Amsterdam Business School
MSc Finance – Asset Management Track
Master Thesis
Liquidity and other drivers of
corporate bond yields
Author:
Michelangelo Marrani
Supervisor:
Ms. Derya G¨
uler
Acknowledgements
This thesis marks the end of an experience that I will keep in my heart for the rest of my life. It was not an easy journey, and I could not have successfully reached the end of it without the help and support of many. First, I wish to thank all those teachers who inspired me during this year, and especially my supervisor, who helped me improve this thesis with constructive and priceless suggestions. Second, I want to thank my lovely family, which I hope I made proud of me. And last but not least, I want to thank all the people I have the honor of calling friends for having encouraged me and never letting me feel alone.
Statement of originality
This document is written by Michelangelo Marrani, who takes full responsibility for the contents of this document. I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it. The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.
Abstract
Stock and bond markets have been extensively analyzed during the course of the last decades, but most of these researches treated these two markets separately. Only a few papers inspected the connections between equity and fixed income markets, but all of them focused on the role of Treasury bonds, rather than cor-porate bonds. This thesis extends this literature by adding strong evidences of cross-market links between stocks and corporate bonds. Using a sample of more than 100,000 corporate bonds issued from July 2002 to December 2015, this re-search provides two main findings: first, variables derived from the stock market can explain changes in corporate bond yield spreads; second, movements in bond market variables can predict changes in the stock market and vice versa. The evi-dences provided by this thesis are useful for both asset pricing and asset allocation purposes.
Contents
1 Introduction 1
2 Theoretical Framework 3
2.1 What is liquidity? . . . 3
2.2 Why does liquidity matter? . . . 4
2.3 Related Literature . . . 5 3 Hypotheses 10 4 Data 10 4.1 Data Sources . . . 10 4.2 Variables Construction . . . 12 4.2.1 Yield Spread . . . 12
4.2.2 Levels and Innovations in Liquidity . . . 14
4.2.3 Other Variables . . . 14
4.3 Sample and Descriptive Statistics . . . 16
4.3.1 First Hypothesis . . . 16
4.3.2 Second Hypothesis . . . 19
5 Methodology 21 5.1 First Hypothesis . . . 21
5.2 Second Hypothesis . . . 22
5.2.1 Vector Autoregressive Model . . . 22
5.2.2 Granger Causality . . . 25
5.2.3 Impulse Response Functions . . . 25
6 Results and Discussion 25 6.1 First Hypothesis . . . 25
6.2 Second Hypothesis . . . 28
6.2.1 Granger Causality . . . 28
6.2.2 Impulse Response Functions . . . 30
7 Robustness Checks 32 7.1 First Hypothesis . . . 32
7.2 Second Hypothesis . . . 33
References 36
Appendix A 38
1
Introduction
What is liquidity? A market or a security are said to be “liquid” when there are enough buyers and sellers to ensure that small buy or sell orders do not affect prices in a relevant way. The term can also mean how easy it is to buy or sell a particular asset: in general, stocks are among the most liquid securities as they can be easily traded on the market, while real estate can be very illiquid. While considering an investment opportunity it is crucial to account for this aspect and thus it is logical to expect that some sort of “liquidity premium” is incorporated into the price of an asset, making the price of a liquid security higher than the price of an illiquid one. This effect on prices, and thus returns, represents the foundation of what will be discussed later in the text.
Liquidity has always played an important role in the research concerning the pricing of stocks. Among the most famous papers in this field it is possible to count the work conducted by Amihud and Mendelson (1986) and the one carried out by Acharya and Pedersen (2005) in which stock returns are inspected while accounting for liquidity in the market. Amihud and Mendelson (1986) were the first to relate stock returns to the indisputably important feature of liquidity, providing a game-changing point of view which would have affected the research in asset pricing for the following years. In particular, they find that returns increase with the bid-ask spread, a well-known measure of liquidity. Acharya and Pedersen (2005) inspect the importance of liquidity by means of a brand-new Capital Asset Pricing Model adjusted for liquidity. With this tool, they explain that the level of liquidity in the stock market is relevant in explaining the so-called “excess return”: the difference between the return of a stock and the return of a risk-free asset.
By using this framework, they are also able to study the effect of liquidity risk1,
which explains the 1.1% of returns in the cross-section. Moreover, they study the liquidity sensitivity to the market return, an effect which no literature had faced before.
More recently, the importance of liquidity has been inspected also in the en-vironment of different markets, such as the option market and the fixed income market. This aspect has been put aside for too long, as fixed income is usually considered as a predictable and well defined type of investment, while derivatives are difficult to analyse and face a problem of data availability, as they are often not
1Liquidity risk is defined as the risk that a certain security cannot be traded quickly in the
traded in a regular market. Since the beginning of the subprime crisis, when fixed income markets have been stressed more than many others, this topic has gained more relevance among academics and regulators, leading to a broader literature. As Dick-Nielsen et al. (2012) demonstrate, the corporate bond market experienced a rise in liquidity premia which peaked in correspondence of the Lehman Brothers default. Moreover, they show that bonds issued by financial institutions became more illiquid during the subprime crisis. These findings show that corporate bonds are affected by systematic events as much as stocks are and that the way their prices are defined is not as trivial as some may think, since it involves many risk factors.
My research is inspired by the challenge proposed by P`astor and Stambaugh
(2003), who studied whether changes in the aggregate liquidity of the stock market affect stock returns. At the end of their paper, they invite future researchers to inspect the same effect in different markets, such as the fixed income market, and this is what my research tries to test. Contrary to the methodology followed by most of the authors who wrote about fixed income, I seek to inspect whether some form of systematic liquidity risk, derived from the stock market, can affect yield spreads of the corporate bonds issued. Very often in literature risk measures and return measures are derived from the same market, but an important aspect worth analyzing is also the connection between the stock market and the bond market, which is widely underrated. In this context, it is utterly important to inspect whether corporate bond market and equity market interact and whether one can predict the other. The results of this study could justify phenomena such as flight-to-quality and flight-to-liquidity: situations during which investors prefer to invest in safer or more liquid securities.
First, the bound between stock and bond markets is analyzed by means of an OLS regression. Results show that both stock market liquidity and volatility can explain changes in corporate bond yield spreads. Robustness tests with different measures of liquidity and yield spreads partially confirm these findings. Second, this research employs a Vector Autoregression to inspect whether there is any
lead-lag relationship2 among market variables that can explain how the corporate bond
market reacts to shocks in equity market and vice versa. Results show that said lead-lag relationships exist and these findings are validated by robustness tests
2Variable X and variable Y are bounded by a lead-lag relationship if when something affects
X, Y is affected after some time as well. In this case X is the “leader” variable and Y is the “lagged” variable.
that employ different samples of corporate bonds. This kind of bounds between stock and corporate bond markets has not been stressed enough in literature, where typically these markets are treated separately. The aim of this thesis is to contribute to fill this gap in literature and try to understand more deeply the link between two different types of security. In practice, knowledge about this relation could lead to predictability of movements in stock and bond returns and increased performances in asset allocation strategies.
The thesis is structured as follows: Section 2 describes the theoretical frame-work in which the thesis is set. Section 3 describes the research question and the hypotheses necessary to answer it. Section 4 lists the variables in use and their characteristics. The methodologies employed are described in Section 5. Section 6 provides results and discusses them. Section 7 presents some robustness tests and Section 8 concludes.
2
Theoretical Framework
In this Section a deeper explanation of the theory behind this research is provided. A description of the notion of liquidity is followed by a brief exposition concerning the agents that are involved in defining its levels. Thereafter, the role of liquidity in the pricing of assets is analysed together with an overview of the results provided by the existing literature.
2.1
What is liquidity?
The way financial markets actually work is far from the idea that many have of a frictionless and perfect market. When approaching a trading platform, even the less skilled investor can notice how easy it is to buy or sell a security, as if on the other side of the transaction there is always a counterparty ready to trade. Markets do not really work like this: all investors are not present and ready to trade at the same time, and do not have the same set of information available, so they have different opinions about the fundamental value of a security. Market agents submit buy and sell orders according to their valuations and their needs, thus prices offered for the same asset could be very different among traders. This disagreement over prices and the absence of enough people willing to trade are only two of many determinants of illiquidity: when the highest price investors are willing to pay for a security (bid price) is lower than the lowest price investors are willing to accept to sell the same security (ask price), then such security is said
to be illiquid. The most common measure of illiquidity is indeed the difference between the bid and the ask prices: the bid-ask spread. The wider the spread, the more illiquid the market. There are many other proxies for illiquidity, as the Roll’s estimate (Roll, 1984) and the Amihud illiquidity ratio (Amihud, 2002), and all can be computed from market data. In general, a liquid asset is easy to trade as there are many investors who wish to buy or sell it, while it could take some time before a sell order on an illiquid asset is satisfied. Real estate market is a good example of illiquidity: houses and apartments are usually sold only after a deep inspection of their characteristics and this process could take months.
In order to guarantee a certain level of liquidity in the market, so that such market can be more efficient, market makers act as intermediaries and fulfill orders that otherwise would stay unmatched. A market maker is a company that quotes a bid and an ask price at which it is willing to trade a security and gains upon the difference in these prices. As they are always ready to trade, they provide enough liquidity not to cause high oscillation in security prices, preventing the market from freezing and too much uncertainty. In fact, market illiquidity can be a proxy for uncertainty in the market: it is sufficient to notice that during the financial crisis the average bid-ask spread in the stock market doubled from the beginning of 2008 to six months after the failure of Lehman Brothers in September 2008. This very characteristic of measure of general uncertainty is employed in this research to inspect whether illiquidity in the stock market can lead to a phenomenon of “flight-to-liquidity” and make bonds (generally considered as much safer securities than stocks) more desirable.
2.2
Why does liquidity matter?
The relevance of liquidity, comes from its impact on asset prices, which in turn affect returns. Much of the modern portfolio theory has focused its efforts in finding the determinants of asset prices and trying to explain why they move in a certain direction. One of the cornerstones of this research is represented by the Capital Asset Pricing Model (CAPM). This model builds on the work of H. Markowitz and states that the expected return of a security or a portfolio is equal to the rate of return on a risk-free asset plus a premium for the risk undertaken. Such a model is useful to price securities and compute the profitability
of an investment. Although this is the most famous model, it is also one of
the most criticized, as it makes strong assumptions which are distant from the situation in a real market. Such assumptions are, for instance: the fact that all
investors at the same time have access to all information, that they can borrow and lend unlimited amounts of money at the risk-free rate and that there are no transaction or taxation costs. Despite its questionable efficiency, the CAPM stays the starting point for many other researchers, such as Acharya and Pedersen,
that in 2005 theorize a liquidity-adjusted CAPM3. They find a positive relation
between the required return of a security and the covariance between its illiquidity
and market’s4 illiquidity. This means that when the illiquidity of a stock moves in
the same direction and with a similar magnitude than the illiquidity of the market, investors demand higher returns to bear this “liquidity risk”. Their paper confirms that liquidity plays a role in defining stock prices, and that a premium for liquidity risk is indeed required. Starting from this evidence, my thesis inspects the role of liquidity in both stock and bond markets, trying to find connections useful for further research concerning the determinants of security price movements.
2.3
Related Literature
The hypothesis that bond and stock markets are closely linked has been proven true since 1993 by Fama and French, who inspected how different combinations of variables are able to capture common movements in the returns of bonds and stocks. In 1992, Fama and French theorize their famous model to describe the determinants of excess returns of common stocks. They find that leverage, book-to-market ratio, size and earnings-price ratio can explain the movements of average returns for the period which spans from 1963 to 1990. In their other paper of 1993, they expand the previous research by applying their model to bond returns and using variables different from those mentioned before. They first use a linear
regression model to explain stock returns by means of their three-factor model5,
which captures stock features such as size and book-to-market ratio of the firm. Second, they perform a similar regression for bonds, but with different variables
3While the standard CAPM explains the excess return of stocks by means of their correlation
with the market excess return, the liquidity-adjusted CAPM is a pricing model that seeks to explain the excess return of stocks by means of a measure of liquidity and a measure of liquidity risk.
4The “market” is defined as a theoretical portfolio that includes every type of asset available
in the market. The S&P 500 is widely used in practice to approximate the “market” in asset pricing models.
5The Fama French three-factor model explains stock excess return from the market by means
of three factors, namely MKT, HML and SMB, that are built from portfolios of securities sorted by market capitalization and firm size. They indeed argue that these two features of a firm are important in stock pricing.
which consider common risk factors such as changes in the term structure of interest rates and in the likelihood of default. For stocks, they find that average returns and size are negatively related, while returns and book-to-market ratio have a strong positive relation. In case of bonds, the results are not strong: there is little evidence that long-term corporate bonds perform better than Treasury bonds and that bonds with a lower rating have higher returns. After having drawn these conclusions, Fama and French expect common variations in stock and bond returns. They ask themselves whether returns in one of these two markets can be explained by means of those factors which belong to the other market. In order to run this analysis, they use linear regressions in which they include both stock-related and bond-stock-related factors. They ultimately find that returns of corporate
and Treasury bonds6 are not affected by stock-related factors, but stocks and
bonds, however, are connected through two factors: changes in the term structure of interest rates and likelihood of default.
Ten years later, Chordia, Sarkar and Subrahmanyam (2005) strengthen these conclusions by suggesting that common movements in liquidity and volatility in the two markets are driven by common factors, such as monetary policies and monetary flows to the stock and bond sectors, which influence both stocks and bonds. They state that there is reason to believe that effects across stock and bond markets exist: for instance, if investors choose to allocate their assets with a certain lag in time, accordingly to shocks in one of the two markets, then trading behaviour in one market could forecast trading behaviour in the other, with direct effects on liquidity. The same can happen with respect to liquidity and volatility: if macro shocks in liquidity and/or volatility affect the stock market before the bond market, it is possible to predict how liquidity and volatility will change in the bond market. This reasoning is mirrored in my thesis in the use of a variable derived by the stock market and inserted in the environment of corporate bonds. To draw their conclusions, Chordia, Sarkar and Subrahmanyam make use of a Vector Autoregressive model. This methodology allows them to inspect bi-directional causalities between the four explanatory variables they choose (volatility, liquidity, returns and order imbalances) while considering lead-lag relationships. They find that a shock to the bond spread produces an increase in the stock spread, where the “spread” is defined as the difference between the highest price at which investors buy and the lowest price at which they sell a security and represents the most common measure of liquidity. Also, a shock in stock returns causes a fall in bond
6A Treasury bond is a bond issued by the U.S. Government, in this case, for public finance
spreads, which, conversely, rise in case of shocks in stock volatility and spread. These commonalities represent a strong basis for what my thesis illustrates and they encourage further research about the link between liquidity in fixed income and stock markets.
Moreover, Goyenko and Ukhov (2009) used a Vector Autoregressive analysis to show that stock and bond illiquidity interact: an increase in stock illiquidity lowers bond illiquidity (which is consistent with the hypothesis of flight-to-liquidity) while a higher illiquidity in the bond market leads to an increase of illiquidity in the stock market as well. They start from the abovementioned work driven by Chordia, Sarkar and Subrahmanyam (2005) and develop a study that shows how bond market and stock market are bounded via illiquidity: the level of illiquidity in one market can predict the level of illiquidity in the other one. In particular, they focus on Treasury bonds. In addition to using a Vector Autoregressive model, they
also employ impulse response functions7 as Chordia, Sarkar and Subrahmanyam
(2005) did in their paper: this method allows them to inspect the effect of a one-time increase of a variable on the present and future values of other variables. By means of this analysis they find further evidence of cross-market effects of liquidity. One of the most important findings they make is that it is important to study the liquidity of bonds over different maturities: in fact, the role of short-term bonds is far more relevant in cross-market relations if compared to the one of medium-and long-term bonds.
Although it has been extensively demonstrated that liquidity plays a significant role into corporate bond yield spread, the best part of these analyses focuses on the variations over time of the yield spread of those bonds which are traded in an over the counter market, and use typical measures of liquidity. Friewald et al. (2012) provide a good example of this: they consider the Amihud measure, the price dispersion measure, the Roll measure and the Zero-return measure as proxies for liquidity and try to explain the variation in yield spreads by means of these variables. The Amihud measure in a certain day is computed as the average value of the ratio between the absolute value of returns and the trading volume of the security considered. The higher the Amihud measure, the higher the price impact of a trade, which is the very definition of liquidity. The price dispersion measure is a proxy that computes how much traded bond prices are dispersed around the
market consensus valuation8. The wider the dispersion, the higher the transaction
7Impulse response functions are further described in Section 5.2.3.
costs and the lower the level of liquidity of an asset. The Roll measure considers the “bounce” in prices provoked by trades and can be used to extrapolate the bid-ask spread directly from market movements. Ultimately, the zero-return measure is a dummy variable which assumes a value equal to one if the price of the security does not change over several trading days and zero in the opposite instance. To run their analysis they make use of both a panel regression and a Fama-MacBeth regression: this allows them to inspect changes of the yield spread both over time and over the cross-section. They find that liquidity explains 14% of the variations in the yield spread over time, and that this effect is even stronger during the periods of GM/Ford crisis and subprime crisis. Moreover, they find that liquidity plays an important role in pricing speculative grade bonds (those with a rating around BB), meaning that bonds with higher credit risk also suffer of higher liquidity risk. The abovementioned measures are computed by looking at the transaction data taken from the Trade Reporting and Compliance Engine (TRACE). My thesis partially employs this methodology, but instead of using several bond-derived measures of liquidity, such as the price dispersion measure or the zero-return measure, it
involves a new proxy proposed by P`astor and Stambaugh (2003). They create a
new measure to investigate whether market-wide liquidity influences stock returns. Specifically, they build a variable called “innovations in liquidity”, and then use it to investigate the effect of this variable and the three factors by Fama and French on excess returns. Their analysis covers a period of 40 years, from 1960 until 2000, and enhances the role of liquidity over time by focusing on those periods of time which are affected by market downturns. Said paper mainly provides an asset pricing model, but the key element which will be involved in this thesis is their new variable. Such variable will be employed in the bond market in order to test whether it affects corporate bond yield spreads.
A study close to the one I am about to conduct has been led by Lin, Wang
and Wu (2011), who use both the Amihud (2002) and the P`astor and Stambaugh
(2003) methods to inspect how market-wide liquidity risk affects expected corpo-rate bond returns by means of the Fama-MacBeth regression. The most innovative feature of their paper consists in incorporating the measure of innovations in liq-uidity into the asset pricing model proposed by Fama and French (1993). They find that liquidity risk has a statistically significant positive effect on corporate bond excess returns: if liquidity risk increases, investors demand higher returns. This work from Lin, Wang and Wu (2011) represents one of the most relevant con-tributions to the existing literature in this field: apart from employing the methods already mentioned, they also perform univariate and multivariate portfolio sorts
and employ and extended Acharya-Pedersen model, all in order to inspect the liquidity effect and confirm their findings. As expected, they demonstrate that liquidity risk is a priced factor in corporate bond markets and plays a role in defining the cross-sectional variations in their returns. The main difference be-tween this paper and my thesis relies on the fact that their measure of illiquidity is still derived from the bond market to inspect that very same market, while mine is derived from the stock market. Moreover, they use expected return rather than yield spread as dependent variable. The same difference holds with respect to the works of Bao, Pan and Wang (2011), Houweling at al. (2005) and Chen et al. (2007), who analyse the role of liquidity on bond valuation, confirming the abovementioned findings concerning liquidity premia and the role of credit risk.
de Jong and Driessen (2012) take into account the role of stock market in bond pricing and try to solve the credit spread puzzle: the fact that spreads on corporate bonds are way larger than the amount that can be explained by illiquidity, default probabilities and taxes. This paper proves that a component of stock illiquidity is in fact priced in bonds, and this evidence gives even more relevance to the topic I’m about to inspect. As already demonstrated, excess returns are larger for those bonds which have a lower rating. Moreover, they give evidence that corporate bond returns are exposed to Treasury bond market liquidity. In order to validate their findings they apply the same methodology both for the U.S. market and the European market and they indeed draw the same conclusions. Even though they
take inspiration from P`astor and Stambaugh (2003), they do not use the original
measure of “innovations in liquidity” which I intend to use in my research. Ultimately, a strong evidence of stock-bond return correlation comes from the job driven by Baele et al. (2010), who employ a large number of factors and models to find any proof of co-movement between these securities. Such factors are: output gap, inflation, interest rate, risk aversion and cash flow growth. They employ three different Vector Autoregressive models and they find that those models which include in their specifications measures of economic uncertainty and risk aversion perform better than those which do not. This confirms that variations in risk premia are important factors to include in any model that seeks to explain correlations between returns of stocks and bonds.
3
Hypotheses
This Section clarifies what the hypotheses under consideration are and how they will be tested. This thesis contains one research question which is answered by means of two hypotheses.
In this thesis, I answer to the following question: “Do corporate bond market and equity market interact and can one predict the other?”. In order to find a solution to this problem, I test two hypotheses.
Hp 1 : A measure of market-wide liquidity derived from the stock market can explain movements in corporate bond yield spreads.
Hp 2 : Variables such as returns, volatility of returns and liquidity of both markets are linked via lead-lag relationships.
The evidence provided by this research can be useful both for asset pricing and asset allocation purposes. In fact, proof of the effect of stock market liquidity on corporate bond yield spreads can underline the importance of stock market variables into bond pricing models. In general, asset pricing models seek to ex-plain excess returns of a specific security, and in doing so they employ variables that belong to the same market of the security under analysis. Instead, if the Hy-pothesis 1 was proven true, this would provide evidence that cross-market effects must be taken into account when developing an asset pricing model. On the other hand, if Hypothesis 2 was not rejected, this would mean that the evolution of some variables in both markets can be predicted and this finding can be a useful tool for asset allocation strategies.
4
Data
4.1
Data Sources
The main database employed for the purposes of this research is TRACE (Trade Reporting and Compliance Engine) provided by the FINRA (Financial Indus-try Regulatory Authority). FINRA, then the NASD (National Association of Securities Dealers), introduced this system in July 2002 with the aim of increas-ing price transparency in the U.S. corporate bond market. The system collects and divulges information concerning market transactions in bonds publicly traded over-the-counter. Such information involves price, yield, time and volume of each
transaction occurred. The release of this instrument has passed through three
phases. Phase I started in July 2002: at the end of that year, the TRACE
database included about 520 securities, most part of which was represented by investment-grade bonds with an original issue size greater or equal to $1 billion. Phase II started in March 2003. The database started including also those bonds with a par-value of at least $100 million and a rating of A− or higher. Moreover, 120 investment-grade securities rated BBB and with issue size lower than $1 bil-lion were added. During this phase, the total number of bonds in the database increased to 4,650. On October 2004 TRACE was further expanded (Phase III). The database at that time provided information on 99% of all publicly traded corporate bonds, with more than 30 thousand issues recorded. Many studies have been conducted until now on the effect of this database on trading activity and trading behaviour, but there is little evidence of any impact (e.g., see Goldstein, Hotchkiss and Sirri, 2007, and Duffie, 2012). What most matters is that this system provides enough data to make an actual comparison between information available in the stock market and in the bond market, thus lowering transaction costs and allowing for more aware investment decisions. Though, it is worth men-tioning that all this public information could have disadvantaged dealers: if a dealer submits large buy or sell orders to the market in order to provide liquid-ity, having the information on price and quantity public could place a maximum on the resale price they can ask. This concern led to the decision by FINRA of limiting trade size reports at $1 million for high-yield bonds and $5 million for investment grade bonds.
The sample I downloaded from this source includes 104,055 bonds from 7,446 different issuers. The time spans from July 2002 until December 2015 for a total of almost 140 million daily transactions. From this database it is possible to obtain information about the yield of every single bond exchanged, which in turn will allow me to compute the difference in yield between corporate bonds and Treasury bonds. Moreover, the volume of transaction, also provided in TRACE, proves to be relevant in the analysis.
Treasury yields are taken from the Federal Reserve Boards H.15 release, avail-able through the WRDS platform. Every business day, the Federal Reserve releases data concerning yields of many securities, such as: commercial papers, Eurodol-lar deposits, bank prime loans, interest rate swaps, some corporate bonds and Treasury bills and bonds for all maturities. The type of Treasury bonds selected for this research are not inflation indexed as TIPS (Treasury Inflation Protected
Securities), mainly employed for edging purposes. The indexing to inflation could bias the regression and create unnecessary complications to the interpretation of results.
The VIX index, a proxy for aggregate volatility in the stock market, is taken
from the WRDS platform as well as the P`astor and Stambaugh liquidity measure9.
The last database employed is Datastream: data on aggregate U.S. corporate bonds are taken from here.
4.2
Variables Construction
4.2.1 Yield Spread
In fixed income pricing, yield spreads are preferred to excess returns: in fact excess returns in case of bonds can be biased by the differences in credit risk and maturity of each security. Yield spreads, instead, are a useful instrument to compare bonds with different characteristics. The yield spread is defined as the difference between the rates of return of two different instruments with similar characteristics: usually with same maturity, but different credit risk. It is often
computed as the difference in their yields to maturity10. If investments X and Y
have the same maturity and a yield to maturity of 5% and 3% respectively, the “yield spread of X over Y” is 2%. This means that investment X is riskier than investment Y. A similar reasoning can be followed if X and Y are bonds which do not have the same maturity, but are comparable under other aspects. In this
case, the yield spread represents the term premium11required for investing. When
inspecting bonds, the second security to which compare the first one is usually a Treasury bond with the same maturity. A Treasury bond is considered to be a riskless investment; hence it can easily represent a benchmark to which compare any other bond. Another commonly used benchmark is the swap curve: a term
9Although available in WRDS, the VIX index is computed by the Chicago Board Options
Exchange.
10A standard “coupon bond” is made of a set of cash payments (coupons) before maturity
and a “principal” payment at the end. The yield to maturity of a coupon bond is the discount rate at which the sum of all future cash flows from the bond (coupons and principal) is equal to the price of the bond.
11When two bonds have different maturities, the longer one usually has a higher yield to
maturity. This is because the longer bond is more likely to be affected by changes in the term structure of interest rates which, in turn, can affect the return. For this reason, investors who buy bonds with long maturities demand lower prices (and thus higher returns and yields). The difference in yields of long-term and the short-term bonds is called “term premium”.
structure of fixed coupon rates of quoted interest rate swaps. For the ends of this research, on-the-run Treasury yields to maturity will be employed as benchmark measure.
Yields of corporate bonds are taken from the TRACE database, using daily
data for the period which spans from July 1st 2002 to December 31st 2015. As
Dick-Nielsen (2009) points out, when dealing with this database it is first necessary to filter the observations. While TRACE becomes the most used database to inspect U.S. corporate bond market, his work represents the first attempt to deeply explain what problems could occur while manipulating this daily records and how to solve them properly. He defines three filters that, if not applied, could bias the results derived from this database and define a too liquid market. In TRACE is indeed common to find several wrong transactions which are rectified by means of cancellations, corrections or reversals. According to Dick-Nielsen, the first thing to do is of course deleting true duplicates: if two or more reports have the same message sequence number, then some are duplicates, as the message sequence number is unique for each report. Second, it is necessary to delete reversals, i.e. reports in the opposite direction that the original (wrong) transaction, which must be deleted as well. Third, as last step, corrections and cancellations should be handled: if the report is a correction, only the original should be deleted, while if it is a cancellation both must be erased. Although these instructions seem easy to follow, some problems arise when dealing with reversals in TRACE data from WRDS, as WRDS does not provide information about the filling date of a report, which would be useful in order to match reversals and original reports. If this is the case, as in this research, he provides a special methodology: if the identical match of a reversal cannot be found, only the reversal is to be deleted, while if more than one trade matches the reversal, then all of them are erased not including any “as-of” trade. This process allows to delete more than 4 million observations, which represents 3% of the dataset. It is also worth mentioning that as the number of bonds in the database increases over time, the number of errors decreases accordingly, maybe due to a better performance of the mechanisms of automation which are employed to report these transactions.
For each observation, the time to maturity is then computed as the difference between the maturity year and the year in which the transaction occurred. By means of this new variable it is possible to match each corporate bond with a Treasury bond which has a similar time to maturity and compute the yield spread between these two securities. The matching is approximated to the nearest
Trea-sury bond maturity, which could be of 1, 2, 3, 5, 7, 10, 20 or 30 years. Results are then trimmed at 5%, thus cutting off the first and the last five percentiles of the distribution in order to exclude any potential outlier. Finally, observations are averaged for each month.
4.2.2 Levels and Innovations in Liquidity
This measure is firstly proposed by P`astor and Stambaugh in 2003. It represents
a widely-used proxy for liquidity in the U.S. stock market. In fact, its computa-tion involves daily data taken from both the New York Stock Exchange (NYSE) and the American Stock Exchange (AMEX).When included in an asset pricing model together with the three factors by Fama and French, their measure is use-ful to explain stock returns. This means that market-wide liquidity is indeed a
priced factor. “Innovations in liquidity” is built on the OLS estimate of γi,t in the
following regression:
rei,d+1,m = αi,m+ βi,mri,d,m+ γi,msign ri,d,me V oli,d,m+ i,d+1,m (1)
Where ri,d,m is the return on stock i on day d of month m; rei,d,m = ri,d,m −
rmkt,d,m where rmkt,d,m is the value-weighted market return on day d of month m;
sign rei,d,m is a function whose outcome is equal to 1 if the excess return in day
d is positive and −1 if it is negative; V oli,d,m is the dollar volume for stock i on
day d of month m. For each month (or quarter) it is possible to find a measure
of market-wide liquidity by simply taking the average γ, namely ¯γm. This is what
will be next addressed as “level of liquidity”. From (1), following appropriate
steps, it is also possible to compute the “innovations in liquidity” factor Lt for
each month as the error term of regression (2), divided by 100:
∆¯γt = a + b∆¯γt−1+ c mt−1 m1 ¯ γt−1+ ut (2) Lt = 1 100ut (3)
These variables are directly available on a monthly basis on the WRDS platform.
4.2.3 Other Variables
VIX Index : The CBOE (Chicago Board Options Exchange) Volatility Index was first introduced in the paper of Whaley (1993) and is a measure to estimate the implied volatility of the S&P 500 index over the following 30 days. It is computed from a variety of options which have the S&P 500 as underlying security. It is
commonly addressed as the “fear index”, even though a high VIX does not always mirror a market downturn, but the expectation that the S&P 500 could sharply move either upwards or downwards.
Trade Size: In the TRACE database, each report also provides information about the size of a transaction. The largest transactions are usually made by market makers: financial firms that buy or sell large amounts of securities to keep the market liquid enough. When a market maker buys a bond, it will then sell it charging a higher price and gain the difference, called “margin”. Before the introduction of TRACE, investors could not see the actions of market makers, so the did not know how much margin they charged in the sale. With TRACE, instead, investors can identify a large transaction made by a market maker, read the price, and bargain on the resale price if the margin is too high. For this reasons, following increasing concern from market makers about the effects on their margins that a real-time disclosure as TRACE would have caused, the information on trade size has been limited: for high yield and unrated bonds, transactions greater than $1 million are censored and reported as “1MM+”; for investment grade bonds, transactions greater than $5 million are reported as “5MM+”. In this research it is assumed that in these cases the size of the transactions is exactly $1 million and $5 million respectively. Daily data are then averaged on a monthly basis.
Stock Return and Volatility: The variable that describes stock returns is the monthly return on the CRSP NYSE/AMEX value-weighted market index,
avail-able through the WRDS platform. The same measure has been employed in
Chordia, Sarkar and Subrahmanyam (2003), Goyenko and Ukhov (2009) and Bao, Pan and Wang (2011). This index includes securities such as common stocks and certificates listed on NYSE, AMEX, and Nasdaq National Market. Volatility in the stock market is computed as the standard deviation of daily returns of the same CRSP NYSE/AMEX value-weighted market index.
Bond Return and Volatility: Data about corporate bonds are taken from Datas-tream. In order to inspect this market, this research makes use of another index, namely the Bloomberg Barclays Intermediate U.S. Corporate Bond Index. This index is a specification of the broader Bloomberg Barclays U.S. Corporate Index that only contains those U.S. corporate bonds whose maturity is greater or equal to 1 year and less than 10 years. Furthermore, it only includes investment grade debt with $250 million or more par amount outstanding, issued by U.S. industrial, utility, and financial institutions. This index has been used also in Bao, Pan and
Wang (2011) to compute returns on the aggregate corporate bond market. Datas-tream provides the time series of daily prices for this index, and from those prices it is possible to compute the monthly percentage return as the average of daily returns, as follows: Bond returnm = 1 D D X d=1 Pd− Pd−1 Pd−1 (4) Where m is a month that goes from July 2002 to December 2015, D is the total number of days in a month, d is a day in month m and P is the price of the index. Volatility in the bond market is computed as the standard deviation of daily returns of Bloomberg Barclays Intermediate U.S. Corporate Bond Index in each month.
Bond Illiquidity: Bond illiquidity is computed by means of the Roll’s estimate. Firstly introduced in Roll (1984), this measure provides a method to estimate the effective bid-ask spread from prices. A security is exchanged either at the bid price or at the ask price, following a sell or a buy order, respectively. Roll asserts that the price then bounces back and forth between bid and ask until it stays at one end or the other of the spread. This effect is called “bid-ask bounce”. Starting from this theory and other assumptions, the effective spread at time t is computed as:
Rollt = 2p−cov (∆pt, ∆pt−1) (5)
Where ∆pt is the change in prices from t − 1 to t and cov is the covariance.
4.3
Sample and Descriptive Statistics
4.3.1 First Hypothesis
Table 1 contains summary statistics about the variables employed in this research.
The sample considered includes monthly data from July 1st 2002 until December
31st 2015. First, in order to build the yield spread variable, daily transactions
data of 104,055 corporate bond from 7,446 different issuers were collected. This led to a total of almost 139 million observations of which almost 4.5 million were canceled as described in Section 4.2.1.
Yield spread ranges from a minimum value of -0.33 to a maximum of 5. As shown in Panel A of Figure 1, the minimum is reached at the very beginning of the series. When this variable assumes negative values, it means that on average the Treasury yield to maturity is higher than that of corporate bonds. This would mean that in that period Treasury bonds were more profitable and risky than
Table 1: Summary statistics for first hypothesis
Table 1 reports summary statistics of the variables involved in the research: yield spread, in-novations in liquidity, levels of liquidity, VIX index and trade size. The yield spread is the difference between corporate bond yields to maturity and yields to maturity of Treasury bonds with the same maturity, averaged each month. Innovations in liquidity is the P`astor and Stam-baugh measure for monthly aggregate liquidity in the stock market as computed in Equation (3). Levels of liquidity is the monthly level of aggregate liquidity in the stock market as computed by Equation (1). The VIX index is the CBOE volatility index. The trade size is the average amount of dollars exchanged in the bond market. All variables are built on monthly basis.
Yield Spread Innovations in Liquidity Levels of Liquidity VIX Index Trade Size Smallest -0.3365 -0.2281 -0.3337 10.8176 231537 Largest 4.9992 0.1324 0.1343 62.6395 676272 Observations 162 162 162 162 162 Mean 2.0839 0.0029 -0.0261 20.0607 375729.4 Std Deviation 0.8260 0.0574 0.0701 9.0348 100555.2 Variance 0.6822 0.0033 0.0049 81.6285 1.01e+10 Skewness 0.7623 -1.0403 -1.4906 2.1102 0.7534 Kurtosis 4.2512 5.8589 6.9293 8.6163 2.6882
corporate bonds. Although this could indeed be the case in some circumstances, it is necessary to remember that the database from which corporate yields are taken (TRACE) has passed through three phases, and has started covering 99% of the secondary corporate bond market only from 2004. Already from August 2002 yield spreads are always above zero, consistently with the most general market behavior of corporate bond yields higher than Treasuries’. Thus, the negative spreads observed in the first month of the sample could simply be generated from a lack of data. Moreover, the peak in spreads is reached in December 2008 and the sharpest rise is the one from August to September of that very year. This is not surprising, given the default of Lehman Brothers in September 2008. Coherently with theory, the awareness that even banks “too big to fail” could go bankrupt, together with the effect that this had on the credit system in the U.S., led to a rise of corporate bond yields to compensate for the high risk implied in these securities.
As shown in Panel B of Figure 1, the variable of innovations in liquidity displays the typical shape of a stationary distribution. It moves around its mean of 0.0029
0 1 2 3 4 5 Yield spread 2002m1 2004m1 2006m1 2008m1 2010m1 2012m1 2014m1 2016m1 Panel A −.2 −.1 0 .1 .2 Innovations in liquidity 2002m1 2004m1 2006m1 2008m1 2010m1 2012m1 2014m1 2016m1 Panel B 10 20 30 40 50 60 VIX 2002m1 2004m1 2006m1 2008m1 2010m1 2012m1 2014m1 2016m1 Panel C 200000 300000 400000 500000 600000 700000 Trade size 2002m12004m12006m12008m12010m12012m12014m12016m1 Panel D
Figure 1: Time series plot
Figure 1 plots the variables Yield spread, Innovations in liquidity, VIX index and Trade size over time. On the x axis, “m1” stands for the first month of each year, i.e. January. The yield spread is the difference between corporate bond yields to maturity and yields to maturity of Treasury bonds with the same maturity, averaged each month. Innovations in liquidity is the P`astor and Stambaugh measure for monthly aggregate liquidity in the stock market as computed in Equation (3). The VIX index is the CBOE volatility index. The trade size is the average amount of dollars exchanged in the bond market.
following shocks in the general level of liquidity in the stock market. Its variance is 0.0033 and much of it is determined by the period that goes from the second half of 2007 to the beginning of 2009, when levels of aggregate liquidity were low as well. Coherently, the second and the third lowest points in the variable called “levels of liquidity” were both reached in 2008. Besides, this variable has the characteristics of a stationary distribution and its variance is 0.0049.
The distribution over time of the VIX index, displayed in Panel C of Figure 1, has a peak at the end of 2008, as expected. At that time, it was rational to have bearish outlooks about the movements of the S&P 500, given the tough situation in which financial markets were heading. Once more, much of the variation over time is due to the period corresponding to the crisis.
Ultimately, the movements in the average trading size seem to be divided in three parts: from July 2002 until the end of 2004, from 2005 to the first half of 2008 and from the second half of 2008 until the end of the sample. Once more, it is worth remembering that the TRACE database suffered of a lack of data until the beginning of the third phase of its development, which could explain why the average trading volume is so high in that period. More specifically, this could be the effect of reporting much more trading activity from market makers than from other agents. Later, this measure drops with the onset of the financial crisis and remains stable until the end of 2015. This shows that during the crisis smaller amounts of bonds were bought or sold, either because of a lack of confidence in these instruments or the will of holding them for safety reasons.
4.3.2 Second Hypothesis
Consistently with the previous analysis, the sample considered spans from July 2002 to December 2015 with monthly frequency. Table 2 displays summary statis-tics of the measures employed. As far as stocks are concerned, the variable that describes their returns is the return on the CRSP NYSE/AMEX value-weighted market index. Its values go from −19% to +11.5%, with a negative skewness of −0.841 which makes the left tail of the distribution longer than the right one. Volatility in the stock market is computed as the standard deviation of daily returns of the same CRSP NYSE/AMEX value-weighted market index. As pre-dictable, the peak of this time series, which is equal to 0.0496, is reached in October 2008, right after the filing for Chapter 11 by Lehman Brothers in the month of September of the same year. As measures of equity market aggregate liquidity, this research employs the abovementioned “innovations in liquidity” and “level of
liquidity” by P`astor and Stambaugh (2003). As Table 3 shows, liquidity in the
stock market is negatively correlated with bond market volatility (−0.2892) and illiquidity (−0.3475).
As far as the fixed income market is concerned, bond returns are much less volatile than stock returns: they vary from a minimum of −0.28% to a maximum of 0.18% and their variance is close to zero. Their volatility touches a peak at the end of 2008, as well as stock volatility. Table 3 shows that the correlation between volatilities across markets is 0.6813, which highlights a strong linkage in return volatility between stocks and corporate bonds. As corporate bonds are risky securities, even though less than stocks, it is logical to expect that returns become more volatile in both markets in times of general uncertainty as at the beginning
Table 2: Summary statistics for second hypothesis
Table 2 reports summary statistics of the variables involved in the inspection of the second hypothesis: S RET is the monthly return on the CRSP NYSE/AMEX value-weighted market index, S VOL is the volatility of daily returns of the same index, PS I is the P`astor and Stam-baugh (2003) “innovations in liquidity” as described by Equation (3), PS L is the P`astor and Stambaugh (2003) “level of liquidity” as described by Equation (1), B RET is the monthly return on the Bloomberg Barclays Intermediate U.S. Corporate Bond index, computed as described in Equation (4), B VOL is the volatility of daily returns on the same index and B ILLIQ is the measure of illiquidity in the bond market as measured by the Roll’s estimate in Equation (5).
S RET S VOL PS I PS L B RET B VOL B ILLIQ
Smallest -0.188 0.0031 -0.2281 -0.3337 -0.0028 0.0948 0.0086 Largest 0.1152 0.0496 0.1324 0.1343 0.0018 0.7299 0.7134 Observations 162 162 162 162 162 162 162 Mean 0.0049 0.0104 0.0029 -0.0261 6.15e-7 0.2283 0.181 Std Deviation 0.0426 0.0069 0.0574 0.0701 0.0006 0.0963 0.1105 Variance 0.0018 0.0001 0.0033 0.0049 3.61e-7 0.0093 0.0122 Skewness -0.841 2.7747 -1.0403 -1.4906 -0.669 1.5383 1.4905 Kurtosis 5.3623 13.0166 5.8589 6.9293 6.9272 7.2571 6.861
of the financial crisis. This is consistent with the findings of Goyenko and Ukhov (2009) and Fleming et al. (1998) who, instead, inspect the connections between Treasury bonds and U.S. stocks and point out that correlations in volatility are indeed strong and positive. Moreover, stock volatility is highly correlated with bond illiquidity (0.445).
Ultimately, bond illiquidity spans from a minimum of 0.0086 to a maximum of 0.7134, reached in September 2008, as predictable from the already mentioned reasons. The description of these variables and their correlations makes it possible to argue that many of the measures employed covariate across markets, and are probably subject to the same determinants and react in similar ways to the same events, as it was in the case of the financial crisis.
Table 3: Correlation matrix
Table 3 reports the correlation matrix for the variable involved in the inspection of the second hypothesis: S RET is the monthly return on the CRSP NYSE/AMEX value-weighted market index, S VOL is the volatility of daily returns of the same index, PS I is the P`astor and Stam-baugh (2003) “innovations in liquidity” as described by Equation (3), PS L is the P`astor and Stambaugh (2003) “level of liquidity” as described by Equation (1), B RET is the monthly return on the Bloomberg Barclays Intermediate U.S. Corporate Bond index, computed as described in Equation (4), B VOL is the volatility of daily returns on the same index and B ILLIQ is the measure of illiquidity in the bond market as measured by the Roll’s estimate in Equation (5).
S RET S VOL PS I PS L B RET B VOL B ILLIQ
S RET 1 S VOL -0.4543 1 PS I 0.2429 -0.3402 1 PS L 0.2108 -0.4361 0.7205 1 B RET 0.2870 -0.0300 0.1045 0.0320 1 B VOL -0.2720 0.6813 -0.2892 -0.3493 -0.1886 1 B ILLIQ -0.3208 0.4450 -0.3475 -0.3139 -0.3448 0.6813 1
5
Methodology
5.1
First Hypothesis
The inspection of the effect that liquidity in the stock market can have upon yield spreads of corporate bonds is firstly approached by means of the following time-series Ordinary Least Squares (OLS) regression:
Y ield spreadt= β0+ β1Liquidityt+ β2V olatilityt+ β3T rade sizet+ t (6)
Where Y ield spreadt is the dependent variable, indicating the monthly average
difference between yields of corporate bonds and Treasury bonds, β0 is a constant,
Liquidityt represents one of the liquidity factors already mentioned, that is
“Inno-vations in liquidity” or “Level of liquidity”, V olatilityt stands for the VIX index,
T radesizet is the monthly average trade size and t is the error term.
The factor of major interest is β1: the effect of stock market liquidity on yield
spreads. The VIX index is a variable derived both from the stock market, as it mirrors current expectations about movements in the S&P 500, and the derivatives market, as it is computed by means of calls and puts prices whose underlying asset is the S&P 500. Thus, the effect of the VIX index on yield spreads is still part of
the research question. Moreover, the average trade size is added to the regression as potential explanatory variable of the yield spread, as a higher trade size could imply a higher level of liquidity in the bond market and, consequently, a lower yield due to a lower liquidity risk.
5.2
Second Hypothesis
5.2.1 Vector Autoregressive Model
The second hypothesis is inspected by means of a Vector Autoregressive model. A Vector Autoregression, henceforth VAR, is an econometric method introduced by Christopher Sims in 1980 that seeks to identify linear interdependencies among several time series. VAR models are an expansion of univariate autoregressive (AR) models, where a variable is explained by means of its own lagged values. Christopher Sims highlighted the efficiency of VAR models in macroeconomic re-search, showing that they do not need any underlying economic theory to be per-formed and are much less demanding in terms of restrictions that the structural models previously in use in literature. The only theoretical requirement needed to perform a VAR is the hypothesis that a set of variables is able to affect each other over time.
The first thing to settle is the number of lags to consider. This step is essential to inspect the stationarity of the time series and finally set up the VAR. Table 4 shows four information criteria used for this purpose, namely FPE, AIC, HQIC, and SBIC, together with the log likelihood (LL) and a sequence of likelihood ratio
Table 4: Lag selection criteria
Table 4 reports lag selection criteria to use in the VAR(2) model: LL is the log likelihood, LR is the LR statistic, FPE is the final prediction error, AIC is the Akaikes information criterion, HQIC is the Hannan and Quinn information criterion and SBIC is the Schwarzs Bayesian information criterion. The best number of lags according to each criterion is marked with a ‘*’.
Lag LL LR FPE AIC HQIC SBIC
0 3178.43 1.5e-25 -57.2606 -57.2606 -57.2606
1 3335.99 315.12 3.1e-26 -58.7993 -58.5159 -58.1015*
2 3395.21 118.43 2.3e-26* -59.0932* -58.5264* -57.6976
3 3425.17 59.917* 2.5e-26 -59.0157 -58.1665 -56.9233
tests (LR). Each of the five criteria chooses a number of lags to maximize the log likelihood and they differ from each other for the penalization that the inclusion
of an additional lag would cause. In this case, L¨utkepohl (2005) versions of the
information criteria are employed. The likelihood ratio statistics (LR) compare the values of the likelihood function under the null and the alternative hypothesis of the VAR model. Although strictly speaking it is not an information criterion, it is widely used in literature to determine the right number of lags, and thus
included in this analysis. Letting LLj be the value of the log likelihood with j
lags, the LR statistic for lag order j is computed as:
LRj = 2 (LLj − LLj−1) (7)
The final prediction error is a criterion that simulates various specifications of the model and, comparing them, indicates the best number of lags to use. In
L¨utkepohl (2005) it is defined as:
F P E = |X | T + Kp + 1 T − Kp − 1 K (8) Where P
is the maximum likelihood estimate of Et,
0
t
12
, T is the number of observations, K is the number of equations and p is the number of lags. AIC, SBIC and HQIC are all very similar and just treat differently the addition of a lag parameter. Their differences are easier to understand by looking at their computation: AIC = ln|X | + 2 TpK 2 (9) SBIC = ln|X | +ln (T ) T pK 2 (10) HQIC = ln |X | +2ln [ln (T )] T pK 2 (11) SBIC tends to underestimate the optimal number of lags, while AIC tends to do the opposite. Because of these concerns, it is useful to compare all the criteria. In Table 4, the optimal lag for each parameter is indicated with a ‘*’ and it is possible to notice that three out of four criteria suggest to use two lags.
Having established that the number of lags must be equal to two, the second step of the analysis consists in inspecting the stationarity or non-stationarity of all the variables employed. This verification is important to define which kind of VAR model should be used. In fact, if all the variables are stationary, then a standard VAR should be performed. Conversely, if all the variables are non-stationary,
12E [·] is the expected value and
then a Vector Error Correction Model (VECM) would be more appropriate. In this research, the order of integration of the variables in use is inspected by means of the Augmented Dickey-Fuller (henceforth, ADF) test. The ADF test starts by performing the following regression:
∆yt= β0 + δyt−1+ γ1∆yt−1+ · · · + γp∆yt−p+ ut (12)
where ∆yt are changes in y, β0 is a constant and δ is the coefficient to test. Under
the null hypothesis, δ = 0, the series has a unit root and thus it is non-stationary. Under the alternative hypothesis, δ < 0, the series does not have a unit root and thus it is stationary. To test the null hypothesis, the following test statistic, called “ADF statistic”, must be employed:
ADF stat = δˆ
SE ˆδ
(13)
where ˆδ is the OLS estimate of δ in regression (12) and SE stands for “standard
error”. In the dataset of this thesis, the null hypothesis is rejected for all the variables considered, namely stock and bond return, volatility and liquidity. They are stationary and hence a standard VAR model can be applied.
A VAR model is a generalization of the easier Autoregressive (AR) model. An AR(p) model seeks to explain the dependence of a variable through its own p lagged values and a stochastic term. In the case of, e.g., p = 2, the AR model is as follows:
yt= c + ϕ1yt−1+ ϕ2yt−2+ t (14)
As far as a VAR(p) model is concerned, multiple variables and their lagged values are involved in the explanation of all the other measures. In this case, the linear equation (14) now becomes a system of equations that can be visualized by means of vectors and matrixes, hence the name “Vector Autoregression”. When p = 2 and the variables are six, the VAR(2) is as follows:
y1,t y2,t .. . y6,t = c1 c2 .. . c6 + ϕ11,1 ϕ11,2 · · · ϕ11,6 ϕ12,1 ϕ12,2 · · · ϕ12,6 .. . ... . .. ... ϕ1 6,1 ϕ16,2 · · · ϕ16,6 y1,t−1 y2,t−1 .. . y6,t−1 + ϕ21,1 ϕ21,2 · · · ϕ21,6 ϕ22,1 ϕ22,2 · · · ϕ22,6 .. . ... . .. ... ϕ2 6,1 ϕ26,2 · · · ϕ26,6 y1,t−2 y2,t−2 .. . y6,t−2 + 1,t 2,t .. . 6,t (15)
Where in this case y’s are: stock returns, stock return volatility, stock liquidity, bond returns, bond return volatility and bond illiquidity. The number of lags
chosen for this research is two, ckare constant terms, ϕ1k,k are the factors associated
with the first lags, ϕ2k,k are the factors associated with the second lags and k,t is
the vector of disturbances.
The interpretation of this VAR(2) model can be derived by means of two tools: Granger causality tests and impulse response functions.
5.2.2 Granger Causality
The Granger causality test was introduced in Granger (1969) and is a method employed not to find any causality relation among variables, although the name
may suggest so, but to inspect whether past values of one variable, e.g. y2, are
useful to predict current values of another variable, e.g. y1. The null hypothesis
of this test is that the estimated coefficients (in this example ϕp1,k on the lagged
values of y2 are jointly zero. If the null hypothesis is rejected, then it is possible
to infer that y2 Granger-causes y1 and thus y2 can predict y1.
5.2.3 Impulse Response Functions
Even though Granger causality results are useful to determine lead-lag relation-ships among variables, they do not take into account the dynamic properties of each measure and how they evolve through time and affect each other. For this purpose, it is possible to compute impulse response functions (IRFs) from the VAR(2) model employed. There are several types of IRFs and for the aims of this research orthogonalized IRFs will be used, as in Chordia, Sarkar and Subrah-manyam (2003) and Goyenko and Ukhov (2009). In case of orthogonalized IRFs, the impulse is represented by a one-time increase of one standard deviation in the
endogenous y2 variable and the response is the effect on the dependent variable
y1.
6
Results and Discussion
6.1
First Hypothesis
Table 5 reports the results of various regression specifications. In column (1) only innovations in liquidity is regressed against yield spread, in order to inspect its single effect. The point estimate is −2.95 and is significant at 10% level. As expected, innovations in stock market liquidity have an impact on the level of average monthly yield spread: an increase of one in the measure of innovations in
liquidity leads to a decrease of −2.95 percentage points in the average yield spread. Given that the measure of innovations in liquidity ranges from a minimum of −0.223 to a maximum of 0.13 and that the mean of yield spread is 2.08, a decrease
of 1 in the P`astor and Stambaugh measure of liquidity would mirror a significant
positive shock in the aggregate liquidity in the stock market which would influence corporate bonds, making them less risky and thus lowering the difference between their yields and those of Treasury risk-free bonds. Moreover, this regression shows a predictive power of 3.6%, expressed by its adjusted R-squared.
Results of column (1) seem to prove true the first hypothesis of this research, but column (2) does not confirm this finding. Employing a different measure of
Table 5: OLS Regression
Table 5 reports OLS estimates of regression (6). The sample covers 162 months from July 2002 to January 2015. Yield Sread is the difference between corporate bond yields and Treasury yields, PS I is the P`astor and Stambaugh (2003) “innovations in liquidity” as described by Equation (3), PS L is the P`astor and Stambaugh (2003) “level of liquidity” as described by Equation (1), VIX is the CBOE VIX index and VOL is the monthly average volume of transactions in the bond market.
(1) (2) (3) (4) (5)
Yield Spread Yield Spread Yield Spread Yield Spread Yield Spread
PS I -2.95∗ -0.63 -3.62∗∗∗ -1.91∗∗ (-1.78) (-0.51) (-3.19) (-2.12) PS L -2.03 (-1.49) VIX 0.048∗∗∗ 0.034∗∗∗ (6.64) (6.83) VOL -0.0000061∗∗∗ -0.0000054∗∗∗ (-14.71) (-12.48) cons 2.09∗∗∗ 2.03∗∗∗ 1.13∗∗∗ 4.38∗∗∗ 3.46∗∗∗ (32.29) (32.59) (9.00) (23.90) (15.11) N 162 162 162 162 162 adj. R2 0.036 0.024 0.277 0.581 0.697 t statistics in parentheses ∗ p < 0.10,∗∗ p < 0.05,∗∗∗ p < 0.01
aggregate liquidity, such as “levels of liquidity”, results in a point estimate that is not statistically significant. It is indeed not possible to deny that the effect of the stock aggregate level of liquidity on corporate yield spreads is different from zero. The difference in these outcomes derives from the substantial difference between the two measures. Although they are derived from the same regression (1), and they both are liquidity measures, the variable “innovations in liquidity” captures an aspect that “levels of liquidity” neglects: the unanticipated innovations. In
more statistical terms, this is evident if we look at (3): Lt is computed as the
average of the error terms in (2), and these error terms contain all those elements that are not identified as explanatory variables in regression (2) but affect in
some way the variations in the average γ at time t (∆¯γt). What these outcomes
demonstrate is that corporate bond investors do not react to anticipated changes in the aggregate level of liquidity of the stock market, but yields drop if an unpredicted positive change in aggregate liquidity takes place. What eventually takes place when investors see an unexpected rise in aggregate liquidity is a higher average return in the corporate bond market: when stock market liquidity rises, yields fall because their quality is now higher; as a result of this higher quality, risk aversion of investors falls, thus increasing the demand of bonds which, in turn, leads to higher prices and returns. This positive relation between liquidity factors and bond returns is consistent with previous findings by Lin, Wang and Wu (2011).
Columns (3) and (4) show estimates for two regressions built on regression 6 with the addition of one of the remaining variables. Both specifications show that both VIX index and trading volume can explain variations in yield spreads. The most comprehensive analysis is thus represented in column (5), where all variables are aggregated. The fifth specification in fact confirms that innovations in stock market liquidity play a role in defining yield spreads and, additionally, provides the effects of VIX and bond trading volume. In particular, VIX is positively related to yield spreads while trading volume has a negative effect on them. The role of VIX is especially significant to the end of this research: it is a measure that expresses uncertainty and is derived from the stock market. Evidence shows that an increase of one unit in the VIX index leads to an increase in corporate yield spreads of 0.034, with a level of significance of 0.1%. It is thus arguable that when the stock market becomes volatile and investors disagree more about the future value of the S&P 500, the risk associated with a corporate bond investment rises and drives up yields while lowering prices and returns. On the other side, the monthly average trading volume has a negative relation with yield spreads. Intuitively, the bond trading volume can be seen as a measure of bond market
liquidity: the higher the number of bonds exchanged on average in a certain month, the higher the liquidity in the market. It is then logical to expect that higher liquidity in the bond market is a result of lower risk aversion with respect to bonds due, in turn, to higher quality. When investors perceive that bonds are high quality securities, the level of risk of these financial instruments falls as well as yields and yield spreads. The point estimate for the effect of average trade size on spreads is equal to 0.0000054, meaning that just one more dollar spent (on average) would cause a decrease in spreads of 0.054 basis points. That is of course a very small number, but considering that the average trade size, as reported in Table 1, is equal to $375729.4 and the average yield spread is 2.08%, it is utterly comprehensible that just one more dollar of transaction can only make a small difference.
Overall, the explanatory power of specification (5) equals 69.7%. This adds a relevant +42% to specification (3) where only innovations in liquidity and VIX are considered. This provides evidence that the combination of variables chosen can sufficiently describe the variation in yield spreads. Unexpected aggregate liquidity changes in the stock market and expectations about future movements in the S&P 500 are two stock-related measures that can affect the bond market, as this research wanted to demonstrate. Much of the previous literature which aims to explain expected returns, both with respect to stocks and bonds, only focuses on measures derived within the same market and neglects cross-market influences. What this research wants to show is that corporate stocks and bonds are more closely linked than expected, and that their mutual interaction should be taken into account when inspecting any particular feature of one market or the other. Having proved true the predicted effect that these two markets influence each other, the next subsection analyses in more detail what are the major drivers of such influence and whether these drivers have any predictive power.
6.2
Second Hypothesis
6.2.1 Granger Causality
Table 6 shows χ2 statistics of pairwise tests between the six variables of this
analysis, together with the associated p-values. The maximum significance level of 10% is employed in the interpretation of results, which means that if a p-value is lower than or equal to 0.1, the null hypothesis of the test is rejected. Table 6 shows twelve Granger causality tests in which the null hypothesis is rejected, half of which involves two variables that are not in the same market. More specifically, corporate
