LIQUIDITY, STOCK RETURNS AND INVESTMENTS

Transcription

1 Spring Semester 12 LIQUIDITY, STOCK RETURNS AND INVESTMENTS A theoretical and empirical approach A thesis submitted in partial fulfillment of the requirement for the degree of: BACHELOR OF SCIENCE IN INTERNATIONAL BUSINESS ADMINSTRATION Of TILBURG UNIVERSITY By MICKEY DERKS Supervisor: Dr. Miao Nie School of Economics and Management Department of Finance Joint 2012

2 Abstract This bachelor s thesis consists out of two major parts; the first section reviews existing literature, elaborating on liquidity, its measurements and how it affects stock valuation estimates. We make use of Amihud s ILLIQ-measurement to proxy liquidity and both the CAPM and Fama-French three-factor model to estimate the required return of a stock. After having elaborated on these variables we created an intuitive relation between the market premium ((Rm)-Rf), Liquidity, Liquidity Risk, and the Fama-French factors. We will support the thesis that higher returns are expected for less liquid stocks. In the second section we conduct our own research on the correlation between the mentioned variables using U.S. data from securities listed on the NYSE, AMEX and NASDAQ obtained from the Center for Research in Security Prices (CRSP). After conducting the statistical analysis we conclude that indeed a statistically significant correlation can be found and that less liquid stocks outperform liquid stocks, arguably as a consequence of liquidity costs, as such these costs and liquidity factors should as such be considered in order to make a rational investment decision. 1

3 Table of Contents Abstract... 1 Contents Introduction Problem Statement Literature Review Liquidity Determinants of Liquidity Measuring Liquidity and its Consequences Amihud s ILLIQ-Measure Capital Asset Pricing Model (CAPM) Fama-French Three factor model Relations Empirical Research Hypothesis Research Data Methodology Results Conclusions Critique Appendix I Overview data AMEX, NYSE & NASDAQ II Overview data SML, HML & Rf III Statistical formulas References

4 1 Introduction The purpose of this thesis is to examine the relation between liquidity and stock returns and to see whether statically significant conclusions can be drawn for rational investors, to aid them with their rational investment decision-making that will be partially based on this relation. The original topic assigned to this thesis was the predictability of stock returns, but due to the many variables that have an influence on stock returns, we have decided to focus and narrow the scope down to see the effect of a single variable liquidity on asset pricing. Because we also want to focus on the predictability aspect of the original assigned topic and its investment consequences, we will elaborate on how rational investments can be made with the liquidity information of stocks. Liquidity is first of all defined as the ability of an asset to be converted into cash quickly and without price discounts. We start with a review of existing literature on liquidity. As the liquidity literature is vast, we will focus primarily on the most prominent, widespread and general accepted papers and literature on liquidity and asset pricing (with stocks in particular), as the content is simply too large and expanding too rapidly. By using the most prominent papers we will obtain a coherent, logical literature review, which covers the most important aspects of liquidity, which will help us understand the later constructed relation better. This thesis covers the origins of liquidity, measures of liquidity and the intuitive relations. We will then elaborate on the expected returns of an asset where we will make use of the capital asset pricing model (CAPM) and the Fama-French three factor model. Finally, we will do our own research with data to see if we can find a statistically significant correlation by using U.S. stock data, obtained from the Center for Research in Security Prices (CRSP). 3

5 1.1 Problem Statement Before we elaborate on the empirical data and research design, we should have a clear theoretical understanding of the concepts and theoretical relations that can be identified. This leads us to several sub questions that will require an answer and will subsequently provide us the theoretical tools to compose a theoretical capitulatory answer to our research question. The theoretical analysis in this thesis will answer the following sub questions: What is liquidity? How can we measure liquidity? How do we predict the expected return on a security? How does liquidity affect stocks? These questions will form the base of our literature review section in this thesis. The conclusion of these sub questions will subsequently help us to answer our main problem statement as it provides us a theoretical relation between liquidity and stock returns. 4

6 2 Literature Review In this section we first elaborate on the concept of liquidity and general asset pricing. We show which sources affect liquidity and how we can measure (or proxy) liquidity. Next, we will review models which can help us predict the expected return of a stock; for this purpose we will make use of both the CAPM and Fama-French three factor model as these models have proven to be relatively accurate at predicting variations within portfolios. With this knowledge we will attempt to establish a theoretical and intuitive relation between the liquidity of a stock, and its respective price and thus potential future returns. 2.1 Liquidity For this thesis we will use the definition of liquidity as defined by Damodaran (2006). Liquidity is defined as the ability of an asset to be converted into cash quickly and without any price discounts as a result of transaction costs. In the case of illiquidity, one cannot immediately sell an asset. If one buys such a security and as a result of buyers remorse want to reverse the decision and sell the respective security, this will then come at a cost. The cost of this illiquidity is the cost of the remorse 1. It becomes clear that for a heavily traded stock on a major exchange, these costs have to be relatively small as a purchased stock can almost immediately be sold after it was acquired (there are costs however like the bid-ask spread and brokerage fees). For stocks from smaller (private) businesses these costs are naturally higher, as it will take more time, effort and money to find a buyer. Hence different stocks have different degrees of illiquidity costs attached to them. Investors as such value liquidity over illiquidity. The concept of liquidity applies not only on stocks, but also on virtually all asset classes that can be traded. Common examples are bonds, real assets or even businesses. For this thesis we will only look at the different liquidities among stocks. It must be noted that no matter how illiquid an asset is, it can still be sold, as long as the seller of the security is willing to accept a lower price. As a result 1 Damodaran on valuation: Security analysis for investment and corporate finance (2006) 5

7 assets should not be categorized as being either liquid or illiquid but rather ranked on a continuum of liquidity 1. The consensus conclusion is that investors demand higher returns when investing in more illiquid stocks. Put another way, investors are willing to pay higher prices for more liquid investments relative to less liquid investments. If we want to categorize stocks on the continuum of liquidity we ought to know what factors influence liquidity and how we can measure or proxy the approximate levels of liquidity. 2.2 Determinants of liquidity The costs of illiquidity, defined as the costs of buyer s remorse, are made up out of several sources that create illiquidity. All these sources of illiquidity impose costs to the holder of the assets. These costs of illiquidity should be reflected in the asset prices, as the investors should require a compensation for holding them. The compensation for the costs and risk associated with illiquidity should be reflected in a higher expected return 2. Though not directly relevant for the main research question under investigation in this thesis, we believe it should be included for a better understanding of the mathematical analysis in subsequent sections. In the following section we will review the most important sources of liquidity as presented by Amihud, Mendelson and Pedersen (2006). Exogenous transaction costs Exogenous transaction costs are comprised by those sources of illiquidity that are being incurred for the specific transaction. These costs can be the costs for conducting a transaction like brokerage fees, processing costs and transaction taxes but also more stock specific factors like the bid-ask spread. For every security that is being traded the buyer and/or seller incurs some form of transaction costs, additionally the buyer of a security will need to anticipate these costs when he buys the security and the costs of this illiquidity throughout the entire period which he holds the security. 2 The sources of liquidity are presented by Amihud, Mendelson and Pedersen (2006) 6

8 Demand pressure and inventory risk Demand pressure refers to the fact that even in today s markets, not all agents (being buyers and/or sellers of securities) can be present in the market at any given time. This means that those who want to sell a security quickly might not have a willing buyer available to them to conduct the transaction (and vice versa for buyers). For this reason there are so called market makers who are creating liquidity in the market by buying securities at any given time to subsequently at a later point in time sell the respective security. This means that the security from the point of acquisition to the point where it is being sold is considered an asset in inventory. The market maker as a result, has a risk due to the fluctuating prices of the security during the time in which the security is in its inventory this risk has a price, which is being imposed and transferred to the seller of the security. Search frictions When the demand pressure is minimized by the presence of many agents in the market, it might still be difficult to actually locate the counterparty to trade the security with, especially when an agents tries to sell a particularly large number of securities it might be difficult to find a willing buyer. Once a willing buyer has been identified they also need to agree over the price of the security. This comes at a cost, which we will refer to as search frictions. It is intuitively clear that in the case of stock, large numbers of shares or shares of the smaller companies will incur bigger search frictions, as the market for these stocks is relatively small. The introduction of digital central marketplaces has greatly facilitated and reduced these search frictions but they are still present in every trade. As mentioned in section 2.1 of this thesis, no matter how illiquid a security is, it can still be sold. The respective security however, will need to be traded at a discount to widen the scope of the market and willing buyers. We can thus conclude that the seller of the security has a tradeoff between quick trading at a discount and incurring significant search frictions for his security. In addition, Amihud, Mendelson and Pedersen also point out that due the volatility in liquidity, risk-averse investors may require a compensation for being exposed to liquidity risk. 7

9 Information asymmetry Finally, trading a security may be costly because there exist information asymmetries among traders. In the case one party has more information than the other, due to private information that has not been made public, trading with the informed counter party will result in a loss on investment. This information asymmetry can be any kind of information, which is relevant for the transaction and has a potential future effect on the stock price. We see that liquidity costs and risks affect the required return by investors. Illiquidity affects corporations and the allocation of real resources. Liquidity plays an important role for investors as it influenced when and if they can get out of the market at the desired time and price. Liquidity should be taken into account whenever they choose to take a position. As such liquidity will have an expected effect on the expected returns of a stock. 2.3 Measuring Stock Liquidity and its consequences As discussed in section 2.2, securities have several sources that significantly affect its respective liquidity. In general we can say that the more independent variables there are (especially the qualitative ones), the more difficult it becomes to construct a single measure of a dependent variable (i.e. liquidity). This results in a problem, as it is difficult to construct a single measure that captures the exact liquidity of a security. It is important to realize that there is as such not a single perfect measure of liquidity. This problem has a big impact on our research question as it greatly reduces the significance level at which we can reject the null-hypothesis, as there will always be a certain error in the measurement. This also means that the effect of liquidity becomes harder to detect and this should be taken into account for the empirical research. Researchers have agreed that several concepts could provide information, which is correlated with the respective liquidity. Examples include share classes, IPOs, Turnover rates, Bid-ask spreads etc. It is generally agreed that some of these measurements do particularly well as a proxy of the real unknown level of liquidity. 8

10 2.4 Amihud s ILLIQ-Measure We have decided to focus on a particular measurement of liquidity namely Amihud s ILLIQ-measure (ILLIQ). We employ the ILLIQ-measurement for several reasons but mainly because Amihud (2002) has shown that the ILLIQ measure strongly correlates with the bid-ask spreads and price impacts. ILLIQ is therefore a useful measure of liquidity but it has the important advantage that its containing variables are widely available (data returns on trading volume and returns), unlike the data which is required for example bid-ask spreads for longer periods of time. Though used on a global scale and preferred for our thesis, we still have to take into account, that the subsequent measurement remains a noisy estimate of the true unknown parameter. In Amihud model he suggests the following measure for daily measure for liquidity: the absolute price change divided by trading volume for stock i on day d.!!,!!!,! This formula shows how a unit of volume has an effect on the returns of that security. For heavily traded stocks a relatively large change in traded volume is required to make a change in returns and vise versa for smaller traded stocks. This is how the formula proxies the daily liquidity. This measure is however very noisy as there are many more independent variables that have an effect on the returns of a stock. This means we will therefore have to average the results on a weekly, monthly or annual base. We have decided the average the results on a monthly base. We will derive the following formula:!!!""!#!,! = 1!!!!,!!!!!!,! 9

11 Where!! represents the number of trading days in a month t. With this measure we can arguably proxy the illiquidity of a security over a longer period of time at a level that is significant enough to make statistical conclusions. 2.5 Capital Asset Pricing Model (CAPM) To establish a relation with the level of liquidity and the returns of a stock we will need a good measure the appropriate rate of return of an asset. The original capital asset pricing model (CAPM, Sharpe 1964) offers a way to approximate the appropriate rate of return of an asset when that asset is added to a well-diversified portfolio given that asset s non-diversifiable risk. The model takes into account the asset s sensitivity to non-diversifiable risk (market risk, β) in the financial industry, as well as the expected return of the market and the expected return of a theoretical risk free asset 3. Solving for!!! yields: With:!!!!! β! =!!!!!!!! =!! + β! (!!!!! ) β! =!"#(!!,!! )!"#(!! ) Basically this model states that the return of an asset should equal the risk free rate (e.g. U.S. Government bonds) plus a market premium multiplied with its respective Beta representing the stock specific sensitivity to market returns. 3 Sharpe, William F. (1964). Capital asset prices: A theory of market equilibrium under conditions of risk, Journal of Finance. 10

12 2.6 Fama and French three-factor model Like the CAPM, the Fama-French Three-factor model (FF3) uses the same Beta to compare a security (or portfolio) to the market as a whole (though the value for the FF3 beta will differ from the CAPM beta due to the added independent variables). However, Fama and French (1993) have added several other variables in other to obtain a better r-squared fit (variation explained by regressing the independent variables on the dependent variable). Fama and French basically made a distinction in their model between value and growth stocks, as it are two common methods for assessing investment opportunities. The difference between growth and value is based on where the returns on an investment would be derived from. In the case from a growth stock, an investor expects a continuing growth in the value of the stock over time, which should therefore increase his returns. In the case of a value stock the investor believes the stock is currently underpriced in the market and hopes to make a quick return when the market corrects the price. Fama and French realized this key difference in stocks and decided to incorporate it within their model as an extension on the CAPM model. As such the three-factor model uses three variables, which will result in a relative improvement of 20-percentage point. As such it has better predicting powers and is therefore also used in our thesis next to the CAPM. In order to make the distinction in the model between value and growth, in addition to the β!, Eugene Fama and Kenneth French added the small minus big market capitalization (SMB) and high minus low market-to-book ratio (HML), essentially creating a distinction between the two classes of stock. SMB and HML measure the historic excess returns of small caps and value-stocks over the market as a whole. This leads us to the Fama-French three-factor model:!!! =!! + β!!!!! +!!!"# +!!!"# With:!! =!h!"#!"#$%!"#$%&!"#$!! β! =!h!""!"#$%& β!! &!! =!"#$$%&%#'()!"!"#!"#!"#!"#$%&'(!"#$!"#$%&!"#!"$$%&' 11

13 The Fama-French three-factor-model explains roughly 90% of the diversified portfolios returns 4. The regression from liquidity on the Fama-French model will be the most complex regression conducted in this thesis. 2.5 Relations As was briefly mentioned there is an intuitive relation between the liquidity and its respective expected return of an asset. We mentioned that heavily traded stock on a major exchange will be more liquid compared to smaller private businesses stock, and will have lower costs attached to it (i.e. buyers remorse costs ceteris paribus). In other words we expect that there should be a correlation between liquidity and returns as liquidity has a direct effect on the returns of an asset. The consensus conclusion is that for stocks that are less than perfectly liquid, investors will incur costs of illiquidity when liquidating a position. Rational investors would be expected to demand a return premium reflecting the expected costs of illiquidity. As we measure illiquidity this means that the beta coefficient should theoretically be positive, as an increase in illiquidity will yield higher expected returns. 4 Fama, Eugene F. French, Kenneth R. (1992). "The Cross- Section of Expected Stock Returns 12

14 3 Empirical Research Now that we have identified and clarified all relevant variables and have touched upon an intuitive relation between all independent variables and the expected returns, we should find out whether the theoretical established correlation can statistically be shown by applying several regression techniques. We start by formulating our intuitive relation in a testable hypothesis. 3.1 Hypothesis Research In order to test our hypothesis and see if a significant correlation can be found between liquidity and returns we apply an advanced regression of liquidity and the CAPM on returns and liquidity and FF3 on returns. In addition to these we will use the normal CAPM and FF3 regression as a base, which will serve as a comparison. This will give us a total of four regressions with the FF3-liquidity as the most complex regression analysis, which should therefore explain most of the variation in returns. We will then determine the significance of the betas and the combined betas with an F-test and see if liquidity is jointly significant and hence influences returns. In addition the r-squares will be determined to see how much of the variation in returns is explained by the models. Moreover, besides the illiquidity measure (LIQ) we have added a market liquidity measurement (!"#! ), which should improve the model, as total liquidity risk will be accounted for. CAPM:!! = β! +β! (!!!! )!! = β! +β! (!!!! ) + β! (LIQ) + β! (LIQ) + β! (!"#! ) FF3:!! = β! +β!!!!! + β!!"# + β!!"#!! = β! +β!!!!! + β!!"# + β!!"# + β! (LIQ) + β! (!"#! ) 13

15 3.2 Data The input for our statistical analysis comes from the monthly U.S. stock returns on the three major exchanges (NASDAQ, NYSE & AMEX). In order to obtain the data from these major exchanges we need a verified reliable and unbiased source of data; we have decided to collect our data rom the Center for Research in Security Prices (CRSP) database. This database is widely used and renowned for its accurate and complete lists of history data from U.S. stocks. The data comprises data from 1925 to It is however important to utilize several filters which will (greatly) reduce the number of stocks. This is important as the measurements of liquidity already are noisy estimates and without filters, this noise would be increased. So have we removed the penny stocks (by requiring a minimal share price of $2) and stocks that have a lifespan on the market of less than five years. In addition, a minimum cap of 10 million USD was implemented and full information had to be available. We also excluded the Real Estate Investment Trusts (REITs), American Depository Receipts (ADRs), Warrants, Exchange Traded Funds (EFTs), and closed-end funds from our study. Once we look at the last 10 years of data points that meet these requirements, we still have data points left to do our statistical analysis with. However, due to the complexity of the regression (normal linear regression does not suffice as the data points belong to different shares) and limitations in terms of time, resources and software we decided to focus on a sample size of 400 shares (as will be elaborated on in the methodology). Please refer to Appendix I for summary statistics of the stock population database. The SML, HML data and risk free rates (monthly U.S. Government Bond), which are required for the FF3 model, are obtained directly from the Fama-French database offered by the Wharton University Database (Appendix II). We use the monthly return of the S&P 500 index as our market return, which is widely available. 14

16 3.3 Methodology To test our hypothesis that was defined in section 3.1, we will need a solid statistical methodology to make sure our findings are statistically correct, significant and meaningful. Our methodology will start by identifying all variables required for our regression, and will consist out of the following subsequent steps: ILLIQ estimation: for each stock in the NYSE, AMEX and NASDAQ liquidity will be determined; the market liquidity will also be determined. SMB, HML, Market return and Rf (U.S. Treasure Bill rate) are given and will be matched with each stock. The market premium can be calculated as the market return and risk free rate are given. Once all the relevant variables have been identified, simple linear regression over the variables is not possible. This is because there are over 6000 different stocks per annum in the dataset meaning we would no longer factor for changes in liquidity alone but also stock specific properties (as will be elaborated on later in this section). What we can do to prevent this is to run an individual regression for each stock on the mentioned variables. This makes sure this statistical problem is solved. We then aggregate these results and take the mean and run a second regression over the betas. This should provide accurate predictions and a statistical model, which is useful. We will illustrate this methodology with an example; we take the fist stock from the 01/31/2011 (is 2011) sample (A.E.P. Industries Inc.). When we run the regression for market premium on returns we obtain: Model Summary Model R R Square Adjusted R Square Std. Error of the Estimate a a. Predictors: (Constant), MarketPrem 15

17 Model 1 Sum of Squares ANOVA a df Mean Square F Sig. Regression b Residual Total a. Dependent Variable: returns b. Predictors: (Constant), MarketPrem Model 1 Coefficients a Unstandardized Coefficients Standardized Coefficients B Std. Error Beta (Constant) MarketPrem a. Dependent Variable: returns t Sig. We see that the market premium (Rm-Rf) does a relative good job at predicting the returns of this specific stock where 60.7% of the variation in returns is explained by the model. In addition we see that the model is useful by running an F-test which yield a P-value of meaning that the model is (very) useful as it passes our 95% confidence requirement (and would even pass with 99% confidence). Moreover the Market-premium variable is useful which we can tell by the t-value which is (also).003 and therefore significant at our required 95% confidence level 5. It is obvious that whenever another stock would be added with different returns and a similar market premium (which is the same for all stocks from time period t) the results of the linear regression would statistically be (falsely) reduced in usefulness, as stocks are being compared with other stocks and not with their own returns over time (which is required for our regression). In other words the regression line would shift to the mean value of the data points, essentially increasing the sum of squared errors (SSE) leading to errors in the standard error (SE), which yields extreme and unreliable values for both T-tests and F-tests, making simple linear regression futile. 5 Please refer to the Statistical Formulas (p.27) for the statistical formulas used in order to derive these numbers. 16

18 Therefore all betas, t- and f- values for all variables must be individually computed, aggregated and compared before we can make a statistically relevant conclusion. Unfortunately, our main statistical program (SPSS) has no (easy) option to do this automatically for us (for as far as we know). This means all the mentioned results for all variables in all four of the regressions have to be manually computed. Over a sample size of nearly this becomes virtually impossible. We therefore decided to regress each individual stock chosen by a random sample and to have a cross sectional regression over the betas, t- and f-values. This will allow us to make a statistical conclusion about the entire population. We considered a sample size of 400 to be sufficient. For all our conclusions we will use a minimum confidence level of 95%. We must note that this is indeed a limitation, which should be removed by further studies of contemporary data analysis to provide more confidence in the derived conclusions. 3.4 Results After regressing the required variables on returns and determining the aggregated means of the results with a cross sectional regression as explained in section 3.3, we obtain the following results for the following models:!! =!! +!! (!!!! ) R2 Overall Regression Descriptives Statistic Std. Error Mean % Confidence Interval for Mean Lower Bound Upper Bound % Trimmed Mean Median Variance.084 Std. Deviation Minimum Maximum

19 Ftest Ttest Range Interquartile Range Skewness Kurtosis Mean % Confidence Interval Lower Bound for Mean Upper Bound % Trimmed Mean Median Variance Std. Deviation Minimum Maximum Range Interquartile Range Skewness Kurtosis Mean % Confidence Interval for Mean Lower Bound Upper Bound % Trimmed Mean Median Variance Std. Deviation Minimum Maximum Range Interquartile Range Skewness Kurtosis We see that the market premium (!!!! ) is useful and significant in the model. Primarily because we see that 95% of the R-squares of the complete sample will be located in the [ ; ] interval. As such we can conclude that 95% of the complete population under investigation has a minimum R-squared of (approximately 43% of the variation explained by the model), for the CAPM model. In addition we see that the F-test data tells us that the model is extremely useful with 95% of the data having a minimum F-test score of The likelihood of obtaining this F-score is extremely small meaning that our model is certainly useful 18

20 at the 95% confidence level. The T-test confirms this one more by having a mean of and 95% of the T-scores being higher than Taking this value as a p-value translates into a model, which is significant with at least 95% of confidence. Now we have the base values of our CAPM regression we can add other variables.!! =!! +!! (!!!! ) +!! (LIQ) +!! (!"#! ) R2 Ftest TtestLIQ Overall Regression Descriptives Statistic Std. Error Mean % Confidence Interval for Mean Lower Bound Upper Bound % Trimmed Mean Median Variance.029 Std. Deviation Minimum Maximum Range Interquartile Range Skewness Kurtosis Mean % Confidence Lower Bound Interval for Mean Upper Bound % Trimmed Mean Median Variance Std. Deviation Minimum Maximum Range Interquartile Range Skewness Kurtosis Mean % Confidence Lower Bound Interval for Mean Upper Bound

21 TtestLIQm TtestMrktPrm 5% Trimmed Mean Median Variance Std. Deviation Minimum -.06 Maximum Range Interquartile Range 5.28 Skewness Kurtosis Mean % Confidence Lower Bound Interval for Mean Upper Bound % Trimmed Mean.2039 Median.0820 Variance Std. Deviation Minimum Maximum 2.49 Range 4.00 Interquartile Range 1.59 Skewness Kurtosis Mean % Confidence Interval for Mean Lower Bound Upper Bound % Trimmed Mean Median Variance Std. Deviation Minimum Maximum Range Interquartile Range Skewness Kurtosis By adding the liquidity factors into the regression on top of the original CAPM, we see that the model now yields an R-square average of with 95% of the data points in the population being bigger than Recall that the prior CAPM 20

22 model (without the liquidity factors) had a mean R-squared of and a 95% confidence lower bound As we can see the liquidity factors significantly improve as the explained variance in returns increases. Moreover the model is useful (see F-test score) at 99% and so is the individual liquidity factor. It must be noted that the market liquidity factor is not significant in the model, which is arguably a cause of the smaller sample size while the factor itself is based on the entire population, which therefore has a relatively small effect on an individual stock return, which is as such not reflected well in the model.!! =!! +!!!!!! +!!!"# +!!!"# R2 Ftest Overall Regression Descriptives Statistic Std. Error Mean % Confidence Interval for Mean Lower Bound Upper Bound % Trimmed Mean Median Variance.034 Std. Deviation Minimum Maximum Range Interquartile Range Skewness Kurtosis Mean % Confidence Interval Lower Bound for Mean Upper Bound % Trimmed Mean Median Variance Std. Deviation Minimum Maximum Range Interquartile Range

23 TtestSMB TtestHML TtestMrktPrm Skewness Kurtosis Mean % Confidence Interval Lower Bound for Mean Upper Bound % Trimmed Mean Median Variance Std. Deviation Minimum Maximum 6.16 Range 8.20 Interquartile Range 2.29 Skewness Kurtosis Mean % Confidence Interval Lower Bound for Mean Upper Bound % Trimmed Mean.2812 Median.7530 Variance Std. Deviation Minimum Maximum 3.29 Range 9.13 Interquartile Range 2.27 Skewness Kurtosis Mean % Confidence Interval for Mean Lower Bound Upper Bound % Trimmed Mean Median Variance Std. Deviation Minimum Maximum Range Interquartile Range Skewness Kurtosis

24 In the Fama-French model, we see that on average 63% is explained by the model, and in 95% of the cases a minimum of 56% is explained. We see once again an improvement when we compare this model to the original CAPM. The model is useful which we derive from the F-test, which gives us a 95% confidence level lower bound of 4.72, which is well above the required level for usefulness at 95%. It must be noted however that the individual t-test scores for SMB and HML is very low and the overall score is quite volatile. This means that the SMB and HML does not have the same predicting power for every stock and that major differences can be witnessed causing the average to be somewhat in the middle. The complete model however has proved to be useful.!! =!! +!!!!!! +!!!"# +!!!"# +!! (LIQ)+!! (!"#! ) For clarity purposes only the data discussed will be summarized in the table below. R2 Ftest Overall Regression Descriptives Statistic Std. Error Mean % Confidence Interval for Mean Lower Bound.9361 Upper Bound % Trimmed Mean.9540 Median.9730 Variance.001 Std. Deviation Minimum.87 Maximum 1.00 Range.13 Interquartile Range.07 Skewness Kurtosis Mean % Confidence Interval Lower Bound for Mean Upper Bound % Trimmed Mean

25 TtestLIQ BetaLIQ Median Variance Std. Deviation Minimum 9.27 Maximum Range Interquartile Range Skewness Kurtosis Mean % Confidence Interval Lower Bound for Mean Upper Bound % Trimmed Mean Median Variance Std. Deviation Minimum -.79 Maximum Range Interquartile Range 4.63 Skewness Kurtosis Mean % Confidence Interval for Mean Lower Bound Upper Bound % Trimmed Mean Median Variance Std. Deviation Minimum 1.20 Maximum Range Interquartile Range Skewness Kurtosis Finally, we see that by adding the liquidity factor to the Fama-French model we obtain a total explanation in the variation of return of 95% on average with 95% of the data being explained for over 93% or more. Meaning the FF3 + liquidity model. We can see that the complete model is also useful with an extremely high f-score of 24

26 73.52 on average. Moreover we have included the Beta and t-scores for liquidity to show that this variable is useful and we obtain a positive coefficient for the illiquidity measure. 3.5 Conclusions CAPM CAPM & LIQ FF3 FF3 & LIQ 95% Confidence lower bound R-squared As we can tell by the mean r-squares and lower bound of the 95% confidence interval of the four regressions we indeed find proof that liquidity significantly explains a lot of the variation in an individual stock s return. So do we see a minimum increase in R-squared of 20 percentage points by adding the liquidity variable. Not only have we shown that Liquidity does a good job at improving the models but also that the variable is significant by looking at the t-scores of the variable which is significant at 95% confidence. We can also see that the FF3 model with the liquidity factors explains most of the variation of all 4 regressions, which has to be, expected as it basically the CAPM with more variables. An increase in variables can never reduce the explaining power of the model and this is therefore statistically logical. We also see that the beta value for liquidity has a mean value of Is can be said that there is a negative value ( ) in the 95% confidence interval, but this is a result of the extreme volatility in the beta value. The mean value and the fact that it is heavily skewed to the left assures us that a positive beta value is to be expected. This positive beta indicates that for an increase in!!,! (increase in illiquidity of the!!,! stock) the expected return of the stock indeed increases. It can be said that the expected return on a stock listen on the AMEX, NASDAQ or NYSE increases on the long run whenever the liquidity of that stock goes down (or vise versa when the illiquidity goes up). 25

27 As mentioned there are several explanations as for why less liquid stocks perform better in the long run. Investors prefer liquidity over illiquidity. This creates the liquidity premium, which requires higher expected return for a stock that is more illiquid ceteris paribus. The valuation of illiquidity is not expected to disappear in the future. Liquidity will continue to be valued high, and illiquid stocks will still come at a discount. As liquidity increases, valuations increase (and vice versa). Thus the investor in less liquid stocks gets lower valuations, effectively buying stocks at a discount. This means that the investor who chooses to invest in less liquid stocks will obtain the gains for any increases in liquidity and the change in valuation that comes with it. This theory holds in practice as we have shown in this thesis. As such we have demonstrated that liquidity can be used as an investment style. Less liquid stocks indeed outperform the more liquid stocks in the long run. An investor can create a liquidity factor by simply creating his own long-short liquidity stock factor and compare it to the market mean. Selecting stocks in this manner should yield a higher return in the long run as opposed to stocks, which have been selected on market, size, value and growth factors alone. As liquidity increases, valuations increase and vise versa. Investors in less liquid stocks therefore get a lower valuation, effectively buying the stock at a discount. The investor in less liquid stocks will as such obtain the gain from any future increases in liquidity. 3.6 Limitations As noted there are several limitations that should be taken into account. First of all: the limitation of liquidity. We can only proxy liquidity and never get the true value. This creates noise and poor estimates in some of the stocks, which is incorporated, in the final regression. This deteriorates the results and decreases the overall reliability of the tests. There is however no real solution for this problem and as our conclusions all used a minimum of 95% confidence a decrease in confidence would still yield fairly accurate predictions. 26

28 The second limitation is that the NYSE, AMEX and NASDAQ are all generally very liquid. This means that their variation in liquidity is kept to a minimum, which makes comparing extreme levels of liquidity differences difficult. It would be these data points that can be most important in terms of statistics and are not incorporated in our analysis as those stocks are not being traded on the mentioned markets. The third limitation is our statistical analysis as we were forced to limit our scope to 400 stocks only. It is obvious that an increase in the number of stocks under investigation would significantly increase the reliability of the results. However, due to the bounded resources it was not possible to conduct a more extensive research at the time of writing this thesis. Lastly, the survivorship bias poses a problem for our statistical analysis. Basically because the highly illiquid stocks that are making consecutive negative bottom line results tend to die out over time, and will as such not be incorporated within our sample. This means that there is a selection prior to our statistical analysis that tends to select only those illiquid stocks that make high returns. This also means that the main negative of aspect of higher expected returns, higher risk, is not well reflected in our sample, making our results less trustable. We encourage any future researchers on the topic of contemporary liquidity on today s stock market to improve this investigation on the importance of this relation and to exclude some of the limitations that we encountered in his thesis. 27

29 Appendix I Overview data used from NASDAQ, NYSE & AMEX 6 II Overview data HML, SML & Risk free rate 7 Descriptives Statistic Std. Error Mean Shares Outstanding Price or Bid/Ask Average 95% Confidence Interval for Mean Lower Bound Upper Bound % Trimmed Mean Median Variance Std. Deviation Minimum Maximum Range Interquartile Range Skewness Kurtosis Mean Lower 95% Confidence Bound Interval for Mean Upper Bound % Trimmed Mean Median Variance Std. Deviation Given at end of year, monthly data omitted in table for simplicity but was used for the derivation of our statistical conclusions. 7 Given at end of year, monthly data omitted in table for simplicity but was used for the derivation of our statistical conclusions. Rounded off at integers (complete numbers were used in analysis). 28

30 Returns Volume LIQUIDITY Minimum 2.05 Maximum Range Interquartile Range 8.75 Skewness Kurtosis Mean Lower % Confidence Bound Interval for Mean Upper.0833 Bound 5% Trimmed Mean.0444 Median.0206 Variance.042 Std. Deviation Minimum -.40 Maximum 1.52 Range 1.91 Interquartile Range.14 Skewness Kurtosis Mean 8 9 Lower % Confidence Bound Interval for Mean Upper Bound 3 5% Trimmed Mean Median Variance Std. Deviation 07 Minimum Maximum 1.08E Range 0 Interquartile Range Skewness Kurtosis Mean

31 MarketRetMinRF HML 95% Confidence Lower Bound Interval for Mean Upper Bound % Trimmed Mean Median Variance.000 Std. Deviation Minimum Maximum Range Interquartile Range Skewness Kurtosis Mean Lower 95% Confidence Bound.0046 Interval for Mean Upper Bound % Trimmed Mean.0107 Median.0284 Variance.002 Std. Deviation Minimum -.17 Maximum.11 Range.28 Interquartile Range.04 Skewness Kurtosis Mean Lower 95% Confidence Bound Interval for Mean Upper Bound % Trimmed Mean

32 SMB RiskFreeRate Median Variance.001 Std. Deviation Minimum Maximum Range Interquartile Range Skewness Kurtosis Mean Lower 95% Confidence Bound Interval for Mean Upper Bound % Trimmed Mean Median Variance.001 Std. Deviation Minimum Maximum Range Interquartile Range Skewness Kurtosis Mean % Confidence Interval for Mean Lower Bound Upper Bound % Trimmed Mean Median Variance.000 Std. Deviation Minimum Maximum Range Interquartile Range Skewness Kurtosis

33 III Statistical Formulas Regression=! (!!!)! SSR =!!! Residual=! (!!!! )! SSE =!!! Total=! (!!! )! SST =!!! R! =!! =!!"!!" = 1!!"!!" Usefulness= (!!" )/!!!" F =!!"/(!!!!! ) Reject!!!!;!;!!(!!!) Usefulness single var=!! T=!! /!!!!!! Reject!!!!!;!!! or t!!;!!!!!!! =!! =!!! 1 =! 2 32! (!!!! )!!!!

34 References Books: Amihud, Yakov and Haim Mendelson, 1986, Asset Pricing and the Bid-Ask Spread Journal of Financial Economics. Amihud, Yakov, Haim Mendelson and Lasse Heje Pedersen, 2005, Liquidity and Asset Prices. Brennen and Subramanyan, 1996, Market Microstructure and Asset Pricing: On the Compensation for Illiquidity in Stock Returns. Damodaran, 2006, Damodaran on Valuation: Security Analysis for Investment and Corporate Finance 8 Fama, Eugene F.; French, Kenneth R., 1992, The Cross-Section of Expected Stock Returns. Gert Nieuwenhuis, 2009, Statistical Methods for Business and Economics. Graham, Benjamin, and David Dodd, 1940, Security Analysis. Idzorek, Thomas, James X. Xiong, and Roger G. Ibbotson, 2010, The liquidity style of mutual funds. Pastor, Lubos, and Robert Stambaugh, 2003, Liquidity risk and expected stock returns Journal of Political Economy. Silber, William L., 1991, Discounts on Restricted Stock: The Impact of Liquidity on Stock Prices. Yakov Amihud and Haim Mendelson, Financial Markets And portfolio Management, Stocks and Bond liquidity and effect on Prices and Financial Policies. Research Papers: Partially available on: urce=bl&ots=8ub8ma51ch&sig=2qpotepe0382mjlnjobyt4- vgzw&hl=nl&sa=x&ei=rxvzt47bj5oa0awy7kxrdq&ved=0cegq6aewaw#v=onepage&q&f=false 33