled managers are also closely related to the
unconditional one. The χ2-test values show no significant differences here. In contrast,
the subsample of low-skilled managers differs from the unconditional sample at the 5%
level for winners and even at the 1% level for losers. This indicates more controversial
behavior for the minority of less-skilled managers, who seem to secure their wins if
possible and cut their losses during bad tournaments.
In the next step, we take a closer look at years of extreme risk-shifting. Therefore,
we aggregate the five years with the highest risk decreasing by losers and those with
the highest risk increasing by losers whilst winners acting vice versa. These periods
are classified as years dominated by unemployment risk and years dominated by
compensation incentives. Hence, we follow Kempf et al. (2009) in their explanation for
different risk-shifting tendencies in special periods. The five compensation incentive
dominated years are 1992, 1995, 2006, 2014, and 2017, identified in Table A.1 in
Appendix A.1 as those years where the highest RARs are given for mid-year losers. The
years dominated by the risk of unemployment are 1993, 2000, 2001, 2004, and 2016;
these are the years where losers have extremely low risk adjustment ratios at mid-term.
Table 5.2 highlights the differences between years dominated by compensation incen-
tives (Panel A) and years dominated by unemployment risk (Panel B). Overall, both
panels show similar patterns of winners increasing the risk level in the second half as
well. Whilst Panel B has no significant difference between skilled and unskilled winners,
Panel A emphasizes the overconfidence of managers with higher skill levels, who are
44
5.3 Empirical Results
Table 5.2: Risk Transitions Based on Extreme Risk-Shifting
Values in % Winner Loser
Risk tercile high medium low high medium low
Panel A: Compensation Incentive Dominated
High Skill
high 57.5 35.0 7.5 72.0 28.0 0.0
medium 12.5 52.5 35.0 47.8 34.8 17.4
low 10.0 30.0 60.0 8.7 34.8 56.5
Low Skill
high 43.3 40.0 16.7 47.1 41.2 11.8
medium 26.9 42.3 30.8 25.0 39.3 35.7
low 12.5 16.7 70.8 8.9 17.9 73.2
χ2 5.17* 12.3***
Panel B: Unemployment Risk Dominated
High Skill
high 75.9 24.1 0.0 71.2 15.4 13.5
medium 31.0 57.1 11.9 4.0 64.0 32.0
low 0.0 26.4 73.6 0.0 22.2 77.8
Low Skill
high 77.8 18.5 3.7 67.5 25.0 7.5
medium 25.8 61.3 12.9 18.9 32.4 48.6
low 4.1 26.5 69.4 2.3 22.7 75.0
χ2 0.79 9.68***
Notes: This table presents risk transitions for aggregated data of the five years dominated
by compensation incentives and unemployment risk, respectively. The selection of the years
is based on risk adjustment ratios for each year. χ2-values representing homogeneity test
statistic for transitions of high- and low-skilled managers. ***, **, and * indicate significance
at the 1%, 5%, and 10% level (two-tailed tests), respectively.
45
5 Managerial Behavior in Fund Tournaments - the Impact of TrueSkill
seeking more risk past the interim period. The difference between skilled and unskilled
managers is significant at the 10% level.
More importantly, the skill level of individual managers affects their decision making
in a losing scenario. Skilled managers seem to rely on their prior performance and
increase their risk level dramatically with transitions of 47.8, 8.7, and 34.8 in years
dominated by compensation incentives. Instead of cutting their losses, they attempt
to catch up by investing very self-confident. Less skilled managers behave differently
and cut their losses. They have decreasing transitions of 41.2, 11.8, and 35.7. Here,
the skill level of each manager seems to determine his behavior dramatically. The
difference between these types of skills is significant at the 1% level. In years dominated
by unemployment, the majority of all managers decrease their risk level in the second
half of the year, if they lose. Only very few skilled managers try to increase their risk
level at this stage and instead, a few managers with less skill start to gamble for a win.
Those few seem to go all in before their funds are closed permanently. The difference
between skilled and unskilled managers is again significant at the 1% level.
Ammann and Verhofen (2007) introduce another dynamic Bayesian network approach
to analyze the impact of prior performance. They find similar results given as risk-
increasing behavior after years of good performance and decreasing risk-taking after
bad years. Within their following work, Ammann and Verhofen (2009) even highlight
the same patterns of winning managers increasing their risk in the following period and
loser acting vice versa. Our findings are in line with their results and even underlining
the impact of prior performance - measured as investor’s belief about the individual
manager’s skill.
5.3.3 Robustness Tests
Underlying Performance Measure The most important parameter within the TrueSkill
setup seems to be the choice of the underlying performance measure to calculate monthly
rankings, which are the start of further skill calculations. We test for the impact of
different performance measures by repeating our analysis with monthly active returns
of all participating managers. Table A.2 in Appendix A.2 shows very similar results
to our previous analysis indicating high-skilled managers to increase their risk most
of the time regardless of their performance in the first half of the year. These results
46
5.3 Empirical Results
underline the high correlation between different performance measures as shown by
Eling and Schuhmacher (2007). We calculate the Spearman rank-order correlation
coefficients inclusive a two-sided p-value for a hypothesis test with the null hypothesis
of non-correlation between the data series for three different measures. Table A.3
in Appendix A.2 outlines the strong and significant correlation between the Sharpe
ratio, Information ratio and active return rankings. We conclude that the choice of the
underlying performance measure does not affect our initial results significantly.
Skill Thresholds The results could be driven by the choice of quantiles that classify
managers into their skill level. In the main specification, we classified the top 20% as
highly skilled and the bottom 20% as low-skilled which leaves a 20-60-20 split. Other
reasonable splits, e.g. 10-80-10, lead to the same conclusions as we show in untabulated
results.
Risk Adjustment Ratio Approach Our next robustness test deals with the general
tournament behavior regardless of the individual skill of each manager. Therefore,
we replicate the contingency table approach introduced by Brown et al. (1996) based
on the risk adjustment ratios. The results presented in Table A.1 in Appendix A.1
are in line with our results of skill-driven investments, indicating a different trend of
individual behavior in tournaments in recent years. Winners have higher RARs in most
of the years, which is in contrast to earlier findings of Brown et al. (1996). Still, this
demonstrates that our findings are in line with previous methodologies.
Hyperparameter of the Prior Distribution In our empirical analysis, we set the
initial p∏rior distribution of the fund managers’ skills as described in Section 2.2 as
f(s) := ni=1N (µ , σ2i i ) with µ = 25 and σ = µii i 3 ≈ 8.33. Please note that the average
skill level µi is not of much interest in absolute terms since all managers are assumed
to start with the same initial skill. Since we do not define a unit to measure the skill
other than using the Gaussian’s parameters µi and σi, the relative belief of two fund
managers given by their skill distribution is of higher relevance. In that terms, it does
not make much difference whether we start with a level of 10, 100, or the standard level
of 259 as proposed by Herbrich et al. (2006), which originates from TrueSkill’s early
comparability with the ELO ranking.
47
5 Managerial Behavior in Fund Tournaments - the Impact of TrueSkill
To underline the low impact of the initial priors on our results, we vary the relation
between µi and σ , i.e. σ ∈ {µi , µii i 2 4 }. The results are qualitatively similar to our base
case σ = µii 3 , see Tables A.4 and A.5 in Appendix A.2.
The neglectable impact of the priors is in line with the theoretical expectation about
their impact: With sample size n −→∞ the difference between two posteriors based on
different Gaussian priors tends towards zero. The same holds for larger prior variances
σi, as outlined for example by Ley et al. (2017).
Different Benchmark Indexes Within our analysis, we use a risk-adjusted approach
to determine the rankings of each manager used for the TrueSkill algorithm. In fact,
our measure of choice is the information ratio as a market model adjustment measure
where the benchmark is the MSCI North America. Given the different setup of mutual
funds and their long-term purposes, e.g. equity-only, long-only, multi-asset, and so on,
our chosen benchmark might not be appropriate for every mutual fund in the universe.
Nevertheless, we restrict our fund sample to growth-oriented US equity mutual funds as
earlier researchers before (Brown et al., 1996; Taylor, 2003; Kempf and Ruenzi, 2008b).
The categorization is based on the widely accepted classification by Morningstar, which
leads to a quite homogeneous sample. We qualify this putative sample restriction by
similar arguments used in earlier research.
However, Morningstar specifies two benchmark indexes for each of its categories. The
primary index for all three categories used in this study is the S&P500 which correlates
almost perfectly with the MSCI North America. The secondary benchmark index differs
for each category.10 We repeat our analysis benchmarking each fund on its secondary
benchmark index and report the results in Table A.6. Overall, the conditional transition
matrices differ stronger from the unconditional transition matrix than in our baseline
case. In line with our previous findings, we find a tendency that winning managers
increase their risk more than losers and that managers classified as low-skilled seem to
adjust their risk less than managers classified as high-skilled.
Regression Approach On the basis of the conditional transition matrix approach, our
results suggest that the risk-shifting tendencies are significantly different for low- and
high-skilled fund managers and, beyond that, that high-skilled managers tend to increase
their risk-levels to a higher extent compared to low-skilled managers. We acknowledge
48
5.3 Empirical Results
that conclusions like these have to be interpreted with caution due to unobservable
covariates that might influence the results. To mitigate the effect of omitted variables
and provide further empirical evidence for our conclusions, we formulate the following
regression model:
∆σ H L First Halfi,t = β1Ranki,t ×Di,t + β2Ranki,t ×Di,t + β3σi,t + i,t (5.3.1)
where the dependent variable, ∆σ = σSecond Half−σFirst Halfi,t i,t i,t , is the change in standard
deviations of fund i’s returns from the first to the second half of the year t. Ranki,t
denotes the rank of the fund manager with respect to all other managers scaled to the
interval [0, 1] (1 being best). High respectively low manager skill is denoted by D∗i,t
with ∗ ∈ {H,L}. In a further specification, we replace Ranki,t with dummy variables
indicating that a fund manager ranked in the top 20% respectively bottom 20% of all
active managers analogous to the main analysis. For all specifications, we include time
and fund-company fixed effects. The latter control, for example, for all time-invariant
characteristics attributable to a manager’s company that may influence the results.
We present the results of four specifications in Table 5.3, two each using either the
information ratio or active returns to estimate the managers’ skill levels via TrueSkill.
All specifications indicate that high-skilled fund managers significantly increase their risk
after performing well in the first half of the year. Contrary, we find the opposite signs
for any coefficient associated with risk-shifting of less-skilled fund managers. Equality
tests reject the null hypotheses DWin×DH = DWin×DL and DLoss×DH = DLoss×DL.
The explained variation in risk-shifting amounts to ≈ 75%, which is a common value in
fund tournament studies. Overall, the results support our conclusions drawn from the
conditional transition matrix approach and provide further insights on the channels
that foster the results.
Comparison to ELO Last, we compare TrueSkill with another popular skill measure
- the ELO rating, most known from the world of chess. The ELO ranking system is
used in competitive chess as well as various unofficial rankings, e.g. online gaming or
football tournaments. It is much simpler in its calculations and therefore not capable
to adapt teams playing each other. Figure 5.4 shows the skill development of three
random managers of the whole period sample measured by TrueSkill and ELO. Both
49
5 Managerial Behavior in Fund Tournaments - the Impact of TrueSkill
Table 5.3: Regressions of Risk-Shifting on Different Skill-Levels
Information Ratio Active Return
(1) (2) (3) (4)
Rank × DH 0.005* 0.006*
(1.889) (1.940)
Rank × DL −0.003 −0.003
(−0.584) (−0.689)
DWin × DH 0.003* 0.005**
(1.747) (2.060)
DWin × DL −0.003 −0.003
(−1.042) (−0.986)
DLoss × DH 0.003 0.003
(0.909) (0.917)
DLoss × DL −0.004 −0.008***
(−1.580) (−2.854)
σFirst Half −0.361*** −0.363*** −0.362*** −0.365***
(−5.116) (−5.124) (−5.114) (−5.143)
Fixed effects Yes Yes Yes Yes
Observations 5155 5155 5157 5157
Adjusted R2 0.747 0.747 0.747 0.748
Notes: This table presents results of a regression of fund managers’ performance in the first
half of the year on their risk-shifting in the second half, conditional on their estimated skill
levels (high, low). Rank denotes the rank of the fund manager scaled to the interval [0, 1] (1
being best) and D indicate dummy variables for high or low skill, or for ranking among the
top 20% best or worst managers. Robust standard errors are clustered by year. ***, **, and *
indicate significance at the 1%, 5%, and 10% level, respectively.
50
5.3 Empirical Results
ratings are based on monthly matches between all managers participating in a given
year’s tournament. The skill levels are normalized to make them comparable since the
absolute level differs between both systems. Due to our premise of being included in
a year’s tournament if and only if there is more than one year of tracking record, the
managers seem to start with different levels, but in fact, they started all with the same
setup, initially. The ELO ratings vary rapidly on a high frequency, whilst the TrueSkill
ratings are adjusting themselves much slower and only react to unexpected outcomes.
Figure 5.4: Comparison of Skill Developments by TrueSkill and ELO
Notes: This figure shows the temporal development of skill ratings based on TrueSkill and
ELO for three randomly chosen fund managers from our sample. ELO is a method for
calculating the relative skill levels, commonly used in chess. The bottom figure compares the
rolling standard deviations of the two methods highlighting the stable skill belief estimated
via TrueSkill.
The third panel of Figure 5.4 underlines the differences in volatility by representing
the rolling standard deviation over 12 months of the normalized ELO and TrueSkill
ratings, respectively. Hence, the average TrueSkill standard deviation is at 0.106 and
therefore much lower than the one of ELO given as 1.907. A good skill measure should
offer low volatility to establish a stable belief about the skill level of an individual
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5 Managerial Behavior in Fund Tournaments - the Impact of TrueSkill
manager in the long-term. The time stability of TrueSkill shows its potential to classify
managers into skill levels and derive skill-based behavior from it.
Summarizing the results of the robustness checks, our results about the impact
of skill are in line with theories in behavioral finance and psychology, showing the
overconfidence of outperforming managers in their investment decisions. Taylor and
Brown (1988) find evidence of people having unrealistically positive views of themselves
which leads to the described self-confidence not only after being among the winners for
a couple of competitions and De Bondt and Thaler (1995) detect a positive correlation
between high confidence and above average-trading frequencies.
5.4 Conclusion
Our results highlight the self-confident behavior of skilled managers by holding or
increasing their portfolio risk in almost every situation compared to those with less
skill. Applying the TrueSkill algorithm to display investors’ beliefs about the individual
skill level of fund managers, we present a way to model the positive correlation of prior
performance and new investment decisions.
The impact of good performance in recent years seems to lead to an overconfident
investment style of managers, who are shifting their portfolio risk towards the higher
tercile of the peer group in the second half of the year. Only a few managers classified
with less skill increase their risk in a losing situation. We demonstrate the robustness of
our results regarding the choice of the performance measure to rank the managers each
month as well as the usability of TrueSkill as an adequate representation of investor’s
belief about a manager’s skill.
52
6 Do Firms Hedge in Order to Avoid
Financial Distress Costs? New Empirical
Evidence Using Bank Data
The following is based on Hahnenstein et al. (2020).
6.1 Introduction
From Smith and Stulz (1985) there has been a continuously growing literature dealing
with the questions why and how firms should use derivative instruments to reduce the
variability in their income cash flows. Economic theories of corporate hedging are based
on taxes, agency costs and costs of bankruptcy or – in a wider sense – costs of financial
distress (FDC) as the main market imperfections, whose management may result in
value increases for the firm‘s stakeholders.11 Bartram (2000) and Aretz and Bartram
(2010) provide a comprehensive overview of rationales for corporate hedging and the
empirical literature testing these economic theories which have up to now not been
able to yield final conclusions regarding the explanatory power of the three different
economic arguments:
• Tax reasons are due to rules that give rise to convexity in the taxation schedule and
have been very thoroughly scrutinized for the U.S. tax system (see e.g. Graham
and Smith (1999)) but have overall accomplished only weak empirical support.
• Variables used to test whether corporate hedging can lower the agency costs
resulting from shareholder-bondholder or shareholder-manager conflicts lead to
fairly mixed results with only a small number of proxies (e.g. R&D expenditures)
showing the predicted effects.
53
6 Do Firms Hedge in Order to Avoid Financial Distress Costs?
• There are quite many studies that contribute to the existing empirical evidence
with regard to FDC, see e.g. Nance et al. (1993), Mian (1996), Géczy et al. (1997)
and Chen and King (2014). Various proxy variables – like e.g. the (long-term)
debt ratio, the short-term liquidity ratio and the interest coverage ratio, credit
ratings or credit spreads – have been employed as substitutes for the expected
FDC, which cannot be directly observed, and could sometimes be identified as
statistically significant. However, there is up to now no paper in the hedging
literature that tries to estimate FDC, which is the economic explanatory variable
according to finance theory, directly on the basis of a structural model (cf. e.g.
Glover (2016), who uses a similar approach to analyze the firm’s capital structure
decision).
In this paper, we provide a fresh perspective on the question if firms hedge in order to
avoid ex ante expected FDC. We estimate FDC as a fraction of an asset-or-nothing put
option and further introduce a trade-off between FDC and transaction costs to calculate
optimal hedge ratios. Using a proprietary dataset stemming from a major German bank
comprising OTC derivative deals from 189 German middle-market companies, we find
both variables to explain a significant share of the observed cross-sectional differences
in hedge ratios based on both nominal and market values. To be more specific, we
contribute to the literature in two main aspects:
Firstly, we come up with a novel estimation approach for the ex ante expected FDC
as an explanatory variable that is much more closely tied to corporate finance theory
than the previous literature. The basic idea is to construct an empirical measure of
the differences in ex ante expected FDC that can be directly used as an explanatory
variable for the observed cross-sectional differences in samples of companies’ derivatives
usage. Our estimation approach is based on the Merton (1974) structural model of
debtholder default and makes use of the analytical formula for an “asset-or-nothing put
option”, which allows us to compute FDC as a fraction of the value of total assets.
We are able to show empirically that this measure of firm-specific ex ante FDC is
superior to the use of the traditional proxy variables, like e.g. the debt ratio, the interest
coverage ratio and the short-term liquidity ratio, whose choice is rather ad hoc and is not
based on a solid theoretical underpinning. Moreover, the new explanatory variable lends
itself to be further exploited within an equilibrium setting that leads to an additional
54
6.1 Introduction
type of explanatory variables: In order to determine its optimal hedge ratio, each firm
trades off its ex ante expected FDC with the expected transaction costs (TC) that serve
as a counterweight to prevent the firm from hedging as much as possible. As there are
no systematic deviations in the transaction costs for derivatives hedging between firms
(except for size issues), the cross-sectional differences between optimal hedge ratios in
equilibrium should mainly be driven by differences in the firm-specific, marginal benefit
of a further FDC reduction. We calculate both the absolute amount of expected FDC
and the optimal hedge ratio for each firm in such an FDC-TC equilibrium and use them
as our leading explanatory variables in a couple of linear regression models that build
the empirical part of our paper.
Secondly, we enrich the existing knowledge about corporate hedging by testing our
explanatory variables on a rich dataset from a major bank providing thus far unavailable
information on the firm level. We analyze the usage of over the counter (OTC) interest
rate and exchange rate (FX) derivatives in a sample of German middle-market companies
on the basis of a proprietary dataset that stems from a major German bank. The
credit risk in this market segment is generally non-investment grade so that the proper
assessment of the risk of bankruptcy or default is crucial for the banks servicing this
segment. From the firms’ viewpoint, the threat of financial distress therefore should
have a relevant magnitude.
The dataset contains all the existing derivative contracts that these companies had
entered into with the bank at fiscal year-ends 2015 and 2014 and it provides us with
single-contract level data. Hence, our dataset is different from those used in previous
empirical studies in two main respects: On the one hand, our information on the
derivative contracts used is far more granular, but on the other hand, the firms might
have also entered into OTC derivatives contracts with other banks, which stay hidden
from our eyes. As the probability that a firm does derivatives business with more than
one bank is supposed to increase with company size, we pay special attention to the size
variable as a control variable. Because of the construction of our proprietary dataset, we
are able to exploit the bank‘s internal rating data and the resulting regulatory “Basel
II” PDs for the firms in the sample as well as the corresponding balance sheet data that
are processed as a part of the rating system. Since the previous studies have been fairly
restricted in terms of data availability, access to a large set of companies‘ derivative
contracts opens up a new perspective on corporate hedging, namely that of a financial
55
6 Do Firms Hedge in Order to Avoid Financial Distress Costs?
intermediary.
While the literature dealing with expected financial distress as an incentive for
corporate hedging mostly covers the US market (see again Aretz and Bartram (2010) for
an overview), there are only a few studies taking the hedging activities of German firms
into account. In an extensive international study, Bartram et al. (2009) analyze 7319
non-financial companies from 50 countries covering over 80% of the overall global market
capitalization of non-financial firms. Surprisingly, their empirical findings frequently
run counter to the usual theoretical predictions: Regarding the FDC hypothesis, for
example, they find derivative users to be larger and more profitable. However, they
also find derivative users to have significantly higher leverage and fewer liquid assets, as
suggested by the common theory. Carroll et al. (2017) investigate the determinants of
derivative usage for non-financials firms from 11 European countries including Germany.
They also find leverage to be a significant determinant of overall derivative usage.
In line with these findings for European countries, Marsden and Prevost (2005) also
find a positive relationship between the probability of financial distress and the use of
derivatives for a sample of New Zealand listed companies. In an earlier study, Prevost
et al. (2000) compare and contrast risk management practices of firms in New Zealand
to those of firms in the considerably larger, more developed US, UK, and German
markets. Among other findings, the authors conclude that New Zealand companies
have many of the same reasons and objectives for using derivatives as firms in the much
larger American and European economies.
Bodnar and Gebhardt (1999) were the first to compare the derivative usage of US
and German companies by analyzing two surveys from both countries and matching
companies based on size and industry: Their findings suggest that German firms are
overall more likely to hedge and that US and German firms partly differ in their primary
goal of hedging, their choice of instruments, and by the influence of their market view
when taking derivative positions. In a more recent study, Henschel (2006, 2010) finds
German SMEs’ hedging decisions to be clearly orientated towards the owner-manager
and that a direct link between risk management and business planning is seldom found.
In an earlier study for Germany, Glaum (2002) surveyed 74 non-financial firms’ exchange
rate risk management purposes: He finds that highly levered firms rather hedge and do
not take bets or use selective risk management in line with the FDC hypothesis. Hence,
it seems reasonable to take the analysis of the FDC hypothesis a step further when a
56
6.2 Modeling Framework
new proprietary data source, namely the single-contract files of the derivative use of
the corporate middle market clients of a large German bank, is available. As the FDC
hypothesis has been studied for other European markets as well (e.g. Judge (2006)
finds strong evidence supporting the FDC hypothesis in a sample of 400 UK firms) we
produce results that are also quite relevant in an international setting. Furthermore,
we do not only contribute to the knowledge about the derivatives usage in the largest
economy of the Eurozone, but the modeling approach presented generally allows for a
refined analysis of the FDC hypothesis that seems applicable in other countries as well.
It could similarly be applied to other national or international datasets whenever they
entail a measure for the firm’s PD that can be combined with leverage in an exogenous,
Merton-style capital structure model. An example of such a potentially fruitful usage in
future research would be e.g. the publicly available data from the European high-yield
bond markets.
The paper is structured as follows: In Section 2, we introduce our modeling framework
and present the theoretical basis for our approach. Section 3 contains the empirical
analysis, while the final section concludes.
6.2 Modeling Framework
6.2.1 Estimation of the FDC Function
We consider a capital market with several firms, which are all-equity financed in the
beginning. Each firm can choose to issue debt and to enter into derivative contracts.
The stochastic future market value of their operating cash flows in t is called Vt and
we assume for the sake of analytical tractability that Vt follows a geometric Brownian
motion ( )
σ2
µ VV − 2 t+σVWt
Vt = V0e , (6.2.1)
where µV denotes the drift-term, σV > 0 the volatility and Wt a Wiener-process.
(Hence, Vt)follows for fixed t a lognormal distribution with parameters µ = ln(V0) +
σ2 √
µ VV − 2 t ∈ R and σ = σV t. For reasons of simplicity, we limit this analysis on a
one-period setting and denote V := V1 and σ = σV . For V ∼ LN (µ, σ2) the well-known
57
6 Do Firms Hedge in Order to Avoid Financial Distress Costs?
probability density function, which is defined on ]0,∞[, is given by
1 − (ln(V )−µ)2f(V, µ, σ) := √ e 2σ2 (6.2.2)
2πσV
and the corresponding cumulative distribution function is given by
∫ V
F(V, µ, σ) := f(x, µ, σ)dx. (6.2.3)
0
Following Merton (1974), we use a structural model of default. The firm can issue debt
in t = 0. We denote the total amount, which is payable in t = 1 and which comprises
both principal and interest, by D > 0. Assuming the drift term µV , V0, D and the
probability of default are given, t(he v)olatility σ can be determined by solving 1 ln V0 + µ 2D V − 2σPD = Φ −  ,
√ σ ( ( ) ) (6.2.4)
⇔ σ = Φ−1 (PD) + Φ−1 (PD)2 + 2 ln V0 + µV
D
where Φ denotes the cumulative distribution function of the standard normal distribution.
Equation (6.2.4) provides a way to estimate the volatility of the asset value distribution,
which is not a directly observable parameter, when the drift parameter µV , the firm’s
leverage, and an appropriate PD estimate are given.
We use the regulatory (“Basel II”) probability of default (PD) of the companies to
calculate the volatility parameter. The Basel II PD contains all the available information
about the probability of a potentially distressed situation from a debtor’s point of view
and is a reliable source. We regard the Basel II default definition as an adequate way to
distinguish between “bad” states of the world, in which the firm incurs costs of financial
distress and those “good” states, where it does not.12
FDC arise when a firm is in or close to default. In the literature on corporate
finance and corporate hedging, several alternative specifications of FDC functions can
be found.13 Following the mainstream literature on corporate hedging, we model FDC
as a percentage portion of future firm value. We denote the FDC parameter, which we
will assume to be firm-specific in the empirical study later on, by η ∈]0, 1[ (Brennan
and Schwartz, 1978; Leland, 1998; Ross, 1996). Assuming general risk-neutrality, which
58
6.2 Modeling Framework
is rather common (cf. Castanias (1983), Bradley et al. (1984), Kale and Noe (1990) or
Kale et al. (1991)) in this type of analysis, the market value of FDC in t = 0 emerges as
∫ D
FDC := ηV f(V, µ, σ)dV = ηE0 [V | V < D] . (6.2.5)
0
According to option pricing theory, we can interpret the conditional expected value
E (V | V < D) as the value of an asset-or-nothing put option. Given an underlying
S = V and an strike price K = D, this binary option has the payoff
S if S ≤ K, or 0 if S > K. (6.2.6)
Following Cox and Rubinstein (1985), its value is given by the first term of the Black-
Scholes formula (Black and Scholes, 1973), which equals the unprotected present value
of the underlying asset price conditional upon exercising the option. Cf. also Rubinstein
and Reiner (1991). Hence, we can rewrite Equation (6.2.5):
 ( ) − ln V0 + rf + 12σ2DFDC = ηV e r0 fΦ −  . (6.2.7)
σ
With the exception of η, the parameters in (6.2.7) are either directly observable market
data or can be calculated from the firm’s PD via (6.2.4). We assume η to be a firm-
specific parameter that can be inferred from a capital structure equilibrium which
is given by a traditional static trade-off model. We introduce this capital structure
model in Appendix B.1. For the optimally levered firm η is given by Equation (B.1.5)
in Appendix B.1, so that we can estimate the absolute amount of FDC for all firms
according to (6.2.7).
6.2.2 Implications of an FDC-induced Hedging Equilibrium
In this subsection, we analyze the testable implications of an equilibrium situation,
in which all firms choose their optimal level of volatility considering the effects of a
volatility reduction on the present value of their specific FDC.
Hedging is regarded as a means to reduce the volatility σ in the distribution of future
asset value V . We assume that all firms have unrestricted access to a set of derivative
59
6 Do Firms Hedge in Order to Avoid Financial Distress Costs?
contracts which are traded in frictionless market. Without any market frictions, the
value of any hedging contract is zero at the time when it is written. By assuming an
economy of risk-neutral agents, this translates into a zero effect on expected future
cash flows. Hence, the effect of hedging on FDC is given by the following first partial
derivative from (6.2.7) using the asset-or-nothing put option pricing formula:
 ( )  ( ) 
D
∂FDC −  ln − rf − 1 2= rf V0 2σηV0e φ  ln V0 + rD f 12 − > 0 (6.2.8)∂σ σ σ 2
1
if 2D < eµ = V0erf− 2σ .
Obviously, the FDC function of each firm is strictly increasing in firm value volatility
σj , if its probability of default PDj is below at least 50%. Hence, by costlessly reducing
σj, hedging reduces the ex ante expected FDC and increases firm value in these cases.
Since we assume a capital structure optimum independent of the hedging policy, the
change in σ in Equation (6.2.8) stems directly from the use of a derivative instrument
on the asset value distribution and shall not be misinterpreted as a change of σ resulting
from a shift in leverage or PD that would have an indirect impact via Equation (6.2.4)
and that would lead to a different comparative statics result.
What is new in our paper is that we give Equation (6.2.8) a cross-sectional interpre-
tation, which we will state later on as a part of our main Proposition: The marginal
benefit of hedging by reducing FDC differs across firms: It depends on the firm’s
individual FDC parameter η, on its size V0, on its exogenously given optimal debt ratio
Dj/Vj and on the corresponding probability of default PDj , which enters (6.2.8) via σj
(see (6.2.4)). Hence, the differences among firms’ marginal benefits of FDC hedging can
be measured using a set of firm parameters, that are either directly observable or that
can otherwise be empirically estimated.
As a counterweight to FDC that prevents firms from transferring 100% of its hedgeable
risk to the financial markets, we introduce transaction costs TC into the model, cf.
Hahnenstein and Röder (2007) for a hedging model with transaction costs. The
transaction cost function we assume is
TC = γV (σ∗0 − σ) , (6.2.9)
60
6.2 Modeling Framework
where σ∗ denotes the optimal firm value volatility after hedging. Thus, the difference
(σ∗ − σ) denotes the reduction in volatility that goes along with the optimal hedging
policy. The parameter 0 < γ < 1 denotes the marginal transaction costs of hedging,
which we assume to be constant across all firms. It can be interpreted as a basis point
fee payable for each EUR of the derivative nominal that leads to the intended decrease in
the absolute firm value volatility. The optimal (percentage) volatility σ∗ that minimizes
the sum of FDC and transaction costs is given by the following first-order condition as
a unique solution:
∂FDC + ∂TC =! 0⇔ ∂FDC
∂FDC
− !γV ∂σ0 = 0⇔ =! γ (6.2.10)
∂σ ∂σ ∂σ V0
In our empirical study, we use the minimum firm-specific value of ∂FDC in our sample
∂σ
(“marginal utility of the marginal hedging company”) as a proxy for γ, as this value
indicates, at which transaction cost level hedging is still profitable.
The second-order condition for a minimum of total costs is met, which can be
easily seen by taking a look at Formula (6.2.7). As the cumulative standard normal
distribution function φ(x) is convex for all x smaller than zero, the sum of FDC and
transactions costs is con(vex)in σ as a sum of two convex functions, if
ln D − 2r − 1σ∗ 2
V0 f 2 1 ∗< 0⇔ D < V0erf+ 2σ∗ , (6.2.11)σ
which does not imply further restrictions as we already restricted D to be less than
V r −
1σ2 r + 1σ2 ∗
0e f 2 < V0e f 2 (cf. Appendix B.1). Hence, σ indeed minimizes the sum of ex
ante expected FDC and transaction costs.
As the parameter σ represents the asset volatility of the unhedged company and as
the optimal level of volatility after hedging is given by σ∗, we can now define the firms’
optimal hedge ratios in equilibrium as
∗ σj − σ
∗
H := jj . (6.2.12)σj
. The optimal hedge ratio of the firm is defined as the optimal volatility reduction
that satisfies the first-order condition (6.2.10) as a percentage of the volatility before
hedging. We state the results of this section in the following proposition which lays the
61
6 Do Firms Hedge in Order to Avoid Financial Distress Costs?
basis for our discussion of its testable empirical implications later on.
Proposition. In the hedging equilibrium, each firm j = 1, ..., n chooses its optimal
hedge ratio H∗j , so that the sum of ex ante expected (FDC and tra)nsaction costs is
∗
minimized. For a given set of firm-specific parameters Dj , PD∗ ∗
V0 j
, ηj and a non firm-
j
specific transaction cost parameter γ, there is a unique set of optimal hedge ratios
H∗j . The cross-sectional variation in the firms’ hedging behavior can be attributed to
differences in the marginal b(enefits of hed)ging ∂FDC , which result from differences in∂σ
∗
the firm-specific parameters Dj , PD∗ ∗j , ηj .V0j
We illustrate the empirical implications of the model - with the help of a numerical
example - on the basis of our empirical dataset in Appendix B.2. However, it is already
clear that we aim at explaining the cross-sectional differences in the observed hedging
behavior in our sample by the differences in H∗j , which will be our leading explanatory
variable (besides the absolute amount of FDC). H∗j can be interpreted as the optimal
relative (percentage) volatility reduction that minimizes FDC and that combines a set
of empirically observable company parameters in a theoretically rigid approach.
6.3 Empirical Results
6.3.1 Data Description and Methodology
The dataset is obtained in an anonymized form from a major German bank in line
with the rules of banking secrecy. In particular, all company names were erased
beforehand. The bank has been in the corporate lending business for decades and is
considered systemically relevant for the banking system. Within its internal organization,
specialized departments are dedicated to sell all types of derivatives to its corporate
clients.
The data contains all the OTC derivative contracts the bank entered with about 500
non-financial, non-public sector counterparties at year-end 2014 and 2015. The dataset
includes detailed counterparty information such as legal entity identifier, client group
identifier (for client groups consisting of more than one legal entity, i.e. typically one
parent company with more than one subsidiary), industry code, current counterparty
credit rating and corresponding Basel II PD, the precise product type and both the
62
6.3 Empirical Results
current fair value and the nominal amount of the contract. The interest rate risk-
related product types included in our sample are interest rate swaps, forward deals
as well as caps and floors. The currency risk-related products contain FX swaps,
cross-currency swaps, and FX options in about 20 currencies. The sample does not
include exchange-traded products as these derivatives are not bilateral with the bank
being the contractual counterparty.
The so-called Basel II PDs are the bank’s internal PD estimates, which are used
for both bank internal processes like loan approval and margin calculation, but also
for the purpose of calculating regulatory capital requirements. The PD estimation is
based on a quantitative logit model, which is calibrated to the actual default time series
observed within the banking group by means of maximum likelihood estimation and
which has been granted a supervisory permission as an IRBA (internal ratings-based
approach) model under the Basel II regulation. See e.g. the overview in Engelmann
and Rauhmeier (2006) for this typical type of Basel II rating models.14 The logit
model makes use of the firms’ equity/total assets ratios, interest coverage ratios and
liquidity ratios, but also employs some other balance sheet and cash flow figures, which
constitute the individually transformed input variables and which are then combined
within a non-linear function. Hence, the [0;1]-scaled PD function that contains all the
balance-sheet data relevant to predict corporate default risk seems to be the natural
starting point to estimate the firms’ ex ante financial distress costs.
The German market for OTC derivative sales to corporates has been subject to
further regulation during the last years (e.g. through the new EU directives EMIR and
MiFID II). However, there were no external restrictions that limit the banks‘ contractual
freedom and competition between banks has been close. As we are not aware of any
particular internal restrictions either (e.g. bank internal rules that would require a
certain hedging policy if a downgrade crosses a rating class border or which are related
to its lending standards), we are convinced that the observed contracts are the outcome
of a primarily demand-driven process of contract conclusions.
We remove counterparties with missing information, natural persons, and public-
private partnerships and limit our analysis to non-defaulted firms with an annual
turnover between EUR 5 million and EUR 1 billion as the bank uses different risk
management and client approaches for very small and large firms. Hence, our typical
sample firm is a German middle-market corporate (“Mittelstand”) who has its shares
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6 Do Firms Hedge in Order to Avoid Financial Distress Costs?
not listed on a stock exchange, so that we cannot measure the hedging benefit on
shareholders’ equity directly like e.g. in Allayannis and Weston (2001). Moreover, most
of these firms rely on bank financing and private debt rather than on issuing tradable
bonds, which is in line with the observation that they are not externally rated by the
international rating agencies and that their financial statements are prepared according
to local GAAP and not according to IFRS. Many firms in the sample produce goods
for worldwide export, so that they need to manage their FX exposure from future
revenues in various foreign currencies, actually more than 20 in the sample. However,
unlike the sample firms analyzed by Allayannis et al. (2001) or by Bartram et al. (2010),
they usually do not represent truly multinational corporates in the sense that they
run production facilities across international subsidiaries and can thereby apply an
arsenal of non-financial hedging instruments. Since our sample firms are on average
much smaller and since they form a homogeneous group insofar as they have their
production facilities mainly concentrated within Germany or countries of the Eurozone,
we think that they will not greatly differ in their use of alternative FX risk management
tools, like e.g. operational hedging, pass-through or foreign-currency denominated debt.
Hence, although we do not have data on these potential other hedging instruments, we
think that the derivative use we can observe in the bank data is a very good proxy for
the sample firms’ overall hedging approach.
In examining the relation between hedging and FDC we estimate the linear regression
model
yi,t = Xi,tβ + αi + γt + i,t, (6.3.1)
where yi,t is the hedge proxy variable andXi,t the design matrix whose rows correspond
to the independent variables. αi denote the industry fixed effects, γt the time fixed
effects and i,t the error term.
The approach taken is commonly based on linearly regressing a binary variable for
the hedger/nonhedger characteristic of the firm on a set of pre-specified explanatory
variables. Instead of using a binary regression, other papers (see e.g. Allayannis et al.
(2001) and Graham and Rogers (2002)) have used a metric variable for the hedge ratio
that is based on mandatory accounting disclosures of derivative use.
We construct three variables as hedge ratio proxies for each firm, namely the sum of
64
6.3 Empirical Results
all notional values of the derivative contracts divided by (a) its book value of total assets
and (b) its total debt. The third hedge ratio proxy (c) is constructed as the sum of all
fair values of the derivative contracts divided by the firm’s book value of total assets.
While we use (a) as our main hedge proxy, we introduce (b) and (c) in the robustness
section. A potential disadvantage of aggregating nominal values as a proxy for derivative
use arises from neglecting other characteristics of the contracts (for example different
maturities, long/short positions, ...). In fact, we are in the lucky position of having
access to the detailed single-contract data of all the derivatives the firms have with
the particular bank at year-end. But, since we cannot distinguish between differences
in the firm’s overall open position that underlies the hedging contract at the point in
time it was closed, it does not make sense to net between long/short resp. buy/sell
positions. Another similar justification for aggregating nominals is given by Carroll
et al. (2017), who refer to Hentschel and Kothari (2001): Although one would expect
financial firms to hold offsetting positions to run a ’balanced book’, non-financial firms
have no obvious reason to hold offsetting derivatives positions. So we simply chose to
interpret an additional EUR of the aggregated notional amount as an indicator of more
hedging activity. Other existing studies suggest using a dummy variable indicating the
use of derivatives (e.g. Bartram (2019)) as an alternative measure. Unfortunately, this
approach is not applicable in this study since our dataset comprises only companies
being involved in hedging activities. However, the use of aggregated nominal values
is commonly used in many other recent studies (see e.g. Allayannis et al. (2001);
Borokhovich et al. (2004); Gay and Nam (1998); Graham and Rogers (2002); Hentschel
and Kothari (2001); Howton and Perfect (1998); Lel (2012)) and we regard it as the
best alternative for our dataset.
Due to the nature of our sample, we solely study firms that are involved in hedging
activities, which may introduce a self-selection bias. This potential issue could be
bypassed by adding firms to the sample which are not involved in hedging activities
and using a Tobit regression instead of OLS to account for the introduced zeros in the
dependent variable. For example, Guay and Kothari (2003) also report regressions of
hedging activity on proxies for hedging incentives based on a sample of firms using
derivatives. In untabulated results, their results are similar for a Tobit specification
that included non-derivatives users. Besides sample selection biases, Bartram et al.
(2011) also discuss other empirical challenges such as endogeneity and omitted variables.
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6 Do Firms Hedge in Order to Avoid Financial Distress Costs?
In an attempt to mitigate these concerns, they use a matching method that controls
for the differences in the likelihood of using derivatives. By this, they attempt to find
“similar” firms, where to the extent possible the “similar” firm differs only in its choice
not to use derivatives. Unfortunately, we cannot conduct a comparable analysis using
our data, since a large sample of non-derivative users comparable to ours is required to
obtain meaningful matches.
The book value of total assets is our main proxy for company size. Besides size,
we include profitability, short-term liquidity, interest coverage ratio, and debt ratio
as commonly used control variables in our regression model. These variables were
included in the bank’s datafile and were inter alia used to produce the internal rating
valid at year-end, which is also used for the regulatory capital calculation under the
Basel II internal ratings-based (IRB) approach. Cf. Table 6.1 for a detailed overview.
Descriptive statistics on the dataset, winsorized at 1% and 99%, are shown in Table 6.2.
It is commonly argued that firms with higher debt ratios, lower interest coverage,
lower profitability, and less liquidity are more likely to use derivatives. Moreover,
bankruptcy costs are typically assumed to be less than proportional to firm size, hence
smaller firms should be more likely to hedge (Gruber and Warner, 1977). In that sense,
these variables can be interpreted as traditional proxy variables for FDC. Empirical
evidence on international datasets, which also include observations from Germany,
however, has up to now found only mixed support for these hypotheses (see e.g. again
Bartram et al. (2009)). The corresponding part of the correlation matrix for our dataset
in Appendix C (see Table B.1 in Appendix B.3) is in line with this literature and
supports all of these predictions, since all the correlations with the observed derivative
use (variables named Notionals and MV) have the hypothesized signs, albeit some are
just above the 10% significance level.
For the variable H, which is the optimal hedge ratio we predict on the basis of the
marginal FDC, we obtain highly significant negative correlations with interest coverage,
profitability, and short-term liquidity and a positive correlation with leverage, which is
plausible, as it indicates that our new explanatory variable H is quite in line with the
predictions for the commonly applied traditional proxy variables concerning observed
hedge ratios. It should be noted that a similar interpretation of the “correct” sign for
the correlation coefficients between the absolute FDC variable and the control variables
is not valid. The reason for this is that we estimate the absolute FDC as a fraction
66
6.3 Empirical Results
of firm value in case of bankruptcy, which by construction leads to a high correlation
between firm size and the absolute amount of FDC. Hence, the interpretation of the
univariate effects is misleading here.
Our final sample consists of 189 medium-sized firms giving us a representative sample
of hedging in the German middle-market corporates (“Mittelstand”). Table B.2 in
Appendix B.4 shows the industry distribution of the dataset. Due to the size of the
companies and their relationship with the major bank, we are likely observing that the
vast majority of the firms’ hedging activity is channeled through this bank. Cf. Machauer
and Weber (2000) for more insights into the German bank/firm relationship and Harhoff
and Körting (1998) find most German small and medium-sized companies to trade with
less than two banks. However since we cannot ensure that other hedging activities take
place, we consider the hedge ratio as a random variable that asymptotically approaches
the real (unobservable) hedge ratio with decreasing firm size. Therefore using company
size as a control variable in the empirical analysis is both crucial for us and common in
previous studies.
6.3.2 Absolute FDC and Hedging
We begin with testing whether there is a positive relationship between the absolute
amount of expected FDC and the observed hedge ratios. To reiterate, the absolute
amount of expected FDC is defined as a fraction of an asset-or-nothing put option.
Table 6.3 shows the result of the estimated regression model (6.3.1) of the absolute
FDC on the scaled sum of notional values of derivative contracts held by each company,
considering control variables and both industry and time fixed effects.
Strikingly, the size of a company, measured by the natural logarithm of the book value
of total assets, significantly negatively influences the observed hedge ratios. Several
explanations come to mind with regard to this finding, especially as it differs from
what Graham and Rogers (2002) document. We already mentioned earlier that the
probability that a firm engages in derivatives business with more than one bank increases
in firm size (Harhoff and Körting, 1998; Ongena and Smith, 2000). Hence, there is
an increased probability of missing notional values of derivative contracts for larger
companies due to contracts with other banks. Second, one could think of the simple
mathematical explanation that, since the observed hedge ratios are scaled by total
67
6 Do Firms Hedge in Order to Avoid Financial Distress Costs?
Table 6.1: Variable Definitions
Variable Definition
Dependent variables
Notionals Sum of gross notional values of derivative contracts held by the
company over book value of total assets.
MV Sum of absolute fair values of derivative contracts held by the com-
pany over book value of total assets.
NotionalsDebt Sum of gross notional values of derivative contracts held by the
company over book value of debt.
Independent variables
FDC Expected absolute financial distress costs as constructed in Section
6.2.
H Theoretical hedge ratio comprising marginal benefits of FDC reduc-
tion through hedging as constructed in Section 6.2.
Debt ratio Book value of liabilities over total assets.
Interest coverage ratio Operating income over interest expenses.
Profitability Operating income over sales.
Short-term liquidity Cash and cash equivalents over book value of liabilities.
Size The natural logarithm of book value of total assets.
Notes: Table 6.1 shows the definitions of the variables used throughout this study. Independent
variables are constructed by balance sheet data used by the bank to obtain the probabilities of
default according to its approved regulatory model under the Basel II internal ratings-based
approach.
68
6.3 Empirical Results
Table 6.2: Descriptive Statistics
Variable Min Mean Max SD
Dependent variables
Notionals 0.0025 0.0978 0.7222 0.1202
MV 0.0000 0.0034 0.0276 0.0051
NotionalsDebt 0.0042 0.1540 1.1480 0.1936
Independent variables
FDC 293.75 10843.04 54822.34 12033.89
H 0.2578 0.4060 0.6020 0.0697
Debt ratio 0.2182 0.6722 0.9680 0.1554
Interest coverage ratio -5.8189 6.8798 56.9224 10.9794
Profitability -0.0318 0.0423 0.1750 0.0419
Short-term liquidity 0.0012 0.1392 1.5209 0.2341
Size 15.6570 18.1677 20.1976 1.1577
Notes: Table 6.2 shows the descriptive statistic of the dataset. Notionals respectively
NotionalsDebt is the sum of gross notional values of derivative contracts held by the company
over book value of total assets respectively liabilities. MV is the sum of absolute fair values of
derivative contracts held by the company over book value of total assets. FDC is the absolute
amount of expected financial distress costs and H the theoretical hedge ratio comprising
marginal benefits of FDC reduction through hedging as constructed in Section 6.2. Debt ratio
is defined as book value of liabilities over total assets. Interest coverage ratio is calculated
as operating income over interest expenses and Profitability as operating income over sales.
Short-term liquidity denotes cash and cash equivalents over book value of liabilities. Size is
measured as the natural logarithm of book value of total assets.
69
6 Do Firms Hedge in Order to Avoid Financial Distress Costs?
assets, an enlargement of the denominator can lead to lower hedge ratios if they do not
increase proportionally.
Besides the size variable, the only other variable with significant explanatory power
in both specifications is the absolute amount of FDC. Its coefficient is significantly
positive at the 1% confidence level with a magnitude of 2.9008. Including industry
and time fixed effects, this value increases to 2.9132. This indicates that German
middle-market companies indeed hedge in order to avoid ex ante expected FDC: The
higher the absolute FDC, the greater is the incentive to reduce a portion of firm value
volatility by using derivatives.
Regarding the remaining control variables, we cannot observe significant links to the
dependent variable. This is surprising as these variables are commonly used proxies for
hedging incentives such as expected financial distress. However, one evident reason for
this is that the FDC variable already explains the fraction of the observed hedge ratio
variations rooted in expected FDC. Obviously, our direct estimation approach yields
better empirical results than the use of the traditional proxy variables.
Regarding our results, we recognize the issue of endogeneity to which empirical studies
of corporate hedging are generally prone (cf. Aretz and Bartram (2010)). Our approach
to tackle this challenge is to treat the underlying FDC function as exogenous, insofar
as it is designed to measure differences across firms that result from their operating
business models when it comes to a situation of financial distress. This FDC function
forms the common basis for both the company’s hedging and leverage decisions in our
modeling approach and is estimated from empirically observable variables.
6.3.3 Marginal FDC Benefits and Hedging
We repeat the previous regression model estimation but replace the absolute amount
of FDC with the theoretically optimal hedge ratio H estimated from marginal FDC
reduction benefits as an explanatory variable. To reiterate, H results from trading off
expected FDC with the transaction costs of hedging these.
H can be interpreted as another perspective on FDC incentives to hedge. While the
significantly positive coefficient for FDC in Table 6.3 implies a higher hedging extent
for firms facing higher expected FDC, a significantly positive coefficient here would
indicate that the cross-sectional differences between the empirical hedge ratios can be
70
6.3 Empirical Results
Table 6.3: Regressions of Hedging Extent on Hedging Incentives including the
Absolute Amount of Expected FDC
Dependent variable Notionals
FDC (in million) 2.9008*** 2.9132***
(3.321) (3.313)
Debt ratio 0.0485 0.0426
(0.991) (0.837)
Interest coverage ratio 0.0002 0.0002
(0.384) (0.360)
Profitability 0.0229 0.0190
(0.104) (0.086)
Short-term liquidity −0.0015 0.0060
(−0.071) (0.246)
Size −0.0739*** −0.0742***
(−5.571) (−5.625)
Intercept 1.3753***
(6.261)
Fixed Effects No Yes
Adj. R2 0.2458 0.2348
Observations 321 321
Notes: Table 6.3 displays the results of the estimated model yi,t = Xi,tβ + αi + γt + i,t
where Xi,t denotes the design matrix whose rows correspond to the independent variables,
αi denote the industry fixed effects, γt the time fixed effects, and i,t the error terms. Below
the coefficients the robust t-statistics are given in parentheses. Notionals is the sum of gross
notional values of derivative contracts held by the company over book value of total assets.
FDC is the absolute amount of expected financial distress costs as constructed in Section 6.2.
Debt ratio is defined as book value of liabilities over total assets. Interest coverage ratio is
calculated as operating income over interest expenses and Profitability as operating income
over sales. Short-term liquidity denotes cash and cash equivalents over book value of liabilities.
Size is measured as the natural logarithm of book value of total assets.
Significance levels: *** p < 0.01, ** p < 0.05, * p < 0.1.
71
6 Do Firms Hedge in Order to Avoid Financial Distress Costs?
explained by firm-specific, marginal benefits of FDC reduction. Albeit we expect a
positive coefficient for the hedge ratio variable H, our expectation about deviations
in the magnitude and significance of its coefficient compared to the coefficient of the
absolute FDC variable is ambiguous. By construction, H comprises information about
both the absolute height of FDC and the marginal benefits of hedging these costs.
Hence, we believe that the model estimation will result in a higher regression coefficient
for H than FDC. On the other hand, a decrease in the coefficient might indicate a
superior role of the absolute expected FDC height in the hedging decision process.
A similar pictures emerges when comparing the results in Table 6.4 to those obtained
in the previous regression (cf. Table 6.3). The marginal FDC variable H possesses
significant explanatory power in both specifications. The coefficient of H is significantly
positive at the 5% confidence level with a magnitude of 0.44 and 0.50 respectively. This
finding is in line with our expectations. The economic impact of our results is quite
pronounced. A one unit increase in H (FDC) FDC corresponds to a 0.50 (2.91) unit
increase in the absolute hedge ratios. Thus companies on average compensate for higher
bankruptcy costs by a steep increase in their hedging activities. This magnitude can
be explained by the relatively low absolute FDC in our sample as well as the practical
implementation costs of higher hedging activity.
6.3.4 Robustness Checks
Extent of Derivative Use and Number of Banking Relationships In the attempt
to control for the case that larger firms are more likely to possess derivative contracts
whose notional values are not included in the dataset, we gradually downsize the dataset
with respect to the companies’ size and rerun the conducted analysis. More specifically,
we build two subsamples based on dismissing (a) the largest 25% and (b) the largest
50% of the firms. We expect both the coefficients of the FDC variables and their
significance to increase. The results are documented in Table 6.5.
In line with the expectation, both the FDC and the H coefficient increase each
time a smaller subsample is taken into consideration. While for H the coefficient
almost doubled compared to the main specification, the FDC coefficient jumps from a
magnitude of 2.9132 to 19.7861 when only considering companies having a size below
the median firm size of the sample. However, only the coefficient of H is becoming
72
6.3 Empirical Results
Table 6.4: Regressions of Hedging Extent on Hedging Incentives including the
Marginal Benefits of FDC Reduction
Dependent variable Notionals
H 0.4443** 0.5033**
(2.222) (2.361)
Debt ratio −0.0915 −0.1105*
(−1.626) (−1.835)
Interest coverage ratio 0.0001 0.0002
(0.216) (0.347)
Profitability 0.1606 0.1734
(0.702) (0.753)
Short-term liquidity −0.0023 0.0080
(−0.106) (0.313)
Size −0.0480*** −0.0484***
(−7.398) (−7.343)
Intercept 0.8460***
(7.055)
Fixed Effects No Yes
Adj. R2 0.2470 0.2419
Observations 321 321
Notes: Table 6.4 displays the results of the estimated model yi,t = Xi,tβ + αi + γt + i,t
where Xi,t denotes the design matrix whose rows correspond to the independent variables,
αi denote the industry fixed effects, γt the time fixed effects, and i,t the error terms. Below
the coefficients the robust t-statistics are given in parentheses. Notionals is the sum of gross
notional values of derivative contracts held by the company over book value of total assets. H
is the theoretically optimal hedge ratio estimated from marginal FDC benefits as explained
in Section 6.2. Debt ratio is defined as book value of liabilities over total assets. Interest
coverage ratio is calculated as operating income over interest expenses and Profitability as
operating income over sales. Short-term liquidity denotes cash and cash equivalents over book
value of liabilities. Size is measured as the natural logarithm of book value of total assets.
***, **, and * indicate significance at the 1%, 5%, and 10% level, respectively.
73
6 Do Firms Hedge in Order to Avoid Financial Distress Costs?
slightly more significant. The FDC coefficient, on the other hand, is less significant for
the two subsamples than for the full sample. We believe that this finding is rooted in
the relation between the size of a company and our construction of the expected FDC.
Our FDC function is defined as a fraction of firm value (see Formula (6.2.5)) in the case
of bankruptcy. By gradually downsizing the sample with respect to company size, we
not only control particularly for size, but also for the absolute amount of FDC. Hence,
both size and FDC lose some significance following this procedure, whereas H does not.
Different Measures of the Extent of Derivative Use So far, we scaled the sum of
notional values of derivative contracts by the book value of total assets for each company.
Commonly, the amount of debt is used alternatively for scaling the notional values to
construct a measure of the extent of derivative use (Graham and Rogers, 2002). Hence,
we replace the dependent variable used in the main analysis by one scaled by debt.
This specification can be interpreted as a measure for hedging per dollar of debt.
As the left columns in Table 6.6 show, we obtain similar results to the previously
conducted analysis. In this case though, the coefficients of the marginal and absolute
FDC variables are significant at 10% and 5% level respectively. Nonetheless, the debt
ratio again seems to negatively influence the observed hedge ratios. Graham and Rogers
(2002) use the amount of debt as a measure of financial distress and find their results
to be robust against this alternative measure of hedging extent. We argue that this
specification induces a negative coefficient on the debt ratio because debt is in the
denominator of the dependent variable. The correlation matrix B.1 in Appendix B.3
fosters this impression as NotionalsDebt is the only dependent variable being negatively
correlated to the debt ratio.
As a third proxy of a firm’s hedging extent, we utilize the fair values of the derivative
contracts. They specify the amount that the contract holder would receive or pay to
liquidate these contracts and have also been utilized as proxies for derivative use in
earlier studies (see for example Berkman and Bradbury (1996); Spanò (2007); Choi
et al. (2013)). A serious disadvantage of using fair values compared to notional values
as a hedging proxy is their dependence on the price movements of the corresponding
underlying. For example, many fair values of derivative contracts are zero at origination,
although their hedged notional values might be large. In our sample though, derivative
contracts are already held for 2.5 years on average. Thus, we believe that the sum
74
6.3 Empirical Results
Table 6.5: Regressions of Hedging Extent on Hedging Incentives for Different Sub-
samples
Dependent variable Notionals
Sample ≤ 75% quantile(Size) ≤ 50% quantile(Size)
FDC (in million) 5.7843* 19.7861*
(1.936) (1.674)
H 0.6190** 0.8816**
(2.545) (2.527)
Debt ratio 0.0874 −0.1194 0.2053* −0.0804
(1.143) (−1.576) (1.858) (−0.724)
Interest coverage ratio 0.0005 0.0005 0.0011 0.0014
(0.561) (0.549) (0.748) (0.904)
Profitability −0.0090 0.2228 −0.2073 0.1142
(−0.035) (0.837) (−0.431) (0.235)
Short-term liquidity 0.0131 0.0002 0.0487 0.0494
(0.393) (0.005) (0.572) (0.647)
Size −0.0869*** −0.0634*** −0.1288*** −0.0890***
(−4.369) (−6.080) (−3.557) (−4.693)
Fixed effects Yes Yes Yes Yes
Adj. R2 0.1853 0.2193 0.1312 0.1843
Observations 241 241 161 161
Notes: Table 6.5 displays the results for two subsamples, namely by excluding the 25%
respectively 50% largest companies in the sample. The estimated model is yi,t = Xi,tβ +
αi + γt + i,t where Xi,t denotes the design matrix whose rows correspond to the independent
variables, αi denote the industry fixed effects, γt the time fixed effects, and i,t the error terms.
Below the coefficients the robust t-statistics are given in parentheses. Notionals is the sum of
gross notional values of derivative contracts held by the company over book value of total
assets. FDC is the absolute amount of expected financial distress costs and H the theoretical
hedge ratio comprising marginal benefits of FDC reduction through hedging as constructed
in Section 6.2. Debt ratio is defined as book value of liabilities over total assets. Interest
coverage ratio is calculated as operating income over interest expenses and Profitability as
operating income over sales. Short-term liquidity denotes cash and cash equivalents over book
value of liabilities. Size is measured as the natural logarithm of book value of total assets.
***, **, and * indicate significance at the 1%, 5%, and 10% level, respectively.
75
6 Do Firms Hedge in Order to Avoid Financial Distress Costs?
of the absolute fair values of derivative contracts serves as a suitable alternative to
test our hypothesis. The correlation between the constructed hedge proxies based on
fair values respectively notional values amounts to 0.642 (cf. Table B.1 in Appendix
B.3). Table 6.6 summarizes the results. They are similar to those obtained in the main
specifications. Both the coefficient of the marginal and absolute FDC variables are
significant at 5% and 1% level respectively supporting our hypothesis that firms hedge
in order to avoid FDC.
76
6.3 Empirical Results
Table 6.6: Regressions of Alternative Proxies of Hedging Extent on Hedging Incentives
Dependent variable NotionalsDebt MV
FDC (in million) 3.2730** 0.1254***
(2.407) (3.055)
H 0.5878* 0.0169**
(1.788) (2.003)
Debt ratio −0.1923** −0.3691*** 0.0025 −0.0031
(−2.124) (−3.721) (1.003) (−1.128)
Interest coverage ratio −0.0003 −0.0002 0.0000 0.0000
(−0.269) (−0.235) (−1.375) (−1.298)
Profitability −0.1110 0.0702 0.0045 0.0095
(−0.270) (0.166) (0.393) (0.802)
Short-term liquidity 0.0989 0.1016 0.0014 0.0015
(1.383) (1.388) (1.256) (1.179)
Size −0.1034*** −0.0743*** −0.0023*** −0.0013***
(−5.200) (−7.191) (−4.300) (−4.138)
Fixed effects Yes Yes Yes Yes
Adj. R2 0.2328 0.2376 0.0988 0.0925
Observations 321 321 321 321
Notes: Table 6.6 displays the results for two alternative proxies of hedging extent. The
estimated model is yi,t = Xi,tβ + αi + γt + i,t where Xi,t denotes the design matrix whose
rows correspond to the independent variables, αi denote the industry fixed effects, γt the time
fixed effects, and i,t the error terms. Below the coefficients the robust t-statistics are given in
parentheses. NotionalsDebt is the sum of gross notional values of derivative contracts held
by the company over book value of liabilities and MV is the sum of absolute fair values of
derivative contracts held by the company over book value of total assets. FDC is the absolute
amount of expected financial distress costs and H the theoretical hedge ratio comprising
marginal benefits of FDC reduction through hedging as constructed in Section 6.2. Debt ratio
is defined as book value of liabilities over total assets. Interest coverage ratio is calculated
as operating income over interest expenses and Profitability as operating income over sales.
Short-term liquidity denotes cash and cash equivalents over book value of liabilities. Size
is measured as the natural logarithm of book value of total assets. ***, **, and * indicate
significance at the 1%, 5%, and 10% level, respectively.
77
6 Do Firms Hedge in Order to Avoid Financial Distress Costs?
6.4 Conclusion
Exploring if firms hedge in order to avoid ex ante expected FDC, we expand a Merton
(1974) framework to estimate FDC as a fraction of an asset-or-nothing put option.
Our empirical FDC estimates are based on information about the firms’ regulatory
“Basel II” PDs, which are available on bank-level, as well as on publicly accessible
accounting figures. Assuming that firms choose their debt levels optimally, we determine
the firm-specific financial distress cost parameters in a trade-off between FDC and
tax savings. The resulting absolute amount of firm-specific FDC is our first novel
explanatory variable, which is assumed to be positively correlated with the observed
hedging activity. We then introduce a second trade-off between FDC and transaction
costs to calculate optimal hedge ratios. We expect these optimal hedge ratios, which
result from the marginal FDC benefits in equilibrium and which are our second novel
explanatory variable, to be accompanied by higher observed hedge ratios, too.
Testing our model empirically we obtain granular data from a major bank. The
dataset contains all OTC derivatives the bank entered with about 189 middle-market
counterparties (non-financial, non-public sector) at fiscal year-end 2014/2015. Regress-
ing our FDC variables on the observed hedge ratios we find a significantly positive
relation when considering several control variables and fixed effects. Although it is quite
probable that most firms in our dataset mainly trade derivatives with only a very small
number of banks, we admit that the observed hedge ratios might not cover the full
extent of derivative use of the considered companies. Since larger firms are more likely
to possess derivative contracts whose notional values are not included in the dataset,
we gradually downsize the dataset with respect to the companies’ size in the robustness
section. In line with our expectations, the magnitude of the coefficients for both FDC
variables increases. Furthermore, our results are robust to different constructions of the
empirical hedge ratio and different choice of model parameters.
Our contribution is a novel measure of firm-specific ex ante costs of financial distress
based on an option-theoretical approach, which we then extend to an equilibrium setting.
Moreover, our empirical analysis differs from previously used datasets and enriches
the literature by the perspective of a financial intermediary on the corporate hedging
decision. We are able to support the hypothesis that firms hedge in order to avoid ex
ante expected financial distress costs.
78
6.4 Conclusion
Concerning implications of our findings for corporate policy as well as for banks’
derivative sales policies, we think that a quantitative analysis of the FDC impact should
become a more explicit part of firms’ hedging decisions. For the German middle-market
companies we have focused on in this paper, such an analysis should expediently be
linked to the annual Basel II rating process, whose main outcome is a forward-looking
one-year default probability. In this context, our modeling approach to estimate ex
ante expected FDC could be integrated for benchmarking purposes. From a financial
intermediate’s perspective, such quantitative information should be embraced as a new
anchor point for offering customized consulting and marketing activities in financial
derivatives.
From a research point of view, currently only very little seems to be known about
the role of the sell-side for corporate derivatives use. In order to gain more insight
about the policies of the different sell-side market participants and their views on the
corporate decision process, more survey work seems necessary in a first step.
The partial equilibrium two-stage estimation approach presented in this paper is
applicable in future research projects or the above mentioned quantitative business
analyses whenever data about the firms’ debt-equity ratio and a corresponding PD
proxy variable is available. The PD proxy variable, which we took from our bank
dataset as a regulatory Basel II PD, could easily be replaced by mean historical default
rates per rating class if a sample of externally rated firms was analyzed. Moreover, such
a proxy variable could also be constructed from observed credit spreads, if the firms
under analysis have issued bonds, which are traded in a liquid debt market, or if there
are even liquid CDS quotes. In order to have a substantial magnitude of FDC in the
sample, a natural choice would be the European market for high-yield bond issuers,
where both external ratings and credit spreads are available.
79

A Appendix for Chapter 5
A.1 General Tournament Behavior
To detect general tournament behavior, we follow Brown et al. (1996) by using their
contingency table approach: The performance of every manager i is given as the
information ratio IRiM against the MSCI North America in the first 6 months of
the year to identify mid-term winners and losers as those above or below the median
information ratio, respectively. All managers hold an equally weighted portfolio of their
funds j ∈ {1, ..., n} included in the tournament with a portfolio return of rportik .
In order to calculate the information ratio fo∏r eac(h fund gi)ven as IR = active premiatracking error ,we determine the cumulative return as RTN 6i = k=1 1 + rportik − 1, which finally leads
to the formula for the information ratio of manager i at the end of June
= √ R∑TNi −IRi ( RTNb ) (A.1.1)1 6 rport 26−1 k=1 ik − rbk
The variable RTNb is the cumulative return of benchmark returns rbk of month
k ∈ {1, ..., 6}. The risk adjustment ratio (RAR) of manager i for the given tournament
year with interim assessment date in June is:
√√√√∑ √12 ( port √− )2 √√∑r r̄ 6 porti(12−6) (r − r̄i)2RAR = k=6+1 ik k=1 iki (12− 6) 1 ÷ 6 1 (A.1.2)− −
with r̄i(12−6) and r̄i representing the mean portfolio return of fund manager i before
and after the assessment date, respectively. This variable measures the risk adjustments
of a given portfolio within the two periods of the year’s tournament by comparing the
portfolio’s volatility in both periods. Thus, we rank the RAR in a similar way as the
IR and determine high RAR as those above the median and low RAR as those below
the median for the first and second period, respectively.
81
A Appendix for Chapter 5
Table A.1: Contingency Tables of Annual Tournaments following Brown et al. (1996)
Freq. in % Winner Loser
Year No.a High Low RAR High Low RAR χ2-valueb p-value
RAR RAR
1992 94 19.2 30.9 30.9 19.2 5.1489 0.1612
1993 97 33.0 17.5 17.5 32.0 9.6907 0.0337
1994 109 22.9 27.5 27.5 22.0 1.1284 0.7702
1995 135 20.2 30.6 29.9 19.4 5.8806 0.1176
1996 153 25.8 25.2 24.5 24.5 0.0728 0.9949
1997 165 27.9 22.4 22.4 27.3 1.7636 0.6229
1998 189 26.1 24.5 23.9 25.5 0.2128 0.9755
1999 220 25.0 25.0 25.0 25.0 0.0000 1.0000
2000 245 33.1 17.1 17.1 32.7 24.208 0.0000
2001 269 34.2 16.0 16.0 33.8 34.985 0.0000
2002 325 25.9 24.3 24.3 25.5 0.2552 0.4625
2003 350 27.1 22.9 22.9 27.1 2.5714 0.4625
2004 362 30.4 19.6 19.6 30.4 16.807 0.0008
2005 340 23.0 26.8 27.1 23.0 2.1563 0.5406
2006 328 20.2 30.0 30.0 19.9 12.927 0.0048
2007 311 30.4 20.1 19.7 29.8 12.877 0.0049
2008 296 27.8 22.0 22.4 27.8 3.6983 0.2959
2009 259 24.9 25.7 25.3 24.1 0.1362 0.9872
2010 231 26.0 24.2 24.2 25.5 0.2208 0.9742
2011 215 27.0 23.3 23.3 26.5 1.0558 0.7878
2012 208 27.4 22.6 22.6 27.4 1.9231 0.5885
2013 196 24.5 25.5 25.5 24.5 0.0816 0.9930
2014 193 21.2 29.0 29.0 20.7 4.9896 0.1726
2015 185 31.3 18.9 18.9 30.8 10.957 0.0120
2016 169 33.1 17.2 17.2 32.5 16.633 0.0008
2017 148 21.0 29.1 29.1 21.0 3.8919 0.2734
a Number of managers within the given tournament year.
b Based on the null hypothesis of every cell receiving 25% of the distribution.
Notes: This table presents the annual contingency tables of risk adjustment ratios (RAR)
for the whole dataset covering the years 1992-2017. Managers who perform at the end of
the interim period above (below) the median are classified as winners (losers). The same
methodology applies to the classification of high (low) RARs.
82
A.2 Tables – Robustness Tests
A.2 Tables – Robustness Tests
Table A.2: Risk Transitions for Managers of Different Skill Levels - Active Return
Values in % Winner Loser
Risk tercile high medium low χ2 high medium low χ2
Panel A: Unconditional
high 62.6 25.9 11.5 61.1 24.9 14.0
medium 31.0 44.7 24.2 25.5 40.9 33.7
low 12.1 32.2 54.8 10.2 27.9 61.9
Panel B: High Skill
high 64.1 26.4 9.5 66.5 24.1 9.4
medium 35.5 45.0 19.5 26.9 43.8 29.4
low 16.2 34.7 49.1 5.96* 14.2 29.1 56.7 7.18**
Panel C: Medium Skill
high 61.7 25.2 13.1 59.3 25.3 15.4
medium 29.4 44.4 26.2 26.1 41.6 32.3
low 11.1 32.2 56.7 1.92 10.3 29.4 60.3 2.06
Panel D: Low Skill
high 63.5 27.6 8.8 61.0 24.5 14.5
medium 31.0 45.7 23.4 22.2 36.1 41.7
low 9.7 34.8 55.5 0.64 7.3 22.8 69.9 13.63***
Notes: This table shows the risk-shifting tendencies from the year’s first-half tercile to the
tercile in the second half of the year for the full aggregated data set. Panel A represents the
whole sample, whilst Panel B to D show the transitions for different skill levels. Each manager
is classified as being a winner (loser) if his performance measured by the active return lies
above (below) the median at the end of the interim period. χ2-values testing H0-hypotheses
of conditional transitions being equal to the unconditional. ***, **, and * indicate significance
at the 1%, 5%, and 10% level (two-tailed tests), respectively.
83
A Appendix for Chapter 5
Table A.3: Rank Correlations Based on Different Performance Measures
Performance measures Sharpe ratio Active return Information ratio
Sharpe ratio 1
Active return 0.983*** 1
Information ratio 0.943*** 0.963*** 1
Notes: The ranks are calculated over a time period from 1992 to 2017. ***, **, and * indicate
significance at the 1%, 5%, and 10% level, respectively.
84
A.2 Tables – Robustness Tests
Table A.4: Risk Transitions for Different Hyperparameters: σ = 12.5
Values in % Winner Loser
Risk tercile high medium low χ2 high medium low χ2
Panel A: Unconditional
High 62.4 25.7 11.8 61.3 25.0 13.7
Med 30.0 44.8 25.2 26.6 40.9 32.5
Low 11.1 32.9 56.0 11.1 28.2 60.7
Panel B: High Skill
High 62.6 26.7 10.8 65.8 23.8 10.4
Med 29.2 46.6 24.2 26.8 42.5 30.7
Low 13.0 31.3 55.7 6.35** 13.5 31.8 54.7 0.66
Panel C: Medium Skill
High 62.7 25.0 12.3 61.0 25.4 13.6
Med 32.1 42.7 25.2 27.2 42.9 29.9
Low 11.5 34.7 53.7 2.75 12.2 29.4 58.3 2.63
Panel D: Low Skill
High 61.1 27.4 11.5 57.6 25.3 17.2
Med 23.9 49.4 26.7 24.8 34.8 40.5
Low 7.3 29.6 63.1 22.25*** 6.8 22.5 70.8 8.58**
Notes: This table shows the risk-shifting tendencies from the year’s first-half tercile to the
tercile in the second half of the year for the full aggregated data set for a different choice of
hyperparameter for the prior distribution of the fund managers’ skills, e.g. σ = 12.5. Panel
A represents the whole sample, whilst Panel B to D show the transitions for different skill
levels. Each manager is classified as being a winner (loser) if his performance measured by
the active return lies above (below) the median at the end of the interim period. χ2-values
testing H0-hypotheses of conditional transitions being equal to the unconditional. ***, **,
and * indicate significance at the 1%, 5%, and 10% level (two-tailed tests), respectively.
85
A Appendix for Chapter 5
Table A.5: Risk Transitions for Different Hyperparameters: σ = 6.25
Values in % Winner Loser
Risk tercile high medium low χ2 high medium low χ2
Panel A: Unconditional
High 62.4 25.7 11.8 61.3 25.0 13.7
Med 30.0 44.8 25.2 26.6 40.9 32.5
Low 11.1 32.9 56.0 11.1 28.2 60.7
Panel B: High Skill
High 64.7 23.9 11.4 64.1 22.2 13.8
Med 26.9 46.2 26.9 23.7 41.0 35.3
Low 11.2 36.2 52.6 4.36 11.8 32.4 55.9 3.42
Panel C: Medium Skill
High 61.8 26.3 11.9 60.8 25.9 13.3
Med 33.1 43.1 23.8 27.0 43.6 29.4
Low 12.0 32.6 55.3 3.50 11.9 29.6 58.5 1.98
Panel D: Low Skill
High 61.8 26.1 12.1 60.5 24.9 14.6
Med 23.9 48.3 27.8 27.7 33.0 39.4
Low 8.0 29.3 62.6 16.75*** 9.0 22.3 68.8 7.26**
Notes: This table shows the risk-shifting tendencies from the year’s first-half tercile to the
tercile in the second half of the year for the full aggregated data set for a different choice of
hyperparameter for the prior distribution of the fund managers’ skills, e.g. σ = 6.25. Panel
A represents the whole sample, whilst Panel B to D show the transitions for different skill
levels. Each manager is classified as being a winner (loser) if his performance measured by
the active return lies above (below) the median at the end of the interim period. χ2-values
testing H0-hypotheses of conditional transitions being equal to the unconditional. ***, **,
and * indicate significance at the 1%, 5%, and 10% level (two-tailed tests), respectively.
86
A.2 Tables – Robustness Tests
Table A.6: Risk Transitions for Managers of Different Skill Levels - Secondary
Benchmarks
Values in % Winner Loser
Risk tercile high medium low χ2 high medium low χ2
Panel A: Unconditional
High 62.4 25.7 11.8 61.3 25.0 13.7
Med 30.0 44.8 25.2 26.6 40.9 32.5
Low 11.1 32.9 56.0 11.1 28.2 60.7
Panel B: High Skill
High 68.7 23.6 7.7 67.5 23.4 9.1
Med 31.6 43.1 25.4 24.2 38.8 37.1
Low 16.5 34.6 48.9 8.65** 12.8 31.5 55.7 10.52***
Panel C: Medium Skill
High 60.3 25.7 14.0 57.8 27.1 15.1
Med 29.2 44.7 26.1 28.2 43.1 28.7
Low 10.8 33.0 56.1 5.78* 10.4 29.2 60.4 1.35
Panel D: Low Skill
High 60.9 28.7 10.3 65.0 20.7 14.3
Med 30.7 46.5 22.8 24.0 36.3 39.8
Low 6.5 30.8 62.7 9.90*** 12.3 22.6 65.1 5.61*
Notes: This table shows the risk-shifting tendencies from the year’s first-half tercile to the
tercile in the second half of the year for the full aggregated data set for a different choice
of benchmarks for the funds in the sample. While in the baseline specification all funds are
benchmarked against the MSCI North America, they are now benchmarked against different
Russel indexes, see Section 5.3. Panel A represents the whole sample, whilst Panel B to D
show the transitions for different skill levels. Each manager is classified as being a winner
(loser) if his performance measured by the active return lies above (below) the median at the
end of the interim period. χ2-values testing H0-hypotheses of conditional transitions being
equal to the unconditional. ***, **, and * indicate significance at the 1%, 5%, and 10% level
(two-tailed tests), respectively.
87

B Appendix for Chapter 6
B.1 Estimating the Financial Distress Cost Parameter
η from a Capital Structure Equilibrum
In order to provide a counterweight to the FDC given by (6.2.7), we introduce a
corporate tax function. Interest rates are tax-deductible, which provides an incentive
for borrowing at the corporate level by generating a tax shield. Taxable income in t = 1
stems from operating earnings (EBIT), which go along with an increase (or decrease)
in total assets from V0 to V . It is reduced by multiplying the nominal interest rate
i > 0, which is assumed to be constant across all firms for the sake of simplicity, times
the debt level that is chosen by the company. The corporate tax rate is denoted as
τ ∈ ]0, 1[. The resulting tax shield is only valuable in those future states of the world,
where the taxable income is not negative anyway. The whole idea of tax shield modeling
is to create an analytically tractable counterweight to financial distress costs for the
firms’ capital structure decision. It is not meant as an analysis to further explore tax
reasons for corporate hedging, which we do not regard as relevant under the German
tax system. Cf. Graham and Rogers (2002) for such an analysis for the US. The market
value of the tax claim in t = 0, which is identical with the expected value of the random
cash flow in t = 1, emerges as
∫ ∞
T := τ (V − V0 − iD)f(V, µ, σ)dV. (B.1.1)
V0+iD
The shareholders decide about the optimal debt amount by maximizing the market
value of their residual claims in t = 1. Since the debtholders are fairly compensated for
the default risk, the maximization of shareholder wealth is equivalent to minimizing the
sum of the market values of taxes and financial distress costs. This yields the first-order
89
B Appendix for Chapter 6
condition
∂T + ∂FDC =! 0. (B.1.2)
∂D ∂D
Applying the product rule to the derivative of the tax function leads to
∂T = −τi+ τiF (V0 + iD, µ, σ) < 0. (B.1.3)
∂D
Applying Leibniz‘ rule, the first partial derivative of Equation (6.2.5) with respect to D
is given by
∂FDC = ηDf(D,µ, σ) > 0. (B.1.4)
∂D
When the firm has chosen its unique optimal debt level D = D∗ according to the
first-order condition (B.1.2), the following condition must hold for the financial distress
cost parameter η∗:
− ∂T = ∂FDC ⇔ τi− τiF (V ∗0 + iD , µ, σ) = η∗D∗f(D∗, µ, σ)
∂D ∂D
∗ (B.1.5)
⇔ ∗ = τi− τiF (V0 + iD , µ, σ)η
D∗f(D∗, µ, σ) .
To ensure the optimal debt level D∗ is indeed minimizing the sum between the market
values of taxes and financial distress costs, the second order condition needs to be met:
∂2T + ∂
2FDC
2 2 = τi
2f(V0 +
∂ !
iD, µ, σ) + ηf(D,µ, σ) + ηD f(D,µ, σ) > 0. (B.1.6)
∂D ∂D ∂D
It is sufficient to take the second part of this Equation into account, as the second
derivation of the tax function ∂2T 22 = τi f(V0 + iD, µ, σ) is greater or equal to zero∂D
regardless of the chosen debt level D. Simple mathematical operations show for the
second derivation of the financial distress costs function that
∂2FDC
2 = ηf(
∂
D, µ, σ) + ηD f(D,µ, σ)
∂D ∂D ( )
η − (ln(D)−µ)
2 1 − (ln(D)−µ)2= √ + −√ − l√n(D)− µ
(ln(D)−µ)2
e 2σ2 η e 2σ2 e− 2σ2
2πσD 2πσD 2πσ3D
(ln(D)−µ)2
= √ η e− 2σ2 (µ− ln(D)) > 0
2πσ3D
(B.1.7)
90
B.2 Construction of the FDC Variables
if D < eµ = V erf− 12σ20 . Thus, if D∗ is less than the median of the respective lognormal
distribution, in other words if the probability of default of the company is less than at
least 50%, D∗ is indeed a cost minimum. We summarize these intermediate results in a
Lemma:
Lemma. In the exogenous capital structure equilibrium, each firm j = 1, ..., n chooses
its optimal debt level D∗j , so that the sum of ex-ante expected corporate taxes and
financial distress costs is minimized. For any given set of firm-specific distribution
parameters (µj, σj) and of non firm-specific parameters for interest and tax rates, there
is a unique set of firm-specific financial distress cost parameters η∗j , which characterizes
this capital structure equilibrium.
In the empirical part of this study, we set the the total average effective tax rate on
company profits to 30% which comprises both federal tax and municipal tax components
(BdF, 2016) and the interest rate i to 1.94% for 2014 and to 1.68% for 2015, which
were the effective rates for new loans to non-financial firms granted by German banks
in December of the respective year according to Beier and Bade (2017).
B.2 Construction of the FDC Variables
We want to motivate the construction process of the FDC variables with some graphical
illustrations. Given the empirical book values of debt and total assets of the 189 firms
in our sample and their probabilities of default according to the internal ratings-based
approach of the bank, we can numerically solve for the firm value volatility following
the Merton (1974) framework (cf. Equation (6.2.4)). For example, we pick a random
observation from the dataset with book values of debt and totals assets amounting to
about EUR 115 million and EUR 228 million respectively, and a probability of default
of 0.53%. For this firm one obtains a firm value volatility of about 25.5%.15 Note that
we have set rf = 0 without loss of generality.
We now utilize our FDC model and are able to calculate η∗j for each firm j =(1, ...18)9
following the closed formula (B.1.5). As shown in Appendix B.1, this setting D∗ ∗j , ηj
is optimal in the sense of trading off expected financial distress costs and tax savings.
Figure B.1 visualizes this choice for the exemplary firm mentioned above.
91
B Appendix for Chapter 6
Figure B.1: Visualization of the First Optimization Problem
6,900,000
6,850,000
6,800,000
6,750,000
6,700,000
6,650,000
0 50,000,000 100,000,000 150,000,000
Debt
Notes: Visualization of the first optimization problem (see Equation (B.1.2)). The solid black
line shows the sum of financial distress costs and taxes in dependence of debt D. The red
dotted line shows the optimal amount of debt D∗, given the FDC parameter η∗.
Thereupon, we are able to determine the expected FDC determined by multiplying
the FDC parameter η∗ with the value of the asset-or-nothing put option (cf. Equation
(6.2.7)). To prevent firms from fully hedge all of their risk, we introduce transaction
costs as described in formula (6.2.9). The marginal transaction costs parameter γ
is set to 0.4 basis points, which is the minimum firm-specific value of the derivation
of the FDC function (cf. Equation (6.2.7)) with respect to the firm value volatility.
By numerically solving Equation (6.2.10), we obtain the optimal firm value volatility
σ∗ that minimizes the ex-ante expected FDC with respect to transaction costs. For
the exemplary firm mentioned at the beginning of this section this optimization is
illustrated in Figure B.2.
Finally, we are able to determine the theoretically optimal hedge ratio H∗ according
to Equation (6.2.12) which is driven by the marginal benefits of FDC reduction. Figures
B.3 and B.4 show H∗ in dependence of the debt ratio and PD. The black points highlight
the obtained values H∗j based on the observations as of 2015 from the dataset.
92
FDC + T
B.2 Construction of the FDC Variables
Figure B.2: Visualization of the Second Optimization Problem
4,000
3,000
+
2,000
1,000
0
0% 5% 10% 15% 20%
σ
Notes: Visualization of the second optimization problem (see Equation (6.2.10)). The solid
black line shows the sum of financial distress costs and transaction costs in dependence of the
firm value volatility σ. The red dotted line shows the optimal firm value volatility σ∗.
Figure B.3: Contour Plot of the Theoretical Hedge Ratio
Notes: Contour plot of the theoretical hedge ratio H∗ in dependence of the debt ratio DV
and probability of default PD. The black points highlight the obtained values H∗j based on
the observations as of 2015 from the dataset. Note that the x-axis is for the sake of clarity
logarithmically scaled.
93
FDC  TC
B Appendix for Chapter 6
Figure B.4: 3D Plot of the Theoretical Hedge Ratio
Notes: Theoretical hedge ratio H∗ in dependence of the debt ratio DV and probability of
default PD. The black points highlight the obtained values H∗j based on the observations as
of 2015 from the dataset.
94
B.3 Correlation Matrix
95
B.3 Correlation Matrix
Table B.1: Correlation Matrix
atio y
bt o overa
ge r uidit
s s e i lity m liq
Notio
nal ional DV ot ebt r
at terest
c bi
rofita ort-t
er
M N D In P Sh Size FDC
MV 0.642*** 1
(14.960)
NotionalsDebt 0.915*** 0.601*** 1
(40.516) (13.417)
Debt ratio 0.082 0.065 -0.145*** 1
(1.478) (1.169) (-2.617)
Inte. cov. ratio -0.089 -0.091 0.010 -0.424*** 1
(-1.590) (-1.633) (0.171) (-8.357)
Profitability -0.087 -0.048 -0.039 -0.287*** 0.564*** 1
(-1.556) (-0.866) (-0.697) (-5.348) (12.205)
Short-term liqu. -0.051 -0.022 0.121** -0.486*** 0.499*** 0.423*** 1
(-0.913) (-0.393) (2.169) (-9.927) (10.281) (8.331)
Size -0.403*** -0.238*** -0.359*** -0.109* 0.146*** -0.016 0.003 1
(-7.874) (-4.378) (-6.873) (-1.958) (2.634) (-0.289) (0.047)
FDC -0.300*** -0.145*** -0.253*** -0.232*** 0.042 0.065 0.019 0.736*** 1
(-5.607) (-2.616) (-4.661) (-4.259) (0.752) (1.171) (0.340) (19.392)
H 0.221*** 0.190*** 0.029 0.653*** -0.487*** -0.469*** -0.457*** -0.147*** -0.113**
(4.057) (3.452) (0.524) (15.391) (-9.958) (-9.491) (-9.177) (-2.662) (-2.035)
Notes: Table B.1 shows the Pearson correlation coefficients of the dependent and independent variables. Below the coefficients
the t-statistics are given in parentheses. Notionals respectively NotionalsDebt is the sum of gross notional values of derivative
contracts held by the company over book value of total assets respectively liabilities. MV is the sum of absolute fair values of
derivative contracts held by the company over book value of total assets. Debt ratio is defined as book value of liabilities over
total assets. Interest coverage ratio is calculated as operating income over interest expenses and Profitability as operating
income over sales. Short-term liquidity denotes cash and cash equivalents over book value of liabilities. Size is measured as
the natural logarithm of book value of total assets. FDC is the absolute amount of expected financial distress costs and H the
theoretical hedge ratio comprising marginal benefits of FDC reduction through hedging as constructed in Section 6.2.
B Appendix for Chapter 6
B.4 Industry Distribution
Table B.2: Industry Distribution of the Dataset
Industry sector Absolute Relative (%)
Consumer NonDurables 62 19.31
Consumer Durables 10 3.12
Manufacturing 97 30.22
Oil, Gas, and Coal 0 0.00
Chemicals 12 3.74
Business Equipment 1 0.31
Telephone and Television 0 0.00
Utilities 2 0.62
Wholesale, Retail, and Some Services 102 31.78
Healthcare, Medical Equipment, and Drugs 0 0.00
Money Finance 0 0.00
Other 35 10.90
Notes: Table B.2 shows the industry distribution of the dataset. Originally, firms were
classified by the German "WZ2008" code, which we mapped to Fama French 12 industry
classes obtained from Kenneth French’s homepage using three crosswalks, namely WZ2008 to
ISIC Rev. 4, ISIC Rev. 4 to NAICS 2017 and NAICS 2017 to SIC. We checked each mapping
to avoid unreasonable mappings.
96
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