Risk and Return Characteristics of Risk Arbitrage in Western Europe

MASTER’S THESIS

Authors: Supervisor: Lise K. Kristensen Niklas Kohl Charlotte B. J. Reventlow

Presented to the faculty of the Department of Finance for the degree of Master of Science in Advanced Economics and Finance (Cand.oecon) at Copenhagen Business School - May 17, 2016 No. of standard pages (characters): 105.5 (240,011) ii iii

Abstract

Historically, the risk arbitrage literature has documented substantial abnormal returns in the US, the UK, Australia and Canada for the seemingly market neutral risk arbitrage strategy. However, some recent literature has questioned and rejected market neutrality and even finds evidence indicating a non-linear relationship between excess risk arbitrage returns and excess market returns. The combination of rapid capital inflows into hedge funds, the large abnormal returns and the question of market neutrality motivates a study of the risk and return characteristics of risk arbitrage in Western Europe. Using a sample of 2167 Western European stock and cash transactions during 1990-2015, we analyze the risk and return characteristics of risk arbitrage. First, we benchmark the risk arbitrage returns against linear and non-linear asset pricing models. Further, we examine the impact of portfolio construction and discuss quantitative arbitrage investing vis-a-vis discretionary portfolio selection. Moreover, in order to examine the potential of developing an optimal exit timing strategy, we conduct an autocorrelation analysis. Finally, we discuss the implications of the results for practitioners. While we find risk arbitrage to earn abnormal returns, the trend appears to be decreasing over time. Further, we find that risk arbitrage is not a market neutral strategy, which is predominantly driven by the characteristics of cash transactions. The abnormal returns are robust to the portfolio construction approach, asset pricing models, diversification constraints and constant transaction costs. Finally, we find potential for the development of an optimal exit timing strategy. The results lead to the following implications for practitioners. First, despite the abnormal returns, risk arbitrage is not an entirely market neutral strategy. Hence, in order to achieve market neutrality, practitioners should actively short the market. Second, practitioners should consider the portfolio construction approach, as it impacts the risk and return characteristics. Third, practitioners should consider developing an optimal exit strategy in order to decrease completion risk and increase abnormal returns. iv

Contents

Abstract iii

1 Introduction1 1.1 Motivation...... 1 1.2 Research Question...... 3 1.3 Contribution to research...... 3

2 Theoretical Framework of Risk Arbitrage5 2.1 Introduction to Risk Arbitrage...... 5 2.1.1 M&A Transaction Characteristics...... 6 2.1.2 The Risk Arbitrage Trade...... 7 The Arbitrage Spread...... 7 The Mechanics of the Risk Arbitrage Trade...... 10 2.2 Sources of Abnormal Returns and Risk in Risk Arbitrage.. 12 2.2.1 Portfolio Construction...... 14 2.2.2 Transaction Costs and Other Practical Limitations.. 14 2.2.3 Market Risk...... 16 Market neutrality...... 16 Market-Neutrality of Risk Arbitrage...... 16 2.2.4 Risk Arbitrageurs as Providers of Liquidity...... 18 2.2.5 Hedging the trade and the portfolio...... 19 2.3 Modeling Excess Return and Risk Exposure...... 19 2.3.1 Linear Approach...... 20 2.3.2 Non-Linear Approach...... 20 2.3.3 Testing for Non-Linearity...... 22 2.3.4 Contingent Claims Analysis...... 22 Contingent Claims Analysis Using Black-Scholes-Merton 22 2.3.5 Effect of Market Returns on the Probability of Trans- action Failure...... 23 2.4 Autocorrelation during the Risk Arbitrage Trade...... 25

3 Review of Risk Arbitrage Risk and Return Literature 28 3.1 Abnormal Returns of Risk Arbitrage...... 28 3.1.1 Early Research...... 28 3.1.2 Later Research...... 29 3.2 Market Risk Exposure of Risk Arbitrage...... 32 3.2.1 Testing the Beta using Linear Models...... 33 v

3.2.2 Testing the Beta using Non-linear Models...... 34

4 Methodology of Data and Portfolio Construction 38 4.1 Data Description...... 38 4.1.1 Sample Selection Criteria...... 38 4.1.2 Sources of Data...... 40 4.2 Portfolio Construction...... 43 4.2.1 Daily Return Series...... 44 4.2.2 Calendar Time Versus Event Time...... 45 4.2.3 Value-Weighted Average Return Series (VWRA)... 46 4.2.4 Equal-Weighted Average Return Series (EWRA)... 48 4.2.5 Practitioner Arbitrage Portfolio Return Series..... 49 Practitioner Arbitrage Portfolio Return Series (PA).. 49 Practitioner Arbitrage Portfolio post Transaction Costs Return Series (PATC)...... 50 4.2.6 Transaction Cost...... 50 Transaction Cost Estimation Procedure...... 51 4.2.7 Credit Suisse Risk Arbitrage Returns...... 52

5 Empirical Analysis and Results 54 5.1 Annualized time series of monthly returns...... 54 5.2 Benchmarking Risk Arbitrage Returns against Linear Asset Pricing Models...... 57 5.2.1 All market conditions...... 57 Abnormal Risk Arbitrage Returns: Comparison of Port- folio Returns...... 59 Abnormal Risk Arbitrage Returns: The Effect of Trans- action Costs...... 59 Abnormal Risk Arbitrage Returns: Comparison of Cash vis-a-vis Stock Portfolios...... 60 The Relationship between Market Excess Returns and Risk Arbitrage Excess Returns...... 62 5.3 Risk Arbitrage in Months of Low Excess Market Returns.. 63 5.4 Benchmarking Risk Arbitrage Returns against a Non-Linear Asset Pricing Model...... 67 5.5 Testing for Non-Linearity...... 72 5.6 Contingent Claims Analysis...... 75 5.7 Sensitivity Analysis of the Diversiﬁcation Constraint.... 77 5.8 Effect of Market Returns on the Probability of Transaction Failure...... 82 5.9 Hedge Fund Returns...... 83 5.9.1 Characteristics of Risks and Returns...... 83 5.9.2 Correlation between CSRA and portfolio returns... 86 5.10 Autocorrelation...... 88 vi

6 Discussion 91 6.1 Implication of Results for Practitioners...... 91 6.2 Impact of Consideration Type on Risk and Return...... 93 6.3 Impact of Discretionary Portfolio Selection...... 94 6.3.1 Passive versus Active Arbitrageurs...... 95 6.3.2 Impact of Regulation on Discretionary Portfolio Se- lection...... 97 6.4 Impact of Modeling the Cross-Country Nature of the West- ern European Markets...... 99 6.5 Impact of Takeover Regulation...... 100 6.6 Questions for Further Research...... 101 6.6.1 Transaction Cost Estimation...... 101 6.6.2 Discretionary Portfolio Formation based on the Be- havior of Arbitrageurs...... 102 6.6.3 Investment Strategy based on Autocorrelation.... 102 6.6.4 Market Speciﬁc Takeover Regulation...... 103 6.7 Overview of Main Conclusions...... 103

7 Conclusion 104

Bibliography 106

A Risk Arbitrage Cumulative Returns 110

B Risk Arbitrage Cumulative Returns - Logarithmic Scale 112

C Transaction Costs by Year 113

D Time Series Regressions of Risk Arbitrage Returns on Common Risk Factors 115 vii

List of Figures

1.1 Increasing Hedge Fund Assets Under Management.....2

2.1 Median Arbitrage Spread...... 8 2.2 Median Arbitrage Spread by Period...... 10 2.3 Risk Arbitrage Trade Time Line...... 12 2.4 Piecewise Linear Model...... 21 2.5 Bid-Ask Bounce...... 26

4.1 Distribution of Transactions in our Sample, by Country, 1990- 2015...... 43 4.2 Returns of Active Transactions...... 47 4.3 Transaction Costs...... 52

5.1 Risk Arbitrage Cumulative Returns (1990-2015)...... 57 5.2 Scatterplot of PATC Returns...... 71 5.3 Scatterplot of CSRA Index Returns...... 84

A.1 Risk Arbitrage Cumulative Returns (1994-2015) - Cash and Stock...... 111

B.1 Risk Arbitrage Cumulative Returns (1990-2015) - Logarith- mic Scale...... 112 viii

List of Tables

3.1 Literature Overview Panel A...... 36 3.2 Literature Overview Panel B...... 37

4.1 Sample Selection Criteria...... 38 4.2 Sample Summary...... 42

5.1 Annualized Time Series of Monthly Returns...... 55 5.2 Time Series Regressions of Excess Risk Arbitrage Returns on Common Risk Factors – All market contitions...... 58 5.3 Time Series Regressions of Excess Risk Arbitrage Returns on Common Risk Factors - During Periods of Excess Market Re- turns below -3%...... 64 5.4 Time Series Regressions of Excess Risk Arbitrage Returns on Common Risk Factors - During Periods of Excess Market Re- turns below -5%...... 66 5.5 Piecewise Linear Regressions: Excess Risk Arbitrage Returns Vis-a-Vis Excess Market Returns...... 69 5.6 Piecewise Linear Regressions: Excess Risk Arbitrage Returns Vis-a-Vis Excess Market Returns...... 70 5.7 Piecewise Linear Regressions: Excess Risk Arbitrage Returns Vis-a-Vis Excess Market Returns, split in two periods.... 72 5.8 Davies Test for Non-Linearity...... 73 5.9 Contingent Claims Analysis Results...... 76 5.10 Annualized Time Series of Monthly Returns - Sensitivity of the Diversification Constraint...... 78 5.11 Linear Diversification Constraint Sensitivity Analysis.... 79 5.12 Non-Linear Diversification Constraint Sensitivity Analysis. 81 5.13 Effect of Market Returns on the Probability of Transaction Failure...... 82 5.14 Time Series Regressions of Excess Risk Arbitrage Returns for the CSRA Index, using CAPM and the Fama and French three- factor model...... 85 5.15 Piecewise Linear Regressions: Excess Risk Arbitrage Returns for the CSRA Index Vis-a-Vis Excess Market Returns..... 86 5.16 Correlation Between CSRA and Portfolio Returns, 1994-2015 87 5.17 Autocorrelation Results...... 89 ix

6.1 This table shows an overview of our six hypotheses, their conclusions and the strength of the evidence...... 103

C.1 Transaction Costs 1980-2015...... 114

D.1 Time Series Regressions of Risk Arbitrage Returns on Com- mon Risk Factors...... 116 1

Chapter 1

Introduction

“Give a man a ﬁsh and you feed him for a day. Teach him how to arbitrage and you feed him forever” Warren Buffett, 1988

Risk arbitrage or merger arbitrage is a type of event-driven investment strategy that bets on the completion of proposed takeovers of public companies, deriving its proﬁts from the spread between the offer price and the price of the target stock shortly after the transaction has been announced. The strategy generally yields moderate positive returns as transactions reach completion. However, in the instances where a transaction fails, the arbitrageur is faced with potentially catastrophic losses. In order to secure protection against such losses, the arbitrageur diversiﬁes the risk arbitrage portfolio across a number of transactions in the market.

1.1 Motivation

According to Wyser-Pratte, 2009, prior to the 1970s, risk arbitrageurs were mainly found at large ﬁnancial institutions on Wall Street and public knowledge about the strategy was limited. However, a surge in M&A activity in the 1960s and 1970s generated interest among investors. As a consequence, risk arbitrage became a popular strategy until the stock market crash in 1987, where the risk arbitrage community as a whole suffered large losses. With the rapid growth of hedge funds throughout the late 1980s and 1990s, a category of hedge funds specializing in event-driven strategies (e.g. risk arbitrage) has emerged (Ackermann, McEnally, and Ravenscraft, 1999). Ackermann, McEnally, and Ravenscraft, 1999 study 2-, 4-, 6- and 8-year sample periods ending in December 1995 consisting of 547, 272, 150 and 79 hedge funds and earning annual returns of 11.1% ,15.8%, 14.7% and 17.9% respectively. The study ﬁnds that event-driven funds on average have more assets under management as well as generate superior returns. In fact, Ack- ermann, McEnally, and Ravenscraft, 1999 state that: 2 Chapter 1. Introduction

“The only category [of hedge fund classiﬁcations] that shows above-average returns and below-average variance is event driven”

These attractive characteristics have led the literature to examine risk arbitrage further. Specifically, early US studies (1990s) find large annual abnormal returns in excess of 100%. More recent US papers, e.g. Mitchell and Pulvino, 2001, find annual abnormal returns of 4%-10%, depending on the model used. This is similar to studies from other markets, e.g. the UK, where Sudarsanam and Nguyen, 2008 report annual abnormal returns of 6%. Historically, risk arbitrage has been considered a market neutral strategy, but in recent years, researchers have questioned this characteristic (Hall, Pinnuck, and Thorne, 2013, Mitchell and Pulvino, 2001). While studies have been conducted in the US, Canada, the UK and Australia, to the best of our knowledge, no research exists across multiple European markets.

FIGURE 1.1: Increasing Hedge Fund Assets Under Management The ﬁgure plots the annual global hedge fund assets under management in USD billion. Source: BarclayHedge via Statista 2016

The abnormal returns documented by literature have attracted investors all over the world, leading to large capital inflows into risk arbitrage. For instance, during recent years, hedge funds have experienced a large increase in assets under management, see figure 1.1. Particularly, risk arbitrage has 1.2. Research Question 3 increased in popularity, reaching an annual increase in net demand1 of almost 50% in 2014, coming from 30% the year before. All in all, this has led to net inflows of capital into risk arbitrage hedge funds of $4.2 billion during the period 1990-2013 (Ferreira and Bousarsar, 2014). The sharp increase in both capital inflows and overall demand suggest that risk arbitrage offers attractive risk and return characteristics to investors. The combination of rapid capital inflows into hedge funds, the large abnormal returns and the lack of research in Europe motivates a study of the risk and return characteristics of risk arbitrage in Western Europe.

1.2 Research Question

The main objective of this thesis is to provide answers to the following research question and supporting questions:

How can the risk and return characteristics in risk arbitrage be modeled in Western Europe during the period 1990-2015?

i How do the risk and return characteristics of risk arbitrage depend on excess market returns?

ii How does the consideration type of cash vis-a-vis stock impact the market neutrality characteristic?

iii How has the risk and return characteristics of risk arbitrage developed over time?

iv How does the portfolio construction approach impact the risk and return characteristics of risk arbitrage?

1.3 Contribution to research

In order to answer these questions, we make a number of contributions to existing research on the topic of the empirical risk and return characteristics of risk arbitrage. First, to the best of our knowledge, we construct the ﬁrst large-scale risk arbitrage sample spanning Western Europe. With the exception of limited research in the UK (Sudarsanam and Nguyen, 2008, Sudarsanam and Nguyen, 2009), we are not aware of any other studies characterizing the nature of risk and return in risk arbitrage in Europe. Fur- thermore, previous literature has purely considered single market studies, whereas we contribute to research by analyzing risk arbitrage returns across 25 European markets.

1Net demand is deﬁned as the percentage of investors reporting increasing demand minus the percentage of investors reporting decreasing demand 4 Chapter 1. Introduction

Moreover, we contribute to research by providing an overview of existing empirical literature with an emphasis on recent developments. Specif- ically, we dedicate significant attention to studies conducted on markets outside the US and, further, pay special attention to studies researching the question of market neutrality. Additionally, we provide a unique level of detail on the impact of consideration type on the market neutrality assumption as well as the risk and return characteristics over time. An additional contribution is the comprehensive approach applied to portfolio construction in order to ensure comparability with earlier research as well as the development of more realistic portfolio modeling techniques. Finally, we use an autocorrelation analysis of changes in target mid prices during the transaction window to examine the possibility of developing an optimal exit timing strategy. To the best of our knowledge, this analysis is novel to the risk arbitrage literature. In conclusion, using a sample of 2167 cash and stock transactions, this study is the first to document that risk arbitrage yields abnormal returns in Western Europe. Specifically, we find this strategy to yield annual abnormal returns in the range of 5% to 13% depending on the benchmarking model. We further conclude that the excess returns do not linearly depend on the excess market returns. Particularly, we find this relationship to be driven by cash transactions as these are not market neutral, in contrast to stock transactions. Further, we find the abnormal returns to be decreasing over time. Finally, the portfolio construction approach has an overarching impact on these findings. The study is organized as follows. Chapter2 introduces the theoretical framework of risk arbitrage by introducing the risk arbitrage trade, the sources of risk and returns in risk arbitrage as well as considering the appropriate asset pricing benchmark models. Chapter3 reviews the existing literature on risk arbitrage with an emphasis on the magnitude of abnormal returns and findings on the market neutrality of the returns. Chapter4 is devoted to data and four different approaches to portfolio construction. Chapter5 presents the results and the empirical analysis of these. Chapter6 unfolds several discussion topics based on the implications and limitations of our results as well as questions for further research. Finally, chapter7 summarizes the main conclusions of our study. 5

Chapter 2

Theoretical Framework of Risk Arbitrage

2.1 Introduction to Risk Arbitrage

Event-driven investments attempt to profit from specific events, both at a corporate level and at a market-wide level. Regardless of the type of event, investors generally follow two portfolio construction principles. First, the event specific risk is isolated and the most general risks are hedged out (e.g. market risk, interest rate risk and credit risk). Second, the investment is diversified across many events in order to minimize the idiosyncratic event risk (Pedersen, 2015). The focus of this study is the classic risk arbitrage trade. In risk arbitrage, the returns are derived from the price movements between the announcement of a merger or acquisition and the success or failure of the proposed transaction. Thus, the profit or loss of such a strategy depends on the completion of the transaction. In financial theory, an arbitrage opportunity describes a trade that de- livers positive profits in the future with a positive probability and does not require any cash to make the investment (Pedersen, 2015). Formally, a portfolio, θ, is an arbitrage opportunity if it satisfies one of the following conditions:

i π · θ = 0 and CT θ > 0

ii π · θ < 0 and CT θ ≥ 0

Where π ∈

2.1.1 M&A Transaction Characteristics Before providing a more detailed explanation of the mechanics of risk arbitrage, it is important to review the characteristics of typical mergers and acquisitions. It is especially important to understand transaction characteristics whenever they may have an impact on the probability of success. Many mergers and acquisitions take place between private companies, e.g. a secondary sale of a company between two private equity funds. How- ever, risk arbitrage requires that the target company is publicly traded and thus, only public takeover candidates are of interest here (Pedersen, 2015). There are tree main variables in an M&A transaction: (i) the type of acquirer, (ii) the stance of the management of the target and (ii) the type of consideration. Acquirers generally fall within the two broad categories. First, there are strategic acquirers seeking synergies and second, financial buyers engaged in leveraged buyouts (e.g. private equity funds). Financial transactions are more likely to fail than transactions with strategic buyers. Pedersen, 2015 hypothesize that this is the case due to a higher reliance on external financing among financial buyers and, thus, a higher risk that the acquirer is unable to finance the proposed transaction. Please notice that in some markets, takeover regulation requires that financing is secured by the acquirer before the offer is made (Sudarsanam and Nguyen, 2008). Second, management tends to either be friendly or hostile. Transac- tions often start with confidential negotiations between the target and the acquirer. As a result, a friendly management is interpreted as a sign of successful negotiations and a hostile management is interpreted as a sign of a breakdown in negotiations. According to Schwert, 2000, most friendly and hostile transactions do not differ in economic terms except for the fact that the negotiation process of hostile transactions use publicity as a bar- gaining tool. Schwert, 2000 further finds that hostile takeover attempts are significantly less likely to reach completion. This holds even if there is a competing bidder. Third, the consideration type varies widely from transaction to transaction. Some of the simplest consideration types are cash only transactions where a predetermined cash price is offered per target share and stock swaps where an amount of the shares of the acquirer are exchanged for shares in the target company. These shares can either be exchanged based 2.1. Introduction to Risk Arbitrage 7 on a fixed exchange ratio or a floating exchange ratio. More complex considerations may include a mixture of cash and stock considerations or op- tionality features (Pedersen, 2015).

2.1.2 The Risk Arbitrage Trade Risk arbitrage is an investment strategy targeting companies that have received acquisition offers. The arbitrageur invests after the announcement of the proposed transaction, seeking to proﬁt from the spread that typically exists between the offer price and the trading price immediately following the announcement. Whether this proﬁt will be realized depends wholly on the success or failure of the proposed transaction.

The Arbitrage Spread The arbitrage spread, also called the speculation spread, is the spread between the offer price and the trading price at the time of investment. It is the compensation ultimately received by the arbitrageur if a given transaction reaches completion. The arbitrage spread is defined as follows (Pedersen, 2015): offer T P − Pt+1 St = T (2.1) Pt+1 T where Pt+1 is the stock price of the target the day after the transaction announcement and P offer is the offer value per share of the target. It is important to notice that risk arbitrageurs invest after the announcement. An- ticipating which companies will receive acquisition offers and the timing of such offers is incredibly difficult and may be a signal of insider trading (Pedersen, 2015). Thus, to avoid any confusion, we restrict our study to investments made post announcement. One way to view the arbitrage spread is as compensation for the risk taken by the arbitrageur. In general, it can be said that the greater the risk of failure, the greater is the compensation required by arbitrageurs. According to Jindra and Walkling, 2004, the bid can be viewed as an indicator of the value placed on the target by the acquirer and the arbitrage spread can be viewed as the market’s valuation of the transaction given the offer price and the transaction characteristics. Furthermore, Jindra and Walkling, 2004 find that arbitrage spreads are significantly, positively related to offer premiums, pre-offer price run-ups, hostile target managements and rumors preceding the offer. These transaction characteristics are all known at the time of the announcement. The largest source of risk in risk arbitrage is completion risk2. There- fore, a large arbitrage spread can be interpreted as large uncertainty about the likelihood of transaction success. Samuelson and Rosenthal, 1986 test

2Other risks include market risk, exchange risk etc. 8 Chapter 2. Theoretical Framework of Risk Arbitrage whether the price movements of the target stock in cash only transactions is a good predictor of transaction success. They find that market prices are well-calibrated and, thus, represent the current market odds of success. Further, they find that the probability predictions of the market improves as the time of convergence approaches. Finally, they find that the larger the increase in relative stock prices (i.e. decrease in arbitrage spread), the more likely it is that the transaction will succeed. Following Mitchell and Pulvino, 2001, we plot the median arbitrage spread against the number of trading days until resolution for successful and failed transactions respectively, see figure 2.1. While our sample does not result in as clear a distinction between successful and failed transactions as that of Mitchell and Pulvino, 2001, we still see a clear separation between the respective median arbitrage spreads. As expected from Samuelson and Rosenthal, 1986, the arbitrage spreads for transactions that will ultimately be withdrawn are wider than the arbitrage spreads for transactions that will ultimately be completed, reflecting the market odds of success. For successful transactions, the arbitrage spread appears to continuously decrease towards zero as the resolution date approaches. For failed transactions, the arbitrage spread appears to be much more volatile, which may simply be the result of a much smaller sample of failed transactions vis-a-vis successful transactions. Finally, the arbitrage spread of a failed transaction increases rapidly at the announcement that the offer is withdrawn, which is to be expected.

FIGURE 2.1: Median Arbitrage Spread The ﬁgure plots the median arbitrage spread for all transactions against the number of trading days until transaction resolution. The transaction resolution date is deﬁned as the date of the transaction termination announcement or the consummation date for failed and successful transactions respectively. 2.1. Introduction to Risk Arbitrage 9

Based on a sample of 2,182 American transactions, Jetley and Ji, 2010 find that the arbitrage spread has declined by more than 400 bps since 2002. Jetley and Ji, 2010 name the following potential causes for the decline: capacity constraints over time (i.e. increased competition due to inflows of capital), reduced transaction costs and changes in the risk characteristics of risk arbitrage in the form of lower losses in case of transaction failure. These lower losses may be caused by lower offer premiums, as offer premiums serve as a crude indicator of losses suffered in case of transaction failure. Figure 2.2 plots the median arbitrage spread by transaction outcome against the number of trading days until resolution. The sample is divided into two time periods, namely 1990-2002 and 2003-2015. The figure supports the findings of Jetley and Ji, 2010 by highlighting a clear decrease in returns between the periods before and after 2002. Finally, it is worth to notice that while the arbitrage spread is usually the main source of compensation in risk arbitrage, it is not the only source of compensation received by the arbitrageur. In cash transactions, the arbitrageur also receives dividends on the long position. In stock transactions, the arbitrageur receives dividends on the long position which are offset by dividends paid on the short position. Finally, the arbitrageur receives interest on the proceeds from the short sale (Mitchell and Pulvino, 2001)3. These profits should be considered in light of transaction costs from going long and hedging the position, short-selling costs and funding costs (Pedersen, 2015).

3While large funds such as hedge funds typically do earn interest of the proceeds from the short position, individual investors typically do not. In an unreported analysis, Mitchell and Pulvino, 2001 ﬁnd that the annualized returns are reduced by 2% if short proceeds do not earn interest. 10 Chapter 2. Theoretical Framework of Risk Arbitrage

(A) Successful transactions

(B) Failed transactions

FIGURE 2.2: Median Arbitrage Spread by Period The ﬁgure plots the median arbitrage spread split by the transaction outcome and by time period against the number of trading days until transaction resolution. The transaction resolution date is deﬁned as the date of the transaction termination announcement or the consummation date for failed and successful transactions respectively.

The Mechanics of the Risk Arbitrage Trade The mechanics of a risk arbitrage trade are relatively simple and easily illustrated with an example. The life of a risk arbitrage trade commences with the announcement of the proposed transaction. This announcement usually takes place when the markets are closed. Alternatively, trading is 2.1. Introduction to Risk Arbitrage 11 halted while the announcement is made. In order to persuade the owners of the target company to sell, the acquirer sets the offer price at a premium relative to the current trading price of the target stock. The arbitrageur considers transaction terms such as the price, the arbitrage spread, the probability of success, the duration etc. If the arbitrageur deems that the transaction poses an attractive investment opportunity, a long position is taken in the target stock and, depending on the consideration type, a hedge may be placed (Pedersen, 2015). Suppose that a target’s stock trades at $50 prior to the announcement of an offer of $65 per share. Immediately after the announcement, the target’s stock price jumps to $62 per share. Thus, the resulting deal spread is $3/$62 = 4.8%. If the transaction is successful, the arbitrageur stands to earn the arbitrage spread. If the transaction fails, the price is likely to drop to the pre- offer level of $50 and the arbitrageur stands to lose $12. If adverse information about the target is uncovered during the due diligence process, the price may drop even further than its pre-announcement level, exacerbating the potential losses to the arbitrageur. The price may also drop below the pre-announcement level if a price run-up existed in the target’s stock due to acquisition rumors in the market. During the period between the announcement and the completion date, the offer price may be renegotiated or a competing firm may make a counter-offer (Pedersen, 2015). Thus, the expected profit of the trade depends on the profit in case of completion, the losses in case of withdrawal and finally, the probability of completion. Calculating the expected profit can be quite complex in real life, especially for transactions with complex consideration structures. The conclusion of the trade can generally be divided into four categories. The most common category is completion of the proposed transaction. However, the trade can also be concluded by a higher offer from a competing bidder, a renegotiation of the transaction or a failure of the transaction (Pedersen, 2015). Figure 2.3 shows a stylized time line for a risk arbitrage trade. This figure serves as a simple summary of the mechanics of the risk arbitrage trade discussed above. 12 Chapter 2. Theoretical Framework of Risk Arbitrage

FIGURE 2.3: Risk Arbitrage Trade Time Line The ﬁgure shows a stylized time line for a risk arbitrage trade. The red line represents failed transactions and the green line represents successful transactions. In the period leading up to the transaction announcement day, there will often be rumors about the announcement, causing the price to slowly increase. After the announcement date, the price will jump, but not all way up to the offer price. This creates the arbitrage spread. During the transaction window, the arbitrage spread will either increase or decrease, depending on the outcome of the transaction. Source: Own creation

2.2 Sources of Abnormal Returns and Risk in Risk Ar- bitrage

As mentioned above, the arbitrage spread can be viewed as the compensation the risk arbitrageur receives in return for taking the completion risk. As a consequence, the arbitrage spread should on average be positive in order to compensate for the losses associated with failed transactions. The break-even arbitrage spread is deﬁned as the arbitrage spread that results in average proﬁts of zero (Pedersen, 2015). Looking at empirical results, however, it appears that risk arbitrage has yielded positive returns on average. For an overview over historical returns, see table 3.1 and table 3.2. These positive returns are henceforth referred to as abnormal returns. The abnormal returns measure the value added by the risk arbitrage strategy in excess of the risk-free rate, the market exposure and other relevant factors. This stands in contrast to excess returns, which are simply returns in excess of the risk-free rate. Based on the positive abnormal returns observed in the US, Canada, Australia and the UK (Mitchell and Pulvino, 2001, Baker and Sava¸soglu, 2.2. Sources of Abnormal Returns and Risk in Risk Arbitrage 13

2002, Karolyi and Shannon, 1999, Hall, Pinnuck, and Thorne, 2013, Su- darsanam and Nguyen, 2008), we form the following hypothesis:

Hypothesis 1: We expect to observe positive abnormal risk arbitrage returns in Western Europe during the period 1990-2015.

The extraordinary abnormal returns reported in literature seem to indicate that large market inefficiencies exist in the pricing of target stocks. Whether or not these inefficiencies are acceptable depends on the market perception of the reader. On one side, scholars like Fama and Malkiel, 1970 argue that markets are efficient and that prices are constantly updated to reflect all new information. On the other side, scholars like Shiller, 2003 argue that markets are not efficient and that they are subject to a number of behavioral biases that drive prices away from their fundamental values. We adopt the point of view that markets are "efficiently inefficient" as argued by Pedersen, 2015. Pedersen, 2015 agues that the market is

“inefficient to an efficient extent: just inefficient enough that money managers can be compensated for their costs and risks through superior performance and just efficient enough that the rewards to money management after all costs do not en- courage entry of new managers or additional capital”.

Hence, a certain magnitude of abnormal returns may be justiﬁed in order for risk arbitrageurs to engage in risk arbitrage and, thereby, offer liquidity to those that do not wish to trade on the completion of the transaction. As we see it, there are three main explanations for the abnormal returns reported in the risk arbitrage literature:

i The abnormal returns documented in literature cannot be realized in reality due to transaction costs and other practical limitations

ii The abnormal returns compensate risk arbitrageurs for exposure to non-linear market risk

iii The efficiently inefficient school of thought supports the argument that risk arbitrageurs are compensated for their provision of liquidity at an efficiently inefficient level

The following section will ﬁrst discuss the importance of the portfolio construction approach. Second, we will examine the three main explanations offered for the sources of abnormal returns in risk arbitrage. Third, the section will touch upon additional risks to be considered by the arbitrageur in the form of completion risk and portfolio hedging. 14 Chapter 2. Theoretical Framework of Risk Arbitrage

2.2.1 Portfolio Construction One of the most critical questions to be answered by the risk arbitrageur is how to construct the portfolio of risk arbitrage trades. Given the universe of transactions available at any given time, the arbitrageur must decide which transactions to trade on, the number of transactions to trade on and the maximum weight in any one trade. In this study, all portfolio construction approaches are based on quantitative and automatic investment criteria. However, in reality, the risk arbitrageur may invest on a discretionary basis with a more concentrated portfolio based on careful analysis. This type of transaction selection is difﬁcult to simulate without hindsight bias and, thus, the empirical analysis of this study is based on quantitative trade selection. The critical importance of the portfolio construction approach leads us to form the following hypothesis:

Hypothesis 2: We expect to observe that the portfolio construction approach has an impact on the risk and return characteristics of the risk arbitrage strategy.

The different portfolio construction approaches applied in this study are developed in chapter4 and a detailed discussion of the effect of introducing discretionary portfolio selection is provided in Chapter6.

2.2.2 Transaction Costs and Other Practical Limitations While it is easy to blame overall market inefficiencies for the abnormal returns generated by risk arbitrageurs, an alternative explanation presented by Mitchell and Pulvino, 2001 is that the extraordinary returns reported by the literature are impossible to replicate in real life. As risk arbitrage returns are mostly reported without considering transaction costs or only considering direct transaction costs, the abnormal returns may be significantly smaller in reality. Mitchell and Pulvino, 2001 model indirect trading costs in the form of the costs of the market impact of the trade as well as direct transaction costs consisting of brokerage fees, transaction taxes and other surcharges. As risk arbitrage involves high turnover trading, these direct and indirect transaction costs can be substantial. Mitchell and Pulvino, 2001 find that 1.5% of the annualized returns predicted in their study can be attributed to direct transaction costs, 1.5% can be attributed to indirect transaction costs and 2.5% of returns can be attributed to limitations on the position sizes due to illiquidity in the stock of the target and/or acquirer. This leads us to the following hypothesis: 2.2. Sources of Abnormal Returns and Risk in Risk Arbitrage 15

Hypothesis 3: We expect to observe a substantial impact of transaction costs on the magnitude of the abnormal risk arbitrage returns.

Another practical limitation stems from agency problems. Shleifer and Vishny, 1997 argue that inefficiencies are created by agency problems inherent in the relationship between the arbitrageurs who contribute with highly specialized knowledge and the investors who contribute with resources. In situations without agency problems, arbitrageurs invest more aggressively when prices move further away from fundamentals, which might very well be the case during a risk arbitrage trade. In situations with agency problems, the investors may not understand that prices sometimes move further apart before convergence, which in many cases requires extra margin to be posted. The investors may then refuse to post additional capital or in the worst case scenario, withdraw some or all of the capital invested at the worst possible time. As a consequence, capital flows become dependent on past performance. In other words, the arbitrageurs may become capital constrained when they have the best opportunities (i.e. the largest arbitrage spreads). The consequence of this adverse scenario is more cautious investing, which in turn limits the push towards efficient markets. Further, in order to avoid extensive outflows of capital at a later point in time, risk averse arbitrageurs may liquidate their positions before they are pushed to liquidate by investors. Shleifer and Vishny, 1997 further argue that the theoretical assumption that markets include many well-diversified arbitrageurs may not hold in reality since engaging in risk arbitrage requires specialized knowledge that not all investors possess. Thus, much risk arbitrage capital may be concentrated in the hands of few arbitrageurs and the abnormal returns may be viewed as compensation for idiosyncratic risk that has not been diversified away. Another possible explanation for the substantial abnormal returns from the risk arbitrage strategy is to view the returns as compensation for costly information acquisition by the arbitrageur when assessing whether or not a transaction is likely to reach completion. Larcker and Lys, 1987 show that arbitrageurs are able to acquire information that is superior to that of the general public and that they are rewarded with positive abnormal returns. These results are based on the underlying assumption that prices are sufficiently noisy to warrant costly information searches. Finally, many studies report the portfolio returns in event time rather that calendar time. Thus, the returns from the transactions are annualized before they are averaged across all transactions, which leads to the assumption that the portfolio can earn event-time returns continuously. This may lead to wildly unrealistic returns in excess of 100% on an annual basis (Dukes, Frolich, and Ma, 1992, Jindra and Walkling, 2004). Please see chap- ter4 for a more thorough treatment of the effect of using event time versus 16 Chapter 2. Theoretical Framework of Risk Arbitrage calendar time.

2.2.3 Market Risk Market risk as a source of abnormal returns implies that arbitrageurs are compensated for being correlated with market movements. However, risk arbitrage has traditionally been viewed as a market neutral strategy. Thus, it is important to explore the notion of market neutrality before exploring the relationship between market risk and risk arbitrage returns.

Market neutrality In all simplicity, being market neutral means that the performance of a certain investment strategy or portfolio is not related to the performance of the overall market. More formally, market neutrality can be deﬁned as (Peder- sen, 2015):

β = 0 (2.2) From the equation presented above, it can be seen, that knowing the beta of a given strategy enables the investor to adjust for potential market exposure by hedging the strategy according to the ratio speciﬁed by the beta itself (Pedersen, 2015). This potential market exposure may be linear or non-linear depending on the strategy in question. For illustrative purposes, assume that a linear strategy has a beta of 0.5. In order to hedge this strategy, the investor needs to short 0.5 dollars of the market portfolio for every dollar invested in the strategy. Among the appeals of market neutral returns are the diversiﬁcation potential it offers to prospective investors and the fact that a market neutral strategy has an equal chance of generating positive returns in appreciating markets as well as depreciating markets. Hedge funds are famous for providing such market neutral strategies, although many hedge funds are not truly market neutral (Pedersen, 2015).

Market-Neutrality of Risk Arbitrage While early literature assumed that risk arbitrage was market neutral or at least linearly related to market risk (Bhagat, Brickley, and Loewenstein, 1987, Jindra and Walkling, 2004, Maheswaran and Yeoh, 2005), later literature has documented a non-linear relationship between risk arbitrage and market risk (Mitchell and Pulvino, 2001, Hall, Pinnuck, and Thorne, 2013, Branch and Wang, 2008). Please see table 3.1 and table 3.2 for an overview over the literature. Specifically, risk arbitrage returns seem to be uncorre- lated with market movements in flat and appreciating markets, while the risk arbitrage returns seem to be positively correlated to market movements 2.2. Sources of Abnormal Returns and Risk in Risk Arbitrage 17 in depreciating markets. For now, we assume that depreciating markets are defined as months, where the market returns fall below -4%. It is worth to notice that the risk arbitrage portfolio returns are not directly correlated with the market, as it is assumed that the arbitrage portfolio is well-diversified. Instead, the risk arbitrage portfolio returns can be indirectly correlated with the market because the rate of transaction failure is directly correlated with the market. In flat and appreciating markets, overall market events will not affect the rate of transaction failure. Thus, the returns of the risk arbitrage portfolio depend on idiosyncratic events and, in theory, have a market beta close to zero. In depreciating markets, the rate of transaction failure increases and, as a result, the returns of the risk arbitrage portfolio are exposed to market movements with a positive market beta (Pedersen, 2015). This leads us to the following hypothesis:

Hypothesis 4a: We expect to observe a non-linear relationship between the excess risk arbitrage returns and the excess market returns.

The degree of the correlation with the market depends on the consideration type. Cash only offers are highly likely to be withdrawn during sudden, severe depreciating markets, as the offer price may suddenly appear too high relative to market prices. Fixed exchange ratio stock swaps may be less sensitive to this effect, as the relationship between the target stock price and the acquirer stock price may remain the same during depreciating markets. Please note that this naturally depends on the beta of both stocks. Furthermore, it is often more difﬁcult to obtain ﬁnancing in depreciating markets, which increases the rate of transaction failure (Mitchell and Pulvino, 2001). This leads us to the following hypothesis:

Hypothesis 4b: We expect to observe a stronger non-linear relationship between the excess risk arbitrage returns and the excess market returns for cash transactions vis-a-vis stock transactions.

Further, based on the decrease in arbitrage spreads between the periods 1990-2002 and 2003-2015 illustrated in ﬁgure 2.2, we form the following hypothesis:

Hypothesis 4c: We expect to observe decreasing abnormal risk arbitrage returns over time.

We expect the above hypothesis to hold regardless of the relationship between the excess risk arbitrage returns and the excess market returns. In conclusion, we expect the large abnormal returns reported by early literature to compensate the arbitrageur for the exposure of the risk arbitrage portfolio to non-linear market risk. Speciﬁcally, we expect a stronger 18 Chapter 2. Theoretical Framework of Risk Arbitrage non-linear market exposure for cash transactions and we expect the abnormal returns to decrease over time.

2.2.4 Risk Arbitrageurs as Providers of Liquidity The primary risk assumed by the arbitrageur is completion risk. This is true regardless of transaction characteristics such as the consideration type, the stance of management and the type of buyer. Existing shareholders in the target stock may not want exposure to completion risk and, thus, the existing shareholders start exiting their positions in the target. If the existing shareholders keep their position and the deal is successful, they will be able to earn the arbitrage spread just like the risk arbitrageur (Pedersen, 2015). However, presumably the pre announcement shareholders initially bought the target stock based on a view that the future performance of the company would result in higher stock prices and/or significant dividends. After the announcement, the stock price no longer depends on the fundamentals of the company, but on the probability of transaction success. The shareholders may not feel confident in their ability to evaluate the completion risk. Additionally, the shareholders are not necessarily diversified across a number of risk arbitrage transactions and are, as a result, more vulnerable to transaction failure than the well-diversified risk arbitrageurs. All of these arguments lead to a selling pressure on the target’s stock (Pedersen, 2015). The consequence of this selling pressure is a widening arbitrage spread, which generates positive risk arbitrage returns on average. From the seller’s point of view, the arbitrage spread can be interpreted as an insurance against transaction failure. By selling the stock after the initial post announcement price jump, the existing shareholders earn the increase in the stock price, but pay the remaining arbitrage spread in insurance against transaction failure when they exit the position (Pedersen, 2015). Before and after the transaction, there will be a set of natural shareholders of the target stock. These shareholders invest in the company because they believe in the future prospects of the stock. During the transaction window, the natural shareholders suddenly become sellers of the stock. Thus, an urgent demand for liquidity is created. The risk arbitrageur meets this demand for liquidity by buying the stock from the natural shareholders. As long as the risk arbitrageur is sufficiently diversified across a number of risk arbitrage trades, the failure of some transactions should not be detrimental to the overall returns of the risk arbitrage portfolio (Pedersen, 2015). Following the market view of Pedersen, 2015, the arbitrage spread is determined to reach an efficiently inefficient level, where the arbitrageurs are compensated for their liquidity provision, but entry of additional risk arbitrage capital is discouraged. Thus, the magnitude of the arbitrage spread depends on the availability of risk arbitrage capital. During times when the 2.3. Modeling Excess Return and Risk Exposure 19 number of transactions is high (low) relative to the amount of risk arbitrage capital, the expected returns increase (decrease). The efficiently inefficient level is supposedly also reached when comparing arbitrage spreads across transactions, as the supply of liquidity will be largest for the transactions that are the most likely to succeed. Thus, the arbitrage spreads will be narrower for these transactions and wider for the transactions deemed more likely to fail. This leads us to the following hypothesis:

Hypothesis 5: We expect to observe that risk arbitrageurs are compensated for providing liquidity during the transaction window

2.2.5 Hedging the trade and the portfolio Risk arbitrage is an investment intended to earn the arbitrage spread while carrying the risk of transaction failure. In other words, the purpose of risk arbitrage is to carry the completion risk while hedging out all other significant risks and diversifying across many transactions. Hedging a risk arbitrage trade depends on the consideration type. In a cash transaction, the offer price per share of the target’s stock is independent of the value of the stock of the acquirer. Therefore, no hedge specific to the transaction is needed. In a fixed-exchange ratio stock swap, the price per share of the target’s stock depends on the value of the acquirer’s stock. The most natural way to hedge the trade is to short the amount of shares of the acquirer that is stipulated by the exchange ratio, also called the hedge ratio (Pedersen, 2015). To illustrate, if the acquirer offers 2 of its own shares in exchange for 1 of the target’s shares, then the arbitrageur shorts 2 of the acquirer’s shares for each share he is long in the target. Our sample consists of cash transactions and fixed-exchange ratio stock swap transactions. However, it is worth to note that hedging becomes increasingly complex as the complexity of the consideration structure increases. If the risk arbitrage strategy violates market neutrality, it is not necessarily enough to hedge the transactions at an individual level. If the risk arbitrageur only wants to carry completion risk, it is necessary to hedge the market exposure at a portfolio level. Pedersen, 2015 suggests hedging the overall risk arbitrage portfolio against market exposure by shorting equity index futures or going long index put options. These types of portfolio hedges should be sized according to the overall value of the risk arbitrage portfolio and the composition of the portfolio across transaction types.

2.3 Modeling Excess Return and Risk Exposure

Modeling the excess return and risk exposure of conducting a risk arbitrage strategy lies at the heart of this study. Whether to benchmark monthly risk 20 Chapter 2. Theoretical Framework of Risk Arbitrage arbitrage returns against linear models or non-linear models depends on the relation between the excess risk arbitrage returns and the excess returns of the overall market. As literature has shown varying evidence on this relationship, we provide both linear and non-linear benchmark models in the following section. The following models do not take transaction costs into account, as transaction costs are better modeled at the portfolio level. That does not mean, however, that transaction costs are not important. In fact, Mitchell and Pulvino, 2001 demonstrate that taking transaction costs into account signiﬁcantly impacts the estimated abnormal returns of risk arbitrage. The application of transaction costs is treated at the end of this chapter.

2.3.1 Linear Approach Following the point of view of early literature on the proﬁtability of risk arbitrage, we suggest benchmarking the estimated risk arbitrage returns against the CAPM and the Fama and French, 1993 three-factor asset pricing model:

CAPM: (RRiskArb − Rf ) = α + βMkt(RMkt − Rf ) (2.3)

Fama and French:

(RRiskArb−Rf ) = α+βMkt·(RMkt−Rf )+βSMB ·SMB+βHML·HML (2.4)

RArb is the monthly return from a portfolio that is invested in risk arbitrage, Rf is the risk-free rate, α is the average monthly abnormal return of the strategy, β is the correlation with the market, RMkt is the market return as provided by the Kenneth French Database. SMB and HML are the Fama and French factors. SMB is deﬁned the difference in returns between small stocks and big stocks and HML is deﬁned as the difference in returns between stocks with high book-to-market ratios and stocks with low book-to-market ratios.

2.3.2 Non-Linear Approach Mitchell and Pulvino, 2001 provide evidence that the payoff structure of risk arbitrage in the US strongly resembles that of writing uncovered index put options 4. In other words, whenever the excess market returns are

4Wolﬁnger, 2005 deﬁnes writing uncovered put options as accepting “... an obligation, for a limited time, that grants the other party the right to force you to buy stock at an agreed-on price (strike price).” Please notice that uncovered positions are positions in which the investor does not own the underlying asset. 2.3. Modeling Excess Return and Risk Exposure 21 above the strike (the threshold), the payoffs are modest and stable. How- ever, whenever the excess market returns are below the strike (the threshold), the payoffs are negatively related to the excess market returns. This leads to an expectation that the payoff structure of risk arbitrage in Western Europe also resembles writing uncovered put options. To test this expectation, we follow the methodology of Mitchell and Pul- vino, 2001 by estimating the following piecewise linear CAPM-type model:

RRiskArb − Rf = (1 − δ)[αMktLow + βMktLow(RMkt − Rf )] (2.5) +δ[αMktHign + βMktHigh(RMkt − Rf )] where δ is a dummy variable equal to one if the excess market return is above a speciﬁed threshold level and zero otherwise. To ensure continuity at the kink, the following restriction is imposed:

αMktLow + βMktLow(T hreshold) = αMktHigh + βMktHigh(T hreshold) (2.6)

Mitchell and Pulvino, 2001 arrive at a threshold equal to -4.0% by minimiz- ing the sum of squared residuals. For comparability, we report results with a threshold of -4.0%. Additionally, we reestimate the threshold by minimiz- ing the sum of squared residuals of our own sample. Mirroring a depiction by Mitchell and Pulvino, 2001, ﬁgure 2.4 provides a graphical illustration of the piecewise linear model as expressed by equations 2.5 and 2.5. The model is depicted assuming a negative threshold.

FIGURE 2.4: Piecewise Linear Model The ﬁgure provides a graphical illustration of the piecewise linear model as expressed by equations 2.5 and 2.6, depicting a negative threshold. 22 Chapter 2. Theoretical Framework of Risk Arbitrage

2.3.3 Testing for Non-Linearity In order to test whether there is evidence for the piecewise linear model, we want to test for non-linearity. This can be done by testing whether there is evidence for a change in the slopes, βlow and βhigh. To do this, we apply the Davies test. The Davies test checks for a non-zero difference in the slope parameter of a segmented relationship by stating the following null hypothesis:

H0 : ∆β = 0 (2.7) To test this null hypothesis, we estimate a linear fitted model. We use equation 2.3 for the fitted model. We could also have applied a segmented fitted model, but the Davies test is most reliable when the segmented model is estimated within the model rather than pre-estimated. Therefore, we use a linear fitted model. Thereafter, we find the "best" breakpoint estimated by the model and test for the significance of this breakpoint by testing whether the difference in slopes is significantly different from zero (Davies, 1987).

2.3.4 Contingent Claims Analysis As mentioned above, research has indicated that risk arbitrage as a strategy is neither linear nor market-neutral (Mitchell and Pulvino, 2001,Hall, Pinnuck, and Thorne, 2013). Thus, it does not appear appropriate to model the excess returns using linear models such as the CAPM or the Fama and French three-factor model. Instead, alternative approaches such as contingent claims analysis are needed.

Contingent Claims Analysis Using Black-Scholes-Merton Simply stated, contingent claims are claims that are dependent upon some other event and the outcome of this event. A type of contingent claim asset is a simple ﬁnancial put or call option. Due to this option structure, the Black-Scholes-Merton option pricing formula can be used in contingent claims analysis to calculate the price of an asset. As we hypothesize that the payoff structure of risk arbitrage in Europe resembles writing an uncovered index put option, we will present the theory of contingent claims analysis using the formula of a put option and, in order to clarify, we will further include the context in which this analysis is used. In essence, we attempt to replicate a $100 investment in a portfolio devoted to risk arbitrage by investing in a portfolio based on a risk-free bond (long position) and an index put option (short position). In order to determine whether risk arbitrage generates abnormal returns, we need to calculate the cost of replicating such a portfolio. To estimate this, we need the price of a put option. Therefore, we ﬁrst calculate the strike price, which for 2.3. Modeling Excess Return and Risk Exposure 23 consistency reasons is based on the -4% threshold suggested by Mitchell and Pulvino, 2001:

Strike price: $100(1 + T hreshold + rf ) (2.8) Second, we need to estimate the cost of replicating the risk arbitrage portfolio. This includes pricing the index put option. To do this, we use the Merton, 1973 version of the Black-Scholes-Merton option pricing formula:

Cost of replicating portfolio:

$100(rf + αMktHigh) −rT − βMktLow · [Ke N(−d2) − S0N(−d1)] (2.9) 1 + rf where 2 S0 σ ln( K ) + (rf + 2 )T d1 = √ (2.10) σ T

2 S0 σ √ ln( K ) + (rf − 2 )T d2 = √ = d1 − σ T (2.11) σ T

rf is deﬁned as the sample average of the monthly risk-free rate, αMktHigh is the abnormal return estimated for the total PATC portfolio using the piecewise linear regression and applying the -4% threshold, βMktLow is the "ﬁrst slope" (the slope in depreciating markets) of the same regression and K is the strike price. N is the cumulative normal distribution function and S is the spot price. Finally, T is the time to maturity and σ is the variance of the return of the underlying asset, which we calculate as the standard deviation of the monthly market returns multiplied by the square root of 12. For consistency purposes, our risk return analysis of risk arbitrage using this approach will be based on monthly returns, similar to the other analyses we perform. It could be questioned whether or not this will affect the results when testing for non-linearity. However, literature shows that the choice between monthly and annual returns only has a very limited impact on the results (Mitchell and Pulvino, 2001).

2.3.5 Effect of Market Returns on the Probability of Transaction Failure Due to the significant potential losses associated with transaction failure, it is of great interest to arbitrageurs to try to predict whether or not a transaction will be successful. In a model utilizing only information available shortly after the announcement and before the position is established, Branch 24 Chapter 2. Theoretical Framework of Risk Arbitrage and Wang, 2009 find that the most significant variables in predicting transaction outcomes are the target’s stock price run-up, the resistance of the target, the arbitrage spread, the relative size of the target and bidding competition. The non-linear relationship between excess risk arbitrage returns and excess market returns is largely caused by an increase in the probability of transaction failure during depreciating markets. Hence, we want to test if the probability of transaction failure is a decreasing function of the market returns in the three months leading up to the resolution date. In order to test the effect of market returns on the probability of transaction failure, we estimate the following probit model as specified by Mitchell and Pulvino, 2001:

F ail = α + β1RMkt + β2RMkt−1 + β3RMkt−2 + β4LBO (2.12) +β5Cash + β6P remium + β7Size + β8Hostile where F ail is a dummy variable equal to one if a transaction is withdrawn and zero otherwise, RMkt is defined as the monthly market return for the month wherein the transaction resolution date falls, RMkt−1 is the monthly market return for the month before the resolution date, RMkt−2 is the monthly market return two months before the resolution date, LBO is a dummy variable equal to one if the acquirer is private and zero otherwise, Cash is a dummy variable equal to one if the transaction is paid 100% in cash and zero otherwise, P remium is defined as the target stock price one day after the announcement divided by the target stock price four weeks prior to the announcement, Size is the logarithm of the equity value of the target and Hostile is a dummy variable equal to one if the stance of management is hostile. We make no assumptions about independence of transactions occurring within the same year. Please note that Mitchell and Pulvino, 2001 define transaction failure as any transaction where the arbitrageur loses money. Hence, in addition to transactions where the offer is withdrawn by the acquirer, Mitchell and Pulvino, 2001 include transactions where the offer price is lowered as failed transactions, although these transactions are in fact completed. In order to be consistent throughout this study, we only include transactions that are ultimately withdrawn in our sample of failed transaction5. We utilize the probit model as specified by Mitchell and Pulvino, 2001 with minor modifications in order to ensure comparability of our results. However, it bears mentioning that more complex models have subsequently been developed to estimate the determinants of the probability of transaction failure. The key motivation for the development of more sophisticated modeling techniques is the low observed frequency of failed transactions

5From the perspective of the arbitrageur, any investment that yields negative returns is a failed transaction. However, from an overall M&A perspective, a transaction that is ultimately completed is a successful transaction. 2.4. Autocorrelation during the Risk Arbitrage Trade 25

(approx. 10%), which impacts the prediction power of failure prediction models. However, these models cannot be used as a benchmark, because this branch of research is interested in the general determinants of transaction failure, whereas we are speciﬁcally interested in the effect of market returns (Branch and Wang, 2009)6

2.4 Autocorrelation during the Risk Arbitrage Trade

Recall that a risk arbitrage trade commences at the announcement of a proposed transaction and ends at the resolution date, which is the day the transaction is either terminated or consummated for failed and successful transactions respectively. We would like to test whether alternative ver- sions of these risk arbitrage trade mechanics could exist. Further, recall that completion risk is the largest risk in the risk arbitrage strategy. Thus, in cases where transactions ultimately fail, it would be advantageous for the arbitrageur to exit the positions before the termination announcement. As a consequence, the risk arbitrage portfolio would be shielded from potentially disastrous losses. Speciﬁcally, we would like to examine the potential for developing an optimal exit strategy. When testing almost any hypothesis within the ﬁeld of risk arbitrage, it is important to limit the use of data to only include data which is publicly available at the period of time leading up to the endogenously determined optimal exit date. Due to this limitation, we use autocorrelation within mid prices to test the potential for creating an optimal exit strategy7. Since we are interested in running autocorrelation regressions, the choice of which price variable to use becomes crucial. Campbell, Lo, and MacKin- lay, 1997 explains why it is of critical importance to use mid prices and not closing prices in our autocorrelation analysis:

“..as random buys and sells arrive at the market, prices can bounce back and forth between the ask and the bid prices, creating spurious volatility and serial correlation in returns, even if the economic value of the security is unchanged.”

Thus, while the closing price of a stock may fall anywhere within the bounds of the bid-ask spread, the choice of mid prices will mitigate the issue of serial correlation, which would otherwise prevail and invalidate our analysis. Figure 2.5 clearly illustrates how closing prices ﬂuctuate between the bounds of the bid-ask spread and why this ﬂuctuation introduces a large

6A commonly used method to increase the number of failed transactions in a sample is to use pair-matched sampling. However this methodology has been proven to yield biased probability estimators in logistic regression models (Branch and Wang, 2009). In order to adjust for the bias, Branch and Wang, 2009 use a weighted logistic regression model. 7We would like to thank our supervisor, Niklas Kohl, for suggesting this analysis. 26 Chapter 2. Theoretical Framework of Risk Arbitrage amount of noise to any autocorrelation calculation. This effect is also known as the bid-ask bounce. As a consequence of the bid-ask bounce, closing prices exhibit negative autocorrelation.

FIGURE 2.5: Bid-Ask Bounce The ﬁgure plots the bid price, ask price, closing price and mid price for the Dan- ish corporation FLSmidth (NASDAQ OMX: FLS). The choice of FLSmidth is for illustrative purposes.

To obtain mid prices, we collect the bid and the ask prices from Datas- tream and average the two:

P t + P t P t = Ask Bid (2.13) mid 2 Due to the nature of prices in general, we expect to ﬁnd autocorrelation within a time series of mid prices. This relationship is most easily explained by a martingale, i.e. a stochastic process {Pt} satisfying the following condition:

E[Pt+1 | Pt,Pt−1, ...] = Pt (2.14) The martingale hypothesis states that the price of tomorrow is expected to equal the price of today given all historical price information. Although we do not expect markets to be fully efﬁcient, there is limited value in testing for day-to-day correlation between prices. Instead, we want to use the 2.4. Autocorrelation during the Risk Arbitrage Trade 27 percentage change in mid prices when testing for autocorrelation:

t t−1 t Pmid − Pmid %∆Pmid = t−1 (2.15) Pmid Since the martingale hypothesis implies that prices will not change from day-to-day, any price changes in a fully efﬁcient market should be random and, thus, have an autocorrelation equal to zero. This leads to the following null hypothesis:

H0: The autocorrelation coefﬁcients of the changes in mid prices equal zero at various lags

Based on the percentage changes in price, we estimate the autocorrelation coefficients of the first six lags and determine the p-values using 99% confidence intervals. This analysis is carried out for each transaction in the sample and subsequently averaged. Contrary to the expectation of the null hypothesis above, we suspect that there may be positive autocorrelation between the day-to-day changes in mid prices during the transaction window. Thus, we state the following hypothesis:

Hypothesis 6: We expect to observe positive autocorrelation in the time series of changes in mid prices during the transaction window.

If the above hypothesis is confirmed, there may be a potential to develop an optimal exit timing strategy. Specifically, it would be interesting to determine if an exit signal can be derived from the changes in mid prices. This signal could potentially be derived from the magnitude of autocorrelation in the change in prices of a given target’s stock or from the duration of unbroken positive autocorrelation. This analysis does not fall within the core risk and return analysis of this study, as it clearly alters the mechanics of risk arbitrage since the risk arbitrageur is no longer invested during the entire transaction window. To the best of our knowledge, the proposed analysis has not previously been conducted in a risk arbitrage setting. Nevertheless, we find this analysis to be worthwhile, as uncovering the potential for creating an optimal exit timing strategy could have a large impact on the way practitioners trade on risk arbitrage. 28

Chapter 3

Review of Risk Arbitrage Risk and Return Literature

There exists a large amount of literature within the topic of risk arbitrage. The majority of these articles are from the US and written during the years 1987-2013. Due to the scope of our paper, we are specifically interested in papers testing the characteristics of the risk and return profile of risk arbitrage. The remainder of this section is organized as follows: Section 3.1 describes the literature which has examined the abnormal returns of risk arbitrage (alpha), starting with the earliest research and ending with the most updated literature within this field. Section 3.2 lists the most relevant literature that has discussed the correlation between the excess returns of risk arbitrage and the market excess returns (beta), starting with the articles using linear models to test this risk return profile and ending with the articles using non-linear models. Finally, we include a table which aims to summarize all papers mentioned in this section.

3.1 Abnormal Returns of Risk Arbitrage

As discussed in the introduction of this paper, earlier literature has shown that risk arbitrage earns large abnormal returns relative to other investment strategies applied by funds like hedge funds (Ackermann, McEnally, and Ravenscraft, 1999). This has led researchers to examine the ﬁeld in order to clarify where the abnormal returns come from.

3.1.1 Early Research Larcker and Lys, 1987 study a sample consisting of 131 cash and stock transactions. Due to data availability issues, they only include the long position and hence, exclude an important part of the transaction, the short position. In retrospect, we know that this is a major limitation of their study, limiting the comparability of their results. However, based on this methodology they ﬁnd average abnormal returns of 5.3% over an average transaction period of 31 trading days. This generates annual abnormal returns of 51.9%, which is a high return for a strategy considered to be market neutral. 3.1. Abnormal Returns of Risk Arbitrage 29

As we will report, later studies have found even larger annual returns. For instance, a few years later, Dukes, Frolich, and Ma, 1992 study a sample of 761 cash transactions between 1971 and 1985 and find daily returns of 47bp, yielding a 50-day return of 25%. Before commenting on the likelihood of such returns, we should point to the fact that early studies, including this one, often use event time rather than calendar time when calculating annual returns, please see chapter4, for a discussion of event time vis-a-vis calendar time. Such discrepancies in methodology lead to a low degree of comparability between early studies and more current literature. Having commented on that, these returns intuitively seem high, compared to an annual excess market return in 1990 of 4.37% 8. The authors concede that such high returns are difficult to repeat on a continuous basis. The same holds for a study by Karolyi and Shannon, 1999 on the Cana- dian market in 1997, using a sample of 37 cash and stock transactions valued at above $50 million. They find annual abnormal returns of 33.9%. Similar to Dukes, Frolich, and Ma, 1992, they also use event time 9. We include the study despite the limited sample size and the use of event time as, to the best of our knowledge, it is the only study that has been performed on the Canadian market. However, as it is difficult to compare the results to the US studies, we only conclude that equivalent to the US market, risk arbitrage also seems to be a profitable strategy in the Canadian market. Other studies have used different approaches to model the risk and return characteristics of risk arbitrage. Jindra and Walkling, 2004 study 362 US cash transactions during the period from 1981 to 1995, using the Fama and French three-factor model. They report monthly abnormal returns of 2.01%, equivalent to annual abnormal returns of 27%, which seems more realistic. However, as the study only includes cash transactions, we can naturally only compare these with our cash sample.

3.1.2 Later Research One of the most inﬂuential papers in this ﬁeld, due to its sample size and depth of research, is written by Mitchell and Pulvino, 2001. They study 4750 US cash and stock transactions during the period 1963-1998 and report monthly abnormal returns, not accounting for transaction costs, of 0.74% (CAPM) and 0.79% (Fama and French). When accounting for transaction costs, the monthly abnormal returns fall to 0.29% (CAPM) and 0.27% (Fama and French). For CAPM, this translates to annual returns of 9.25% and 3.54% respectively. This shows that transaction costs truly are an important factor limiting the abnormal returns in their study. Furthermore, Mitchell and Pulvino, 2001 seek alternative explanations for the abnormal returns by investigating alternatives to the use of linear

8We do not have data prior to 1990 9This is typical for studies written prior to 2001 30 Chapter 3. Review of Risk Arbitrage Risk and Return Literature asset pricing models. They base these alternatives on two factors. One, by splitting the sample into periods when excess market returns fall below -3% and -5% respectively, they find an increasing beta. Two, the R2 significantly increases when the sample is limited to months where the excess market returns fall below -3% and -5%. These two findings indicate that the relationship between the excess returns of risk arbitrage and the excess market returns is not linear. Specifically, Mitchell and Pulvino, 2001 find annual abnormal returns increasing to 36.1% (CAPM) and 27.72% (Fama and French) in periods where the excess market returns fall below -3%. They test this for two portfolios: One is a value weighted returns portfolio (VWRA), see chapter4 for details, and the other is a hypothetical index portfolio, the risk arbitrage index manager (RAIM). RAIM attempts to replicate reality using two primary restrictions. The first is a position limit of 10% for each investment and the second is a limit on the fund’s investments in illiquid stocks. Finally, they estimate a piecewise linear regression suggesting that in flat and appreciating markets, the annual returns are 6.2%. Based on these results, Mitchell and Pulvino, 2001 find that risk arbitrage returns may be better evaluated using contingent claims analysis. Im- plementing this approach yields slightly different returns with monthly abnormal returns of 33bp and annual abnormal returns of -4%. This leads to the conclusion that the abnormal returns from risk arbitrage in fact, represents a premium earned by the arbitrageur as compensation for their liquidity provision. This is especially important and risky during severely depreciating markets. The large sample size of Mitchell and Pulvino, 2001, especially when compared to the small samples of early research makes this a very credible study. However, it is important to remain critical. For instance it is worthwhile to note that the sample consists of 73% cash transactions. Hence, the specific characteristics of cash transactions drive the results. Furthermore, Mitchell and Pulvino, 2001 exclude some of the most extreme observations without explaining why and which criteria they base these exclusions on. This creates a less realistic sample. Despite this, the study remains the most comprehensive within the field. Baker and Sava¸soglu, 2002 study a sample of 1901 cash and stock transactions during the period 1981-1996. They test the returns of risk arbitrage by creating both an equal weighted as well as a value weighted portfolio, see chapter4 for details. By using the Fama and French three-factor model, they report abnormal monthly returns of 0.8% (10% annually). They deduct direct transaction costs, but conclude that they do not have a big impact for the respective period. This is similar to our results. In fact, this model yields similar returns to the model of Mitchell and Pulvino, 2001. Baker and Sava¸soglu, 2002 argue that the abnormal returns exist due to limited 3.1. Abnormal Returns of Risk Arbitrage 31 capital of arbitrageurs, leaving these liquidity providers (arbitageurs) unable to fully close the arbitrage spread. Note that this could be mitigated by the recent rapid capital inflows to hedge funds, see figure 1.1. Mitchell and Pulvino, 2001 also address the liquidity provision hypothesis, while still devoting a great deal of attention to transaction costs and the hypothesis of non-linearity. We will test for all three sources of abnormal returns (outlined in chapter2), but we will use a different approach to transaction costs. Within the literature of risk arbitrage, Ben Branch has contributed with a lot of material. In a list of articles, he and his colleagues research the risk and returns of risk arbitrage. Among these publications is a study of 1309 cash, stock and collar transactions during the period 1990-2000. Branch and Yang, 2006 focus on the differences in risk and return characteristics of cash vis-a-vis stock transactions. Interestingly, the majority of their sample (65%) consists of stock transactions. They find that successful stock transactions tend to generate higher returns than successful cash transactions and conclude that the differences are caused primarily by information asymmetry at the announcement. This conclusion is similar to our findings. A few years later, Branch and Wang, 2008 publish a paper on 187 stock transactions during the period 1994-2003, focusing on the outcome of the transactions and the effect this has on the arbitrage spread. They show that successful transactions have lower spreads than failed transactions and conclude that the market has rational expectations of the deal duration and the end price. In this analysis, they also document annual abnormal returns for different strategies in the range of 11.88%-22.7% for the contingent claims analysis, 9.25%-11.88% for CAPM and 8.6%-10.27% for the Fama and French-three factor model. These results confirm the existence of abnormal risk arbitrage returns. Further, we note that they find a higher return from both their contingent claims analysis as well as their CAPM and Fama and French model compared to Mitchell and Pulvino, 2001. From looking at existing literature, it is clear that the vast majority of the articles are from the US and thus, not necessarily directly comparable with the European market. However, there are also a few articles on other markets, eg the Australian market. Among these is a paper by Maheswaran and Yeoh, 2005 who use a sample of 193 cash transactions from 1991-2000 to construct a value-weighted as well as an equal-weighted portfolio. The CAPM and Fama and French three-factor model both yield statistically significant monthly abnormal returns between 84bp and 1.20% (annual returns between 10.6% and 15.4%). These numbers exclude transaction costs. Inter- estingly, these results are significant only before the mentioned transaction costs are deducted, possibly due to the very limited sample size. In general, the sample size makes the results of this article questionable. Based on a larger sample than the sample of Maheswaran and Yeoh, 32 Chapter 3. Review of Risk Arbitrage Risk and Return Literature

2005, Hall, Pinnuck, and Thorne, 2013 study 431 transactions from 1985- 2008, investigating the returns of risk arbitrage by taking a closer look at the influence of consideration type and strategy. In order to analyze these two factors separately, they perform analyses on the following strategies: long only, long-short and the vanilla strategy 10. This yields abnormal monthly returns of 1.5% (cash: long-only), 1.4% (stock: long-short) and 1.4% (stock: long-only). All returns are statistically significant. These results are ex-transaction costs, as the results become insignificant when Hall, Pinnuck, and Thorne, 2013 deduct them, due to their limited sample size, leading the authors to rely on the ex-transaction cost results. Interestingly, this is one of the studies devoting the greatest amount of attention to the distinction between cash and stock transactions. Finally, Hall, Pinnuck, and Thorne, 2013 state different reasons for the abnormal returns. First, they consider how practical limitations like transaction costs prevent arbitrageurs from earning abnormal returns and hence, why the arbitrage spread prevails. Second, they explain the abnormal returns with compensation for systematic risk and argue in a manner similar to Mitchell and Pulvino, 2001 that the abnormal returns of risk arbitrage do in fact not linearly depend on excess market returns. Finally, we turn to one of the only European studies on this specific topic within risk arbitrage. Sudarsanam and Nguyen, 2008 examine 1105 cash and stock transactions and report abnormal monthly returns for a practitioner portfolio, restricting each position to a maximum weight of 10%, a value-weighted portfolio and an equal-weighted portfolio. They find abnormal returns between 50bp and 60bp, equivalent to annual returns between 11% and 17%. Finally, they also perform a contingent claims analysis which yields monthly returns of 52bp and annual abnormal returns of 6.4%. All in all, they conclude that risk arbitrage is a profitable strategy in the UK. This conclusion is particularly interesting for our study as it indicates that risk arbitrage has similar abnormal return characteristics in the European market as in the US market.

3.2 Market Risk Exposure of Risk Arbitrage

In addition to the seemingly lucrative abnormal returns of risk arbitrage, the strategy was initially believed by both practitioners and academia to be market neutral. However, as already indicated in the previous paragraphs, literature has started to question the notion of market neutrality during the past decade.

10The vanilla strategy is long only for cash transactions but long target and short acquirer for stock transactions 3.2. Market Risk Exposure of Risk Arbitrage 33

3.2.1 Testing the Beta using Linear Models The market neutrality of risk arbitrage should result in beta values of essentially zero, see section on market neutrality in chapter2 for more details. Bhagat, Brickley, and Loewenstein, 1987 are some of the first to focus on the beta values of the risk arbitrage strategy. They study 295 US cash transactions from the period 1962-1980. Based on this sample, they report beta values, using linear asset pricing models, which are higher pre announcement than post announcement. Specifically, they report a beta of 0.94 pre announcement dropping to 0.33 post announcement, concluding that the risk parameters significantly change during this process. Due to the topic of our study, we are mostly interested in the fact that they report post announcement betas for cash transactions, which are significantly different from zero. Additionally, later studies like Jindra and Walkling, 2004 study 392 cash transactions during the period 1981-1995 and similarly report average betas for their sample of 0.71, using linear asset pricing models. These articles are all based on US samples, but similar analyses have been performed in other markets. Maheswaran and Yeoh, 2005 interestingly conclude that risk arbitrage is a market neutral strategy in Australia. They base this conclusion on insignificant beta values of 0.05 and 0.06 for the equal- and value-weighted portfolio respectively. This result is a little surprising as their study is from 2005 and the majority of later articles seem to conclude that risk arbitrage is, to a smaller or larger extent, not a market neutral strategy. However, as mentioned above, the limited size of their sample causes their conclusions to be less reliable. Around the same time, Branch and Yang, 2006 research the difference in beta of cash versus stock transactions using linear asset pricing models. They report betas of 0.121 and -0.221 for cash and stock transactions respectively, arguing that the difference is due to different strategies. Specifically, the arbitrageur has no hedge (short position) in cash transactions. They suggest that cash transactions have non-linear returns and stock transactions have linear returns. However, none of the betas are statistically significant. Thus, they do not make any final conclusions. The idea behind their theory is highly relevant for our study, but will need further research before it can be confirmed. In practice, it is difficult to imagine that an arbitrageur would only long the target in a stock transaction as this would increase the risk of the arbitrage trade substantially. Hence, it is problematic to compare a cash transaction (long-only) with a stock transaction that also only includes a long position. Instead, Branch and Yang, 2006 suggest that the risk and return characteristics of cash vis-a- vis stock transactions may simply be different. This issue can be illustrated with a simple example. Imagine that there are two different assets, one has low volatility and the other has high volatility. These assets may have the same average abnormal returns, but the asset with the highest volatility implies larger deviations 34 Chapter 3. Review of Risk Arbitrage Risk and Return Literature from the average. Hence, you can lose or gain a lot more with this asset. The less volatile asset never earns as large losses or gains, as the deviations from the average are smaller. Therefore, the average abnormal returns of the two assets are not directly comparable. The same people who studied abnormal returns on the UK market, Su- darsanam and Nguyen, 2008, also investigate the value of the beta and conclude that despite the findings of significantly different beta values during appreciating and depreciating markets, risk arbitrage is a market neutral strategy. First, they perform a traditional linear analysis using CAPM and the Fama and French three-factor model. The results from CAPM show a beta of 0.11, which they consider to be market neutral and the Fama and French three-factor model shows a beta of 0.16, which they again consider to be market neutral. This leads them to conclude that, risk arbitrage is a profitable and market neutral strategy in the UK.

3.2.2 Testing the Beta using Non-linear Models The non-linearity of risk arbitrage is yet to be finally confirmed. Therefore, recent papers have started to model the strategy using non-linear models to test this hypothesis. Mitchell and Pulvino, 2001 include the already mentioned piecewise linear regression, which results in two different betas, one for flat and appreciating markets and one for depreciating markets. This approach is used to test whether CAPM and the Fama and French three- factor model truly do fall short of capturing the actual risk that an arbitrageur holds when invested in a transaction. Specifically, Mitchell and Pulvino, 2001 test the relationship between the excess returns of risk arbitrage and excess market returns by first including different degrees of depreciating markets in their linear asset pricing models and show that the beta increases as the market excess return decreases. Their piecewise linear regression indicates that in conditions where the market is either flat or appreciating the beta is in fact zero. Interestingly, the value of beta increases to 0.5 in periods when the market return is below a certain threshold, -4% 11. This leads Mitchell and Pulvino, 2001 to suggest that the risk and return characteristics of risk arbitrage have strong similarities with uncovered index put options, which also generate positive, but limited returns in most periods and in rare cases lead to substantial losses to the investor. The implication of this risk and return characteristic is that the risk arbitrage strategy should not be modeled linearly and hence, Mitchell and Pul- vino, 2001 seek to use contingent claims analysis to capture the true risk. After performing all these analyses, they conclude that the non-linearity is most severe for cash transactions in severely depreciating markets, and

11This threshold is chosen to minimize the sum of squared residuals and is a purely theoretical threshold 3.2. Market Risk Exposure of Risk Arbitrage 35 that overall, the linear models do in fact mask the true risk of risk arbitrage. Finally, they also conclude that the errors in estimating the abnormal returns from benchmarking to linear models are small. Hence, for samples which include both stock and cash transactions, the non-linear models will not result in abnormal returns that are largely different from the abnormal returns estimated by the linear model. Mitchell and Pulvino, 2001 conclude that cash transactions are non- linear and that especially during severely depreciating markets, risk arbitrageurs could lose money from applying this strategy. It remains unknown to what extent their conclusions are applicable to the European market, but it leaves an interesting platform for us to build our analysis on. In addition to the linear models performed by Sudarsanam and Nguyen, 2008, they also apply a piecewise linear regression, similar to the one applied by Mitchell and Pulvino, 2001. In appreciating markets, they report a beta of essentially zero (0.0788) and in depreciating markets, they report an increased beta value of 0.43. However, recall that they conclude in favor of market neutrality. The reasons for not considering these findings as evidence against market neutrality are the following. First, in their study, the definition of a depreciating market is market excess returns below -11.90%. This is almost triple the threshold used by Mitchell and Pulvino, 2001. Sec- ond, they conclude that non-linearity, if applicable, is mostly true for cash transactions. Finally and most importantly, the article aims to prove that the takeover regulation in UK has a large impact on the risk and return characteristics. Specifically, the Takeover Code in the UK prohibits the acquirer from subjectively deciding to withdraw their offer, e.g. due to depreciating markets. In the end, this leads Sudarsanam and Nguyen, 2008 to conclude that in the UK risk arbitrage is a profitable and market neutral strategy. Finally, non-linear models have also been applied in other markets. Hall, Pinnuck, and Thorne, 2013 study the market neutrality of risk arbitrage in the Australian market and conclude that based on the beta values, the excess returns of risk arbitrage depend non-linearly on the market excess returns. They report betas of 0.47, -0.49 and 0.12 for the long portfolio, the long-short portfolio and the vanilla portfolio respectively. They also examine the beta during appreciating and flat markets as well as depreciating markets. They find the beta to be -1.26 in flat and appreciating markets and 0.63 in depreciating markets, which leads them to the conclusion of non- market neutrality. Concluding, as the years have passed and research has increased both in size and scope, it appears that risk arbitrage is not necessarily a market neutral strategy. However, as most of these studies are US-based, it remains unknown whether the European market displays the same characteristics. Chances are that in the European market, risk arbitrage may also lead to un- foreseen large market-correlated losses in depreciating markets, potentially with a significant difference between cash and stock transactions. 36 Chapter 3. Review of Risk Arbitrage Risk and Return Literature h al rvdsa vriwo h eeatltrtr ihnteseictpco ikadrtr hrceitc frs rirg,during arbitrage, risk of characteristics return and risk of topic specific the within literature 1987-today relevant period the the of overview an provides table The T ae aaol 20)U 9119 91CP:() .8 (monthly), 0.78% (1): CAPM: 1901 1981-1996 US (2002) Savasoglu & Baker are returns (all General: 4750 1963-1998 US (annnu- (2001) 28% Pulvino 0.393 & (pre-announcement), 0.517 Mitchell (monthly), 2.01% 361 1981-1995 US (annual) 33.9% (1991/2003) Walking & Jindra 37 1997 Canada pe- (50-day 25% around 0.47% (1999) Shannon & Karolyi 761 1971-1985 US (from 5.32% (annual), (1992) Frohlich Dukes, 51.9% Sample 131 Period 1977-1983 Country US (1987) Lys & Larcker Study ABLE 3.1: ieaueOeve ae A Panel Overview Literature Size 2:07%(monthly) 0.84% , 0.74% (1): (annual) (2): 7.2% (monthly), French: 0.6% (2): Fama (monthly), (annual) 10.8% 1.16%, (annual) RAIM: 4% 2.32% claim: Contingent RAIM: VWRA: FamaFrench: CAPM: VWRA: 2.98%, FamaFrench: CAPM: are 3.54%, returns (all Downturns monthly): (VWRA). (below-3%): RAIM: 1.01% (RAIM), 0.53% Piecewise: 0.27%, FamaFrench: RAIM: CAPM: 0.29%, 0.79%, VWRA: FamaFrench: 9.25%), = (annually 0.74% VWRA: CAPM: monthly): ally) Description Portfolios (daily) riod) Beta resolution) to date transaction Alpha rnh 1:03,() 0.25 Fama (2): 0.37, 0.22, (1): (2): French: 0.3, (1): CAPM: 0.5074, 0.4041 RAIM: RAIM: FamaFrench: VWRA: VWRA: CAPM: CAPM: 0.5532, French: (be- Fama 0.5194, -3%): Downturns low 0.48 (RAIM), (VWRA). 0.1052, 0.49 RAIM: Piecewise: us- 0.1232, French: spreads, RAIM: VWRA: VWRA: Fama arbitrage CAPM: Examine French: 0.0176, CAPM: Fama 0.054, General: portfolio weighted Value (average) 0.71 (post-announcement) fers of- multiple of portfolio arbitrage risk weighted offers, Value first (2): of portfolio arbitrage risk weighted Value (1): price commis- impact) and taxes transfer costs brokerage sion, transaction (both include manager, Hypothetical index arbitrage risk RAIM: trans cost), (ignore returns av- of erage weighted Value VWRA: risk of potential profit Exmamine portfolio weighted Value a eun ftesrtg.Do costs. transaction strategy. include not the of returns mal abnor- use the assess They to French Fama million. value $10 transaction exceeds own the to and seeks transactions 100% bidder cash the US where only ing of underestimated potential is arbitrage profit risk the that and hold not does efficient hypothesis market the that an- conclude risk. They reduce on to date success nouncement of the investigate probability to offers. attempt tender They cash in arbitrage risk potential profit the Examine prices stock noisy to due returns substantial arbi- earn that trageurs conclude evaluate and to returns fac- (CAPM) single model a use tor They more 5%. than of They purchases include positions. only long in- the only clude but transactions, stock information. and cash of consists costly sample Their acquire to traders for exist incentives if Test n rsueaddcesn in capital arbitrage decreasing risk and pressure in- sell- ing risk, is completion return in creasing that predict arbitrageurs. capi- of They limited number and the the tal explain with and profits and French CAPM Fama stock using and transactions cash of of arbitrage returns risk abnormal Examine index uncovered options put selling to similar are arbitrage risk to that returns show results Their analysis. claims linear contingent and piecewise regression CAPM, French, both use Fama They ar- risk bitrage. of characteristics and return risk the characterize to tions transac- stock and cash Examine in Canada strategy profitable a is arbitrage risk evaluate multi-factor that conclude To a and model use they mio. returns valued $50 deals above include only but use transactions, stock They and cash both Canada. in arbitrage 3.2. Market Risk Exposure of Risk Arbitrage 37 Examine profitability of riskbitrage ar- anderates find significant thatturns abnormal it re- before gen- but transaction when adjusting costs, tion for costs, transac- the abnormalare returns no longeralso significant. conclude that They riskis arbitrage a market neutral strategy Examine the performance of risk arbitrage and find thattrage risk arbi- for successfulaction stock trans- generates largerthan returns successfultions. cashinformation They transac- asymmetry. explain Finally thisthey also with findpends that on the the beta considerationand type de- the market conditions Examine arbitrage spreadsfind and thatacquirer’s they stock correlatetransaction volatility duration. with and they identify Further, ationship non in linear rela- riskteristics of return risk charac- two arbitrage different portfolios using forCAPM, both Fama French and a piecewise linear regression Examine risk arbitrage andthat find it earnsand is abnormal market neutral. returns plain this They with ex- the importanceUK of take-over regulation regarding bidder’s ability tothe renege bid on Examine thecharacteristics risk of and riskand find arbitrage return thatabnormal it returns, generates but large not this market neutral. it They is include do transaction not costs (1):Value weighted Equal weighted, (2): They create 7 portfolios:Risk arbitrage (1): cash, (2):arbitrage Risk stock,hedged risk arbitrage (3): targetcash, - (4): Unhedged risk Un- arbitrage target -hedged stock, risk arbitrage (5): acquirer Un- - cash, (6): Unhedgedbitrage risk ar- acquirer -Active stock, risk (7): arbitrage arbitrage (using fundindices) performance 2 portfolios:get for (1): fixed collar,hedging (2): long Delta tar- Value weighted(VWRA),portfolio equal portfolio tioner’s weighted (EWRA), arbitrage(PA): Impose Prac- realistic portfolio restrictions on weights (1) Vanilla portfolio (longcash for and long-short for stock) (2) all deals - long- (3) all long deals shortlong (4) (5) cash deals cash - long deals short (6) share - deals - longdeals (7) - share long short TransactionCAPM: cost (1):Fama 0.049, French: (2): adjusted: -0.075 -0.054, (1): 0.021, (2): (1): 0.121, (2):(4): -0.221; (3): 0.924, 0.657, (5):(7): 1.009, 0.109, (6): 1.028, (1):(beta Piecewise low), linear:(1): 0.140 (beta 1.008 CAPM:French: high), 0.371, 0.469 (1):linear: , (2): Fama 0.596 (beta(beta Piecewise low), high), -0.268 (2):(2): Fama CAPM: French: 0.034 0.049, CAPM: 0.274(EWRA), (VWRA), 0.147 0.109French: (PA),0.199 Fama (EWRA),Carhart’s 0.348 0.154 four-factor0.231 (VWRA), (VWRA), model: (PA), 0.1880.122 (EWRA), (PA),regression: Piecewisehigh), 0.079 0.43 linear (PA) (beta low) (PA) (beta 4-factor model: (1) 0.12(3) (2) -0.49 0.47 (4)0.74 0.13 (7) -0.13 (5)(1): Piecewise-linear: -0.69 0.5 (6) 0.88 (low), (low), -0.47 -0.13 (high) (high) (3):(low), -0.98 (2): -0.13 (high) (4): 0.21 (low), 0.19 (high) (5): -0.78(high) (low), (6): -0.36 1.13 (low), 0.27(7): (high) -0.11 (low), -0.13 (high) CAPM: (1): 0.758%, (2):Fama 0.491%, French:0.484% (1): 0.824%, (2): 1.4%, (3): 1.4%,1%, (4): (6): 0.1%, 0.9%, (7): (5): 0.6%, - nual 13.5%), (1):(annual CAPM: 0.94% =11.88%),French: 0.82% (annual=10.27%), (1)(2) Piecewise Fama linear:nual 2.29% = 22.7%), (an- (2): CAPM:(annual= 0.74% 9.25%),French: 0.69%, (annual= (2): 8.6%) Fama (VWRA),0.49% 0.64%0.82% (PA), (VWRA), 0.62%0.46% (EWRA), Fama (EWRA), (PA), Carhart’s four-factor model: French: 1.32%(EWRA), (VWRA), 0.59% 0.65% (PA),tingent and claims: con- and Value-weighted equal-weighted:0.6% (CAPM and Fama 0.5%(monthly) French), - and(annual) 11% and (CAPMFrench) 17% and Fama ket return, SMB,(Monthly): (1): HML, 1.64%, (2): UMD) (3): 1.51%, 1.71%, (4): 2.06%,(6): (5): 0.98%, 2.46%, (7):linear: 1.21%, (1): Piecewise 3.10% (alpha low) and 0.81% (alpha high),(alpha (2): low)high), 2.98% and (3): 2.73% 0.85% (alpha low)0.71% (alpha and (alpha(alpha high), low) (4):high), and 1.8% 0.02% (5):and (alpha -0.35 1.29% (alpha high), (alpha(alpha (6): 1.8% low) low)high), and (7): 0.31% 0.72% (alpha low)0.02% (alpha and (alpha high) Alpha Portfolios Description Size Literature Overview Panel B 3.2: ABLE StudyMaheswaran & Yeoh (2005) Australia 1991-2000 193 Transaction Country costBranch and Period Yang (2005) adjusted: Sample US 1990-2000 1309 (monthly) CAPM: (1): 1.5%, (2): Branch and Wang (2008) US 1994-2003 187Sudarsanam and Nguyen (2008) (1): Piecewise linear: UK 1.06% (an- 1987-2007 1105 (monthly) CAPM: 0.87% Hall, Pinnuck and Thorne (2013) Australia 1985-2008 431 4-factor model (Excess mar- T The table provides an overview ofthe the period relevant 1987-today literature within the specific topic of risk and return characteristics of risk arbitrage, during 38

Chapter 4

Methodology of Data and Portfolio Construction

4.1 Data Description

4.1.1 Sample Selection Criteria Our sample includes 2167 transactions from the Western European markets during the time period July 1, 1990 to December 31, 2015. The transactions included in the sample are identiﬁed using the Thomson One Banker SDC Platinum Mergers and Acquisitions Database. For an overview over the sample selection criteria, please see table 4.1.

Description Sample Size 1 Thomson One Banker SDC Platinum- Mergers and Acquisi- 283,421 tions Database for Western Europe 2 The target must be public 29492 3 The acquirer must seek to own 90%-100% of common shares 10762 post transaction 4 The transaction must either be classified as "completed" or 7924 "withdrawn" 5 The acquirer must be seeking to purchase at least 5% of 7550 shares in the transaction 6 The consideration structure must be classified as "cash only" 4161 or "stock only" 7 Datastream codes must be available for the target and, in the 4010 case of stock transactions, for the acquirer 8 Risk free rate must be provided by Kenneth French 3878 9 Financial data must be provided by Datastream and the re- 2167 turns must follow specified criteria

TABLE 4.1: Sample Selection Criteria The table provides an overview of the the sample selection criteria used in Thom- son One Banker SDC Platinum Mergers and Acquisitions Database to select our sample of 2167 transactions. 4.1. Data Description 39

First, the sample comprises an initial sample of 283,421 transactions from 25 Western European markets, see figure 4.1 for a complete list of markets. It is possible to collect data from all European Markets from the Thomson One Banker SDC Platinum Mergers and Acquisitions Database. However, for the following reasons, we choose to restrict our sample to Western Europe. One, very few transactions are recorded from the Eastern European markets. Two, the Fama and French factors used to benchmark the risk arbitrage returns generated by the analysis are constructed based on 16 Western European countries. Three, based on these 16 markets, Fama and French, 2012 argue that it is reasonable to consider Western Europe an integrated market, please see chapter6 for further details. Second, we require that the target must be a public company, as that is a key requirement in risk arbitrage. This requirement reduces the initial sample to 29,492 transactions. Third, we require that the acquirer must seek to own 90%-100% of the common shares after the completion of the transaction, reducing the sample to 10,762 transactions. This requirement ensures that the bidder intends to take control of the target and that the arbitrageur is almost guaranteed to be able to exit the position at the time of convergence. Fourth, we require that the transaction must be classified as "completed" or "withdrawn"12, which reduces the initial sample to 7,924 transactions. Fifth, the acquirer must seek to purchase at least 5% of shares in a transaction in order to identify transactions large enough to attract the attention of arbitrageurs. This further reduces the sample to 7550 transactions. Sixth, the consideration structure must be classified as "cash only" or "stock only", reducing the sample to 4,161 transactions. While some studies only include pure cash deals (Jindra and Walkling, 2004, Maheswaran and Yeoh, 2005, Dukes, Frolich, and Ma, 1992), other studies include both cash and stock deals (Mitchell and Pulvino, 2001, Branch and Yang, 2006).The inclusion of multiple transaction types has two main purposes. One, the multiple transaction types increase the sample size, enabling a study of the times series-characteristics of risk and returns. Mitchell and Pulvino, 2001 argue that employing a risk arbitrage strategy corresponds to writing uncovered index put options. This risk characteristic generates modest returns during most economic environments and large negative returns during severe economic downturns. Thus, it is of critical importance that the sample size is large enough to capture the rare instances of large negative returns. Two, including multiple transaction types enables a more realistic simula- tion of risk arbitrage returns, as investors are not constrained to investing in specific types of transactions. It can be argued that including more transaction types such as collar deals and transactions including a combination of cash, stocks and warrants would increase the real life applicability of the results. However, modeling the complexity of such consideration structures

12Other transaction status categories include "pending", "status unknown" and "rumor" 40 Chapter 4. Methodology of Data and Portfolio Construction lies outside the scope of this study. Seventh, in the case of cash transactions, it is required that Datastream codes are available for the target. In the case of stock transactions, it is required that Datastream codes are available for both the target and the acquirer. This further reduces the sample to 4,010 transactions. Eighth, the risk-free rate and Fama and French factors must be available for the duration of the study, postponing the start date of the study from January 1, 1990 to July 1, 1990 and resulting in a final sample size of 3,878 transactions. Ninth, financial data must be provided by Datastream and the returns must follow specified criteria in order to avoid extreme returns. Mitchell and Pulvino, 2001 simply state that transactions where the terms imply wildly unrealistic returns are excluded. In order to approach the issue of extreme returns from a structured perspective, we require that: i In the case of positive returns, the cumulative returns over the life of a trade must not exceed more than 200% in excess of the arbitrage spread. In the case of negative cumulative returns, no such restriction is imposed due to the nature of non-linear risk arbitrage returns. ii The arbitrage spread must not exceed 100%. According to a study by Jetley and Ji, 2010 examining the arbitrage spread of 2182 transactions in the U.S. during the period 1990-2007, the 95th percentile arbitrage spreads never exceeded 74.3% and averaged 38.1% over the time period.

4.1.2 Sources of Data All stock returns and stock prices for both the targets and the acquirers are obtained from Datastream. All deal specific information (e.g. announcement dates, market value four weeks prior to the transaction, consideration type, transaction status, transaction description etc.) is obtained from the Thomson One Banker SDC Platinum Mergers and Acquisitions Database. A potential limitation to our dataset stems from the way the standard default price variable, P, is constructed in Datastream. In order to ensure that prices are comparable over time, the prices are adjusted for subsequent capital actions, e.g. stock splits. The daily cash return calculations are not affected by the price variable, as the cash returns only depend on the Datas- tream return index, RI, see equation 4.2. However, the stock transactions are potentially affected by the price variable. The stock returns themselves are not affected, as they are based on the return index, but the position sizes implied by the adjusted prices may be unrealistic in some instances. This leads to misalignment between the prices and the hedge ratio, see equation 4.3. The two requirements above are constructed to eliminate these transactions from the study. The price adjustments are expected to be larger for the early time periods, as older data is more likely to be significantly al- tered from the original prices. Fortunately, there is no reason to expect that 4.1. Data Description 41 stocks that have experienced a stock split or other capital actions in the past are significantly different from stocks that did not experience such changes. The Fama and French factors are obtained from the Kenneth French database. The European factors are based on data from Austria, Belgium, Denmark, Finland, France, Germany, Greece, Ireland, Italy, the Nether- lands, Norway, Portugal, Spain Sweden, Switzerland and the United King- dom. The monthly and annual risk-free rate for Europe is also obtained from the Kenneth French database. The monthly risk-free rates are con- verted into daily risk-free rates using the following equation:

1 daily monthly 30 rf = rf + 1 − 1 (4.1) In table 4.2, we present a summary of the 2167 transactions in the sample broken down by announcement year, consideration type, success rate, transaction duration and the size of the target and the aquirer respectively. An average number of 85 transactions are announced per year with the largest number of transactions in the late 1990s and the early 2000s. Please notice that there are fewer transactions announced during the first five years of our sample. It is unclear if the lower number of announce- ments are due to a lower number of transactions during those years or if the cause is a growth in the depth of the database over time. The first year included in the database is 1990. The average proportion of cash transactions equals 92%, which is higher than the 73% proportion in the US market found by Mitchell and Pulvino, 2001 and significantly higher than the 54% proportion in the Australian market as found by Hall, Pinnuck, and Thorne, 2013. Initially the sample contained 79% cash transactions. However, due to missing data or unrealistic implied returns, a significant portion of the stock swap transactions were removed from the sample. There is no clear time trend in the number of cash transactions relative to stock transactions. The average success rate is 87%, which supports the notion that risk arbitrage is mostly profitable, as most transactions reach completion or receive a competing offer. The arbitrageur thereby earns the arbitrage spread. Branch and Wang, 2009 report an average success rate of 89% in a study of 1,313 offers during the period 1995-2005 in the US. The average transaction duration is 101 calendar days and appears to be relatively stable over time. Mitchell and Pulvino, 2001 identify an average transaction duration of 59 trading days for successful transactions and 62 trading days for failed transactions in the American market, while Hall, Pinnuck, and Thorne, 2013 identify an average transaction duration of approximately 111 calendar days (3.7 months) in the Australian market. In our sample, successful transactions have an average duration of 103 days, while failed transactions have an average duration of 88 days. 42 Chapter 4. Methodology of Data and Portfolio Construction

Finally, it is interesting to notice that the average size of the target is $856 million relative to the average size of the acquirer of $10,595 million. That gives a relative target-to-acquirer size of 8% as compared to 13% in the latter part of the sample of Hall, Pinnuck, and Thorne, 2013, 25% in the case of Mitchell and Pulvino, 2001 and 6% in the case of Jetley and Ji, 2010.

Year Number of Number of Average Average Average Tar- Average Ac- Mergers Cash Trans- Success Transac- get Market quirer Market Announced actions as Rate tion Equity Value Equity Value Percent of Duration ($ millions) ($ millions) Total Std. dev. Std. dev. 1990 9 100% 56% 235 186 (394) n,a, n,a, 1991 29 100% 79% 79 83 (80) n,a, n,a, 1992 13 100% 85% 79 73 (70) n,a, n,a, 1993 23 100% 96% 111 280 (562) n,a, n,a, 1994 25 100% 96% 90 142 (156) n,a, n,a, 1995 37 100% 81% 73 327 (685) n,a, n,a, 1996 29 83% 93% 98 2,121 (6,586) n,a, n,a, 1997 56 100% 91% 89 1,067 (3,731) 1,870 (2,158) 1998 100 95% 85% 81 777 (3,776) 8,887 (12,196) 1999 169 96% 92% 93 717 (3,983) 3,188 (7,802) 2000 171 95% 88% 89 299 (610) 5,462 (10,073) 2001 105 92% 94% 103 417 (901) 11,880 (16,209) 2002 73 88% 96% 102 228 (333) 20,583 (65,352) 2003 62 81% 90% 118 1,102 (4,984) 13,010 (27,765) 2004 54 89% 80% 101 566 (2,820) 10,507 (29,129) 2005 104 97% 90% 107 969 (1,855) 6,791 (13,266) 2006 103 97% 76% 120 2,064 (4,467) 15,538 (37,840) 2007 134 98% 83% 112 2,051 (5,791) 8,198 (16,006) 2008 135 93% 86% 91 843 (2,815) 16,904 (27,282) 2009 104 85% 87% 109 403 (1,087) 8,618 (22,556) 2010 104 86% 83% 90 922 (5,181) 8,247 (23,810) 2011 127 91% 90% 104 665 (1,901) 7,857 (13,784) 2012 114 89% 89% 116 925 (3,463) 15,463 (29,257) 2013 84 88% 86% 104 678 (1,572) 4,556 (10,389) 2014 130 86% 88% 118 815 (1,836) 6,982 (12,964) 2015 73 86% 86% 97 955 (1,669) 22,757 (86,531)

Complete 2167 92% 87% 101 856 (3,179) 10,595 (30,938) sample

TABLE 4.2: Sample Summary The table provides an overview of the entire sample, split bt years. It shows the development of the listed categories. The average transaction duration is based on calendar days.

To the best of our knowledge, all of the previously published studies on the topic of risk and return characteristics of risk arbitrage have taken their departure in a single market. Since we are taking our departure in 25 different markets, it is worth to consider the distribution of transactions across countries, see table 4.1 below. The United Kingdom is by far the most active transaction market and accounts for 40.3% of the total sample. This is hardly surprising given the fact that the UK is one of the most active M&A markets on a global scale. 4.2. Portfolio Construction 43

According to Statista, in 2014 and the ﬁrst half of 2015, the UK and Ireland combined constitute the largest M&A market in Europe in terms of both transaction value and transaction volume. The UK and Ireland are followed by the Nordics, Germany, France and Benelux, which corresponds well to the distribution of transactions in our sample.

FIGURE 4.1: Distribution of Transactions in our Sample, by Country, 1990-2015 The ﬁgure illustrates the distribution of the 2167 transactions in our sample across countries.

4.2 Portfolio Construction

One of the most critical questions to be answered by the risk arbitrageur is how to construct the portfolio of risk arbitrage trades. Given the universe of transactions available at any given time, the arbitrageur must decide which transactions to trade on, the number of transactions to trade on and the maximum weight in any one trade (Pedersen, 2015). The following section describes how to calculate the daily return series for each transaction in the sample. Subsequently, four different portfolio construction approaches are outlined, namely (1) the value-weighted average return series (VWRA), (2) the equal-weighted average return series (EWRA), (3) the practitioner arbitrage portfolio return series (PA) and (4) the practitioner arbitrage portfolio post transaction costs return series (PATC). All portfolio construction approaches are based on quantitative and automatic investment criteria. 44 Chapter 4. Methodology of Data and Portfolio Construction

In reality, the risk arbitrageur may invest on a discretionary basis with a more concentrated portfolio based on careful analysis. This type of transaction selection is difﬁcult to simulate without hindsight bias and thus, the following is based on quantitative trade selection. Please see chapter6 for further discussion of discretionary trading.

4.2.1 Daily Return Series Following the standard conventions of the risk arbitrage literature, this paper’s analyses of the risk and return characteristics of risk arbitrage is based on a time series of monthly returns. In order to obtain the most accurate return data for each stock, the return index (RI) from the Datastream database has been used. This index adds the discrete quantity of dividends paid to the price on the ex date of the dividend. The daily returns start at the close of the market on the day after the risk announcement and run up to and including the day of resolution. The resolution day is deﬁned as the day when the target is delisted from the market (successful transactions) or the day after the transaction failure is publicly announced (failed transactions). There are two different approaches to calculating daily returns. One for stock deals and one for cash deals. For cash transactions, calculating returns is simple as the arbitrageur only goes long the target stock. In other words, the transaction is not hedged. Therefore in transactions where the consideration type is cash, daily returns are calculated using the following equation:

T T RIit − RIit−1 Rit = T (4.2) RIit−1 T In equation 4.2, RIit is the daily return index for the target on day t and T RIit−1 is the target’s return index on day t − 1. The subscript i refers to the transaction, t refers to the day of the transaction and T refers to the target. In stock transactions, returns are more complicated to calculate as the arbitrageur’s position consists of both a long position in the target and a short position in the acquirer. In other words, the transaction is hedged. Therefore, the returns from the long position are added to the returns from the short position as well as the interest gained from the short position. This position is assumed to earn the risk-free rate. Further, in order to account for the hedge ratio and, hence, how much the arbitrageur shorts, the hedge ratio is multiplied by both the risk-free rate and the returns from the short position. This implies that the numerator is a percentage return. Finally, the numerator is weighted by the position value of the target from the previous 4.2. Portfolio Construction 45 day. In this paper, the position value is deﬁned as the long position only. In order to calculate daily returns, the following equation is used:

T T A A T RIit −RIit−1 A RIit −RIit−1 A Pit−1 · T − ∆ · Pit−1 · A + ∆ · Pi1 · rf RIit−1 RIit−1 Rit = T (4.3) Pit−1

In this equation T refers to the target, A to the acquirer, ∆ to the hedge A ratio and Pi1 refers the closing price of the acquirer one day after the an- T nouncement of the transaction. Pit−1 is the price of the target on day t − 1 A and Pit−1 is the price of the acquirer on day t − 1. Finally rf refers to the daily risk-free rate. This equation is different from the one used by Mitchell and Pulvino, 2001 for stock transactions:

T T T A A A A Pit + Dit − Pit−1 − ∆(Pit + Dit − Pit−1 − rf Pi1 ) Rit = (4.4) P ositionV aluet−1

as our paper uses Datastream’s return index RI rather than the price P and the dividend D. These equations are used to calculate the daily returns. In order to calculate monthly returns, we employ a couple of different approaches.

4.2.2 Calendar Time Versus Event Time In order to minimize pitfalls in regard to calculating the returns, we use calendar time rather than event time. The two approaches both deﬁne the transaction window as the time after the announcement until the transaction is either completed or withdrawn. The event time approach calculates the daily returns over the transaction window for a given transaction and subsequently annualizes the calculated returns. Finally, the annual returns are averaged across all transactions in the portfolio. The main limitation of this approach is that it implicitly assumes that the returns generated during the event window are sustainable on an annual basis. As a consequence of this extrapolation, the returns of the portfolio will be overstated (Sudarsanam and Nguyen, 2008). The calendar time approach is a commonly used approach and was originally introduced by Jaffe, 1974 and Mandelker, 1974. The methodology involves two steps. In step one, the daily returns for a given transaction are compounded into monthly returns during the active transaction months. Thereafter, the monthly returns are averaged across the portfolio subject to a weighting scheme (e.g. value-weighted or equal-weighted). For further details on the calculation of monthly returns, please see "Value-Weighted Average Return 46 Chapter 4. Methodology of Data and Portfolio Construction

Series (VWRA)" below. Finally, the weighted monthly returns are compounded into annual returns. In step two, the monthly portfolio returns are regressed using time series regressions like CAPM, Fama and French etc. In our case, we further include a piecewise linear regression, but the calendar time concept remains the same. Over time, the calendar time approach has become the preferred methodology of many economists, including Fama, 1998 as well as Mitchell and Stafford, 2000. The key difference between the two approaches is the fact that event time extrapolates directly from daily returns to annual returns, while calendar time averages across active transaction months.

4.2.3 Value-Weighted Average Return Series (VWRA) The value-weighted average return series section closely follows the methodology of Mitchell and Pulvino, 2001. In order to obtain monthly returns, the daily returns for active transaction months are compounded. Active transaction months are defined for every individual transaction as months that include at least one trading day between the announcement date and the resolution date. Whenever a transaction is active for less than a month, Mitchell and Pulvino, 2001 use the partial monthly return. Thus, they assume that the position is invested in a zero-return account for the remainder of the month in question. This leads to the assumption that cash is idle until the end of the month wherein the resolution date falls. As it may not be realistic to assume that arbitrageurs will accept a return of zero like Mitchell and Pul- vino, 2001, we use a different approach, assuming that arbitrageurs earn the risk-free rate between the resolution day and the end of the month. Please see figure 4.2 for a graphical representation of this approach. The return of the portfolio for all active transactions in a given month is obtained by calculating the average of the returns of the active transactions weighted by the the relative value of their market capitalizations. The target market capitalizations are obtained four weeks prior to the announcement date in order to avoid the effects of a possible price run-up in the target’s stock price prior to the transaction announcement. Schwert, 1996 shows that approximately half of the premium paid by the acquirer in a transaction is in the form of a pre-announcement run-up in the price. The premium is defined as the sum of the run-up and the offer mark-up. Furthermore, Schwert, 1996 finds that the cumulative average abnormal returns start increasing 42 trading days prior to the announcement of the transaction. However, the largest increase occurs the last 21 trading days leading up to the announcement, which corresponds with the four calendar weeks used in our calculations. Schwert, 1996 provides three potential 4.2. Portfolio Construction 47 causes for the price run-up, namely insider trading, information leakages and toehold positions by the acquirer13.

FIGURE 4.2: Returns of Active Transactions The ﬁgure illustrates the deﬁnition of active transaction months and the use of partial monthly returns for the months wherein the resolution date falls. Source: Own creation

In further support of using the lagged market capitalizations, Ahern and Sosyura, 2014 find that firms engage in active media management in order to influence the relationship between information and stock prices. The active media management strategy depends on the type of consideration offered. For cash deals, there is an incentive for the target to try to increase its share price between the start of the transaction negotiations and the announcement date. For fixed-exchange ratio stock swaps, there is an incentive for both the target and acquirer to run-up their respective stock prices prior to the announcement date. By using a lagged value of the market capitalization, we seek to avoid any biases stemming from different incentives to increase stock prices. The following equation specifies the calculation of the value-weighted average return series: h i Nj V QM (1 + R ) − 1 X i t=m it Rmonthj = (4.5) PNj i=1 i=1 Vi

13A toehold is a position of less than 5% of the outstanding shares taken by the acquirer in the target company. 48 Chapter 4. Methodology of Data and Portfolio Construction

where i indexes active transactions, j indexes months between July 1990 and December 2015, Nj indexes the number of active transactions in a month and t indexes the number of trading days in a transaction month. The main rationale for weighting the arbitrageur’s investment according to market capitalization is to invest a greater portion of the capital in the larger targets, which are assumed to be more liquid vis-a-vis the smaller targets (Mitchell and Pulvino, 2001). There are three noteworthy caveats to this approach. First, the approach ignores the liquidity or possible illiquidity of the acquirer’s stock. Natu- rally, this is only a concern for stock swap transactions, where the arbitrageur needs to short the acquirer’s stock in order to hedge the position against adverse movements in relative prices between the target stock and the acquirer stock. If the acquirer’s stock is illiquid, it may not be possible for the arbitrageur to short the stock. Second, it is assumed that the arbitrageur invests in all announced transactions without regard for the ﬁxed costs that would be incurred by doing so. Third, it is assumed that no transaction costs are incurred. This assumption is clearly highly unrealistic, as there are always transaction costs associated with assuming the initial position (Mitchell and Pulvino, 2001). In conclusion, while the caveats of the VWRA approach are substantial, the portfolio remains a useful benchmark against the results reported in studies from other markets.

4.2.4 Equal-Weighted Average Return Series (EWRA) The equal-weighted average return series (EWRA) is an alternative to the value-weighted average return series (VWRA). In order to obtain the EWRA, the majority of the calculations are equivalent to the calculations for obtain- ing the VWRA and the same assumptions apply in regards to transaction length and alternative investment. In contrast, when it comes to averaging the returns of the portfolio of all transactions in a given month, the approach is different. For EWRA we calculate the average by weighting the total return by the number of transactions in the given period. This can be expressed in following equation: h i Nj QM (1 + R ) − 1 X t=m it R = (4.6) monthj N i=1 where i indexes active transactions, j indexes months between July 1990 and December 2015, Nj indexes the number of active transactions in a month and t indexes the number of trading days in a transaction month. 4.2. Portfolio Construction 49

The main rationale for not weighting the arbitrageur’s investment according to market capitalization is the bias that this can create as transactions performed by large corporations will tend to drive the returns un- proportionally and most importantly, unrealistically. EWRA solves this issue, but creates a range of other issue. Canina et al., 1998 enlighten one of the caveats to this approach in their paper from 1998 where they ﬁnd that the equal-weighted approach can create large biases when compounding over long periods. In our case, we do compound over longer periods, but due to the inclusion of the VWRA portfolio, it is not a signiﬁcant problem and EWRA still serves a purpose as a benchmark to previous studies. Finally, the EWRA approach shares the three main caveats of the VWRA approach.

4.2.5 Practitioner Arbitrage Portfolio Return Series In order to introduce more realistic assumptions to our risk arbitrage portfolio returns, we create two additional return series. Taking departure in the value-weighted average return series, we impose diversiﬁcation and transaction cost constraints.

Practitioner Arbitrage Portfolio Return Series (PA) Following the methodology of Sudarsanam and Nguyen, 2008, the returns are value-weighted with a maximum position size of 10% for a given transaction:

Nj " ! " M ## X Vi Y Rmonthj = min , 0.1 · (1 + Rit) − 1 (4.7) PNj i=1 i=1 Vi t=m

Limiting the size of each position to 10% of the total portfolio ensures that the risk arbitrageur retains a minimum level of diversiﬁcation across the portfolio and thus, will not suffer catastrophic losses as a consequence of a single transaction failure. Mitchell and Pulvino, 2001 impose the same restriction when constructing their risk arbitrage index manager returns (RAIM). They present the limit as a standard rule of thumb employed by risk arbitrage hedge funds. Moore, Lai, and Oppenheimer, 2006 conduct a survey of 21 arbitrageurs in order to determine how arbitrageurs form their portfolios and minimize risk. They ﬁnd that the two most popular means of controlling for portfolio risk is to (i) limit the size of any given position to a certain percentage of the overall portfolio or (ii) to limit the percentage of the portfolio that is allowed to be lost in a single transaction failure. They report both a mean and a median of a position limit of approximately 10%. During months with few active transactions, the portfolio may not be fully invested. In accordance with Sudarsanam and Nguyen, 2008, we assume that the remaining capital is invested at the risk-free rate. In the event 50 Chapter 4. Methodology of Data and Portfolio Construction that a given month does not have any active transactions, the entire portfolio will be invested in the risk-free asset.

Practitioner Arbitrage Portfolio post Transaction Costs Return Series (PATC) In addition to limiting the position size, Mitchell and Pulvino, 2001 also impose a maximum price impact restriction of 5% on both the target and acquirer’s stock for any given transaction. The market impact calculation is based on Breen, Hodrick, and Korajczyk, 1999, but the choice of a 5% level is not motivated. Moore, Lai, and Oppenheimer, 2006 ﬁnd that real arbitrageurs employ dual constraints of a similar nature in addition to various types of portfolio risk controls. In fact, 95% of respondents use options in their risk arbitrage portfolios and 95% of respondents take reverse positions, betting that a given transaction will fail. In accordance with Mitchell and Pulvino, 2001, none of our four portfolios attempt to predict the outcomes of the transactions and thus, we do not allow for reverse positions. We impose dual constraints by including transaction costs in the practitioner arbitrage portfolio post transaction costs return series (PATC). The transaction costs include both direct costs such as brokerage commissions and indirect costs such as the bid-ask spread. The transaction costs are estimated in basis points in order to be able to subtract them directly from the monthly returns before multiplying the individual returns by their respective portfolio weights. Transaction costs are only subtracted in the month when the initial position is taken.

4.2.6 Transaction Cost Transaction costs are costs that are derived from implementing an investment strategy, in this case risk arbitrage. In combination with funding costs, transaction costs represent the majority of total investment costs. In our paper, we will not consider funding costs, as such costs only apply to leveraged positions and we make the assumption that the arbitrageur does not use leverage. However, the transaction costs should be included (Pedersen, 2015). Transaction costs consist of indirect and direct transaction costs. Direct transaction costs include direct costs that the investor will have to pay, e.g. commission. The indirect costs are more complicated and usually consist of the bid-ask spread and market impact (Pedersen, 2015). First, the bid-ask spread is the discrepancy between the price that buyers are willing to pay (what they bid) and the price investors are willing to sell at (what they ask). An implication of this is that in case an investor buys and sells a share in short succession, the investor will lose the bid-ask spread (Pedersen, 2015). Second, the market impact is the price impact from trading a large amount of shares at once. In case the investor wants to buy a substantial portion 4.2. Portfolio Construction 51 of a company’s shares, the price will be pushed up and this will lead to increased costs for the investor. The market impact is the reason why investors often split their transactions into smaller portions. Our portfolios do not split the transaction into smaller portions, which naturally creates a bias as the market impact would be higher for our portfolios. However, considering that the transaction costs are rough estimates and our portfolios are estimates of risk arbitrage returns, we argue that this does not materi- ally affect the applicability of our results (Pedersen, 2015).

Transaction Cost Estimation Procedure In this paper, transaction costs refer to a combined pool of costs including the bid-ask spread, commissions and other costs. There are multiple ways to estimate transaction costs, but what they all have in common is that they are mere estimates. Mitchell and Pulvino, 2001 apply a method from a working paper by Breen, Hodrick, and Korajczyk, 1999 who estimate a liquidity beta based on 11 parameters and then apply this beta to estimate the cost. First, it is out of the scope of this paper to conduct this type of analysis as neither the required data is readily available nor does it lie within the main purpose of this paper to estimate transaction costs. Second, the liquidity beta approach is from 1999 and thus, it would not be sufficient to merely use the estimates provided by Breen, Hodrick, and Korajczyk, 1999. Alternatively, a more recent paper by French, 2008 estimates the transaction costs based on the perception that transaction costs must equal the total revenue from relevant services of brokerage houses and dealers. By using information from FOCUS reports (Financial and Operational Com- bined Uniform Single reports) that registered securities firms are required to file with the SEC on an annual basis, it is possible to extract the rev- enues stemming from both commissions as well as market making. This procedure naturally has its limitations, but it gives a sufficiently realistic estimate of the size of these costs for the purposes of this study. The estimates used by French, 2008 are based on data from the US. Thus, the numbers are not directly applicable to Europe. However, according to Byrne, 2010, the US is no longer the cheapest place to trade and European countries like France and Sweden, in fact, have lower costs today. Based on this analysis, we apply the same estimates of transaction costs as used by French, 2008. As French, 2008 only estimates the costs until 2006 and it is outside the scope of this paper to conduct an updated in depth analysis within this area, we have forecasted the 2007-2015 transaction costs by applying a negative linear trend, please see figure 4.3. Further, please refer to appendixC for a table of linearly estimated transaction costs by year. 52 Chapter 4. Methodology of Data and Portfolio Construction

FIGURE 4.3: Transaction Costs This ﬁgure plots the average transaction costs against the year as estimated by French, 2008.

Further, when applying these transactions costs to our risk arbitrage returns, we have implicitly assumed constant transaction costs. This assumption implies that the transaction costs do not depend on the position size. Often, this is the case when the majority of transaction costs come from the bid-ask spread and commissions. An assumption of constant transaction costs implies that market impact is not a substantial part of the costs. In fact, this is unlikely to be a realistic assumption for risk arbitrage (Mitchell and Pulvino, 2001). This is especially important to consider when the target is a small-cap company vis-a-vis large-cap or when the position size is large. When the target is relatively small, there is a great chance that market impact is substantial. Second, due to the limited span of the transaction window, arbitrageurs are forced to make large investments at once, despite significant market impact. Other literature within risk arbitrage have faced similar challenges in regard to estimating transaction costs. The majority of these studies have completely excluded indirect transaction costs and instead merely included direct costs. See figure 3.1 and 3.2 for an overview of the literature within this field.

4.2.7 Credit Suisse Risk Arbitrage Returns In order to get a realistic perspective on the returns of risk arbitrage, we also investigate the Credit Suisse Event Driven Risk Arbitrage Hedge Fund Index return series (CSRA). The underlying Hedge Fund Index was the ﬁrst average weighted hedge fund index in the industry and is still considered the leading index, which is why we use Credit Suisse as the source for our 4.2. Portfolio Construction 53 real risk arbitrage returns. Credit Suisse restricts the index to only include funds with a minimum of $50 million assets under management and at least 1 year of data and ﬁnancial reports from the present year as a minimum (CreditSuisse, 2016). The Credit Suisse risk arbitrage hedge fund index compiles returns from hedge funds with minimum 85% of the assets under management (AUM) invested in risk arbitrage transactions. Further, Credit Suisse applies fund weight caps in order to minimize concentration issues and intends to minimize survivor bias by not excluding a fund until it is fully liquidated (Cred- itSuisse, 2016). Despite this effort, one should remain cautious as this does not fully eliminate the survivor bias when the methodology is to only select large funds (usually size and survival will be highly correlated). Additional caveats behind this index include that it is based on global data, not European data, which naturally raises questions about the comparability to our study, but not to an extent that invalidates the use of it. It will still provide a useful benchmark. Finally, it is unclear whether the CSRA Index excludes fees and direct transaction costs, which would create additional upward biases. Conclud- ing, the CSRA Index returns represent the returns of the risk arbitrage strategy in reality and despite its biases, it remains indicative of realized returns and enables us to examine the correlation between our theoretical portfolio returns and these real portfolio returns. 54

Chapter 5

Empirical Analysis and Results

This section seeks to examine the risk and returns of risk arbitrage. We ﬁrst summarize the annual returns and illustrate the value of $100 invested during the period 1990-2015 in our four portfolios: VWRA, EWRA, PA and PATC, including the value of $100 invested in the CSRA Index. Second, we examine the abnormal returns from risk arbitrage by benchmarking the returns using linear asset pricing models, CAPM and Fama and French three-factor model (FF) as well as a piecewise linear regression (equation 2.3 through 2.6). Following this, we test for non-linearity in order to gather support for the use of the piecewise linear regression. Based on the results from these models, we apply a contingent claims analysis to estimate the abnormal returns. Following these analyses of abnormal returns, we include a sensitivity analysis of the PATC diversiﬁcation constraint and an analysis of the effect of market returns on the probability of transaction failure. These two sections seek to investigate the factors affecting our abnormal returns. As a way to test whether our own portfolios (VWRA, EWRA, PA, PATC) are reasonable estimators of risk arbitrage performance, we include a comparison between the returns of our portfolios and the returns of an actual risk arbitrage index (CRSA). Finally, we seek to challenge the mechanics of risk arbitrage by testing for autocorrelation within the change in mid prices during the transaction window in order to examine whether there could exist alternative ways for an arbitrageur to earn abnormal returns from risk arbitrage.

5.1 Annualized time series of monthly returns

Table 5.114 presents the annualized time series of monthly returns for the VWRA, EWRA, PA and PATC portfolios. Further, for comparison purposes, the table comprises the CSRA Index, the risk-free rate of return and the market returns throughout the period. As expected, the the VWRA and EWRA portfolios clearly outperform the PA and PATC portfolios with geometric average annual returns for the

f 14 E(R−R ) The Sharpe ratio is defined as SR = σ(R−Rf ) 5.1. Annualized time series of monthly returns 55 Market Return Risk-free rate of return Index Annual Risk Arbitrage Return Series VWRA EWRA PA PATC CSRA 11,58% 15,66%19,22% 13,67% 13,99% 17,31% 16,86% 51,84% 14,28% -208,62% 6,02% 13,52% 63,16% 9,16% 8,78% -1,09% 7,73% 5,66% 16,00% 8,76% 8,77% -1,82% 7,70% 5,80% 16,00% 2,76% 4,03% 0,65% 7,05% 17,97% Annualized Time Series of Monthly Returns 5.1: ABLE Year1990199119921993199419951996 Total1997 -3,21% Cash1998 48,03% -3,21%1999 15,89% 48,03% Stock2000 44,13% 15,89%2001 Total 40,43% 44,13%2002 41,56% 40,43% Cash2003 -9,14% 41,88% 41,56% 14,42%2004 15,59% 35,04% -9,14% Stock -18,08%2005 14,42% 16,30% 15,59% 45,7% -18,08% 12,21%2006 16,48% 11,51% Total 68,56% 14,98%2007 12,21% 26,79% 5,32% 65,2% 40,76%2008 34,01% 14,98% -3,12% Cash 8,75% 22,11% 67,8%2009 40,76% 178,23% -14,56% -2,91% 1,57% 307,0% 22,49%2010 18,47% Stock 43,08% 2,07% 21,17% 21,56% 10,09% 2,69% 7,6% 32,47%2011 22,49% 58,86% 29,60% -59,1% 21,36% 1,57% 21,56% 12,29% Total 27,03%2012 24,19% -14,29% 39,36% 12,54% 2,69% 29,06% -6,45% 0,44% 9,48% 11,22% -10,3%2013 -14,48% 370,21% 38,91% 11,51% Cash 55,16% 12,54% 22,0% 8,22% 7,06%2014 2,23% 189,1% 9,97% 15,20% 13,78% 26,79% 36,28% 11,51% 14,18% -5,16% 24,95% 2,37%2015 Stock -18,24% -218,41% 11,65% 10,40% 13,78% 10,65% -5,3% 42,02% 10,99% 11,92% 23,64% -17,02% -6,89% 20,78% 1,62% 7,54% -11,05% -0,99% 173,6% 11,08% 1,43% 36,44% -80,15% 10,04% 8,55% 33,67% -21,5%Geometric 29,57% 24,08% 3,97% average 20,78% 20,07% 1,40% 2,30% of 6,95% 189,10% 25,89% 1,87% 7,54% 13,51%annual -6,70% -3,69% 33,84% 1,43% returns -1,85% 12,09% -14,22% 13,83% 17,81% 14,37% 14,09% -3,74% 9,92%Annual -23,9% 2,30% 10,33% 12,23% 11,12% -11,40% -5,29% 21,26% 7,80% -6,44% -40,28% 12,09% standard -6,78% 173,56% de- 14,93% 0,8%viation 5,21% 19,92% 23,71% of -3,90% -2,88% 13,26% 1,11% 9,08% 31,88% 11,12% returns -18,74% 35,58% 23,64% -7,3% 5,59% 13,93%Sharpe 31,30% 4,78% ratio -4,67% 43,53% -14,44% -7,10% 13,26% (annual) 15,49% 8,35% -12,2% 14,67% -8,52% 1,37% 13,23%Return 10,50% 29,46% -3,46% -10,60% Index 0,65% 3,33% -57,24% 0,48 12,29% 20,21% 13,91% 20,63% -4,13% 4,86% -5,22% 6,93% -22,8% -10,05% 5,89% 17,07% 5,23% -41,09% 4,68% 21,2% -46,6% 27,20% -1,45% 14,44% -18,88% 1,13% 8,35% 1,65% -4,29% -1,18% 26,74% -25,60% 5,69% 13,53% 6,93% 8,99% 0,74 -2,35% 30,86% -15,65% 4,31% 6,08% 5,26% 14,66% -10,72% 32,74% -4,9% -38,41% 10,12% 2,42% 5,87% 29,04% -10,16% 11,90% 8,25% 1636 3,83% 5,60% 3,08% 24,6% -11,95% 5,46% -7,94% 1,02% -6,07% 14,23% 1,34% -9,1% 0,21 19,1% -1,75% 15,65% 2,36% 3,90% -13,7% -32,04% -1,43% 18,36% 5,60% 8,77% -1,65% -6,28% 4082 -0,01% 2,90% -3,26% 3,51% 3,22% 2,98% 1,20% 8,15% 1,50% -4,84% 0,82 7,14% 5,87% -0,03% 2,30% 9,84% -19,2% 4,66% 10,1% -12,80% 1,60% 8,03% 42,3% 1141 1,24% -6,28% -5,35% 4,80% -8,07% 4,0% 15,22% 1,10 0,94% 18,6% 18,11% 5,26% -6,54% 28,3% -1,61% -6,4% 11,7% -9,73% 12,02% 23,4% 2819 -0,05% 1,00% -4,91% -14,8% -0,11% 6,93% 1,30% -3,38 13,7% 3,15% 0,10% -45,0% 5309 2,80% 35,6% -2,50% 0,78% -5,42% 1,00% 19,3% 0,12% 0,87% 0,40 4,89% -481 0,06% -9,84% 0,04% 35,5% 0,84% -1,31% 0,02% 0,90 -2,53% 444 6,1% 0,02% 21,1% -13,0% 0,41% -0,25 934 28,5% 0,02% 0,36 -6,5% 81 0,85 -0,6% 407 -0,30 0,90 851 71 0 346 0,26 197 568 T The table depicts theReturns (VWRA), annualized Equal-Weighted time Risk seriesCosts Arbitrage (PATC). Finally of Returns the monthly Credit (EWRA), Suisse risk Practitioner Risk arbitrage Arbitrage Arbitrage index returns (PA) (CSRA). and for Practitioner all Arbitrage four Transaction portfolios. Value-Weighted Risk Arbitrage 56 Chapter 5. Empirical Analysis and Results total sample of 11.58% and 13.99% versus 6.02% and 5.66%. The difference between the PA and the PATC portfolios is by definition caused by the introduction of transaction costs. By looking at these results, we already see that the returns appear to be robust for transaction costs. The difference between the VWRA and EWRA vis-a-vis the PA and the PATC portfolios is caused by more realistic trading assumptions, namely limiting the position size taken in a single transaction to a maximum of 10% of the total portfolio. Although the geometric average returns of the PA and PATC portfolios are lower than the VWRA and EWRA returns, the Sharpe ratios of the respective portfolios, also reported in table 5.1, are very similar across three of the four portfolios: VWRA (0.48), EWRA (0.82), PA (0.40) and PATC (0.36). The average annual market returns of 7.05% are higher than the average PATC and PA portfolio returns, but lower than the average VWRA and EWRA portfolio returns. However, due to the low volatility of the PA and PATC portfolios, their Sharpe ratios exceed the market return Sharpe ratio of 0.26. Interestingly, when comparing the returns and the Sharpe ratios for the cash only portfolios with the returns of the portfolios based on both stock and cash considerations, the cash portfolios appear to display superior performance, both in terms of average annual returns and Sharpe ratios. For example, the geometric average return of the cash only PATC portfolio is 8.76% as compared to 5.66% for the total PATC portfolio and a Shape ratio of 0.85 as compared to 0.36. This difference holds across all four portfolios. The Sharpe ratios of the cash only portfolios of the PA and PATC of 0.90 and 0.85 respectively correspond well with a Sharpe ratio of 0.90 for the CSRA Index. The summarized returns in table 5.1 are displayed graphically in figure 5.1. The figure illustrates the value over time of $100 invested in July 1990 in the four total portfolios and the market as well as the value over time of $100 invested in the CSRA Index in 1994. The effect of including more realistic portfolio construction criteria is obvious by comparing the returns from the PA and PATC portfolios with the returns of the VWRA and EWRA portfolios. Investing $100 in the portfolios at the beginning of the period would have yielded the following dollar returns: VWRA ($1636), EWRA ($2819), PA ($444), PATC ($407) and the market ($568). Please see Appendix A for corresponding figures comparing cash and stock returns across the five portfolios. In order to be consistent with the risk arbitrage literature, we have reported the cumulative indices on an absolute scale, please see appendixB for logarithmic scale. 5.2. Benchmarking Risk Arbitrage Returns against Linear Asset Pricing 57 Models

FIGURE 5.1: Risk Arbitrage Cumulative Returns (1990-2015) This ﬁgure illustrates the development over the 1990-2015 period of the value of $100 invested in each of the four portfolios at the beginning of the time period.

In order to examine the abnormal returns from risk arbitrage, we benchmark the returns using CAPM, the Fama and French three-factor model (FF) as well as a piecewise linear regression (equation 2.3 through 2.6). By examining the results of CAPM, FF and the piecewise linear model, we attempt to either reject or conﬁrm hypothesis 1-4c.

5.2 Benchmarking Risk Arbitrage Returns against Lin- ear Asset Pricing Models

5.2.1 All market conditions In table 5.2 we present the results of CAPM and Fama and French three- factor model (FF) for all market conditions. The total market sample con- stitutes 306 months (194 months for the stock transactions). 58 Chapter 5. Empirical Analysis and Results

All Market Conditions 2 α βMkt βSMB βHML Adj. R Sample size VWRA Total CAPM 0.69* 0.293*** 0.067 306 (.306) (.061) VWRA Total FF 0.691* 0.298*** 0.052 -0.005 0.061 306 (.309) (.063) (.138) (.13) VWRA cash CAPM 0.917*** 0.41*** 0.154 306 (.274) (.055) VWRA cash FF 0.877** 0.415*** 0.215(*) 0.135 0.16 306 (.275) (.056) (.122) (.116) VWRA Stock CAPM 2.529* -0.184 0 176 (1.082) (.199) VWRA Stock FF 2.552* -0.276 -0.741 0.448 0.009 176 (1.082) (.205) (.483) (.402) EWRA Total CAPM 0.814*** 0.282*** 0.115 306 (.222) (.044) EWRA Total FF 0.788*** 0.288*** 0.178(*) 0.084 0.12 306 (.223) (.046) (.099) (.093) EWRA Cash CAPM 0.996*** 0.32*** 0.165 306 (.205) (.04) EWRA Cash FF 0.96*** 0.328*** 0.238** 0.116 0.183 306 (.204) (.042) (.091) (.086) EWRA Stock CAPM 1.247 -0.201 -0.002 176 (1.313) (.243) EWRA Stock FF 1.162 -0.329 -0.588 0.83 0.01 176 (1.312) (.249) (.583) (.489) PA Total CAPM 0.222 0.136*** 0.071 306 (.139) (.028) PA Total FF 0.22 0.139*** 0.047 0.003 0.066 306 (.14) (.029) 0.062 0.059 PA Cash CAPM 0.426*** 0.201*** 0.202 306 (.114) (.023) PA Cash FF 0.398*** 0.203*** 0.125* 0.097 0.222 306 (.114) (.023) (.051) (.048) PA Stock CAPM -0.162 -0.052 -0.001 194 (.333) (.061) PA Stock FF -0.156 -0.074 -0.177 0.106 0 194 (.335) (.063) (.149) (.124) PATC Total CAPM 0.193 0.136 0.071 306 (.139) (.028) PATC Total FF 0.192 0.14*** 0.047 0.001 0.066 306 (.14) (.047) 0.062 0.059 PATC Cash CAPM 0.395*** 0.201*** 0.203 306 (.114) (.023) PATC Cash FF 0.367** 0.203*** 0.125* 0.096 0.223 306 (.113) (.023) (.05) (.048) PATC Stock CAPM -0.172 -0.052 -0.001 194 (.333) (.061) PATC Stock FF -0.17 -0.07 -0.18 0.11 0.00 194 (.335) (.063) (.149) (.124)

TABLE 5.2: Time Series Regressions of Excess Risk Arbitrage Returns on Com- mon Risk Factors – All market conditions This table presents the results for excess risk arbitrage returns, using CAPM and the Fama and French three-factor model, see equation 2.3 and 2.4 ***: Significant at the 0.1% level, **: Significant at the 1% level, *: Significant at the 5% level, (*): Significant at the 10% level. 5.2. Benchmarking Risk Arbitrage Returns against Linear Asset Pricing 59 Models

Abnormal Risk Arbitrage Returns: Comparison of Portfolio Returns Starting with the total PATC, we report an alpha of 19.3bp for the CAPM and 19.2bp for FF (2.3% annually). None of these alphas are significant, but when looking at the cash PATC, we report a significant alpha of 40bp and 37bp for the CAPM and FF respectively (around 5% annually). The stock PATC results are very low, -17bp for both CAPM and FF (around - 2% annually). However, these stock PATC results are not significant and hence, we do not draw any firm conclusions based on them. When considering the VWRA and EWRA results, we report significant positive alphas in the majority of the cases.

This leads us to conﬁrm hypothesis 1, stating that we expect to observe positive abnormal returns in the Western Europe during the period 1990-2015.

There could be many reasons why the stock PATC results are not sig- niﬁcant, but two explanations in particular should be mentioned. First, the stock transactions represent only 8% of the sample as there are no stock transactions in the beginning of the sample period. Second, the prices used to calculate returns for stock transactions are, as already mentioned, adjusted for stock splits. When comparing the stock PATC returns numbers to the EWRA and VWRA results, we discover that the values of the alpha are higher across the portfolios, including some very high alphas for the stock samples of both the EWRA and the VWRA. This is expected and similar to results presented in table 5.1. Further, the PA and PATC portfolios are exposed to a downward bias in the beginning of the sample period as there are few transactions in these years. Hence, the arbitrageur has an unnaturally large portion of the portfolio invested at the risk-free rate. This is contrary to the VWRA portfolio, where the arbitrageur is fully invested and where the largest transactions drive the results dis-proportionally due to their large portfolio weights. A similar story can be told about the EWRA portfolio as there are no upper position limits. Hence, the investor will always be fully invested even when there are very few transactions. In this case, however, the large transaction do not dis-proportionally drive the end returns. Overall, this creates larger returns for the VWRA and EWRA portfolios than for the PA and PATC.

Based on this, we conﬁrm hypothesis 2, stating that we expect to observe that the portfolio construction approach has an impact on risk and return characteristics of the risk arbitrage strategy.

Abnormal Risk Arbitrage Returns: The Effect of Transaction Costs When we turn to the difference between the PA and the PATC portfolios, we report lower abnormal returns for the PATC portfolio. Previous studies 60 Chapter 5. Empirical Analysis and Results have claimed that the biggest reason for the estimated abnormal returns are transaction costs and especially, indirect costs (Mitchell and Pulvino, 2001). As discussed in-depth in the transaction cost paragraph, see chapter2, we have estimated these costs independent of the position size, which results in transaction costs that do not have a large impact on the returns. Hence, due to the method of the transaction cost estimation, the costs function as a practical cost that should be deducted to realize realistic results, but not a parameter that explains the large abnormal returns of risk arbitrage as previous papers have found. Pontiff, 2006 goes into depth with the effect of transaction costs and concludes that in combination with other costs of investing, transaction costs (defined by him to include commissions, market impact and brokerage fees) do play an important role for the ability of arbitrageurs to ensure market efficiency. Most papers within risk arbitrage only investigate the effect of direct transaction costs, as the indirect costs are more difficult to estimate. However, Mitchell and Pulvino, 2001 use the methodology of Breen, Hodrick, and Korajczyk, 1999 to conclude that transaction costs (among other practical issues) play a very significant role in explaining the abnormal returns of risk arbitrage. In fact, market impact often represents a large part of transaction costs. This is especially true if the arbitrageur invests a large portion and/or if the target is a smaller corporation (small-cap). Due to the nature of M&A, the target company is often relatively smaller than the acquiring company. This leads to the conclusion that market impact in reality represents a large share of transaction costs within risk arbitrage (Mitchell and Pulvino, 2001). Further, due to the fact that direct transaction costs have fallen in recent years, the relative importance of market impact has increased (Byrne, 2010). In conclusion, due to our assumption of constant transaction costs, our results do not indicate a similar conclusion to that of Mitchell and Pulvino, 2001. Therefore, in our study transaction costs do not provide any reason as to why risk arbitrage has high abnormal returns.

Based on this, we fail to conﬁrm hypothesis 3, stating that we expect to observe a substantial impact of transaction costs on the magnitude of the abnormal risk arbitrage returns.

However, in reality this conclusion may not hold.

Abnormal Risk Arbitrage Returns: Comparison of Cash vis-a-vis Stock Portfolios After having examined the results of the four portfolios relative to each other, we want to compare the alphas of the stock and the cash sample. First, the PATC reports a higher alpha for its cash sample compared to its stock sample, but the results are not significant, whereas the VWRA results 5.2. Benchmarking Risk Arbitrage Returns against Linear Asset Pricing 61 Models show the opposite results and are significant. Therefore, we start by commenting on the VWRA results. According to Branch and Yang, 2006, successful stock transactions should theoretically generate higher returns than successful cash transactions due to information asymmetry. Branch and Yang, 2006 find that the degree of information asymmetry depends on the choice of consideration type and not the underlying strategy (long-short). Myers and Majluf, 1984 generate the information asymmetry hypothesis stating that managers are more well informed than outsiders. Therefore, the decision of the managers to pay in stock can provide information regarding the value of the firm. In relation to the theory of Branch and Yang, 2006, the decision to pay in stock can also indicate that the acquirer believes its stock to be overvalued. As a result, the acquirer wants to pay in stock. In contrast, if the acquirer stockholders believe the firm is undervalued, they prefer to pay in cash. In addition, Branch and Yang, 2006 also hypothesize regarding the value of the target. If the acquirer lacks information about the true value of the target, the acquirer prefers to pay in stock15. Finally, there is also an intuitive aspect. According to Branch and Yang, 2006, the target stockholders often prefer to receive cash. This can increase the chance of the target accepting the transaction and hence attract arbitrageurs. In the end, this may drive down the profitability of risk arbitrage for cash transactions by attracting more arbitrageurs. However, as the target stockholders are only one side of the transaction, this is not per se a very strong argument 16. For a detailed discussion of the reasons, see discussion in chapter6. In summary, there seems to be theoretical evidence to back up why we observe larger alphas for the VWRA stock transactions vis-a-vis. In the case of the PATC portfolio, we report insignificant results indicating the opposite. Due to these opposing results, it is valuable to mention that in fact, literature still questions whether stock or cash transactions earn the largest abnormal returns. The majority of literature believes that stock transactions are the most profitable, but not all. As an example, Branch and Yang, 2006 examined the difference in 2005 and found monthly abnormal returns of 1.5% for cash transactions (long only) and 1.4% for stock transactions (long-short), despite expecting the stock transactions to generate higher returns. When testing whether these results are due to the difference in strategy or consideration type, they find that even if they just long the target in stock transactions, they still find monthly abnormal returns of 1.4% for stock transactions. Based on this dataset, they conclude that there is no significant

15The underlying theory behind this notion is often referred to as "the contingency pricing effect", which essentially refers to the beneficial characteristics that stocks can have for both parties, thereby, creating a signaling equilibrium (Hansen, 1987). 16The acquirer stockholders often prefer stock transactions as the risk of overpayment is lower 62 Chapter 5. Empirical Analysis and Results difference in the behavior of stock and cash transactions, neither with re- spect to alpha nor beta, despite their initial hypothesis and theory stating differently, see discussion in chapter6. Hall, Pinnuck, and Thorne, 2013 also attempt to clarify whether there is truly a significant difference between the return structure of cash and stock transactions. However, due to their limited sample size, they are also unable to conclude anything significant. Hence, the theoretical evidence for our stock PATC results is limited. In conclusion, as we only get significant stock results for the VWRA portfolio, we conclude based on these results that stock transactions seem to experience larger abnormal returns than cash transactions. We explain this finding with information asymmetry. However, as literature still questions the difference, we will return to this discussion in chapter6.

The Relationship between Market Excess Returns and Risk Arbitrage Ex- cess Returns The beta values present important information about the relationship between risk arbitrage and the overall market performance. Table 5.2 presents betas of 0.14 for the total PATC portfolio, 0.2 for the cash PATC portfolio and finally -0.05 for the stock PATC portfolio. The beta for the total PATC portfolio is lower than the beta we find for both the VWRA (0.29) and the EWRA (0.28) portfolios, which can be attributed to the cap of 10% invested in each transaction, limiting the size of each investment. Hence, the arbitrageur is more invested in the risk-free asset with the PATC portfolio than with the EWRA and VWRA portfolios and, despite the lower beta, the PATC portfolio beta is still significantly different from zero.

This provides additional evidence for hypothesis 2, stating that we expect to observe that the portfolio construction approach has an impact on the risk and return characteristics of the risk arbitrage strategy.

Compared to the beta found by Mitchell and Pulvino, 2001, we find much larger betas for our VWRA portfolio (0.29-0.30) compared to their VWRA portfolio (0.02-0.05). This difference could simply be attributed to the structure of the samples, as there are natural differences between the European market and the US market, the time periods etc. Overall, we find that the beta values are significantly different from zero for our total portfolios, including both cash and stock transactions. These findings provide evidence against the market neutrality characteristic of risk arbitrage. Due to the limited size of our stock sample, our beta values for the stock transactions are not significant. However, despite the insignificance, all portfolios report lower beta values for stock transactions than for cash transactions. This is similar to what literature, e.g. Hall, Pinnuck, and 5.3. Risk Arbitrage in Months of Low Excess Market Returns 63

Thorne, 2013 and Mitchell and Pulvino, 2001 expect. Hall, Pinnuck, and Thorne, 2013 emphasize and investigate the influence of the implied long- short strategy of stock transactions and whether this strategy affect the risk- return profile of risk arbitrage. In contrast, Mitchell and Pulvino, 2001 explain the difference with the fact that cash transactions are more likely to fail in general and that more transactions fail during depreciating markets. For now, we conclude that for the entire sample, and for the cash only portfolio, the beta value is significantly different from zero. However, for the stock sample the beta could be close zero, although the results are insignificant. Similar results are obtained when applying the FF model, where the estimates are slightly higher for the VWRA portfolio (0.3 for the entire sample). We will return to the beta value in our paragraph on non-linearity to further conclude on the difference between stock only and cash only portfolios. For now, we conclude that the evidence against market neutrality appears to be stronger for cash transactions than for stock transactions. Finally, for the FF factors, SMB and HML, only the SMB factor is significant and only in a few of the pure cash portfolios. This finding is not surprising as the target is usually a smaller company than the acquirer. We find the SMB factor to be 0.12 for the PATC portfolio. This is similar to the findings of previous studies like Mitchell and Pulvino, 2001.

5.3 Risk Arbitrage in Months of Low Excess Market Returns

By now, we have realized that risk arbitrage appears not to be a market neutral strategy. In order to examine the topic further, we split the sample into subsamples, only consisting of months with depreciating excess market returns. We do this to motivate the use of a piecewise linear regression, instead of the linear models, as we suspect that the use of a linear benchmarking model may not be appropriate. Speciﬁcally, we limit the sample to months where the market return minus the risk-free rate is below -3%, as these periods should display higher alphas and higher betas if the piecewise linear regression is a more appropriate model for risk arbitrage. 64 Chapter 5. Empirical Analysis and Results

Market Return - Rf < -3% 2 α βMkt βSMB βHML Adj. R Sample size VWRA Total CAPM 1.6 0.44* 0.051 63 (1.612) (.212) VWRA Total FF 1.26 0.415(*) 0.234 0.007 0.028 63 (1.712) (.22) (.305) (.274) VWRA cash CAPM 2.799* 0.655*** 0.173 63 (1.333) (.175) VWRA cash FF 2.062 0.577** 0.347 0.317 0.191 63 (1.384) (.179) (.247) (.222) VWRA Stock CAPM 2.884 -0.093 -0.023 42 (5.305) (.669) VWRA Stock FF 3.168 -0.123 -0.662 0.629 -0.038 42 (5.602) (.699) (.955) (.796) EWRA Total CAPM 1.06 0.332* 0.055 63 (1.174) (.154) EWRA Total FF 0.475 0.276(*) 0.31 0.189 0.062 63 (1.228) (.159) (.219) (.197) EWRA Cash CAPM 1.466 0.382* 0.074 63 (1.189) (.156) EWRA Cash FF 0.96 0.328* 0.238 0.116 0.1825 63 (.204) (.042) (.091) (.086) EWRA Stock CAPM 2.295 -0.016 -0.025 42 (5.159) (.646) EWRA Stock FF 0.686 -0.349 0.327 1.809 0.039 42 (5.254) (.66) (.933) (.85) PA Total CAPM 0.562 0.219* 0.071 63 (.694) (.091) PA Total FF 3.1679 -0.123 -0.662 0.629 -0.038 63 (5.602) (.699) (.955) (.796) PA Cash CAPM 1.288* 0.329 0.27 63 (.512) (.067) PA Cash FF 0.849 0.286*** 0.226* 0.153(*) 0.336 63 (.513) (.066) (.091) (.082) PA Stock CAPM -0.285 -0.06 -0.02 42 (1.05) (.131) PA Stock FF -0.304 -0.064 0.005 0.019 -0.073 42 (1.133) (.142) (.201) (.183) PATC Total CAPM 0.527 0.218* 0.071 63 (.693) (.091) PATC Total FF 0.325 0.208* 0.166 -0.045 0.072 63 (.726) (.094) (.13) (.116) PATC Cash CAPM 1.25* 0.328*** 0.27 63 (.51) (.067) PATC Cash FF 0.816 0.285*** 0.224* 0.150(*) 0.334 63 (.511) (.066) (.091) (.082) PATC Stock CAPM -0.294 -0.06 -0.02 42 (1.05) (.131) PATC Stock FF -0.312 -0.063 0.005 0.018 -0.073 42 (1.133) (.142) (.201) (.183)

TABLE 5.3: Time Series Regressions of Excess Risk Arbitrage Returns on Com- mon Risk Factors - During Periods of Excess Market Returns below -3% This table presents the results for excess risk arbitrage returns, using CAPM and the Fama and French three-factor model, see equation 2.3 and 2.4, when the monthly market return is less than -3% ***: Significant at the 0.1% level, **: Significant at the 1% level, *: Significant at the 5% level, (*): Significant at the 10% level. 5.3. Risk Arbitrage in Months of Low Excess Market Returns 65

The lower market return affects the alpha as expected and the monthly abnormal return for the PATC cash portfolio increases to 1.25% monthly and 16% annually. These findings are significant as reported in table 5.3. On the contrary, the alpha for the total PATC portfolio is not significant and neither is alpha for the stock PATC portfolio. This complicates the conclusion and we instead focus on the significant cash results, which can be compared to the significant cash VWRA portfolio (2.8% monthly). This yields a higher annual return of 39%, which is slightly lower than what Mitchell and Pulvino, 2001 report for their VWRA cash portfolio. By limiting the sample to months of low excess market returns, we naturally get a much smaller sample, which now only includes 63 months. Overall, the alphas have significantly increased, compared to the sample with all months included. Similar effects are reported for the values of the beta. Table 5.3 reports a significant beta for the total PATC portfolio of 0.22, for the cash PATC portfolio a significant beta of 0.33 and for the stock PATC portfolio an insignificant beta of -0.06. These estimates are similar to the FF results. When comparing the PATC results to the VWRA results, we find significant beta values that are higher than the PATC values, ranging between 0.42 and 0.66. This strongly indicates that the relationship between excess risk arbitrage returns and excess market returns is not linear and that risk arbitrage returns are correlated with the excess returns of the market. Further, the R2 residuals have increased and, though it remains relatively low, it still indicates that the market risk from risk arbitrage primarily arises in depreciating markets. When including the FF factors, the estimates naturally decrease slightly due to the additional explanatory variables (SMB and HML). The SMB factor remains significant, increasing to around 0.22. The reasons for its significance remains the same as previously discussed. Finally there are even more extreme depreciating markets where the market excess return falls below -5% and even below -7%. Mitchell and Pulvino, 2001 report the -3% level as well as the -5% level, and for comparison reasons, we do the same. For the -5% case, table 5.4 reports monthly returns of 2.06% for the total PATC portfolio and 8.21% for the total VWRA portfolio. Annualized, these numbers yield 28% for the PATC portfolio and 158% for the VWRA portfolio. Both results confirm the impact that depreciating markets have on the alpha, but also raises the question of whether this is possible in reality. As we will discuss further in chapter6, discretionary trading is a more realistic type of strategy. However, in order to ensure comparability with previous literature, the VWRA portfolio as well as the EWRA portfolio are important strategies and are, therefore, included in our analysis. 66 Chapter 5. Empirical Analysis and Results

Market Return - Rf < -5% 2 α βMkt βSMB βHML Adj. R Sample size VWRA Total CAPM 8.21* 1.01** 0.212 28 (3.689) (.351) VWRA Total FF 7.525 0.947* 0.123 0.105 0.15 28 (4.54) (.423) (.456) (.433) VWRA cash CAPM 6.957 1.005* 0.174 28 (4.085) (.389) VWRA cash FF 5.64 0.874(*) 0.221 0.332 0.124 28 (4.985) (.464) (.501) (.475) VWRA Stock CAPM 17.743* 1.102 0.044 19 (9.01) (.816) VWRA Stock FF 11.183* 0.221 1.646(*) 2.894* 0.223 19 (9.594) (.899) (1.12) (1.194) EWRA Total CAPM 2.646 0.48(*) 0.067 28 (2.942) (.28) EWRA Total FF 1.558 0.379 0.194 0.176 0.005 28 (3.599) (.335) (.361) (.343) EWRA Cash CAPM 2.335 0.467 0.06 28 (2.971) (.283) EWRA Cash FF 1.02 0.347 0.237 0.186 0.003 28 3.626 0.338 0.364 0.346 EWRA Stock CAPM 11.278 0.713 -0.017 19 (9.422) (.854) EWRA Stock FF 8.508 0.23 0.796 2.158 0.035 19 (8.06) (.751) (.809) (.767) PA Total CAPM 2.112(*) 0.36** 0.312 28 (1.038) (.099) PA Total FF 1.261 0.292* 0.167 -0.001 0.32 28 (1.223) (.114) (.123) (.117) PA Cash CAPM 1.331 0.334** 0.227 28 (1.174) (.112) PA Cash FF -0.002 0.219(*) 0.251(*) 0.1 0.266 28 1.355 0.126 0.136 0.13 PA Stock CAPM 2.062(*) 0.139 0.071 19 (.994) (.09) PA Stock FF 1.667 0.077 0.107 0.248 0.139 19 (1.132) (.106) (.132) (.141) PATC Total CAPM 2.059(*) 0.357** 0.31 28 (1.036) (.099) PATC Total FF 1.218 0.29* 0.166 -0.005 28 (1.221) (.114) (.123) (.116) PATC Cash CAPM 1.28 0.331** 0.25 28 (1.171) (.112) PATC Cash FF -0.044 0.217(*) 0.249(*) 0.097 28 (1.353) (.126) (.136) (.129) PATC Stock CAPM 2.051(*) 0.139 0.07 19 (.995) (.09) PATC Stock FF 1.657 0.077 0.107 0.246 0.135 19 (1.133) (.106) (.132) (.141)

TABLE 5.4: Time Series Regressions of Excess Risk Arbitrage Returns on Com- mon Risk Factors - During Periods of Excess Market Returns below -5% This table presents the results for excess risk arbitrage returns, using CAPM and the Fama and French three-factor model, see equation 2.3 and 2.4, when the monthly market return is less than -5% ***: Significant at the 0.1% level, **: Significant at the 1% level, *: Significant at the 5% level, (*): Significant at the 10% level. 5.4. Benchmarking Risk Arbitrage Returns against a Non-Linear Asset 67 Pricing Model

Interestingly for the -5% sample, see table 5.4 , the stock returns are now significant and the cash returns are not. This requires a closer look at the stock returns. For the PATC portfolio, the monthly stock returns for the CAPM are significant and realistic at 2.05%. In comparison, the VWRA monthly returns are also significant, but for the CAPM they are as high as 17.74% and for the FF model as high as 11.18%. This translates into annual returns of 610% and 257%, which are obviously not realistic returns for an arbitrageur to expect from this strategy. In regards to the beta for the -5% sample, we again expect to observe an increase. The total PATC portfolio has a significant beta value of 0.36 which is above the estimate for the -3% market conditions. As table 5.4 reports, the stock beta remains insignificant, leading us to only use the cash portfolio and the total portfolio and compare them to the VWRA portfolio which reports significant betas of 1.01 for both the total and the cash transactions. Concluding, the alphas of the -5% sample are very large for the VWRA and seem unrealistic, while the PATC portfolio appears to be more realistic. The beta values have increased across all the portfolios.

This provides preliminary evidence for hypothesis 4a, stating that we expect to observe a non-linear relationship between the excess risk arbitrage returns and the excess market returns.

The reason to extend the analysis to include months with excess market returns of less than -7% is that the estimated piecewise linear regression, which will be discussed below, often suggests breakpoints that are lower than the -4% threshold specified by Mitchell and Pulvino, 2001. However, as appendixD shows, very few of these results are significant. For the significant results of the alpha, the table reports an increase in the total PA portfolio and an increase in the stock PA as well as an increase in the stock PATC, leaving the stock PATC returns at 2.9% monthly (41% annual). The limited significance stems from the very limited sample size for this extreme time period. Concluding, these results suggest that risk arbitrage returns are not linearly dependent on market returns. Rather, in depreciating markets the returns have a higher correlation with the market returns. To further investigate this, we now analyze the returns according to a piecewise linear regression.

5.4 Benchmarking Risk Arbitrage Returns against a Non- Linear Asset Pricing Model

In this section, we build on the results from previous analyses showing that risk arbitrage is not appropriately modeled using linear asset pricing models. Therefore, we use a piecewise linear model instead to test whether there 68 Chapter 5. Empirical Analysis and Results truly seem to be different beta values for depreciating vis-a-vis flat and appreciating markets. Table 5.5 is based on the CAPM, as there is limited value in including the mostly insignificant SMB and HML factors. Based on the non-linear benchmarking model, in this section, we seek to find further evidence for hypothesis 4a. Furthermore, in order to overcome the issue of where to place the threshold in the piecewise linear regression, we include both the -4% threshold for comparability reasons, but also let the data speak for itself and estimate the threshold based on the data. This allows us to compare the results to earlier research and functions as a robustness test in order to ensure that no conclusion is based on an arbitrary threshold. Initially, we look at the regression where the threshold is forced to equal -4%. For the PATC portfolio, we only get significant alphas for the cash PATC portfolio. Overall, the alpha seems to be approximately 1.1% monthly, which corresponds to 14% annually. This means that in most markets, flat and appreciating, the arbitrageur earns 14% annually. In fact, as the alpha in the piecewise linear regression is alpha market high, we expect this alpha to be significantly different from zero, but it is a less important observation than whether the betas are significant. If alpha market high is insignificant, it means that the abnormal returns could potentially equal zero, which makes it a less attractive strategy. However, if the alpha is significant it does not necessarily mean that the strategy is a good strategy, as alpha market low could still be very unattractive. Therefore, we are mostly interested in the beta values. There are now two slopes, one for depreciating markets and one for flat and appreciating markets. The beta market low is 0.3 and the beta market high is 0.1. The beta market low is higher than the beta market high, which indirectly suggest that there is an increased probability of transaction failure during depreciating markets. The estimates for the VWRA portfolio suggest a similar conclusion, although all the numbers are higher.

Based on the more appropriate piecewise linear model, we provide strong evidence for hypothesis 4a, stating that we expect to observe a non-linear relationship between the excess risk arbitrage returns and the excess market returns.

Further, the PATC cash portfolio has a beta market low of 0.32 and a beta market high of 0.16, which are both strongly signiﬁcant. This stands in contrast to the PATC stock portfolio with a beta market low of -0.03 and a beta market high of -0.06, which are both insigniﬁcant.

This provides evidence for hypothesis 4b, stating that we expect to observe a stronger non-linear relationship between the excess risk arbitrage returns and the excess market returns for cash transactions vis-a-vis stock transactions. 5.4. Benchmarking Risk Arbitrage Returns against a Non-Linear Asset 69 Pricing Model

In fact, this result suggests that while cash transaction appear to be non- linearly exposed to market risk, stock transactions appear to be market neutral.

2 αMktHigh βMktLow βMktHigh Adj. R Sample size VWRA Total 1.882 0.477** 0.223** 0.066 306 (1.821) (.209) (.093) VWRA cash 2.148 0.6(*) 0.338(*) 0.154 306 (1.627) (.186) (.083) VWRA Stock 3.767 -0.001 -0.258 -0.011 194 (6.071) (.66) (.308) EWRA Total 1.352 0.365* 0.25(*) 0.111 306 (1.321) (.151) (.067) EWRA Cash 1.386 0.38* 0.297(*) 0.162 306 (1.22) (.14) (.062) EWRA Stock 3.017 0.061 -0.307 -0.011 194 (7.403) (.805) (.375) PA Total 1.121 0.275*** 0.083** 0.077 306 (.821) (.094) (.042) PA Cash 1.204(*) 0.322*** 0.156*** 0.209 306 (.676) (.077) (.034) PA Stock 0.009 -0.027 -0.062 -0.012 194 (1.87) (.203) (.095) PATC Total 1.083 0.273** 0.084* 0.077 306 (.82) (.094) (.042) PATC Cash 1.163(*) 0.32*** 0.157*** 0.21 306 (.673) (.077) (.034) PATC Stock -0.003 -0.027 -0.062 -0.012 194 (1.871) (.203) (.095)

TABLE 5.5: Piecewise Linear Regressions: Excess Risk Arbitrage Returns Vis-a- Vis Excess Market Returns This table presents the results for a piecewise linear regression, see equation 2.5-2.6 relating excess risk arbitrage returns for our four distinct portfolios. The results are presented for a threshold equal to -4%. βMktLow is the correlation of excess returns during depreciating markets. βMktHigh is the correlation of excess returns during flat and appreciating markets. ***: Significant at the 0.1% level, **: Significant at the 1% level, *: Significant at the 5% level, (*): Significant at the 10% level.

Table 5.6 reports results for the piecewise linear regression where the data determines the breakpoint. We see that for the total PATC portfolio, the estimated breakpoint is -7.44% and for total VWRA it is -8%. However, we also see that the cash PATC breakpoint is -4.9%, which is close to the forced breakpoint used in literature and as already discussed, there are various reasons why we have most conﬁdence in the cash estimates. 70 Chapter 5. Empirical Analysis and Results

For the alpha, there is only one signiﬁcant result, which is for the cash PA portfolio of 0.4% monthly return. However, overall the alpha appears to be higher for this regression compared to the forced regression. More interestingly, the beta values are now 0.4 for beta market low and 0.1 for beta market high, again signiﬁcantly different from zero and again lower than the values reported for both the VWRA and the EWRA portfolio.

2 Estimated αMktHigh βMktLow βMktHigh Adj. R Sample breakpoint size VWRA Total -8.003 5.387 0.786*** 0.221* 0.07 306 (3.611) (4.838) (.406) 0.079* VWRA cash -7.568 4.902 0.844* 0.338(*) 0.158 306 (3.615) (4.087) (.348) (.071) VWRA Stock -9.144 16.383 1.165 -0.356 -0.002 194 (4.425) (16.762) (1.306) (.258) EWRA Total 7.222 2.58 0.479*** 0.246(*) 0.112 306 (6.312) (3.157) (.273) (.059) EWRA Cash -7.237 2.38 0.475*** 0.292(*) 0.1621 306 (7.436) (2.916) (.252) (.054) EWRA Stock -8.597 10.357 0.691 -0.333 -0.009 194 (7.977) (19.141) (1.524) (.321) PA Total -7.444 2.572 0.394* 0.092* 0.079 306 (3.099) (2.065) (.176) (.036) PA Cash 13.094 0.445*** 0.215*** -10.26* 0.231 306 (.213) (.112) (.023) (4.527) PA Stock 3.483 -0.28 -0.091 (.068) -0.01 194 (7.51) (.422) (.091) (.249) PATC Total -7.44 2.514 0.391* 0.092** 0.079 306 (3.134) (2.062) (.175) (.036) PATC Cash -4.905 1.387 0.34*** 0.159*** 0.21 306 (3.537) (1.033) (.101) (.031) PATC Stock -9.517 2.089 0.156 -0.076 -0.01 194 (9.222) (5.528) (.422) (.079)

TABLE 5.6: Piecewise Linear Regressions: Excess Risk Arbitrage Returns Vis-a- Vis Market Returns This table presents the results for a piecewise linear regression, see equation 2.5—2.6, relating excess risk arbitrage returns for our four distinct portfolios. The results are presented for thresholds estimated to minimize the sum of squared residuals. βMktLow is the correlation of excess returns during depreciating markets. βMktHigh is the correlation of excess returns during flat and appreciating markets. ***: Significant at the 0.1% level, **: Significant at the 1% level, *: Significant at the 5% level, (*): Significant at the 10% level. 5.4. Benchmarking Risk Arbitrage Returns against a Non-Linear Asset 71 Pricing Model

B (A) PATC Returns, 1990-2002 ( ) PATC Returns, 2003-2015

FIGURE 5.2: Scatterplot of PATC Returns This ﬁgure shows scatterplots of PATC returns plotted using a forced breakpoint set equal to -4%. The PATC returns are segmented into two time periods, namely 1990-2002 and 2003-2015, see equation 2.5-2.6.

Finally, we report the results over the years, segmented into two subsamples: 1990-2002 and 2003-2015. For graphical illustration, please see ﬁgure 5.2. Table 5.7 presents the results from 1990-2002 where PATC’s alpha market high is 1.78 compared to a much lower PATC alpha of 0.52 during the period 2003-2015.

This provides evidence in support of hypothesis 4c, stating that we expect to observe decreasing abnormal risk arbitrage returns over time.

Similarly, the beta market low was 0.26 during the early period and 0.28 during 2003-2015. Finally the table presents a beta market high of -0.05 in the early period, which then rises to 0.17 in the most recent period. Similar results are found for the EWRA portfolio as well as the VWRA portfolio. These results suggest that over the years, arbitrageurs have experienced decreasing returns from investing in risk arbitrage and that the strategy si- multaneously has become less market neutral. The decrease in the alpha is not surprising, as research has shown that arbitrage spreads have decreased since 2002, naturally leading to lower returns (Jetley and Ji, 2010). Additionally, this change may be caused by the composition of our sample. However, as it can be seen in table 4.1, this is not an obvious reason. 72 Chapter 5. Empirical Analysis and Results

1990-2002 2 αMktHigh βMktLow βMktHigh Adj. R Sample size VWRA 3.409 0.445 0.003 -0.005 150 (3.219) (.412) (.017) EWRA 1.434 0.268 0.221 0.019 150 (2.61) (.334) (.145) PA 1.825 0.261 -0.048 0.007 150 (1.291) (.165) (.072) PATC 1.776 0.259 -0.047 0.007 150 (1.289) (.165) (.072)

2003-2015 2 αMktHigh βMktLow βMktHigh Adj. R Sample size VWRA 0.392 0.477* 0.387*** 0.194 156 (2.098) (.217) (.096) EWRA 1.01 0.403** 0.296*** 0.324 156 (1.204) (.125) (.055) PA 0.546 0.283* 0.170** 0.175 156 (1.104) 0.114 0.051 PATC 0.52 0.282* 0.170*** 0.174 156 (1.103) 0.114 0.05

TABLE 5.7: Piecewise Linear Regressions: Excess Risk Arbitrage Returns Vis-a- Vis Excess Market Returns This table presents the results for a piecewise linear regression, see equation 2.5-2.6 relating excess risk arbitrage returns for our four distinct portfolios split into two time periods. The results are presented for a threshold of -4%. βMktLow is the correlation of excess returns during depreciating markets. βMktHigh is the correlation of excess returns during flat and appreciating markets. ***: Significant at the 0.1% level, **: Significant at the 1% level, *: Significant at the 5% level, (*): Significant at the 10% level.

5.5 Testing for Non-Linearity

The predominant motivation for this paper is that parts of existing literature have found evidence that the relationship between market excess returns and the returns from risk arbitrage is not linear. We have therefore analyzed and tested the alternative, a piecewise linear regression, which has yielded results confirming the hypothesis of non-linearity. However, as the results are blurred by transaction costs, model specifications etc, we want to test whether there is clear evidence for the two beta values being significantly different from each other. 5.5. Testing for Non-Linearity 73

Complete Sample p-value Sample size Iterations PATC total 0.059 306 10 PATC cash 0.003 306 10 PATC stock 1 194 10 CSRA Total 0.128 265 10

1990-2002 (1994-2002 for CSRA) p-value Sample size Iterations PATC Total 0.267 150 10 PATC Cash 0.086 150 10 PATC Stock 0.726 58 10 CSRA Total 0.01 108 10

2003-2015 p-value Sample size Iterations PACT Total 0.022 156 10 PACT Cash 0 156 10 PACT Stock 1 134 10 CSRA Total 0.41 156 10

TABLE 5.8: Davies Test for Non-Linearity This table presents results from the test of non-linearity. It is performed by analyzing the potential change in the slope and test if the estimated two slopes are significantly different from each other: First, the p-value is the probability with which the null hypothesis occurs (stating that the two estimated slopes are the same). Second, the sample size shows how many months the analysis is performed over (there are less months with active stock transactions than with active cash transactions). Finally, the iterations are the number of points where the test should be evaluated. ***: Significant at the 0.1% level, **: Significant at the 1% level, *: Significant at the 5% level, (*): Significant at the 10% level.

Table 5.8 reports the results for the PATC portfolio and the CSRA portfolio for the total sample period, 1990-2002 and finally, 2003-2015. For the PATC portfolio, the p-value, which states whether the beta values are significantly different from each other, is 0.059 for the total sample, 0.003 for the cash transactions and around 1 for the stock transactions. This suggests that cash transactions depend highly on the excess market returns, but that the stock transactions on the other hand are market neutral. In order to take a closer look at these results we split the sample into two groups and find that there has been a development over time in the relationship between market excess returns and risk arbitrage excess returns. For the period 1990-2002, we report a p-value of 0.267 for the total sample, a cash p-value of 0.086 and finally a stock p-value of 0.726. In order to compare these results, we then look at the period 2003-2015. During this period we find the cash p-value 74 Chapter 5. Empirical Analysis and Results to have decreased to around zero and the stock p-value to have increased to around 1.

These ﬁndings provide additional evidence in favor of hypothesis 4b, stating that we expect to observe a stronger non-linear relationship between the excess risk arbitrage returns for cash transactions vis-a-vis stock transactions.

Generally, these results indicate three tendencies. First, the cash transactions do not exhibit a linear relationship between market excess returns and risk arbitrage excess returns. Rather, there seems to be evidence of a kink in such linear graph and we can thereby confirm the evidence in favor of a piecewise linear regression for cash transactions. Second, the stock transactions appear linear and it does not seem appropriate to model these with piecewise linear models. Third, there has been a change in the relationship over time. This development may be caused by a list of factors. One, the fact that we simply have better data for the latest period. Two, the financial crisis which hit in 2008 may have affected these returns and caused both consideration type and strategy type to have a stronger impact. This finding is supported by the theory of Branch and Yang, 2006. Based on US data from 1990-2000, they conclude that stock offers do not exhibit an increasing beta during depreciating markets, but that cash offers do. Hence, there seems to be a strong difference between stock and cash returns, which emphasizes the importance for an arbitrageur of distinguishing between these two transaction types. Finally, for the sake of benchmarking, we also report the results from the Credit Suisse risk arbitrage index. However, as this index is based on the global market, it is not directly comparable to the other results. Table 5.8 reports that for the total sample, the strategy is not linear, but that it is, in fact, linear during the latest period. There can be multiple reasons for this. One, we do not have the ability to segment between stock and cash transactions for this index. Thus, it could simply be the case that there are more stock transactions in the latest period, which may have driven the results toward linearity. However, it could also indicate a new tendency in the returns of risk arbitrage. In order to conclude anything for this index, one would have to go deeper into the data and split the index into stock and cash transactions. Unfortunately, we have not been able to gain access to this data. In conclusion, based on these results, we can confirm that risk arbitrage is not a fully market neutral strategy and we therefore conclude that there is evidence supporting modeling the returns using a piecewise linear model. However, since cash transactions seem to be driving the non-linearity of excess risk arbitrage returns, a piecewise model is not necessarily appropriate for a pure stock sample. 5.6. Contingent Claims Analysis 75

5.6 Contingent Claims Analysis

Based on the current reported results, there appears to be support for the theory of Mitchell and Pulvino, 2001 regarding the similarities between the returns of risk arbitrage and the returns of an uncovered index put option. If this is true, then the linear asset pricing models which we have used up until now are not appropriate. Thus, we want to test hypothesis 5, stating that risk arbitrageurs are compensated for providing liquidity during the active period of the risk arbitrage trade. As the piecewise linear model only provides abnormal returns for depreciating and flat & appreciating markets separately, we cannot use this model to test hypothesis 5. Instead, we apply a contingent claims analysis. The use of this model is in line with Glosten and Jagannathan, 1994, Mitchell and Pulvino, 2001 and Sudarsanam and Nguyen, 2008 who also suggest to use a contingent claims analysis to test this hypothesis. To determine the cost of portfolio replication, we first calculate the strike price ($96.22), then we estimate the risk-free rate, using the sample average (22bp) and finally we estimate the market volatility by taking the standard deviation of the monthly market returns (17.3%) and multiplying by the square root of 12. Based on these parameters we determine the cost of replicating the portfolio accordingly and find this to be $100.91. This result is based on the results from the total PATC portfolio when performing the piecewise forced regression, yielding an alpha market high of 1.08% and a beta market low of 0.273. According to this result, see table 5.9, the arbitrageur saves $0.91 by investing in risk arbitrage compared to replicating the portfolio. Thus, this analysis shows that risk arbitrage is a profitable strategy, yielding 91bp in returns per month and 11.5% per year. Alterna- tively, one could apply the results from the PATC cash portfolio as both the alpha market high of 1.16% and the beta market low of 0.32 are significant, contrary to the total PATC sample, where only the beta value is significant. Using this, we estimate the replication cost to be slightly higher and get returns of 96bp per month and 12% per year, see table 5.9.

This supports hypothesis 5, stating that we expect to observe that risk arbitrageurs are compensated for providing liquidity during the transaction window.

In both cases, the results are higher than the results found in previous literature. Mitchell and Pulvino, 2001 ﬁnd 4% annual returns and Su- darsanam and Nguyen, 2008 ﬁnd 6.4% annual returns. The latter is a recent UK study and one could thus, have imagined the numbers to be more similar. 76 Chapter 5. Empirical Analysis and Results

PATC total PATC cash PATC 1990-2002 PATC 2003-2015 Alpha Market high 1.083 1.163 1.776 0.52 Beta Market low 0.273 0.32 0.259 0.282 Threshold -4.00% -4.00% -4.00% -4.00%

Input for Black Scholes formula Strike price 96.22 96.22 96.36 96.10 Price of underlying 100 100 100 100 Time to maturity 0.083 0.083 0.083 0.083 Market volatility 17.31% 17.31% 15.34% 19.05% Price of put 0.616 0.616 0.484 0.738 Risk free rate 0.220% 0.220% 0.365% 0.096%

Cost of replicating portfolio 100.91 100.96 100.33 100.29

TABLE 5.9: Contingent Claims Analysis Results This table presents the results of conducting a contingent claims analysis for the PATC portfolio, the cash only PATC portfolio, as well as the PATC portfolio segmented into the time periods 1990-2002 and 2003-2015, see equations 2.7-2.10

A final interesting analysis when using the contingent claims approach is to test both the early and the late period, 1990-2002 vis-a-vis 2003-2005. Based on the findings of Jetley and Ji, 2010, we expect the cost of portfolio replication to be lower for 2002-2015. Recall that Jetley and Ji, 2010 reports significantly lower arbitrage spreads in the period after 2002. They explain this shift with the following: Capacity constraints over time, reduced transaction costs and changes in the risk profile of risk arbitrage, leading to lower losses in case of transaction failure. The contingent claims analysis using alpha market high and beta market low from the piecewise linear regression during the period 1990-2002 and 2003-2005, respectively, confirms our expectations, see table 5.9. We report replication costs of $100.33 and $100.29 for 1990-2002 and 2003-2015 respectively. This yields annual returns of 4% and 3.5% respectively, indicating that the abnormal returns from risk arbitrage have decreased slightly. This suggests that the role of the arbitrageur as a liquidity provider may decrease over time, at least the compensation of such liquidity provision has slightly decreased in recent years.

This provides additional evidence for hypothesis 4c, stating that we expect decreasing abnormal risk arbitrage returns over time.

In summary, the results of this contingent claims analysis conﬁrm hypothesis 5, stating that arbitrageurs are compensated for their important role as liquidity providers and hypothesis 4c, stating that the abnormal returns are decreasing over time. 5.7. Sensitivity Analysis of the Diversiﬁcation Constraint 77

5.7 Sensitivity Analysis of the Diversiﬁcation Constraint

When creating the PA and PATC portfolios, the key constraint imposed on the arbitrageur ensures that the portfolio reaches a minimum degree of diversification. Recall that, in our case, the maximum position taken in a single transaction is limited to 10% of the overall portfolio value. While this constraint is supported by the literature (Moore, Lai, and Oppenheimer, 2006, Mitchell and Pulvino, 2001), we conduct a sensitivity analysis in order to test whether or not our results are a product of our assumptions rather than driven by the data. As the PA and PATC portfolios are the only portfolios restricted to ensure diversification, it is naturally nonsensical to consider the VWRA and EWRA portfolios in this analysis. Furthermore, as the PA and PATC portfolios closely resemble each other, we only conduct the sensitivity analysis on the PATC portfolio. Thus, diversification constraints of respectively 5%, 15% and 20% are applied to the PATC portfolio. Table 5.10 presents the annualized time series of monthly returns for the original PATC portfolio as well as portfolios with diversification constraints of 5%, 15% and 20%. Further, for comparison purposes, the table comprises the Credit Suisse Risk Arbitrage Index and the risk-free rate of return throughout the period. Decreasing the limit to 5% has a negative effect on the compound annual rate of return, reducing returns for the total sample from 5.66% to 5.35%. Increasing the limit to 15% and 20% increases annual returns to 5.78% and 5.97% respectively. However, the annualized standard deviation of monthly returns decrease (increase) more than the decrease (increase) in annual returns. Thus, the Sharpe ratios appear to be decreasing in the diversification constraints. The original portfolio has a Sharpe ratio of 0.36, while the sensitivity portfolios have Sharpe ratios of 0.53 (5%), 0.30 (15%) and 0.28 (20%). It is important to notice that some of these results may be driven by a lack of transactions in the early sample. Hence, restricting the diversification constraint will result in heavy investments in the risk-free account in the early years and, as a consequence, both the returns and the standard deviations will decrease. Table 5.11 presents the results of benchmarking risk arbitrage returns of the diversification constraint sensitivity portfolios against the CAPM and Fama and French three-factor model. The results show that both the alphas and the betas are increasing in the diversification constraint across all portfolios. As expected, the betas are generally significant for the total portfolios and for the cash only portfolios, while none of the stock portfolios are significant. 78 Chapter 5. Empirical Analysis and Results *:Sgicn tte01 ee,*:Sgicn tte1 ee,* infiata h %lvl *:Sgicn tte1%level. 10% the at Significant (*): level, 5% the at Significant *: level, 20% 1% of the constraint at Diversification Significant 20%: PATC **: 15%, level, of 0.1% constraint the Diversification at Significant 15%: PATC ***: 5%, of constraint Diversification 5%: PATC T eunIdx47817 7 7 0 1 098 3 278 4 197 346 86 1227 439 82 1089 0.02% 419 0.41% 0.02% 0.02% 102 -3.49% -1.31% 4.89% 0.61% 0.06% -7.71% 571 -9.61% 0.04% 0.12% 0.67% 2.95% 2.80% 377 0.10% 6.51% -3.01% -6.01% 0.78% -8.09% 3.15% -2.28% 12.02% -9.91% 0.26% -0.04% 24.59% 0.00% -10.84% 71 -8.02% 10.76% -7.04% 1.83% 0.72% 17.18% 5.26% 4.80% 4.66% 1.60% 1.66% 6.98% -6.29% -5.03% -2.08% -1.6% 2.98% 1.20% 2.36% 851 -7.63% 21.40% -1.94% 9.84% 8.15% -0.40% -3.26% -3.8% 0.66% 8.77% -0.02% 3.51% -9.41% 3.08% -2.5% 5.46% -0.27% 30.33% -6.73% -6.19% 9.05% 5.25% 0.76% 5.41% 17.50% 407 2.90% 29.00% -11.91% 5.65% 0.26% 11.58% 3.99% 35.32% -6.54% -3.3% 7.33% -2.53% 7.89% -14.17% 3.90% 2.33% 3.83% 1.02% 5.60% -8.34% 13.78% 15.09% 0.58% -9.84% 37.89% 17.70% -5.01% -0.8% 0.84% 5.89% -3.71% 7.89% -0.4% 4.86% 23.54% -8.18% 7.34% 1.65% -3.50% -5.42% 17.21% 5.60% 11.90% -6.8% 1.00% 5.26% -11.15% 5.30% 4.68% 5.69% 8.99% 11.39% 0.87% Index 11.23% 14.67% 22.06% -4.77% Return -4.91% -12.32% -35.05% -29.53% 6.49% 3.50% 34.82% 6.93% 6.07% -3.46% 5.60% (annual) 5.59% 1.30% 2.63% 3.90% ratio 13.23% 14.73% Sharpe 65.16% -6.60% -0.11% 18.11% -34.98% 23.80% -1.26% 37.33% returns 16.98% 0.95% 1.4% 19.51% -1.61% 2.90% of 15.98% 5.21% 1.79% 17.20% viation 12.91% -0.05% 10.7% de- 3.85% -6.44% -6.54% 8.33% standard 30.01% -6.70% -8.42% 8.03% 15.93% -1.51% 19.51% -8.07% 16.67% Annual 17.20% 5.0% 15.22% 8.10% 9.09% 1.79% -16.64% 13.83% 8.12% return -25.35% 8.07% -7.39% 26.01% annual -37.62% -6.28% of 0.29% rate 9.1% 20.54% 16.67% 12.57% 3.1% -24.26% 5.87% 12.35% 5.60% 13.65% 12.76% 2.30% 22.63% 50.26% 21.37% -1.75% Compound -1.89% -0.23% 8.07% 14.23% 0.29% 27.07% 8.38% -10.88% 13.13% 32.85% -1.65% 16.72% -10.16% 11.78% -1.72% -0.48% -15.00% 9.00% 2.36% -12.3% 10.12% -10.72% 32.85% 8.56% 13.51% Stock 2015 14.66% -25.7% 7.54% 16.05% 9.20% 26.15% 29.04% 16.42% -5.3% 13.53% 13.97% 2014 11.55% 11.74% -1.04% 4.31% -4.29% 2.50% 17.72% 13.15% 1.94% 22.3% 30.86% Cash 16.42% 15.41% 2013 13.97% 14.49% -18.88% 14.44% 6.31% -4.13% 2.23% 0.30% 7.6% 2012 15.41% 8.72% 7.7% 1.94% 11.75% 20.63% Total -41.09% 0.87% -14.44% 13.91% 19.37% 2011 6.31% 11.16% 6.27% 27.66% -7.10% 23.71% 12.87% 7.3% 2010 35.58% 1.37% 0.87% Stock 8.34% -11.40% 27.66% 10.16% 2009 8.47% 9.08% 8.35% 9.92% 1.11% 10.16% 2008 7.80% Cash 7.98% 7.66% 10.33% 2.77% 25.89% 8.35% 2007 6.95% 13.51% 7.66% Total 2006 2.77% 8.55% 6.93% 1.96% 11.08% 13.44% 2005 11.92% Stock 6.93% 13.44% 2004 1.96% 13.26% 9.97% Cash 2003 11.12% 13.26% 2002 Total 12.09% 11.12% 2001 2.30% 12.09% 2000 Stock 20.78% 2.30% 1999 1.43% 20.78% Cash 1998 1.43% 1997 Total 1996 1995 1994 1993 1992 1991 1990 Year ABLE 5.10: nulzdTm eiso otl eun estvt fteDvricto Constraint Diversification the of Sensitivity - Returns Monthly of Series Time Annualized .608 -0.30 0.85 0.36 -1.82% 8.76% 5.66% CSRA 20% PATC 15% PATC 5% PATC (original) 10% PATC .7 .0 60%55%53%80%1.0 .2 51%1.7 06%1.6 .3 0.65% 4.03% 17.96% 10.62% 11.97% 15.18% 9.42% 10.70% 8.06% 5.34% 5.54% 16.00% 7.70% 8.77% nulRs rirg eunSre estvt fteDvricto Constraint Diversification the of Sensitivity - Series Return Arbitrage Risk Annual .309 -0.36 0.92 0.53 0.12% 7.07% 5.35% .008 -0.26 0.80 0.30 -1.02% 9.82% 5.78% .807 02 .00 0.90 -0.20 0.76 2.76% 0.28 5.80% -0.79% 10.33% 5.97% Index freturn of rate Risk-free 5.7. Sensitivity Analysis of the Diversification Constraint 79

2 α βMkt βSMB βHML Adj. R Sample size Diversification Constraint = 5% PATC 5% CAPM 0.164(*) 0.102 0.103 306 (0.085) (0.017) PATC 5% FF 0.159(*) 0.105*** 0.055 0.014 0.104 306 (0.083) (0.019) (0.035) (0.040) PATC 5% cash CAPM 0.276*** 0.149*** 0.233 306 (0.077) (0.015) PATC 5% cash FF 0.255*** 0.151*** 0.107** 0.071(*) 0.263 306 (0.074) (0.019) (0.035) (0.037) PATC 5% stock CAPM -0.133 -0.041 0.001 194 (0.209) (0.039) PATC 5% stock FF -0.131 -0.053 -0.092 0.068 0 194 (0.211) (0.040) (0.094) (0.078) Diversification Constraint = 15% PATC 15% CAPM 0.208 0.158*** 0.064 306 (0.170) (0.034) PATC 15% FF 0.214 0.162*** 0.019 -0.023 0.058 306 (0.164) (0.037) (0.071) (0.075) PATC 15% cash CAPM 0.475*** 0.23*** 0.177 306 (0.141) (0.028) PATC 15% cash FF 0.446** 0.231*** 0.123(*) 0.097 0.188 306 (0.137) (0.035) (0.062) (0.067) PATC 15% stock CAPM -0.134 -0.055 -0.002 194 (0.400) (0.074) PATC 15% stock FF -0.114 -0.08 -0.253 0.094 0.001 194 (0.402) (0.076) (0.179) (0.149) Diversification Constraint = 20% PATC 20% CAPM 0.227 0.177*** 0.063 306 (0.190) (0.038) PATC 20% FF 0.237 0.180*** 0 -0.04 0.058 306 (0.192) (0.039) (0.086) (0.081) PATC 20% cash CAPM 0.514** 0.25*** 0.165 306 (0.161) (0.032) PATC 20% cash FF 0.487** 0.252*** 0.124(*) 0.092 0.172 306 (0.155) (0.040) (0.072) (0.077) PATC 20% stock CAPM -0.007 0.058 -0.003 194 (0.473) (0.087) PATC 20% stock FF 0.027 -0.086 -0.33 0.082 0.001 194 (0.475) (0.090) (0.212) (0.177)

TABLE 5.11: Linear Diversification Constraint Sensitivity Analysis This table presents the results of time series regressions of risk arbitrage returns for the original PATC portfolio as well as portfolios where the diversification constraint has been fixed at 5%, 15% and 20%, see equation 2.3-2.4. ***: Significant at the 0.1% level, **: Significant at the 1% level, *: Significant at the 5% level, (*): Significant at the 10% level.

As mentioned above, the insignificance of the stock portfolios can be attributed to multiple factors, such as a smaller sample size, an adjusted price variable that may retain some unrealistic properties despite controlling for extreme values and, most likely, market neutrality of stock transactions due to relative pricing of the target and the acquirer. Further, please notice the 80 Chapter 5. Empirical Analysis and Results extremely low adjusted R2 values for the stock only portfolios, indicating extremely low explanatory power of the independent variables. This further supports the notion that stock transactions are likely market neutral. As expected from the results presented above, the HML factor is not significant and the SMB factor is weakly significant for the cash only portfolios. Table 5.12 presents the results of benchmarking risk arbitrage returns of the diversification constraint sensitivity portfolios against the piecewise linear model. As expected, the results are generally strongly significant for all cash only portfolios, significant for the overall portfolios and insignificant for the stock only portfolios. When we set the threshold equal to -4%, none of the alphas are significant. When the thresholds are estimated to minimize the sum of squared residuals, the alphas become strongly significant for the cash only portfolios. However, the estimated thresholds are not necessarily relevant in terms of our analysis and make comparisons across results difficult. In terms of magnitude, the beta market low is higher than the beta market high for all overall portfolios and all cash only portfolios with a threshold equal to -4%. This provides evidence that the non-linear relationship between risk arbitrage returns and excess market returns holds across multiple diversification constraints. The relationship does not hold for any of the stock only portfolios across all sensitivity levels. Thus, the results strengthen the evidence for market neutral stock only portfolio returns. Fi- nally, we report increasing beta values, as we invest a decreasing amount in the risk-free account. This is also fully as expected. Concluding, the sensitivity test shows robust results and both the alphas as well as the values of beta behave as expected. 5.7. Sensitivity Analysis of the Diversification Constraint 81

2 Threshold αMktHigh βMtkLow βMktHigh Adj. R Sample size Threshold = -4% PATC 5% 0.825 0.204*** 0.063* 0.114 306 (0.503) (0.058) (0.026) PATC 5% cash 0.728 0.219*** 0.123 0.237 306 (0.459) (0.053) 0.023*** PATC 5% stock 0.028 -0.017 -0.033 -0.01 194 (1.178) (0.128) (0.059) PATC 15% 1.191 0.31** 0.101* -0.013 306 (1.101) (0.116) (0.051) PATC 15% cash 1.438(*) 0.379*** 0.174*** -0.015 306 (0.837) (0.096) (0.043) PATC 15% stock 0.006 -0.0346 -0.064 -0.008 194 (2.241) (0.244) (0.114) PATC 20% 1.279 0.339** 0.115* -0.013 306 (1.129) (0.129) (0.057) PATC 20% cash 1.573(*) 0.414*** 0.188*** -0.014 306 (0.951) (0.109) (0.048) PATC 20% stock 0.181 -0.031 -0.039 -0.009 194 (2.657) (0.289) (0.319)

Estimated Threshold PATC 5% -5.197 1.091 0.227** 0.066** 0.115 306 (3.120) (0.845) (0.080) (0.023) PATC 5% cash 12.706 0.292*** 0.160*** -3.655 0.255 306 (0.363) (0.077) (0.016) (2.007) PATC 5% stock 3.326 -0.236 -0.074 0.058 0.006 194 (5.630) (0.269) (0.058) (0.153) PATC 15% -7.51 2.951 0.458* 0.108* 0.071 306 (3.257) (2.533) (0.216) (0.044) PATC 15% cash 13.104 0.500*** 0.248*** *-13.96 0.212 306 (0.191) (0.139) (0.028) (5.589) PATC 15% stock -9.17 2.554 0.198 -0.087 -0.01 194 (8.725) (6.209) (0.484) (0.096) PATC 20% -7.623 3.347 0.515* 0.121* 0.07 306 (3.390) (3.002) (0.252) (0.049) PATC 20% cash 13.124 0.544*** 0.272*** **-17.63 0.207 306 (0.166) (0.157) (0.032) (6.320) PATC 20% stock -8.879 3.126 0.242 -0.1 0 194 (8.898) (7.360) (0.574) (0.114)

TABLE 5.12: Non-Linear Diversification Constraint Sensitivity Analysis This table presents the results for piecewise linear regressions of risk arbitrage returns on market returns. The analysis is applied to the original PATC portfolio with a diversification constraint of 10% as well as PATC portfolios with diversification constraints of 5%, 15% and 20% respectively. The results are presented for a threshold equal to -4% as well as for the estimated threshold that minimizes the sum of squared residuals, see quation 2.5-2.6. ***: Significant at the 0.1% level, **: Significant at the 1% level, *: Significant at the 5% level, (*): Significant at the 10% level. 82 Chapter 5. Empirical Analysis and Results

5.8 Effect of Market Returns on the Probability of Trans- action Failure

Table 5.13 shows the results for an estimation of the effect of market returns on the probability of transaction failure. While Mitchell and Pulvino, 2001 find all variables except RMkt−2 and P remium to be significant, we only find significance for the intercept and for Hostile. While the model lacks significance, we still find it worthwhile to comment on the sign and the magnitude of the results vis-a-vis the results of Mitchell and Pulvino, 2001.

Independent Variable Coefﬁcient Estimate Marginal Effect RMkt 0.86226 0.1967 (1.47275) RMkt−1 -0.61975 -0.1414 (1.41173) RMkt−2 -0.44339 -0.1012 (1.37228) LBO 0.11980 0.0273 (0.59533) Cash -0.27527 -0.0628 (0.17038) Premium 0.22046 0.0503 (0.28952) Size 0.01076 0.0025 (0.03502) Hostile 1.59150*** 0.3631 (0.22955) Intercept -1.05419*** -0.2405 (0.26810)

McFadden’s Pseudo R2 0.792 Number of observations 1753

TABLE 5.13: Effect of Market Returns on the Probability of Transaction Failure This table presents the results from a probit model (see equation 2.12) exploring the effect of market returns on the probability of transaction failure. ***: Significant at the 0.1% level, **: Significant at the 1% level, *: Significant at the 5% level, (*): Significant at the 10% level.

As expected, we ﬁnd RMkt−1 and RMkt−2 to have negative signs, suggesting that the probability of transaction failure is a decreasing function of market returns in the two months prior to resolution. Mitchell and Pulvino, 2001 ﬁnds marginal effects of -0.4593 and -0.1662 for RMkt−1 and RMkt−2 respectively. That is a stronger marginal effect than the -0.1414 and -0.1012 5.9. Hedge Fund Returns 83

reported in table 5.13. Surprisingly, we find RMkt to have a positive sign, suggesting that the probability that a transaction will fail is an increasing function of market returns in the month of resolution. This does not seem plausible. However, since this variable is simply the month in which the date of the resolution falls, the returns will reflect the period both before and after resolution, which potentially introduces noise to the model. Confirming the results of Mitchell and Pulvino, 2001, we find that hostile transactions have a greater probability of failure than friendly transactions, with a strongly significant positive result of 0.3631 as compared to 0.1286. Further, although insignificant, we also find that leveraged buyouts have larger probability of failure. However, the effect is small at 0.0273. Contrary to our expectations, the sign of the cash dummy variable is positive, indicating that cash transactions have a lower probability of failure than stock transactions. This is a quite surprising result. Recall that cash transactions quickly seem overpriced, when the value of the overall market decreases, as similar targets can suddenly be purchased at lower prices. However, the effect is small (-0.0628) and insignificant. Mitchell and Pulvino, 2001 find a more expected result of 0.0465, which is significant at the 5% level. To conclude, our probit model generally lacks significance, which limits our interpretation of the results. In general, however, our results correspond well with those of Mitchell and Pulvino, 2001.

5.9 Hedge Fund Returns

In order to examine the validity of our portfolios as benchmarks for the actual returns generated by active risk arbitrage arbitrageurs, we examine the risk and return characteristics of the Credit Suisse Event Driven Risk Arbitrage Hedge Fund Index (CSRA) return series and, further, investigate the correlation structure between the risk arbitrage return series generated by our portfolios and the (CSRA) returns. For details on the construction of the CSRA, please see chapter4.

5.9.1 Characteristics of Risks and Returns In order to compare the return series, ﬁgure 5.3 plots the CSRA monthly returns against the market returns. Please see ﬁgure 5.2 for a plot of the PATC monthly returns against the market returns. Although the CSRA index is a global index, both the CSRA and the PATC portfolio return series are plotted against the European market returns for easy comparability. 84 Chapter 5. Empirical Analysis and Results

(A) CSRA Index with an estimated (B) CSRA Index with a forced thresh- threshold = -8.6% old = -4%

FIGURE 5.3: Scatterplot of CSRA Index Returns This ﬁgure shows scatterplots of CSRA Index returns plotted using an estimated threshold of -8.6% and a forced threshold of -4% respectively, see equation 2.5-2.6

For further analysis, we run both our linear CAPM and FF regressions as well as our piecewise regression on the CSRA return series. The results from the linear analysis are reported in table 5.14. For robustness purposes, we also report the regression results for market conditions where the excess market return is less than -3%, -5% and -7% respectively. As expected, the market beta is significant for all portfolios and increasing in decreasing market conditions. The effect of the SMB factor is significant in all market conditions and in markets with excess market returns of less than -3%. The HML factor is insignificant for all portfolios, which corresponds with the findings of our own portfolios. It is worth to note that the sample size decreases significantly with the depreciating market restrictions, as very few months display negative market returns of that magnitude. Thus, it is not surprising that some of the more extreme tests lack statistical power. 5.9. Hedge Fund Returns 85

2 α βMkt βSMB βHML Adj. R Sample size All Market Conditions CSRA CAPM 0.189** 0.132*** 0.344 264 (.132) (.015) CSRA FF 0.175** 0.138*** 0.09*** 0.018 0.372 264 (.059) (.015) (.025) (.025) Market Return - Rf < -3% CSRA CAPM 0.482 0.174*** 0.233 55 (.317) (.042) CSRA FF 0.167 0.145** 0.153** 0.063 0.312 55 (.279) (.047) (.051) (.039) Market Return - Rf < -5% CSRA CAPM 1.837(*) 0.287** 0.318 23 (.92) (.086) CSRA FF 1.385 0.246* 0.073 0.056 0.266 23 (1.169) (.106) (.112) (.103) Market Return - Rf < -7% CSRA CAPM 1.726 0.278* 0.218 19 (1.312) (.114) CSRA FF 0.663 0.190(*) 0.155(*) 0.044 0.18 19 (.996) (.097) (.084) (.073)

TABLE 5.14: Time Series Regressions of Excess Risk Arbitrage Returns for the CSRA Index, using CAPM and the Fama and French three-factor model This table presents the results for two regressions of excess risk arbitrage returns for the CSRA Index on common risk factors. The results are shown across all market conditions as well as for conditions, where the excess return on the market is lower than -3%, -5% and -7% respectively, see equations 2.3-2.4. ***: Significant at the 0.1% level, **: Significant at the 1% level, *: Significant at the 5% level, (*): Significant at the 10% level.

Table 5.15 presents the results from the piecewise regression. For robustness purposes, we report results holding the threshold of excess market returns fixed at -4% as well as the results that minimize the sum of squared residuals. In line with other results, the estimated threshold is more extreme than the fixed threshold as estimated by Mitchell and Pul- vino, 2001. Regardless of the chosen threshold, both piecewise regressions result in larger beta values in depreciating markets than in flat and appreciating markets. The difference between the two market betas is larger for the estimated threshold. This is a natural result as the estimated threshold is more extreme than the fixed threshold and following the definition of a depreciating market, is more severe. The piecewise regression of the PATC portfolio on excess market returns with a fixed threshold of -4% yields a beta of 0.273 in depreciating markets and a beta of 0.084 in flat and appreciating markets. The difference is larger than for the CSRA Index with market betas of .174 and 0.116. It is important to notice, however, that the CSRA Index comprises a much wider range of 86 Chapter 5. Empirical Analysis and Results transaction types than the PATC portfolio (e.g. collar transactions). We have found the non-linear relationship to be the strongest in the cash only portfolio and evidence that the stock only portfolio is likely to be market neutral. Our distribution of 92% cash transactions, thus, biases the overall results towards non-linear results. We find it unlikely, that the CSRA has a cash weight of equal magnitude. As a result, we expect the effect to be weaker in the CSRA Index.

2 Threshold αMktHigh βMtkLow βMktHigh Adj. R Sample size CSRA -4 0.461 0.174*** 0.116 0.345 264 (.325) (.037) (.017) CSRA -8.602 1.509 0.264*** 0.115*** 0.354 264 (2.53) (.927) (.075) (.014)

TABLE 5.15: Piecewise Linear Regressions: Excess Risk Arbitrage Returns for the CSRA Index Vis-a-Vis Excess Market Returns This table presents the results for a piecewise linear regression, see equation 2.5-2.6 relating excess risk arbitrage returns for the CSRA Index. The results are presented for a threshold equal to -4% and the estimated threshold that minimizes the sum of squared residuals, see equation 2.5-2.6. ***: Significant at the 0.1% level, **: Significant at the 1% level, *: Significant at the 5% level, (*): Significant at the 10% level.

5.9.2 Correlation between CSRA and portfolio returns Following the approach of Mitchell and Pulvino, 2001, we examine the correlation structure between the CSRA monthly returns and the four return series generated by the VWRA, EWRA, PA and PATC portfolios. The purpose of this analysis is to determine the similarities and differences between the theoretical returns and actual hedge fund returns. As mentioned above, a caveat to the comparability is the fact that the CSRA is a global index, while the returns generated by this study are purely European. An additional caveat is the fact that the composition of the index in terms of consideration structures is unknown. Table 5.16 shows the correlation between the CSRA and the four theoretical portfolios under different market conditions and for different durations (monthly versus annual). The table shows that both the monthly and annual returns are positively correlated with the returns of the CSRA index across all market conditions. 5.9. Hedge Fund Returns 87

(A) Correlations using Monthly Returns, Complete Sample CSRA PATC PA EWRA VWRA CSRA 1.00 PATC 0.28 1.00 PA 0.28 1.00 1.00 EWRA 0.27 0.51 0.51 1.00 VWRA 0.18 0.76 0.77 0.41 1.00

(B) Correlations using Monthly Returns. Depreciating Markets CSRA PATC PA EWRA VWRA CSRA 1.00 PATC 0.36 1.00 PA 0.36 1.00 1.00 EWRA 0.26 0.71 0.71 1.00 VWRA 0.23 0.81 0.81 0.47 1.00

(C) Correlations using Monthly Returns. Flat and Appreciating Markets CSRA PATC PA EWRA VWRA CSRA 1.00 PATC 0.20 1.00 PA 0.20 1.00 1.00 EWRA 0.20 0.45 0.45 1.00 VWRA 0.11 0.75 0.75 0.37 1.00

(D) Correlations Using Annual Returns. Complete Sample CSRA PATC PA EWRA VWRA CSRA 1.00 PATC 0.62 1.00 PA 0.63 1.00 1.00 EWRA 0.53 0.47 0.47 1.00 VWRA 0.63 0.83 0.83 0.63 1.00

(E) Correlations Using Annual Returns. Depreciating Markets CSRA PATC PA EWRA VWRA CSRA 1.00 PATC 0.93 1.00 PA 0.93 1.00 1.00 EWRA 0.28 0.28 0.29 1.00 VWRA 0.93 0.96 0.96 0.32 1.00

(F) Correlations Using Annual Returns. Flat and Appreciating Markets CSRA PATC PA EWRA VWRA CSRA 1.00 PATC 0.20 1.00 PA 0.20 1.00 1.00 EWRA 0.60 0.47 0.47 1.00 VWRA 0.34 0.78 0.78 0.66 1.00

TABLE 5.16: Correlation Between CSRA and Portfolio Returns, 1994-2015 88 Chapter 5. Empirical Analysis and Results

Panel 5.16a through 5.16c report the results for the correlation of monthly returns across all five portfolios. Results are reported for all market conditions, depreciating markets and flat and appreciating markets respectively. The threshold between the two market conditions is whether excess market returns are greater than or less than -4% for a given month. The correlation is strongest in depreciating markets with a correlation between the PATC portfolio and the CSRA index of 0.36. This result is to be expected from the hypothesis of non-linearity. However, while Mitchell and Pulvino, 2001 find a correlation close to zero in appreciating markets, we still find a correlation of 0.20 between the CSRA Index and the PATC portfolio. Panel 5.16d through 5.16f report the results for the correlation of annual returns across all five portfolios. The data is segmented into depreciating markets and flat and appreciating markets. In order to ensure an adequate sample size for both market conditions, the annual return threshold is set equal to 0%. This segmentation follows the methodology of Mitchell and Pulvino, 2001. Like Mitchell and Pulvino, 2001, we find stronger correlations over the longer time frame. However, unlike the Mitchell and Pul- vino, 2001, the increase in correlations seems to be almost purely driven by the depreciating market data. The correlation between the PATC portfolio and the CSRA index is 0.93 in depreciating markets and 0.20 in flat and appreciating markets. To conclude, the results of the correlation analysis suggest that the PATC portfolio provides a useful benchmark for evaluating hedge fund returns. This conclusion holds for both monthly and annual time horizons with the caveat that correlation is stronger for depreciating markets and for the annual time horizon.

5.10 Autocorrelation

This section seeks to examine hypothesis 6, stating that we expect to find positive autocorrelation in the time series of changes in midprices during the transaction window. Table 5.17 reports the average correlation coefficient across the 1987 transactions included in the autocorrelation analysis. The table further reports the number of transactions with positive correlation coefficients and the number of transactions with a positive correlation coefficient that is significant at the 1% level. Please recall that we are testing the following null hypothesis:

H0: The autocorrelation coefﬁcients of the changes in midprices equal zero at various lags

If the null hypothesis holds at the 1% signiﬁcance level, we expect 1% of the sample to consist of type I errors. As a result, we expect 1987 · 0.01 = 5.10. Autocorrelation 89

19.87 transactions to show autocorrelation coefficients that are significantly different from zero. Note that these type I errors may display positive or negative autocorrelation. Table 5.17 shows that for the first lag, ρ1, 1502 out of 1987 transactions (76%) have a positive and significant autocorrelation coefficient. For the following five lags, ρ2−6, the number of transactions with significant positive autocorrelation coefficients fall within the range 1503-1511. The magnitude of the coefficients is decreasing in the lags and range from 0.578 for ρ1 to 0.528 for ρ6.

ρ1 ρ2 ρ3 ρ4 ρ5 ρ6 Average correlation coefficient of 0.578 0.581 0.568 0.555 0.539 0.528 all transactions Number of positive and signifi- 1502 1509 1511 1506 1503 1507 cant correlation coefficients Number of positive correlation 1649 1670 1692 1696 1684 1678 coefficients

TABLE 5.17: Autocorrelation Results This table presents the autocorrelation coefficients for the first six lags, represented by ρ1−6. It also includes the sum of all significant and positive autocorrelation coefficients across all transactions for the first 6 lags. And finally it includes the sum of all positive autocorrelaion coefficients for the first six lags. See methodology section for details. ***: Significant at the 0.1% level, **: Significant at the 1% level, *: Significant at the 5% level, (*): Significant at the 10% level.

Thus, there seems to be strong evidence that the changes in mid prices over the past 6 days are good predictors of the changes in the mid price tomorrow. As the autocorrelation coefﬁcients are positive and signiﬁcant for approximately 76% of the transactions in the sample, there is a very good chance that a positive (negative) change in mid price today will result in a positive (negative) change in mid price tomorrow.

This leads us to conﬁrm hypothesis 6, stating that we expect to observe positive autocorrelation in the time series of changes in mid prices during the transaction window.

A positive (negative) change in the mid price of the target’s stock results in a narrowing (widening) of the arbitrage spread, assuming that the arbitrage spread is initially positive. Recall that the arbitrage spread largely can be viewed as compensation for completion risk. Thus, a widening spread may be a signal of an increased probability of transaction failure. Based on the identiﬁed autocorrelation in the time series of changes in mid prices and the fact that there is a relationship between the size of the arbitrage spread and the probability of transaction failure, it may be possible for future researchers or practitioners to derive a strategy for the optimal time of exit. 90 Chapter 5. Empirical Analysis and Results

Thus, the next step is to develop an investment strategy and backtest it on a large sample of transactions. While the results generated by our autocorrelation analysis are very convincing, it is important to remember that there are pitfalls when rely- ing on backtesting. First, the analysis above is based on historical data, which may not be representative of the future. In fact, it seems plausible that once an effect has been identiﬁed, it may start to disappear as investors trade on the knowledge. Second, our sample has been constructed using rigorous criteria, please see chapter4 for details. Thus, our results may not be representative of all types of risk arbitrage transactions for all periods of time. 91

Chapter 6

Discussion

6.1 Implication of Results for Practitioners

The following section is organized according to our seven main hypotheses and their respective conclusions. For each hypothesis, we first explain the conclusion and second explain its implications for practitioners. First, we find evidence that risk arbitrage earns positive abnormal returns in the Western European markets during the period 1990-2015. This has two main implications. First, practitioners should generally expect positive abnormal returns from such a strategy and therefore, should continue to practice risk arbitrage. Second, the conclusion implies that practitioners who previously only have invested in other markets, e.g. the US market, can expect the strategy to be transferable to the Western European market and expect positive abnormal returns this market. Second, we find evidence for the hypothesis stating that in the Western European markets, the portfolio construction approach has an impact on the risk and return characteristics of the risk arbitrage strategy. This implies that the portfolio choice of the practitioner is important. Further, it suggest that practitioners should expect the returns from discretionary trading and different quantitative trading strategies to be different. Hence, an in-depth analysis of returns from different portfolios will be of great value for a practitioner. Third, due to our assumption of constant transaction costs, we fail to confirm our hypothesis stating that transaction costs have a substantial impact on the magnitude of the abnormal risk arbitrage returns in Western Europe. However, we have some reservations against this conclusion, as we know from literature that transaction costs, especially market impact, can play a significant role in other markets, e.g. the US, see discussion of transaction costs below. If we do believe that transaction costs are constant, this conclusion has two main implications for the practitioner. First, it implies that practitioners could buy all desired shares at once, thereby saving time at a crucial state of the transaction. This act would otherwise substantially increase the market impact. Second, if transaction costs are insignificant, the practitioners should engage in a large amount of transactions in order to diversify the risk arbitrage portfolio, eliminating idiosyncratic risk. 92 Chapter 6. Discussion

This is the natural theoretical implication of insignificant transaction costs, which in practice is not fully possible due to other limitations like capital constraints etc. However, an increased level of diversification is possible and recommended. Fourth, we find evidence supporting our hypothesis stating that there exists a non-linear relationship between the excess risk arbitrage returns and the excess market returns in Western Europe. We also confirm that this finding is driven by the cash transactions as we observe stronger non- linearity for these. This finding has the following implications. First, given the fact that the practitioner is already invested in risk arbitrage, the composition of transaction types within the given portfolio should be outlined. This allows the practitioner to re-assess the risk profile of the risk arbitrage portfolio, leading the practitioner to make the necessary hedges and diversification investments. Second, practitioners not already active in risk arbitrage, should invest more in stock transactions. In severely depreciating markets, practitioners should allocate a relatively larger share of the portfolio to stock transactions, despite the more complex nature of such transactions. In case there are externalities preventing this, inability to short etc, the practitioners should hedge the cash transactions "actively" at the portfolio level by performing counter-investments, e.g. short the market. Further, we find evidence in favor of the hypothesis stating that there has been a decrease in abnormal returns over time in the Western European markets. This naturally has implications for both funds as well as individual investors. First, funds active in risk arbitrage and with the risk arbitrage strategy as their predominant source of return should be aware of this and find alternatives. One way to obtain larger spreads is to invest in more complex transactions, including hostile transactions, transactions receiving competing bids, combined cash and stock transactions, etc (Ferreira and Bousarsar, 2014). Second, individual arbitrageurs should re-evaluate why they are active in investments requiring a lot of knowledge and experience. This is especially true considering the need for complexity in order to reap large returns. It may not be profitable for individual arbitrageurs to engage in such complex transactions. Common for both practitioner types is the fact that they should re-evaluate the future of risk arbitrage returns, considering the decreasing return trend. Fifth, we find evidence in support of the hypothesis stating that we expect risk arbitrage practitioners in Western Europe to be compensated for providing liquidity during the active period of risk arbitrage. But we also find that this role has been decreasing in importance in recent years. This leads to the following implications for practitioners. First, most intuitively, this implies that practitioners should look for alternative sources of returns. Alternatively, they could reconsider the mechanics of risk arbitrage. In relation to autocorrelation, we look at the potential to identify an optimal exit 6.2. Impact of Consideration Type on Risk and Return 93 strategy for risk arbitrage and thereby changing the mechanics of risk arbitrage. Such new strategies could provide opportunities for practitioners to adopt the change in their role. Finally, this could be an argument for practitioners to switch from their role of uninformed investors to informed investors with the ability to choose the best transactions for their portfolio and the ability to strategically seek to influence the outcome of the transaction, see discussion paragraph below on discretionary trading for more details. All in all, this implies that the role of small individual investors could diminish over time. Finally, we confirm the sixth hypothesis on the existence of positive autocorrelation in the time series of changes in mid prices in Western Europe. This implies that practitioners may be able to find an optimal exit time strategy and thereby shield themselves from large losses in case of a failed transaction. In conclusion, all these findings have implications for practitioners in Western Europe. Some of these are simple to put into practice and others will require more effort.

6.2 Impact of Consideration Type on Risk and Return

This section seeks to clarify the relationship between cash vis-a-vis stock transactions, their respective betas and alphas as well as the factors leading to these results. The remainder of the section is organized as follows. We start by summarizing the relevant results, then we comment on these results and finally, we discuss the factors determining these results. First, the findings from the VWRA portfolio (CAPM and FF) indicate that most of the time stock transactions have larger abnormal returns than cash transactions. Our PATC results suggest the opposite. However, as they are not significant, we consider the VWRA results to be more reliable. This conclusion is similar to Mitchell and Pulvino, 2001, who also conclude that abnormal returns for stock transactions are larger vis-a-vis cash transactions. Additionally, we conclude that stock transactions have lower betas than cash transactions. Despite the existence of theoretical evidence confirming our findings, we still draw attention to the counter-intuitive negative relationship between risk and return which is implied by the conclusion. Naturally, other risk factors than market risk may explain why we report larger returns for our stock transactions despite the seemingly lower market risk. One factor to note is the practical limitations inherent in hedging the stock transaction. Although, literature does not devote much attention to this limitation, it remains an important factor. In reality, short-selling is not always straightforward. It can be both complicated and costly to short a stock and it is not safe to assume that the arbitrageur can always find a buyer. This inherent risk in short-selling can affect the returns and may be 94 Chapter 6. Discussion part of the explanation why we report larger stock returns despite lower beta values. A second factor could be the increased complexity of stock transactions. Rappaport and Sirower, 1999 discuss the differences between mergers and acquisitions paid for in stock vis-a-vis cash. They state that stock considerations imply less clear roles of the parties, explaining that in a stock transaction, the price variable is just one of many variables on which the outcome depends. In comparison, in cash transactions the price truly is the most cen- tral factor. This may lead to increased risk as the path towards resolution becomes more complex. For instance in stock transactions, it makes a large difference whether the acquirer issues a fixed number of shares or a fixed value of shares. First, in a transaction where the number of shares are fixed, the proportional ownership is also fixed, but the value of the transaction will fluctuate with the stock price. This stands in contrast to a transaction where the value of shares is fixed, but the number of the shares fluctuate with the stock price. Both these structures can add risk (uncertainty) and complexity to the transaction. Despite the overall advantageous risk- and value-sharing nature of stock transactions, the stock transactions imply a larger number of unknown factors, which the arbitrageur cannot control, leading to increased risk (Malmendier, Opp, and Saidi, 2016). Concluding, due to the nature of stock transactions, they carry a larger number of risk factors. Of these risk factors, first, the potential short-selling issues are important, and second, the overall larger amount of variables influencing the outcome of the transaction are important. In combination, they substantially increase the risk of the transaction, potentially explaining why we see larger abnormal returns for stock transactions than for cash, despite their lower beta.

6.3 Impact of Discretionary Portfolio Selection

As hypothesis 2 concerns the effect of portfolio construction, we have em- phasized the importance of carefully considering the portfolio construction approach throughout this study. Further, we have devoted attention to the impact of the chosen approach on the estimated returns. This section aims to provide a different perspective on portfolio selection by considering the impact of discretionary portfolio selection on the performance of the risk arbitrage strategy. Until now, we have implicitly assumed that arbitrageurs exhibit the same investment behavior as the average uninformed investor in the market. Forming the portfolios described in chapter4 neither requires costly private information nor the ability of the investor to inﬂuence the outcome and/or terms of a proposed transaction. Moreover, since the four estimated portfolios rely on the assumption that the arbitrageur invests in all available 6.3. Impact of Discretionary Portfolio Selection 95 transactions, the transactions included in the portfolios become a function of the criteria used to collect the sample dataset. In reality, arbitrageurs are not subject to such strict investment criteria. They are able to choose the transactions they invest in on a discretionary basis and their position sizes are not limited by a diversiﬁcation constraint or a weighting scheme. The key question is whether such discretionary investing outperforms more quantitative portfolio construction strategies that do not utilize private information. To the best of our knowledge, the published empirical and theoretical literature on the topic is limited and the results are varied.

6.3.1 Passive versus Active Arbitrageurs In order to investigate if discretionary portfolio selection outperforms the quantitative portfolio construction approaches we have used in the study so far, we discuss the following. First, the impact of passive investors with the ability to pick superior transactions is discussed based on Larcker and Lys, 1987. Then two types of active investors are discussed based on theoretical models by Cornelli and Li, 2002 and Gomes, 2011. The first type of active investor tries to influence the outcome of the transaction through taking large positions in the target and tendering shares to the acquirer. The second type of investor tries to influence, the offer premium, by delaying the outcome of the transaction. Finally, Hsieh and Walkling, 2005 find evidence in favor of both the passive and the active approach. Notice that the definitions of passive versus active investors are not necessarily mutually exclusive. Larcker and Lys, 1987 focus on the passive investor. In a study of 131 cash transactions during the period 1977-1983 in the US, they find that risk arbitrageurs have the ability to acquire superior knowledge about the outcomes of proposed transactions after the announcement of the transaction vis-a-vis publicly available information. This implies that the presence of risk arbitrageurs signals that a transaction is more likely to succeed than the probability of success implied by market prices. Following the methodology of Cornelli and Li, 2002, Sudarsanam and Nguyen, 2009 argue that holding a large, temporary ownership stake translates directly to influence over the outcome of the proposed transaction. This is the case because arbitrageurs are more likely to tender their shares to the acquirer than the existing, small shareholders. Thus, although the passive arbitrageurs of Larcker and Lys, 1987 do not enter into the risk arbitrage position with an expectation of influencing the outcome of the proposed transaction, the large positions taken to offset costly information acquisition may result in influence over the probability of transaction success. Moving on to the active investor, Cornelli and Li, 2002 model the ability of the active arbitrageur to influence the outcome of a risk arbitrage transaction. They argue that the risk arbitrageur does not necessarily need superior 96 Chapter 6. Discussion information before entering the transaction. In fact, the informational advantage of the active arbitrageur is derived endogenously from the decision to enter into the risk arbitrage trade. The information advantage consists of knowing that the arbitrageur has entered the position. Assuming that ownership of the target is dispersed at the date of the announcement, once a significant amount of shares are held by arbitrageurs, the arbitrageurs have the ability to facilitate the success of the transaction through tendering a fraction of their shares to the acquirer. Further, please notice that it is of critical importance in this model that the arbitrageurs are able to keep their position partially hidden, as this ensures that the price of the target stock will not fully reflect the increased probability of success. Thus, the arbitrageur is able to generate positive abnormal returns. Additionally, competition between arbitrageurs has a negative impact on the risk arbitrage returns, but does not drive the returns all the way to zero. In conclusion, the model by Cornelli and Li, 2002 suggests a positive relationship between the presence of arbitrageurs and the size of the offer premium as well as the probability of transaction success. While Cornelli and Li, 2002 focus on the ability of the active arbitrageur to influence the probability of transaction success, Gomes, 2011 models the ability of the active arbitrageur to influence the terms of the proposed transaction through the ability to take large positions in the target. According to Gomes, 2011, more than 90% of transaction offers in the US and in the UK are any-or-all offers followed by freeze-outs, where the acquirer gains 100% ownership of the target. Exploiting the fact that the acquirer needs to reach a critical amount of shares in order to trigger the freeze-out mechanism, arbitrageurs with temporarily large ownership stakes can delay the takeover process until the acquirer offers satisfactorily high offer premiums. Gomes, 2011 concludes that the threat inherent in this potential behavior of arbitrageurs forces the acquirer to increase the offer premium ex-ante. Hsieh and Walkling, 2005 provide insights to the discussion of whether discretionary arbitrageurs are passive or active. In a study of 608 transactions during the period 1992-1999 in the US, Hsieh and Walkling, 2005 empirically test the implications suggested by the models of Cornelli and Li, 2002 and Gomes, 2011. Hsieh and Walkling, 2005 find that the change in arbitrage holdings is greater for successful transactions, which is consistent with the argument of passive arbitrageurs. However, consistent with the active arbitrageur argument, they also find that the change in arbitrage holdings is positively related to the size of the offer premium, arbitrage returns and the probability of transaction success. Finally, Sudarsanam and Nguyen, 2009 note that the best strategy for the uninformed arbitrageur is to diversify the risk arbitrage investments across a number of transactions with smaller position sizes invested in each transaction. Due to the smaller position sizes, the uninformed arbitrageur does not have the ability or expectation of influencing the outcome of a 6.3. Impact of Discretionary Portfolio Selection 97 transaction. It is the behavior of this type of investor that has been modeled throughout this study. Sudarsanam and Nguyen, 2009 further remark that these uninformed arbitrageurs should earn abnormal returns equal to zero in efficient markets. However, as our study has demonstrated, this is clearly not the case and the uninformed arbitrageur does, in fact, earn positive average abnormal returns. As we have shown, this result is robust to different portfolio construction approaches and the application of constant average transaction costs. Concluding, both theoretical and empirical evidence seem to support the notion that informed passive and active arbitrageurs are able to generate superior returns vis-a-vis the uninformed arbitrageurs.

6.3.2 Impact of Regulation on Discretionary Portfolio Selection The strong performance of informed passive and active arbitrageurs vis- a-vis uninformed arbitrageurs may depend on the takeover regulation in place in a given market. In particular, differences in disclosure thresholds appear to have a large impact on the performance of discretionary risk arbitrage versus quantitative risk arbitrage portfolio construction. In the US, arbitrageurs are required by the SEC to file a statement of intent (Section 13D filing) whenever their ownership stake in the target exceeds 5% of the outstanding shares. This is how Larcker and Lys, 1987 identify arbitrageurs in the US for their study. In the UK, the disclosure threshold is 1% of the target’s outstanding shares, which is how Sudarsanam and Nguyen, 2009 identify arbitrageurs for their study. Furthermore, the timing of the disclosure requirements differ across countries. While arbitrageurs in the UK must notify authorities that the threshold has been exceeded on the next trading day, arbitrageurs in the US are granted 10 days to notify authorities. Sudarsanam and Nguyen, 2009 argue that this difference essentially creates an opportunity for arbitrageurs in the US to build considerable ownership stakes in the target before having to reveal their position to the public. In a study of 653 cash and stock transactions during the period 1997- 2007 in the UK, Sudarsanam and Nguyen, 2009 find that there is a significant negative relationship between the offer premium and the presence of arbitrageurs. This is quite surprising considering the empirical results mentioned above of both Larcker and Lys, 1987 and Hsieh and Walkling, 2005. Sudarsanam and Nguyen, 2009 largely attribute these surprising results to the fact that the disclosure rules in the UK forces arbitrageurs to reveal their positions in the target at an earlier time than in other markets. In other words, the disclosure rules in the UK decreases the degree of information asymmetry. Further, Sudarsanam and Nguyen, 2009 find that offer premiums are significantly lower when the arbitrageurs have to dis- close their presence before the announcement date of a given transaction. 98 Chapter 6. Discussion

This appears to be the case because the acquirer realizes that, since the arbitrageurs are already present, it does not have to increase the offer premium in order to attract arbitrageurs. Please note that positions held by the arbitrageur prior to the announcement seem be in opposition to the notion that arbitrageurs do not enter the risk arbitrage position until after the announcement date. However, also recall that Schwert, 2000 documents that half of the premium paid by the acquirer take the form of pre-announcement price run-ups caused by information leakages, insider trading and toehold positions by the acquirer. The disclosure threshold further has an effect on the composition of the sample. As Sudarsanam and Nguyen, 2009 include arbitrageurs with an ownership stake of 1% of the shares outstanding, their sample includes a wider variety of position sizes. Larcker and Lys, 1987 hypothesize that there may be a positive correlation between the position size and abnormal returns. Specifically, if the private information acquired by the arbitrageur indicates that the stock price of the target is significantly undervalued, the arbitrageur will take a large position, leading to higher abnormal returns. If the stock price of the target is only slightly overvalued, the arbitrageur will enter the trade with a smaller position size, leading to lower returns. Thus, Sudarsanam and Nguyen, 2009 may include more transactions with low expected abnormal returns as a direct result of the lower disclosure threshold in the UK. However, Larcker and Lys, 1987 do not find significant results confirming this hypothesis. 17 Other literature has focused on the ability of arbitrageurs to delay the resolution and thereby, forcing the acquirer to increase the offer premium. In the model by Gomes, 2011, the ease of delaying the resolution of a transaction in order to force the acquirer to increase the offer premium depends on the critical ownership stake required in order to trigger the freeze-out mechanism. In the UK and several other European markets, e.g. Sweden, the freeze-out threshold is a 90% ownership stake, which makes it relatively easy for arbitrageurs to delay the takeover process. In the US, the threshold varies depending on the state of incorporation. In Delaware and other large states, e.g. California, New Jersey and Michigan, the freeze- out threshold is simply 50%, whereas other states such as New York, Ohio and Massachusetts requires supermajority of 2/3. Thus, when the acquirer "only" needs to accumulate 50% of the shares outstanding, it becomes more difficult for the arbitrageurs to acquire enough shares to pose a credible threat to the acquirer (Gomes, 2011). In conclusion, the regulatory scheme in place in the target market appears to have a large impact on whether the performance of informed, discretionary arbitrageurs is superior to that of uninformed arbitrageurs.

17Larcker and Lys, 1987 argue that the lack of significance may be attributed to an artificial truncation of the position sizes as a result of the short-swing profit rule. The short-swing rule stipulates that any owner of more than 10% of the outstanding shares must return any profits made whenever the purchase and sale of the shares fall within a six month window. 6.4. Impact of Modeling the Cross-Country Nature of the Western 99 European Markets

6.4 Impact of Modeling the Cross-Country Nature of the Western European Markets

A potential limitation of this study is the treatment of the European market as one, large, integrated market for risk arbitrage. An argument could be made, that the Western European markets are, in fact, not integrated and are sufficiently different that the cross country differences should be included in the modeling of risk arbitrage returns. To the best of our knowledge, all previous empirical research on the topic of the risk and return characteristics of risk arbitrage has been single market studies. Whenever the studies are centered on markets outside the US, the samples are often small, which adversely impacts statistical power and the ability to draw general conclusions from the generated results. An argument can be made that although the US market is treated as one market in all risk arbitrage studies, the US is very similar to Western Europe in the way that some legislation governing financial markets and stocks is made at the federal level, while other legislation is created at the state level. For example, the SEC regulation 13D filing for ownership stakes above 5% is required of all arbitrageurs at a federal level, while the threshold required to trigger a freeze-out by the acquirer depends on the state of incorporation of the target (Gomes, 2011). Similarly, as most of the countries in our sample are members of the European Union (with Norway and Switzerland as notable exceptions), some legislation is created at the European level, while other legislation remains country specific. Further, Fama and French, 2012 argue that market integration in Europe is a reasonable assumption, as even the countries that have elected not to be formal members of the European Union still participate in most of the open market provisions. From a more theoretical perspective, Griffin, 2002 studies whether global or country-specific Fama and French factors best characterize the time series variations in stock returns. Griffin, 2002 studies monthly returns from the US, Japan, the UK and Canada over the period 1981-1995 and finds that using country-specific factors in the Fama and French three-factor model explains more of the time series variation and leads to more accurate pricing vis-a-vis using global factors. Fama and French, 2012 study the performance of the Fama and French, 1993 three-factor model model and the Carhart, 1997 four-factor model18 over the period 1989-2011 for the following four regions: North American, Europe, Japan and Asia Pacific. Corresponding to the results of Griffin, 2002, they do not find great support for integrated pricing across markets, i.e. support of using the global Fama and French factors. However, Fama

18The Carhart, 1997 four-factor asset pricing model expands the three-factor model by including a momentum factor. Pedersen, 2015 defines a time series momentum strategy as “... a simple trend-following strategy that goes long on a market that has experienced a positive excess return over a certain lookback horizon and goes short otherwise” 100 Chapter 6. Discussion and French, 2012 do find support for the use of regional pricing for the Fama and French three-factor model and mixed support for the four-factor model across the four regions. More specifically, they find support for the use of both the three-factor and four-factor regional models in Europe. In conclusion, we find it reasonable to assume that the Western Euro- pean market is similar to the US market in terms of market integration. Further, based on the empirical evidence provided by Fama and French, 2012, we find the use of regional factors as opposed to country-specific factors to be a reasonable assumption. Finally, country-specific factors are not readily available for the 25 countries in our sample and calculating these lie outside the scope of this study. Thus, we conclude that using regional factors do not greatly limit the validity of our results. However, since Griffin, 2002 only considers large, globally important financial markets in his study, it could be of interest for future researchers to investigate the explanatory power of country-specific factors for smaller markets (e.g. Austria) vis-a-vis regionally based factors in Europe.

6.5 Impact of Takeover Regulation

A topic that is often neglected in the risk arbitrage literature is that of the impact of takeover regulation on the risk and returns of the risk arbitrage investment strategy. While investigating the current takeover regulation and its development throughout the duration of our study falls outside the scope of our study, it is worth to consider the impact that takeover regulation may potentially have on the reported results. Recall that in a UK study, Sudarsanam and Nguyen, 2008 do not find support for a non-linear relationship between risk arbitrage and excess market returns. This result is attributed to the takeover regulation under the UK Takeover Code and, specifically, the restrictions on the acquirers’ ability to withdraw a takeover offer in depreciating markets. According to Su- darsanam and Nguyen, 2008, in the UK, it is not possible to include subjec- tive withdrawal conditions in an offer and it is prohibited to cite a depreciating market as the reason for a withdrawal. Furthermore, the acquirer is required to prove that sufficient financing has been procured before making the official offer. Thus, withdrawals due to insufficient financing are rare, leading Sudarsanam and Nguyen, 2008 to form the following hypothesis:

“Due to the restrictions on the bidder’s ability to withdraw from the bid during market downturn under the UK takeover regulations, the return to the merger arbitrage strategy is related to the market in a linear way and the strategy is market neutral in all market conditions.”

In the US, the takeover regulation is much more lenient towards offer withdrawals. According to Sudarsanam and Nguyen, 2009, acquirers in the 6.6. Questions for Further Research 101

US are allowed to include a "Market Out Condition" in an offer, enabling the acquirer to withdraw its offer in case of material adverse movement in the share price of the target, the share price of the acquirer or general market movements. Furthermore, the acquirer has the option to include conditions in the offer of which the fulﬁllment purely falls to the acquirer. Thus, following the arguments of Sudarsanam and Nguyen, 2008, the US market for risk arbitrage is much more likely to display a risk and return pattern that is sensitive to adverse market movements. While the above mentioned study id centered on a single European market, our study takes its departure in transactions conducted in a variety of Western European nations. Thus, determining the impact of takeover regulation on the results of our study is a more complex procedure that forms a topic for future research. However, in light of our results it is interesting to consider the potential impact of takeover regulation. 875 out of the 2167 transactions in our sample are conducted in the UK. Hence, the lack of ﬂexibility in the withdrawal regulation in the UK may skew our results toward market-neutrality for both stock and cash transactions. The regulatory regime in other countries with many transactions such as France (315), Germany (168), Sweden (160) and Norway (120) are also likely to strongly impact our results.

6.6 Questions for Further Research

6.6.1 Transaction Cost Estimation The methodology used in this paper for estimating transaction costs in Western Europe is based on US data, which is not per se a large problem as we ﬁnd evidence that the difference in transaction costs have decreased during recent years. Additionally, the analysis of French, 2008 is of recent date. Hence, there is no reason to worry about large changes over time. Rather, the most severe disadvantage of this estimation methodology is the exogenous nature of it. This exogeneity stems from the assumption of constant transaction costs. Such an assumption implies that market impact is not a substantial part of transaction costs. In fact, this is unlikely to be a realistic assumption for risk arbitrage. This is especially important to consider when the target is a small-cap company vis-a-vis large-cap or when the position size is large. First, targets are often smaller corporations and hence, there is a great chance that market impact is substantial. Second, due to the limited span of the transaction window, arbitrageurs are forced to make large investments at once, despite signiﬁcant market impact. Concluding, based on the nature of risk arbitrage, constant transaction costs may not be appropriate. Thus, for future research, the assumption of increasing transaction costs seem more appropriate. This assumption 102 Chapter 6. Discussion implies that the main source of transaction costs for large traders is market impact. Such a transaction cost estimation requires the estimate of a liquidity parameter, similar to the beta developed by Breen, Hodrick, and Korajczyk, 1999 in their paper. However, the parameter would need to be based on European factors and include updated numbers. Such analysis has not previously been performed on the European market, but would greatly enlighten the effect of transaction costs and could hence be relevant to investigate further.

6.6.2 Discretionary Portfolio Formation based on the Behavior of Arbitrageurs The PATC is our attempt to create a more realistic portfolio. Naturally, it remains a theoretical portfolio, which most arbitrageurs will not follow. Rather arbitrageurs will attempt to include factors like liquidity, transaction costs, target size, content of the financial press etc, in the attempt to pick the transactions that are most likely to be successful. For instance, Buehlmaier and Zechner, 2013 document that the financial press content can increase the abnormal return of a risk arbitrage trade by more than 12%. This clearly indicates why discretionary trading can make a large difference to arbitrageurs and why it can add value to later studies. For the scope of our paper, modeling such discretionary portfolios are too cum- bersome. However, based on the discussion of discretionary trading, for further research, it would be relevant to develop a portfolio which picks transactions based on some of these factors and base the weighting of each transaction i the portfolio on the risk profile of the arbitrageur. To the best of our knowledge, only Mitchell and Pulvino, 2001 have attempted to replicate such a realistic portfolio allocation and even their portfolio falls short on a list of relevant parameters.

6.6.3 Investment Strategy based on Autocorrelation The autocorrelation analysis confirmed hypothesis 6, stating that we expect the existence of positive autocorrelation within the time series of changes in mid prices. Such finding can be used to develop an investment strategy for the optimal exit time for the arbitrageur. However, due to the scope of this paper we merely examine the foundation for potentially creating such a strategy. In order to develop an optimal exit timing strategy, we would need to find a signal, a critical percentage change in mid prices, indicating the need for action (buy or sell) from the arbitrageur. In further research, there would be a great deal of value in discovering such strategy, as practitioners all over Europe could potentially benefit from it. 6.7. Overview of Main Conclusions 103

6.6.4 Market Speciﬁc Takeover Regulation As discussed above, the impact of takeover regulation on the risks and returns of executing a risk arbitrage strategy is often overlooked in academic literature. To the best of our knowledge, the UK market is the only market in which studies on the impact of takeover regulation has been conducted (Sudarsanam and Nguyen, 2008, Sudarsanam and Nguyen, 2009). Thus, a topic for future research is the impact of takeover regulation on risk arbitrage in other markets with high M&A activity such as the US, Japan, the Nordics, Germany and France.

6.7 Overview of Main Conclusions

Table 6.1 shows an overview of our six hypotheses, their conclusions and the strength of the evidence.

Hypothesis Conﬁrmed Not conﬁrmed Strength of evidence Hypothesis 1 We expect to observe positive abnor- Strong mal risk arbitrage returns in Western Europe during the period 1990-2015 Hypothesis 2 We expect to observe that the portfo- Strong lio construction approach has an impact on the risk and return characteristics of the risk arbitrage strategy Hypothesis 3 We expect to observe a substantial Weak impact of transaction costs on the magnitude of the abnormal risk arbitrage returns Hypothesis 4a We expect to observe a non-linear re- Strong lationship between the excess risk arbitrage returns and the excess market returns Hypothesis 4b We expect to observe a stronger non- Strong linear relationship between the excess risk arbitrage returns and the excess market returns for cash transactions vis-à-vis stock transactions Hypothesis 4c We expect to observe decreasing ab- Medium normal risk arbitrage return over time Hypothesis 5 We expect to observe that risk arbi- Strong trageurs are compensated for providing liquidity during the transaction window Hypothesis 6 We expect to observe positive au- Strong tocorrelation in the time series of changes in mid prices during the transaction window

TABLE 6.1: This table shows an overview of our six hypotheses, their conclusions and the strength of the evidence. 104

Chapter 7

Conclusion

Using a sample of 2167 cash and stock transactions during the period 1990- 2015, we study the risk and return characteristics in risk arbitrage in West- ern Europe. We form four portfolios, VWRA, EWRA, PA and PATC, of monthly risk arbitrage returns. These portfolios are constructed using a variety of weighting schemes and investment constraints. Subsequently these monthly portfolio returns are benchmarked against two linear asset pricing models, namely the CAPM and the Fama and French three-factor model, as well as against a non-linear piecewise model. Further, we formally test for non-linearity and estimate abnormal returns using a contingent claims analysis. Moreover, we compare the monthly returns of our portfolios with the monthly returns of the CSRA Index. Finally, we seek to find evidence to initiate research into the optimal exit time of the risk arbitrage trade. Our study identifies the existence of positive abnormal returns in West- ern Europe. The monthly abnormal returns generated by benchmarking excess risk arbitrage returns against the CAPM fall within the range 19bp- 69bp across the four portfolios. Based on our assumption of constant transaction costs, as opposed to increasing transaction costs, we do not find evidence to support that transaction costs should have a substantial impact on our PATC portfolio returns. This conclusion may change if we were to change the underlying assumption on transaction costs. The results of the linear benchmark analyses, the piecewise linear regression model and the formal test for non-linearity indicate that the assumption of market neutrality only holds for stock transactions, but is vi- olated for cash transactions. The piecewise linear regression model shows a non-linear relationship between excess risk arbitrage returns and excess market returns. Based on this result, we perform a contingent claims analysis. This analysis yields monthly abnormal risk arbitrage returns of 96bp for the PATC cash portfolio. Furthermore, we find evidence suggesting that stock transactions yield higher abnormal returns than cash transactions. Additionally, we find evidence that the abnormal returns have dimin- ished over time. The contingent claims analysis finds abnormal monthly returns for the PATC portfolio of 33bp for the period 1990-2002 and returns of 29bp for the period 2003-2015. Further, our results suggest that the strategy has become less market neutral over time. Chapter 7. Conclusion 105

Through the range of results generated based on the four portfolio construction approaches, we find that the portfolio construction approach is, indeed, of critical importance. The portfolio with the most realistic assumptions, the PATC portfolio, displays the strongest correlation with the CSRA Index. Further, through a discussion of discretionary portfolio selection, we find that our quantitative approach to the formation of risk arbitrage portfolios may understate the potential of risk arbitrage to generate abnormal returns. Finally, we conduct an autocorrelation regression analysis on the time series of changes in daily mid prices. We find strong evidence of the existence of positive autocorrelation and thus, we lay the foundation for future research into the development of an optimal exit timing strategy. All in all, we find risk arbitrage to be a profitable strategy in Western Europe, yielding significant positive abnormal returns. Further, we find evidence of a non-linear relationship between excess risk arbitrage returns and excess market returns, particularly driven by cash transactions. This implies that practitioners earn abnormal returns from risk arbitrage and that, contradictory to previous beliefs, they need to actively short the market in order to obtain market neutrality. 106

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Appendix A

Risk Arbitrage Cumulative Returns

(A) Risk Arbitrage Cumulative Returns (1990-2015) - Cash

(B) Risk Arbitrage Cumulative Returns (1994-2015) - Stock

FIGURE A.1: Risk Arbitrage Cumulative Returns (1994-2015) - Cash and Stock This ﬁgure illustrates the development over the 1990-2015 period of the value of $100 invested in each of the four stock and cash portfolios at the beginning of the time period. 112

Appendix B

Risk Arbitrage Cumulative Returns - Logarithmic Scale

FIGURE B.1: Risk Arbitrage Cumulative Returns (1990-2015) - Logarithmic Scale This ﬁgure illustrates the development over the 1990-2015 period of the value of $100 invested in each of the four cash only portfolios at the beginning of the time period plotted on a logarithmic scale. 113 114 Appendix C. Transaction Costs by Year

Appendix C

Transaction Costs by Year

Year Transaction Costs (bp) 1980 55 1981 46 1982 55 1983 60 1984 49 1985 53 1986 54 1987 53 1988 43 1989 43 1990 38 1991 43 1992 42 1993 44 1994 40 1995 41 1996 38 1997 34 1998 30 1999 30 2000 32 2001 28 2002 30 2003 29 2004 25 2005 21 2006 21 2007 20 2008 18 2009 17 2010 16 2011 15 2012 14 2013 13 2014 12 2015 11

TABLE C.1: This table shows linearly estimated transaction costs based on transaction costs from French, 2008 for the period 1980-2006. 115 Appendix D. Time Series Regressions of Risk Arbitrage Returns on 116 Common Risk Factors

Appendix D

Time Series Regressions of Risk Arbitrage Returns on Common Risk Factors

Market Return - Rf < -7% 2 α βMkt βSMB βHML Adj. R Sample size VWRA Total CAPM 6,38 0,859* 0,153 23 (4,334) (,385) VWRA Total FF 7,365 0,932(*) -0,182 0,067 0,076 23 (5,221) (,458) (,473) (,4) VWRA cash CAPM 5,549 0,888(*) 0,129 23 (4,842) (,43) VWRA cash FF 6,416 0,934(*) -0,1667 0,304 0,076 23 (5,751) (,505) (,521) (,441) VWRA Stock CAPM 24,977 1,64 0,102 16 (11,747) (,998) VWRA Stock FF 20,294 0,851 1,926 3,208 0,347 16 (12,046) (1,036) (1,196) (1,193) PA Total CAPM 2,482(*) 0,388** 0,275 23 (1,43) (,127) PA Total FF 1,457 0,307* 0,188 -0,017 0,271 23 (1,654) (,145) (,15) (,127) PA Cash CAPM 1,786 0,368* 23 (1,585) (,141) PA Cash FF 0,577 0,264 0,219 0,094 0,203 23 (1,836) (,161) (,166) (,141) PA Stock CAPM 2,985* 0,207(*) 0,148 16 (1,286) (,109) PA Stock FF 2,899(*) 0,163 0,123 0,289 0,267 16 (1,434) (,123) (,142) (,142) PATC Total CAPM 2,428 0,385** 0,272 23 (1,428) (,127) PATC Total FF 1,418 0,305* 0,186 -0,02 0,267 23 (1,652) 0,145 0,15 0,127 PATC Cash CAPM (1,723) (,365) 0,208 23 (1,581) 0,14 PATC Cash FF 0,528 0,262 0,217 0,091 0,198 23 (1,834) (,161) (,166) (,14) PATC Stock CAPM 2,983* 0,208 0,149 16 (1,284) 0,109 PATC Stock FF 2,901(*) 0,164 0,122 0,287 0,266 16 (1,435) (,123) (,143) (,142)

TABLE D.1: Time Series Regressions of Risk Arbitrage Returns on Common Risk Factors: This table presents the results for two regressions of risk arbitrage returns on common risk factors when the monthly market return is less than -7%