Funding liquidity risk and the stock exchange
By Leonardo Spina Student Number: 6159303 May 2015 Supervisor Dirk Damsma AcknowledgementsI would like to express my gratitude to everybody that helped me during my bachelor. In particular, I would like to thank the Student Advisors, my thesis Supervisor, the Board of Examiners and the teachers who inspired me, besides my parents, friends and work colleagues.
Statement of Originality
This document is written by student Leonardo Spina, who declares to take full responsibility for the contents of this document.
I declare that the text and the work presented in this document are 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.
Introduction
In valuation theory the main drivers are the different types of risks financial assets imply.In this way, the CAPM and its extensions focus on the economic situation, the market value of a company and its historical performance - among other factors - when valuing stocks.
This and other models have not considered the impact traders have in the market. In fact, according to some academics, investors’ actions may enhance the depth of a crisis due to their borrowing constraints, a key driver for their investment decisions.
This study will try to measure the impact of traders and lenders in the stock market by adding a new explanatory variable to the CAPM.As for the structure, this paper will firstly explain a theoretical framework, according to which models need to measure the traders’ effect on the stock market when they experience capital constraints. The theory will be followed by the description of this constraint as a new priced risk in the stock exchange.
Next, several proxies will be compared in order to identify the best empirical variable that can be obtained for the funding liquidity risk; then, this risk will be put into the 4-factor model of Carhart.
Several multi-variable regressions will test the performance of the new model when adding this innovative factor (FLR) and its results will be analyzed.
The following chapter will sum up the previous steps and will conclude the study.
Finally, the shortcomings of this article will be deconstructed from different perspectives, discussing and offering improvement possibilities.
1) Chapter 1: Literature Review.
Among the basic and most explained methods to price a financial asset in academics, the linear regression Capital Asset Pricing Model (CAPM) is the masterpiece pillar in valuation theory. Although it represents an important formula upon which many professionals base their intuition for making investment decisions, it exhibits some drawbacks as discussed next.
a) Limitations of Capital Asset Pricing Models.
Firstly, it is necessary to provide a review on the formula of the academic tool. The CAPM measures the excess return of the stock market compared to the risk free asset interest rate (usually treasury bills) (Berk and DeMarzo, 2011, pp. 135-136). It has been extended with both the book–to–market (SMB) and market capitalization (HML) regressors to become the Fama and French 3-factor model. Subsequently, the theory has been improved with the momentum (WML) factor by Carhart (Carhart’s 4-factor
More in detail, the mentioned extensions of the CAPM give more insights about anomalies across stocks in the market, therefore leading to major improvements. Although these models present simple ways of calculating stocks’ expected returns, their empirical validity has always been a matter of discussion.
In fact, the comparison published by the renowned Fama and French (2004), provides clear evidence on how limited these asset pricing models are. Despite the statistical improvement among them when adding more explanatory variables, they do not represent a useful tool in practice.
Particularly when measuring the returns of equity mutual funds, Bello (2008) provides more insights in this comparison: the equation of Fama and French predicts significantly better returns than the original CAPM; and -with the additional momentum variable- the equation of Carhart considerably ameliorates the Fama-French 3-factor model’s results. On the other hand, a small improvement comes into sight when considering their statistical explanatory power, approximately an R2 of 71% for all of them. Thus, there is not any remarkable difference among them with respect to their goodness of fit, emphasizing in this way their limitations in calculating returns.
Besides, an alternative methodology using macroeconomic variables has been proposed by Aretz, Bartram and Pope (2005), and compared to the Fama-French and Carhart Models. The results imply that the Macroeconomic Fundamentals model is equally as useful as the previous two. In fact, by studying benchmark portfolios classified by book-to-market, market capitalization and momentum, the authors found evidence that the Fama-French and Carhart indirectly represent macroeconomic information in their equations, such as economic growth, unexpected inflation, raw materials price index, and yet more variables.
As for their performance, the Carhart model statistically proves its superiority once more due to the meaningful momentum factor. It is followed by the Macroeconomic Fundamentals model, which finally outperforms the Fama-French equation. However, the model based on macroeconomic factors shows more stable results when applying different types of tests, making it generally slightly more reliable than Carhart’s.
Whether the variables chosen for the model ought to be directly related to the stock exchange or rather to macroeconomic fundamentals, might seem less of a dilemma when considering the probability of the individual model. For instance, according to Avramov (2002, p. 2), a crucial weakness of the linear pricing models resides in their predictability of stock returns. One underlying reason is found in the uncertainty of the model, which the above asset pricing models do not account for, despite the fact of containing significant explanatory variables.
In order to solve for model uncertainty, Avramov proposes an attractive Bayesian approach, which assigns probabilities to a set of asset pricing models.
Interestingly, by using a ‘weighted forecasting model’, the author encounters an important result: by giving probabilities to the models, the power of some explanatory variables becomes meaningless. Indeed some regressors that were significant within the individual models turn out to be less important when taking the model uncertainty into consideration. Therefore it is more complicated deciding to what extent a variable becomes ‘explanatory’ or important in your model.
Avramov concludes the Bayesian model presents better results than using one single model, but its analysis goes beyond the scope of this paper for the sake of simplicity. All in all, the simple CAPM model presents important shortcomings that have been approached mainly by its extensions in a satisfactory manner. A model containing many probabilistic models or other innovative ideas, such as a macroeconomic model, could also be very complete in explaining the stock market. However the main advantage of the 4-factor model stays its simplicity, its powerfulness relies mainly in the ease of being understood and being recalled from the earliest to the latest stages of our careers.
b) The importance of Funding Liquidity Risk (FLR).
When researching for further improvements of the Carhart model – the 4-factor CAPM model -, some authors make the reader aware of the implication of trading agents to risk, often overlooked.
In particular, Pastor and Stambaugh (2003) and Chordia, Sarkar and Subrahmanyam (2005) claim that this model does not take into account the liquidity effect of trading. In reality – these authors show - trading has a considerable effect on the stock markets due to the liquidity risk of securities or stocks.
The latter is known as the asset liquidity risk, which is described as the ease with which an asset is traded for cash in order to prevent losses (Lee, 2013).
Studies prior to the credit crunch of 2007-08, already consider the effects and complexity of asset liquidity risk. For instance, stocks are closely related to their inherent liquidity and volatility, and so do treasury bonds. However, also these two assets are positively correlated with each other, exposing both to a systemic risk. That is to say. the general economic situation translates into a common effect across different financial asset classes (e.g. stocks, treasury and corporate bonds, forex, options and other derivatives). Also monetary policy enormously affects changes in liquidity of an economy, therefore reducing stocks’ risk when choosing for an expansionary policy during crises (Chordia et al).
In addition to this, Pastor and Stambaugh (2003) found evidence on the commonality effect across stocks and acknowledge a noticeable sensitiveness for aggregate liquidity, especially in small stocks.
The historical global financial crisis of 2007-08 suggested there is a noteworthy shortcoming in the methodology used at the moment. Brunnermeier and Pedersen (2009) propose an extra variable that changes our point of view from the risks in assets to the one related to investors: the funding liquidity risk (FLR).
The funding liquidity is defined as the capacity of traders to raise funds to actually make their investments, thus the FLR increases together with the borrowing constraints of investors. In the exchange markets, leveraged positions depend on capital margin requirements, which change constantly according to the broker’s decision to provide capital (Lee).
The effect of funding liquidity risk is enhanced notably during economic downturns, as stated by Brunnermeier and Pedersen (2009) in their insightful theory. When the financial markets suddenly suffer a liquidity slump - due to circumstances such as the subprime mortgage crisis - then the asset liquidity risk escalates leading to higher required margins. If traders are tight on capital, they have to sell off some of their securities to cover those margins; thus providing enough collateral for their positions. By reducing their positions in the market, investors aggravate the market illiquidity, so the systemic liquidity risk increases even further. Hence brokers and other capital providers need to raise their margins again.
Finally, this mechanism deteriorates more the tightness of capital and causes a higher volatility and uncertainty across different asset classes (stocks, bonds, mutual funds and others), which repeatedly leads to more sell-offs and to a larger market-wide liquidity risk. The phenomenon is known as ‘liquidity spiral’.
In short, besides the ameliorations of the CAPM model that Fama, French and Carhart provided, some other authors studied capital constraints in investments. They found enough evidence of its importance and claim traders’ borrowing limits have a big impact in liquidity. In view of this extended concept of risk, this paper aims at identifying an appropriate measurement for the funding liquidity risk and at adding it to the 4-factor model. Therefore, a new equation with 5 factors will be tested for the significance of the new regressor and its performance.
2) Chapter 2: Methodology
The purpose of this chapter is to select the most convenient measure for our study among different approaches encountered in the literature after describing their advantages and disadvantages. Then, the selection of data and the construction of the FLR will be explained, followed by data contemplated but not selected for the regression in this paper.
a) Seeking a benchmark.
When studying the effect of funding in the financial markets, one encounters a major difficulty, this is, measuring the funding liquidity or its risk. This section concentrates on diverse methods and their contribution to attain the factor we are looking for.
Ideally, the funding liquidity should be measured by the stochastic distribution of its originating risk factors, this is, the required margins. As seen previously, the higher these margins, the more capital constraints faced by speculators.
Since they are specific to each trader - they are determined in a case-by-case basis -, it is enormously complicated to estimate these distributions (Lee, 2013).
Moreover, required margins are not easily accessible because they are not public information, making the task of obtaining them more arduous, even for banks with confidential information (Drehmann and Nikolaou, 2013). Hence, it is necessary to apply proxies to calculate this effect. Besides, required margins are almost an impossible method to implement: they are not only unavailable information, fixed for traders individually and at a fast pace; also, the tool used for calculating them is usually the SPAN (Standard Portfolio Analysis of Risk) of the CME Group. The SPAN implies an algorithmic market simulation based on the Value at Risk (VaR) and it calculates the maximum likely loss of a trader’s portfolio, therefore considering several asset classes, and it is used in more than 50 exchanges worldwide (CME Group, 2010). Because margin requirements are unique to each trader and are calculated according to their capital invested in different assets – stocks, bonds, derivatives or others – including 50 different countries, it follows that the chances of obtaining or computing this information are extremely low.
Next, a possible proxy for the tightness of funding could be the broker-dealer leverage factor, because a higher leverage on their behalf would mean a higher ability to borrow and so to invest. However, according to Fontaine, Garcia and Gungor (2013), leverage and funding liquidity are not always correlated, since their results vary depending on the type of portfolio applied to. For instance, the leverage factor explains roughly 1% of size-sorted portfolios and 85% of book-to-market-sorted portfolios.
Thus, it is not an acceptable general measure.
Instead, the authors value the funding liquidity factor based upon differences among panels composed by on-the-run and off-the-run treasury bonds of similar characteristics. The intuition behind it assumes that a recently issued bond (on-the-run) is more liquid than a similar bond previously issued (off-the-run) with the same maturity and cash flows. The lack of their arbitrage entails a difference in price, apparently revealing frictions in funding. Nevertheless, this link has not yet been tested for value and momentum portfolios, leaving that study uninvestigated.
Another intuition suggests that, when financial institutions are short of capital, they can demand money to the lender of last resort, the central bank. Accordingly, the larger the lack of funding, the more aggressively banks will bid for liquidity by offering higher rates - above the central bank expected marginal rate. It follows that if the banks bid higher prices during the Open Market Operations, then an increase in funding liquidity risk is exhibited.
For this purpose, Dehmann and Nikolaou calculate the so called ‘aggregate liquidity risk insurance premia’ (LRP), which is translated as an average insurance premium of banks against the risk of funding liquidity.
Unfortunately, these tenders are not publicly available, so once more a problem due to lack of data arises. Although the LRP can be proxied with data published by the central bank, large differences occur during crises between the expected and the actual values, in this way, not yielding appropriate results.
In addition, it would be logical to measure the banks’ capital constraints through the money market spreads, such as Euribor-OIS or Libor-OIS. Although they are easily accessible, they present important shortcomings when considering as a proxy. In fact, these spreads are subject to manipulation of interbank rates, they lack essential market information, especially during the crises, and therefore they do not reflect reality in the markets. For instance, such spreads do not expose any information about the volume banks need to obtain to remain liquid; so they remain invalid for our goal.
As previously mentioned, a contagion effect - known as commonality effect - takes place across different stocks and across different asset classes. Besides, systemic liquidity of all assets depends on the speculators’ funding liquidity, especially for assets with higher margins. Therefore, once the markets suffer a slide, traders sell off capital-intensive positions first in order to protect their positions in assets with lower margin requirements.
This practice implies a larger volatility in both market liquidity and returns for the assets which present higher funding sensitivity; in bad times, this implied difference between low-margin and high-margin securities is called ‘flight to quality’ (Brunnermeier and Pedersen, 2009).
It follows that a different approach must be used to tackle the measurement problematic of the funding liquidity in the financial markets. Once again, the point of view changes from the market to the agents, using the consequence of traders’ actions rather than the ways they finance their decisions (Lee, 2013).
Focusing on the flight-to-quality concept and being specific to stocks, speculators prefer to trade in large stocks than in the small ones due to a larger volatility, and so to larger required margins, of the latter. In this way, Lee measures the stocks’ sensitivities to funding changes by calculating the rolling correlations of stock market returns with asset liquidity of large and small stocks, whose definitions used for this paper are explained in the next section.
More specifically:
𝜌𝑙𝑎𝑟𝑔𝑒 = 𝑐𝑜𝑟𝑟(𝑖𝑙𝑙𝑖𝑞𝑢𝑖𝑑𝑖𝑡𝑦 𝑖𝑛 𝑙𝑎𝑟𝑔𝑒 𝑠𝑡𝑜𝑐𝑘𝑠, 𝑠𝑡𝑜𝑐𝑘 𝑚𝑎𝑟𝑘𝑒𝑡 𝑟𝑒𝑡𝑢𝑟𝑛𝑠) 𝜌𝑠𝑚𝑎𝑙𝑙 = 𝑐𝑜𝑟𝑟(𝑖𝑙𝑙𝑖𝑞𝑢𝑖𝑑𝑖𝑡𝑦 𝑖𝑛 𝑠𝑚𝑎𝑙𝑙 𝑠𝑡𝑜𝑐𝑘𝑠, 𝑠𝑡𝑜𝑐𝑘 𝑚𝑎𝑟𝑘𝑒𝑡 𝑟𝑒𝑡𝑢𝑟𝑛𝑠) where illiquidity is estimated by Amihud’s (2002) measure, or else by:
𝐼𝑙𝑙𝑖𝑞(𝑖𝑛𝑡𝑒𝑟𝑑𝑎𝑦)𝑖 = |𝑅𝑒𝑡𝑢𝑟𝑛(𝑖𝑑)𝑖|/[𝐷𝑜𝑙𝑙𝑎𝑟 𝑉𝑎𝑙𝑢𝑒 𝑉𝑜𝑙𝑢𝑚𝑒 (𝑖𝑑)𝑖],
being, for stock i, R(i)id the daily return and DV(i)id the interday dollar trading volume. Finally, the funding liquidity of stock i can be expressed as the difference in the correlations of large and small stocks.
𝑓𝑙𝑟(𝑠𝑖𝑧𝑒) = 𝜌𝑙𝑎𝑟𝑔𝑒 − 𝜌𝑠𝑚𝑎𝑙𝑙
Furthermore, since margin requirements are also decided upon changes in volatility, it also is appropriate to measure funding liquidity with respect to this uncertainty source (Lee). As a result the following formula should be proposed as well:
𝑓𝑙𝑟(𝑣𝑜𝑙𝑎𝑡𝑖𝑙𝑖𝑡𝑦) = 𝜌𝑙𝑜𝑤 𝑣𝑜𝑙 − 𝜌ℎ𝑖𝑔ℎ 𝑣𝑜𝑙
This section has targeted a benchmark for the funding liquidity risk.
The equations provided above contain data that is publicly available and can be calculated effortlessly; also, they consider the commonality across stocks and do not depend on each single trader. Hence, these formulae are the best possible solution for many complex problems other proxies could not tackle.
Although this method may contain some fundamental drawbacks - that will be pointed out in the last chapter – its simplicity represents an enormous advantage in the research of investors’ capital constraints. Thus, this study will continue with the extraction of Lee’s FLR instrument.
b) How to calculate the FLR?
Subsequently, we proceed in the finding of an empirical measure for the FLR with public information and replicable by others.
As for the data, the 4-factors of the Fama-French-Carhart model, the stocks’ returns to form the small-cap, the large-cap and the total market index portfolios, and the S&P 500’s returns were all retrieved from CRSP database. However, the exchange-traded funds’ (ETFs) and Russell’s returns were obtained at Yahoo Finance due to the CRPS account access limits.
For our purpose, it is first necessary to form a portfolio of large-cap and small-cap shares. As explained by Lee, a flight to quality should be evident from small, riskier stocks to the large safe-haven ones in times of turbulence. In other words, these two portfolios are formed to separate shares into two types: the most and the least collateralizable shares, based on the assumption that large stocks require lower margins than small stocks.
In this case the portfolios are based upon the ones presented by the CRSP database, in this way providing the largest sample attainable: CRSP’s US-Large-Cap and US-Small-Cap indices. The first one comprises ‘the top 85% of investable [US] market
capitalization’ including both Mid and Mega capitalization; this is, the listed companies
whose market value belongs to the top 85% of all US stock market indices put together (NASDAQ, Standard and Poor’s 500, etc.). The second one ‘includes U.S. companies
that fall between the bottom 2%-15% of the investable market capitalization’ (CRSP).
The stock returns used are daily from the year 2000 to the end of 2014. Following Lee’s methodology, some observations have been removed whenever the stocks were not active or part of NYSE, AMEX or NASDAQ indices. Also, to avoid result disruptions, the price must be between $1 and $1,000, due to the fact that their return observations would be excessively large or too low respectively; for example, an increase of $0.10 cents on top of a $0.50 share results in a very large 20% growth. Then, stocks with less than 10 days per month of activity were not considered in order to keep reliable and enough observations and to eliminate securities in an unusual trading situation (merged, acquired or officially delisted). Finally, only share codes 10 and 11 (ordinary common shares) were selected for observation since other types, such as certificates and American Depositary Receipts, are not of interest for this study. Once both large-cap and small-cap index components were cleaned up, the illiquidity Amihud measure was calculated daily for each stock and then their equal-weighted average was taken monthly for each portfolio. If their liquidity lay outside the 2.5 and 97.5 percentiles of each portfolio, these outliers were dropped in order to avoid noise. Finally, it was found the return correlation of the large and small stocks portfolios with respect to the S&P 500 market index, representative of the US stock market return and the difference between these two correlations was taken to find the desired variable flr(size). For simplicity, the variable flr(volatility) will be left out of this study, focusing on the size of shares rather than their volatility.
In total, during the calculation of the funding liquidity risk, the observations obtained for the FLR factor are 152, having estimated the correlations of the previous 30 months’ rolling period (Lee). This means that the first FLR factor starts at June 2002, using the correlations of the period comprised between January 2000 and June 2002 (30 months).
c) Accounting for other data sources.
During the data gathering and analysis, several other options were considered in the calculation of the FLR factor. These details are discussed below providing their corresponding rationale.
For instance, monthly returns are easier to obtain than the daily ones and
provide by far less observations to work with. Although this may seem as an advantage, the daily data has been chosen for its reliability. In this way the approximation errors are avoided and also assets with less than ten observations per month are easier to recognize and to remove.
As for the period of time selected, the years between 1990 and 2014 were very tempting to be selected due to a possible larger number of observations. In the beginning, the simulatons carried out included this longer period too, but eventually it has been discarded. The decision for selecting exclusively the period 2000-2014 was made because the number of shares would have halved otherwise, since they did not exist in the 1990s. Many technological and internet companies, which are mainly part of the XXIst century, would have not been taken into account for this study as for the stock activity requirement.
Also, when forming the large-cap and the small-cap portfolios, the components of the S&P500 and the Russell 2000 indices could be respectively a good proxy for these two groups. The first one would result in fewer observations – only 500 stocks at maximum without filter; and the second one would comprehend medium-cap companies, whose shares would not be appropriate for small portfolio.
Finally, the option of using a 20 months rolling window for correlations was contemplated for calculating a larger number of FLR factors, however the 30 months period was deemed as more appropriate according to the literature, since a larger estimated horizon decreases noise. In any case, also simulations using only 20 months rolling periods were conducted.
Conclusively, several of the above factors were actually used for different calculations of the FLR factor to obtain better results. Despite the choices made, based on literature or aimed at larger sample obtainment, the differences were
insignificant for the final regressions. Therefore, the results obtained in the next chapter were based on the initial information explained in part b).
3) Chapter 3: Results
This chapter aims at researching a possible effect of the funding liquidity risk in the U.S. stock market by using the free statistical package R. This package allows the analysis of the excessively large sample of more than 1.6 million observations - the initial sample prior to subset - and by using text files it provides much faster results than other programs. Finally, it is a recommendable tool for manipulating data easily, therefore trial and error calculations and data manipulation become very simple by changing the commands.
Next, the description of the assets contemplated will be followed by the actual results.
a) Assets studied: ETFs and Market indices
As pointed out in the literature review, the Fama and French 3-factor model has been considerably improved by the 4th factor added by Carhart, which provided a larger explanatory power to the excess return of assets or indices (Fama and French, 2004). Following that intuition, this paper compares whether a further improvement in the model takes place by adding a 5th factor, the FLR, focusing on the multiple coefficient of determination R-squared and the adjusted R-squared.
Moreover, this study is focused on six different assets, being the first three popular exchange-traded funds for investors and presented in Table 1 and 3.
These are:
i) the SPY (SPDR S&P 500), one of the most popular ETFs in the U.S., which tracks the S&P 500 Index;
ii) the QQQQ (the Nasdaq 100 Trust), an ETF broadly exposed to the technological sector by tracking the Nasdaq 100 Index;
iii) the IVE (iShares S&P 500 Value), also specialized in the S&P 500 index but with value characteristics, in other words its shares are considered to be underpriced by the market.
In table 2 and 4, three stock market indices have been used for regression, more precisely:
i) the Standard & Poor's 500 Index itself, which is composed by 500 companies according to their market size, liquidity, industry and other criteria, it represents one of the leading indicators of U.S. equities and it reflects the return of the large cap shares;
ii) the Russell 2000, this small-cap index comprehends the bottom 2,000 companies or roughly the 10% of the Russell 3000 index - nearly 98% of the investable US equity market (www.russell.com);
iii) finally the CRSP US Total Market index is composed by around 4,000 companies across mega, large, small and micro capitalizations, providing a very broad horizon of investment possibilities, approximately 100% of the U.S. investable equity market ( http://www.crsp.com/products/investment-products/crsp-us-total-market-index).
The expectations with respect to the above six assets, according to the literature, imply a noticeable effect of the funding liquidity risk on their performance, especially on the S&P 500 because the concept and formula of this risk rest upon this index.
b) Results
Tables 1 and 2 report the regressions’ results related to the above mentioned representative assets.
To briefly refresh the formula, the coefficients represent the ones from the Carhart model, formed by the market excess return on the risk free rate (rmrf), the book–to– market (smb) and market capitalization (hml), and the momentum (wml) factor, extended to the left-handed results by the funding liquidity risk (flr), not present on the right-handed results.
Starting by the ETFs, we observe an flr coefficient close to zero, roughly 0.002 for SPY and IVE and 0.02 for QQQQ. Regardless of its meaningfulness, the FLR has a nearly zero effect on the excess returns of these assets. Since the t-values of this coefficient do not belong to the rejection region in the three cases, it is possible to conclude that the flr coefficient is not meaningful with a 5% significance level.
As for the R2 and the adjusted R2, there is not an increase in the explanation of the dependent variables when the new type of risk is added. In fact, the changes are minuscule –0.0001-, reinforcing the idea of its null effect on the expected returns of the funds. In the case of IVE, the adjusted R2 even increases from 0.9758 to 0.9759 when removing our factor, but such tiny differences ought not to be relevant.
The large coefficients of determination point out the effectiveness of the Carhart model, although the hmd coefficient is surprisingly not meaningful in the case of SPY ETF – its p-value of 0.9 is well above 0.05.
A possible reason for the lack of evidence of the factor studied may be that all the three ETFs track large-cap indices. This fact entails that the FLR should not be of
Intuitively, a liquidity shock should not affect these ETFs returns, unless large enough to affect the whole (both small and large) stock market. As a widespread known fact, such a shock has indeed taken place since 2007 with the sub-prime crisis.
Hence, this explanation should be ruled out and, as the following outcomes will show below, the outcome is not related to the cap-size of indices.
The analysis continues by analyzing the effect of funding shocks in three stock market indices: the S&P 500, the Russell 2000 and the CRSP Total Market Index, being large-cap, small-cap and total-cap related respectively.
For the S&P 500, the flr coefficient is very small (0.0009), almost zero and not significant. Being its p-vale considerably larger than 0.05, the null hypothesis of this coefficient is not rejected.
Also not significant – but less unrelated to our topic - is the Carhart momentum factor, being in both regressions -0.004 and with a not-rejected null hypothesis. On the other hand, the rest of the factors have a heavier weight in the explanation of this index excess return.
Moreover, the R2 and the adjusted R2 do not reflect any improvement or any worsening in explaining the dependent variable. Both measures stick to 0.9976 - with and without FLR -, indicating a well achieved goodness to fit of the model, although it is absolutely not due to the new variable added.
The regression of the Russell 2000 does not present more encouraging numbers.
The flr coefficient equals 0.0013 and the p-value of 0.77 proves it is not meaningful in our model (H0 is not rejected). Once more the momentum factor, the coefficient umd,
has a low weight (0.012) and is not significant.
Also the R2 and the adjusted R2 are almost unaltered when adding our 5th explanatory variable, they remain close to 0.991. Unfortunately, contrary to any expectations, a funding shock is not observable in the small-cap index excess return.
So far, the flight-to-quality phenomenon does not manifest among the less collateralizable shares either.
Lastly, the regressors with respect to the CRSP Total Market Index are estimated.
In this case all the four initial factors have a considerable weight and are significant, being their correspondent p-values much smaller than 0.05, with and without flr. Without success, the flr coefficient itself is nor large (0.0099) nor meaningful for the model; notwithstanding, this time the p-value of 0.0762 is much closer to the 5% significance level than in previous cases. In fact, the goodness to fit measures show a tiny
improvement, the adjusted R2 on the left is 0.9858, a bit larger than the 0.9855 on the right side.
Although these differences seem statistically minuscule, they accord with the theory elaborated in the literature review and with economic rationale, indicating a possible effect of the funding liquidity factor.
All in all, after having investigated the role of the FLR in the six assets proposed and having commented on the ANOVA tables of each of them, one can conclude there is not enough evidence of such a factor impacting in their returns. In other words, there has not been found any trace of the funding liquidity effect when the markets were in trouble, neither has been observed a flight to quality from small to large stocks.
Hence, the influence of borrowing constraints for traders, described by the literature review, has not added any further improvement to the extended CAPM of Carhart.
4) Chapter 4: Conclusion
This paper has presented a new approach in facing asset valuation by providing a new concept of risk.
Starting by an explanation of the CAPM model, a very important pillar in academia, the advantages and disadvantages have been described by going through its rationale, importance and its further expansions by Fama and French, and by Carhart. Among several authors, such as Pastor and Stambaugh and Chordia et al., the concept that in many cases traders’ activity and their funding have not been considered yet for risk stands out. Hence, as a revolutionary point of view, the risk of traders to obtain capital - also known as funding liquidity risk – could play an important role in asset valuation. As a result, an improvement for the model was suggested by adding this funding risk as a factor.
The methodology chapter aimed at finding a benchmark for this new risk.
Authors such as Drehmann and Nikolaou or Brunnermeier and Pedersen suggested formulae that are not easy to calculate or that mainly require unavailable information. The most attainable proxy has been explained by Jaehoon Lee, his model consists of the simple difference between the correlations of large and small stocks. In this paper the formula from the latter author has been used for the study of funding constraints. Using CRSP database, we retrieved the first four factors of the renowned model. Following the steps of Lee and using the advantageous statistical package R, a FLR factor was calculated for the years 2000-2014. Also some different sources for these calculations were discussed, although they were not decisive for the final results.
In the following chapter, the 3 indices and the 3 exchange-traded funds of importance for the regressions – the independent variables - have been described and their relevance highlighted. Next, the results for the regressions were presented and analyzed.
The conclusions based upon the coefficients, t-statistic, the p-values, R2 and adjusted R2, did not provide any evidence of an effect of the FLR, contradicting any expectations or intuition in accordance with previous studies. That is to say, this factor was not significant and its coefficient’s absolute value was not much larger than zero, leading to a null effect in any case.
In the discussion, different reasons will be discussed to explain why the results were not as expected.
First, according to Lee, Pastor and Stambaugh, and Fontaine et al., evidence on FLR was expected to be observed. Second, the new variable and the problematic are thorny enough to present error calculations or methodological obstacles.
Furthermore, it is very difficult to disentangle the change in constraints and in market liquidity since the causes behind them have different sources (Kahraman and Tookes). Finally, other relevant difficulties may arise for a proper calculation of the FLR, such as: risk-diversification via other asset classes and markets, shadow banking, cross-marginal financing of assets (explained below), institutional concentration that enhances contagion - a.k.a. commonality effect - or the asymmetric effect of FLR.
All in all, the CAPM stays, intuitively but also empirically, an important financial model. Even though the effect of financial constraint for traders and investors is of relevance according to theory, the initial model has proven its completeness for studying assets in the real and more complex markets in this study.
Finally, the topic is considerably recent and it results in a complex field; therefore, it ought to be concluded that more research is needed. Some suggestions, improvements and other possibilities are contemplated in the discussion below.
5) Chapter 5: Discussion
In the previous chapter the 5-factor model regression was performed for several assets. There has not been found any evidence of the new risk concept regarding funding constraints. Notwithstanding the plausibility that such factor is non-existent or does not affect the stock market, a description of shortcomings and improvements for the model will follow in this chapter.
Principally, other articles that approached the problematic with a similar model have not been found. This fact may point out that the approach was not the best,
although the topic is very recent, involving many difficulties in the concept’s description, measurement and evidence.
However, having constructed the funding liquidity factor from the beginning, the outcome obtained was compared to the factor presented in the website of Jaehoon Lee; these were considerably different. Despite the lack of similarity, I carried out regressions with Lee’s factor and the results were also not significant.
Furthermore, the first four factors are very influential on the explained variable, leading to think that the funding shocks might be already priced in one of them. It would not be surprising due the large coefficients of determination attained with the 4-factor model; indeed, some R2 results equal 0.90 and others amount to 0.99 indicating a large goodness to fit of this model when excluding the FLR. Autocorrelation ought not to be considered because the t-statistic of the flr coefficient is not inflated with large numbers. Perhaps, taking market efficiency into account, the market excess return coefficient (rmrf) already measures the funding risk. This would be true assuming speculators and lenders have access to all necessary information to take their investment decisions.
In any case, it seems especially arduous to estimate financing constraints and to isolate their potential consequences. In fact, it results complicated to separate the change in constraints and in market liquidity when they are not driven by the same economic forces (Kahraman and Tookes, April 2014).
According to Fontaine et al., also different liquidity channels should have been added in the study. Particularly, it is well known that investors diversify their risks by considering other markets, such as different asset classes (shares, bonds, ETFs, futures, options and other derivatives) in international markets (EU, Japan, Australia). Also other funding sources were not included, e.g. private equity returns and the market liquidity have been found closely correlated; besides, unregulated financing channels as shadow banking were not contemplated.
Along with these, investment decisions should be closely associated to the complex methods used by hedge funds, which achieve their opaque strategies through cross-marginal financing of assets, meaning that they use a large variety of investable assets as collateral to diminish their margin requirements (Brunnermeier and Pedersen).
In addition, the commonality effect – the contagion across assets - may partly depend on institutional ownership. According to Kahraman and Tookes, commonality may increase when stocks are owned by market makers, especially in times when their capital is tight. So the level of liquidity contagion is enhanced by the ownership concentration by one or few institutions with funding constraints.
Interestingly, the lack of the FLR evidence in the study may be explained also by the asymmetric effect of it. This concept consists of small price movements for positive fundamental shocks in the market and large price changes when the shocks are negative. Hence, the liquidity spiral becomes very intense downwards but it does not work equally upwards. This shock asymmetry suggests that the FLR should probably be studied only in recessions (Brunnermeier and Pedersen).
In tables 3 and 4, the study is repeated for the period from August 2007 until December 2014, during the global financial crisis. Looking at their p-values and R2, the results are still not significant, meaning that the asymmetric effect above described is not observable.
Finally, some improvements could be suggested to reach more realistic results. For instance, it would be ideal to observe the performance of the 25-FF portfolios provided by Kenneth French in his renowned website. Since they account for stocks sorted according to both size and value features, thus tackling various types of stocks classes by using a cross-sectional tool.
In fact, in view of the class of stocks, the FLR factor calculated above focused simply on the size of stocks (large vs. small capitalization), while some authors appropriately claim that volatility has a powerful effect on funding and market liquidity.
Certainly, the above mentioned recommendations would lead to better results but their implementation is hard and complex or the required information might not be easily accessible.
References
Aretz, K., Bartram, S. M., & Pope, P. F. (2005). Macroeconomic risks and the Fama and French/Carhart model. EFA 2005 Moscow Meeting.
Avramov, D. (2002). Stock return predictability and model uncertainty. Journal of
Financial Economics, 64(3), 423-458.
Bello, Z. Y. (2008). A Statistical Comparison of the CAPM to the Fama-French Three Factor Model and the Carhart's Model. Global Journal of Finance and Banking
Issues, 2(2).
Berk, J., & DeMarzo, P. (2011). Corporate Finance, Global ed. Essex: Person
Education Limited. Ch. 12, 135-136.
Brunnermeier, M. K., & Pedersen, L. H. (2009). Market liquidity and funding liquidity. Review of Financial studies, 22(6), 2201-2238.
Chordia, T., Sarkar, A., & Subrahmanyam, A. (2005). An empirical analysis of stock and bond market liquidity. Review of Financial Studies, 18(1), 85-129.
CME Group (2010). CME SPAN: Standard Portfolio Analysis of Risk - CME Group http://www.cmegroup.com/clearing/files/span-methodology.pdf (last visit on the 18th of february 2015)
Drehmann, M., & Nikolaou, K. (2013). Funding liquidity risk: definition and measurement. Journal of Banking & Finance, 37(7), 2173-2182.
Fama, E. F., & French, K. R. (2004). The capital asset pricing model: theory and evidence. Journal of Economic Perspectives, 25-46.
Fontaine, J., Garcia, R., & Gungor, S. (2013). Funding Liquidity, Market Liquidity and the Cross-Section of Stocks Returns. Market Liquidity and the Cross-Section of
Stocks Returns (February 28, 2013).
Kahraman, B., & Tookes, H. (2014). Leverage constraints and liquidity: What can we learn from margin trading?
Virtual Appendices
With FLR factor Without FLR factor
SPY Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.0016 0.0005 -3.34 0.0011** rmrf 0.9954 0.0083 120.64 0.0000*** smb -0.1503 0.0139 -10.78 0.0000*** hml -0.0014 0.0134 -0.10 0.9171 umd -0.0167 0.0069 -2.43 0.0165* flr 0.0021 0.0031 0.70 0.4869 Multiple R-squared: 0.9929 Adjusted R-squared: 0.9926
SPY Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.0019 0.0003 -6.17 0.0000*** rmrf 0.9958 0.0082 121.13 0.0000*** smb -0.1505 0.0139 -10.81 0.0000*** hml -0.0011 0.0134 -0.08 0.9360 umd -0.0161 0.0068 -2.37 0.0193* Multiple R-squared: 0.9928 Adjusted R-squared: 0.9926
IVE Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.0024 0.0009 -2.53 0.0126* rmrf 1.0011 0.0163 61.46 0.0000*** smb -0.1184 0.0275 -4.30 0.0000*** hml 0.2656 0.0265 10.01 0.0000*** umd -0.0696 0.0136 -5.14 0.0000*** flr 0.0020 0.0061 0.33 0.7381 Multiple R-squared: 0.9766 Adjusted R-squared: 0.9758
QQQQ Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.0014 0.0016 0.90 0.3695 rmrf 1.1836 0.0426 27.79 0.0000*** smb 0.1536 0.0722 2.13 0.0351* hml -0.7114 0.0692 -10.29 0.0000*** umd -0.0699 0.0351 -1.99 0.0483* Multiple R-squared: 0.891 Adjusted R-squared: 0.888
QQQQ Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.0038 0.0025 1.53 0.1280 rmrf 1.1806 0.0426 27.73 0.0000*** smb 0.1551 0.0720 2.15 0.0330* hml -0.7145 0.0691 -10.34 0.0000*** umd -0.0749 0.0353 -2.12 0.0354* flr 0.0196 0.0158 1.24 0.2172 Multiple R-squared: 0.8922 Adjusted R-squared: 0.8884
IVE Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.0026 0.0006 -4.43 0.0000*** rmrf 1.0014 0.0162 61.79 0.0000*** smb -0.1186 0.0274 -4.32 0.0000*** hml 0.2659 0.0264 10.06 0.0000*** umd -0.0691 0.0134 -5.15 0.0000*** Multiple R-squared: 0.9766 Adjusted R-squared: 0.9759
Table 1 – The effect of funding liquidity risk on three popular ETFs: SPY, IVE and QQQQ. As noticeable in the tables on the left side, the coefficient flr is almost zero and non-significant.
With FLR factor Without FLR factor
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
SP500 Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.0018 0.0003 -6.43 0.0000*** rmrf 1.0015 0.0047 211.38 0.0000*** smb -0.1462 0.0080 -18.26 0.0000*** hml 0.0174 0.0077 2.26 0.0254* umd -0.0042 0.0039 -1.07 0.2870 flr 0.0009 0.0018 0.51 0.6126 Multiple R-squared: 0.9976 Adjusted R-squared: 0.9976
SP500 Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.0019 0.0002 -10.85 0.0000*** rmrf 1.0016 0.0047 212.36 0.0000*** smb -0.1463 0.0080 -18.32 0.0000*** hml 0.0176 0.0077 2.28 0.0238* umd -0.0040 0.0039 -1.02 0.3096 Multiple R-squared: 0.9976 Adjusted R-squared: 0.9976
Table 2 – Two important indices – S&P 500 and Russell 2000 – together with a Total Market Index constructed by CRSP are studied for influence of funding liquidity changes. As seen in the left-sided ANOVA tables, the flr is nearly zero and not meaningful for this model.
CRSP
Tot. Mkt. Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.0041 0.0009 4.67 0.0000*** rmrf 0.9695 0.0149 64.90 0.0000*** smb 0.6987 0.0253 27.67 0.0000*** hml 0.1920 0.0243 7.89 0.0000*** umd -0.1435 0.0124 -11.54 0.0000*** flr 0.0099 0.0056 1.79 0.0762 Multiple R-squared: 0.9862 Adjusted R-squared: 0.9858 CRSP
Tot. Mkt. Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.0029 0.0006 5.16 0.0000*** rmrf 0.9711 0.0150 64.65 0.0000*** smb 0.6977 0.0254 27.43 0.0000*** hml 0.1935 0.0245 7.90 0.0000*** umd -0.1409 0.0124 -11.33 0.0000*** Multiple R-squared: 0.9859 Adjusted R-squared: 0.9855 RUSSELL
2000 Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.0022 0.0007 -3.07 0.0025** rmrf 1.0206 0.0123 83.03 0.0000*** smb 0.8760 0.0208 42.15 0.0000*** hml 0.1441 0.0200 7.19 0.0000*** umd 0.0116 0.0102 1.14 0.2575 flr 0.0013 0.0046 0.28 0.7772 Multiple R-squared: 0.9911 Adjusted R-squared: 0.9907 RUSSELL 2000 Estimate Std. Error t value Pr(>|t|) (Intercept) -0.0024 0.0005 -5.24 0.0000*** rmrf 1.0209 0.0122 83.47 0.0000*** smb 0.8758 0.0207 42.29 0.0000*** hml 0.1442 0.0199 7.23 0.0000*** umd 0.0120 0.0101 1.18 0.2396 Multiple R-squared: 0.991 Adjusted R-squared: 0.9908
With FLR factor Without FLR factor
IVE Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.0031 0.0007 -4.28 0.0000 rmrf 0.9812 0.0178 54.97 0.0000*** smb -0.1803 0.0346 -5.20 0.0000 hml 0.2994 0.0324 9.24 0.0000*** umd -0.0644 0.0151 -4.26 0.000 Multiple R-squared: 0.9835 Adjusted R-squared: 0.9827
SPY Estimate Std. Error t value Pr(>|t|) (Intercept) -0.0014 0.0008 -1.67 0.0989*** rmrf 0.9932 0.0097 102.21 0.0000*** smb -0.1704 0.0187 -9.12 0.0000*** hml 0.0070 0.0175 0.40 0.6878 umd -0.0137 0.0083 -1.66 0.1008 flrsp 0.0032 0.0043 0.74 0.4595 Multiple R-squared: 0.9944, Adjusted R-squared: 0.994
SPY Estimate Std. Error t value Pr(>|t|) (Intercept) -0.0019 0.0004 -4.76 0.0000*** rmrf 0.9943 0.0096 103.70 0.0000*** smb -0.1711 0.0186 -9.19 0.0000*** hml 0.0067 0.0174 0.38 0.7017 umd -0.0127 0.0081 -1.56 0.1222 Multiple R-squared: 0.9943 Adjusted R-squared: 0.9941
QQQQ Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.0014 0.0020 0.69 0.4951*** rmrf 1.1288 0.0490 23.02 0.0000*** smb 0.0872 0.0952 0.92 0.3620*** hml -0.5405 0.0890 -6.07 0.0000*** umd -0.0539 0.0415 -1.30 0.197*** Multiple R-squared: 0.8945 Adjusted R-squared: 0.8895 QQQQ Estimate Std. Error t value Pr(>|t|) (Intercept) 0.0053 0.0041 1.29 0.2008* rmrf 1.1210 0.0495 22.65 0.0000*** smb 0.0928 0.0952 0.98 0.3324*** hml -0.5379 0.0890 -6.05 0.0000*** umd -0.0616 0.0420 -1.47 0.1466*** flrsp 0.0237 0.0217 1.09 0.2771 Multiple R-squared: 0.896 Adjusted R-squared: 0.8898
IVE Estimate Std. Error t value Pr(>|t|) (Intercept) -0.0033 0.0015 -2.20 0.0307 rmrf 0.9816 0.0181 54.09 0.0000*** smb -0.1805 0.0349 -5.17 0.0000 hml 0.2993 0.0326 9.17 0.0000*** umd -0.0640 0.0154 -4.16 0.0001 flrsp -0.0011 0.0079 -0.14 0.8894 Multiple R-squared: 0.9835 Adjusted R-squared: 0.9825
Table 3 – The effect of funding liquidity risk on the previous ETFs: SPY, IVE and QQQQ, from mid-2007 to 2014. The effect is not meaningful.
With FLR factor Without FLR factor
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
SP500 Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.0013 0.0015 0.87 0.3847 rmrf 0.9914 0.0178 55.64 0.0000*** smb 0.6501 0.0343 18.97 0.0000*** hml 0.1379 0.0320 4.30 0.0000*** umd -0.1713 0.0151 -11.32 0.0000*** flrsp -0.0014 0.0078 -0.18 0.8556 Multiple R-squared: 0.9891 Adjusted R-squared: 0.9884
SP500 Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.0015 0.0007 2.13 0.0360* rmrf 0.9910 0.0175 56.53 0.0000*** smb 0.6505 0.0340 19.12 0.0000*** hml 0.1380 0.0318 4.34 0.0000*** umd -0.1718 0.0148 -11.58 0.0000*** Multiple R-squared: 0.9891 Adjusted R-squared: 0.9886 RUSSEL L 2000
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.0016 0.0005 -3.47 0.0008*** rmrf 0.9966 0.0056 177.28 0.0000*** smb -0.1431 0.0108 -13.23 0.0000*** hml 0.0317 0.0101 3.14 0.0023** umd 0.0008 0.0048 0.18 0.8612 flrsp 0.0014 0.0025 0.58 0.5605 Multiple R-squared: 0.9981 Adjusted R-squared: 0.998 RUSSELL 2000 Estimate Std. Error t value Pr(>|t|) (Intercept) -0.0019 0.0002 -8.19 0.0000*** rmrf 0.9971 0.0055 179.98 0.0000*** smb -0.1434 0.0108 -13.33 0.0000*** hml 0.0316 0.0101 3.14 0.0024** umd 0.0013 0.0047 0.28 0.7820 Multiple R-squared: 0.9981 Adjusted R-squared: 0.998 CRSP Tot. Mkt.
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.0001 0.0011 0.05 0.9608 rmrf 1.0142 0.0137 74.22 0.0000*** smb 0.8634 0.0263 32.85 0.0000*** hml 0.1161 0.0246 4.73 0.0000*** umd 0.0048 0.0116 0.41 0.6798 flrsp 0.0125 0.0060 2.09 0.0393* Multiple R-squared: 0.9935 Adjusted R-squared: 0.9932 CRSP Tot. Mkt.
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.0020 0.0006 -3.58 0.0006*** rmrf 1.0183 0.0138 73.85 0.0000*** smb 0.8605 0.0268 32.15 0.0000*** hml 0.1147 0.0250 4.58 0.0000*** umd 0.0088 0.0117 0.76 0.4504 Multiple R-squared: 0.9932 Adjusted R-squared: 0.9929
Table 4 – In this table the same effect is tested for the global crisis period, from August 2007 till December 2014. As seen by the results below, the flr does not make a meaningful difference for the model.
