The role of market and funding liquidity innovations in German stock market returns

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Name

Elsa Feldmann

Date

15

th

December 2017

MSc. Thesis

Title The role of market and funding liquidity innovations in German stock market returns

Abstract

This thesis aims at testing whether specific measures for funding liquidity and market liquidity are able to explain variations in the returns of German stocks. It makes use of the idea that the level of funding liquidity in the aggregate financial market is related to the available volume of arbitrage capital; and exploits noise in the yields of German government bonds to measure a lack of arbitrage capital. Amihud’s measure of illiquidity is used to approximate stocks’ market liquidity. Overall results point to a positive illiquidity premium for stocks with lower market liquidity; and market-wide funding liquidity conditions impacting stocks’ individual market liquidity. While funding liquidity as such is not able to directly explain absolute levels of stock returns, the analysis hints at a “flight-to-quality” during times of reduced available funding liquidity. The thesis concludes by suggesting that further research is needed on the role of funding liquidity in asset returns and points to possible methodological challenges related to the endogeneity of explanatory variables and possible non-linearity.

University University of Amsterdam (Amsterdam School of Economics)

Student number 10827366

Supervisor prof. dr. S.J.G. (Sweder) van Wijnbergen

Program MSc. Economics (Monetary Policy & Banking)

Word count 14,843

Key words Funding liquidity, Noise as an indicator of Illiquidity, Stock

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Statement of Originality

This document is written by Student Elsa Feldmann 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.

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Table of Figures

Figure 1: Evolution of quoted companies per market segments at the Frankfurt Stock Exchange………… 10 Figure 2: Yield spreads in German Government bonds over the period of January 1999 to July 2017……. 18 Figure 3: Evolution of total market capitalisation and number of stock companies included in the dataset.. 19 Figure 4: Evolution of month-end CDAX performance index over the period January 2000 to July 2017... 20 Figure 5: Evolution of month-end CDAX performance index over the period January 2000 to July 2017... 20 Figure 6: Evolution of total market capitalisation and median book-to-market-price ratio across stock

companies included in the dataset………...… 21

Figure 7: Evolution of the monthly returns of the CDAX over the period January 1999 to July 2017…….. 21 Figure 8 Evolution of the EURIBOR over the period January 2000 to July 2017……….…………... 22 Figure 9: Evolution of the variables SIZE and VALUE over the period January 2000 to July 2017………. 23 Figure 10: Evolution of the variable IML in the period from January 1999 to July 2017……… 24 Figure 11: Evolution of average market liquidity and return volatility of stocks from January 1999 to

July 2017………. 35

Table of Tables

Table 1: Fixed effects cross-section regression results on stocks’ monthly returns………. 30 Table 2: Fixed effects cross-section regression results on stocks’ monthly returns………. 32 Table 3: Portfolio matrix sorting by return volatility and market liquidity……….. 33 Table 4: Augmented CAPM single-factor model regression results on ten portfolios sorted by market

liquidity……….………….. 36

Table 5: Augmented CAPM single-factor model regression results on ten portfolios sorted by return

volatility……….. 36

Table 6: Augmented CAPM single-factor model regression results on ten portfolios sorted by market

capitalisation……… 36

Table 7 Augmented CAPM single-factor model regression results on ten portfolios sorted by

book-to-market ratio……… 37

Table 8: Augmented CAPM single-factor model regression results on fifteen portfolios sorted by market

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Table 9: 𝑅𝑅2 measures of the augmented CAPM single-factor model linear regression for portfolios

cross-sorted by market liquidity and return volatility………. 38 Table 10: Augmented Fama-French multi-factor model regression results on ten portfolios sorted by

market liquidity………... 40

Table 11: Augmented Fama-French multi-factor model regression results on ten portfolios sorted

by return volatility………... 40

by market capitalization……….. 40

by book-to-market ratio………... 41

Table 14: Augmented Fama-French multi-factor model regression results on fifteen portfolios sorted

by market liquidity and return volatility………. 41

Table 15: 𝑅𝑅2 measures of the augmented Fama-French multi-factor linear regression for portfolios

cross-sorted by market liquidity and return volatility…...……….………. 43 Table 16: Regression results of 𝐼𝐼𝐼𝐼𝐼𝐼_𝑡𝑡= 𝛼𝛼_{𝐼𝐼𝐼𝐼𝐼𝐼}+ 𝛽𝛽𝐼𝐼𝑀𝑀𝑀𝑀𝐼𝐼𝑀𝑀𝑀𝑀_𝑡𝑡+ 𝛽𝛽𝑆𝑆𝐼𝐼𝑆𝑆𝑆𝑆𝑆𝑆𝐼𝐼𝑆𝑆𝑆𝑆_𝑡𝑡+ 𝛽𝛽𝑉𝑉𝑉𝑉𝐼𝐼𝑉𝑉𝑆𝑆𝑉𝑉𝑉𝑉𝐼𝐼𝑉𝑉𝑆𝑆_𝑡𝑡+ 𝜏𝜏_𝑡𝑡……... 45 Table 17: Regression results of 𝐼𝐼𝐼𝐼𝐼𝐼_𝑡𝑡= 𝛼𝛼_{𝐼𝐼𝐼𝐼𝐼𝐼}+ 𝛽𝛽𝑆𝑆𝐼𝐼𝑆𝑆𝑆𝑆𝑆𝑆𝐼𝐼𝑆𝑆𝑆𝑆_𝑡𝑡+ 𝛽𝛽𝐹𝐹𝐼𝐼𝐹𝐹𝐼𝐼_𝑡𝑡+ 𝜎𝜎_𝑡𝑡……….……….. 46

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Chapter 1 Introduction

1.1 Problem setting

Since the outbreak of the financial crisis in 2008, liquidity risk – the previous orphan child among financial risks – has received much more attention. Consequently, the number of published research papers investigating the causes and consequences of liquidity risk spiralled in the past years. One of the most outstanding papers is that of Brunnermeier and Pederson (2008) on ‘Market Liquidity and Funding Liquidity’. In this paper they provide a model which links an asset’s market liquidity and a trader’s funding liquidity and shows that margins can be destabilizing when market and funding liquidity are mutually reinforcing. They furthermore offer the prediction that exogenous shocks to speculator capital should lead to a reduction in market liquidity; and that the sensitivity of margins and market liquidity to speculator capital is larger for assets that are on average riskier and less liquid. The authors emphasize that their propositions are testable and suggest modelling an exogenous capital shock to study the market liquidity of such an event.

In order to study these effects predicted by Brunnermeier and Pederson, I aim at estimating funding liquidity continuously over the chosen time period, instead of using a specific shock event representing a disruption to funding liquidity. In doing so I make use of the idea and model developed by Hu, Pan and Wang (2013: 1) claiming that the level of funding liquidity

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in the aggregate financial market is related to the available volume of arbitrage capital. In “ordinary” times – when institutional investors have abundant capital that they can deploy to supply funding liquidity – large price deviations are mostly eliminated by arbitrage forces and assets are traded at prices closer to their fundamental values. A lack of sufficient arbitrage capital would limit the force of arbitrage and assets would be traded at prices significantly away from their fundamental values. Following this logic, one can deduct important information about the amount of funding liquidity in the aggregate market when observing temporary mispricings being a key symptom of shortage in arbitrage capital. The results that I obtain when investigating this relationship for the German stock market point to a positive illiquidity premium for stocks with lesser market liquidity as well as a statistically significant relationship between funding and market liquidity. A direct impact of funding liquidity onto the absolute stock return levels was not observed, but could explain differential returns between very liquid and very illiquid stocks.

1.2 Research objective

The objective of this thesis is to test whether specific measures for funding liquidity and market liquidity are able to explain variations in the returns of German stocks. In a first step, I try to establish that there is a measurable illiquidity premium for less liquid stocks, providing evidence for the hypothesis that a stock’s return also depends on its market liquidity. In a second step, I aim at further showing that market-wide funding liquidity then impacts an asset’s market liquidity and its return. The measure for funding liquidity that I construct is based on mispricings of German Government bonds. It is assumed that mispricings, i.e. deviations of the observed market price from the theoretical model price in these bonds occur due to a lack of sufficient arbitrage capital in the market, which is taken as an approximation of market-wide funding liquidity. Previous research for the U.S. market has shown that this measure can help explain cross-sectional variations in returns of hedge funds

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and currency carry trade strategies (Hu, Pan and Wang 2013). This thesis uses the same measure but applies it onto stock returns instead of hedge fund or carry trade returns. For market liquidity, Amihud’s (2002) liquidity measure is used, which captures the daily price response of a stock to one Euro of trading volume in that same stock. It is one of the most widely used measures for market liquidity using the price impact of transactions. The scope of this thesis is restricted to the German market and uses data on German Government bonds as well as German stocks traded at the Frankfurt Stock Exchange.

The hypothesis to be tested is whether funding liquidity and market liquidity represent risks that are priced and where consequently a risk premium can be observed when fitting a capital asset pricing model. Alternatively, even if funding or market liquidity in itself would not be priced risks, they could still represent dimensions which help explain variations in stock returns. Furthermore, it is of interest whether market and funding liquidity move together and if so how large their correlation is. Earlier research came to the conclusions that funding liquidity can explain the cross-section of returns across liquidity- and volatility-sorted portfolios and that excess returns of individual portfolios are negatively correlated with funding liquidity. Similarly, it was shown that market liquidity is priced (Amihud et al. 2015); however, while market and funding liquidity seem to be strongly correlated at times of stressed markets, they otherwise exhibit only little correlation (Adrian et al. 2017).

1.3 Course of investigation

The first chapter introduces the thesis with its motivation, aim and research objective and establishes the course of investigation. The second chapter provides insights into the concepts and definitions of funding and market liquidity and an overview of the German stock market. Chapter 3 explains the method of data selection and the data sources used before setting out the methodology of how the funding liquidity measure has been constructed and the foundations of the factor pricing models applied. The forth chapter

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outlines the estimation techniques for the cross-sectional regressions (4.2), the time-series regressions regarding the impact of funding liquidity on stock returns (4.3) and the time-series regression on differential stock returns (4.4); and also presents the empirical results. The fifth and sixth chapters interpret and discuss the results before chapter 7 concludes.

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Chapter 2 Literature review

2.1 Measures of funding and market liquidity

Very different measures for capturing funding and market liquidity have been proposed over time. They vary due to the different research objectives followed and because of different underlying definitions of funding and market liquidity, but also considerably with regard to the financial market agent that is being investigated. The Bank for International Settlements in one of their Working Papers (Drehman and Nikolaou 2010) defines funding liquidity as “the ability to settle obligations with immediacy” and refer to banks being “able to settle obligations with immediacy”. They suggest to measure aggregate funding liquidity (risk) by measuring the sum of the premia banks are willing to pay above the expected marginal rate which would clear the central bank’s open market operation (auction) times the volume a bank bids, normalised by the expected amount of money supplied by the central bank. Hence, funding liquidity risk here is revealed by the price banks are willing to pay during open market operations and shows that banks with higher funding liquidity risk will bid more aggressively, and the more so, the higher their funding liquidity risk. While the measure that Drehman and Nikolaou (2010) use can exclusively be applied to credit institutions that qualify for central bank auctions, and is therefore not relevant for the course of investigation of this thesis, their paper is interesting in a different regard. They highlight

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the fact that there is a need to differentiate between funding liquidity and funding liquidity risk. While the former is essentially a binary concept, the latter refers to a continuum of infinitely many different values related to the distribution of future outcomes. Furthermore, funding liquidity risk is always forward looking and measured over a specific horizon, while funding liquidity is associated with one particular point in time.

In this thesis, I will use the definitions proposed by Brunnermeier and Pedersen (2008), who define an asset’s market liquidity as “the ease with which it is traded” and a trader’s funding liquidity as “the ease with which they can obtain funding”, where a ‘trader’ represents financial markets’ participants trading in securities such as dealers, hedge funds, investment firms and so on. Applying the above-introduced difference between liquidity and liquidity risk and considering that I am interested in the extent to which the theory brought forward by Brunnermeier and Pedersen (2008) can be observed in real-world financial markets. I intend to estimate (1) to what extend an asset’s ease at which it is traded is priced and therefore is reflected in the asset’s return; and (2) if traders’ funding liquidity risk affects assets’ market liquidity and prices and how this impact differs across assets with different market liquidity risk. The key assumption for the second research aim is that low funding liquidity among traders, i.e. only scarce amounts of capital available to traders, leads to scarce amounts of arbitrage capital deployed in the markets. Temporary asset mispricing, i.e. price deviations or so-called «noise» in prices, is an indication of such shortage in or lack of arbitrage capital. Hence, a measure of price deviations in an asset with a deep market and high market liquidity can be used as an approximation for shortage in arbitrage capital and therefore for funding liquidity of financial markets participants. For the purpose of this thesis the measure for market liquidity will be based on Amihud’s measure (2002); for funding liquidity it will follow the work of Hu, Pan and Wang (2013); both measures will be set out in chapter 3.

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Approaches of researchers such as Longstaff (2004) for the U.S. market and Schuster and Uhrig-Homburg (2013) for the German market have shown that funding liquidity can be captured by measuring the differences in illiquidity premia between government bonds and bonds issued by agencies that are guaranteed by the government. Both papers extract the term structure of illiquidity premia from the spread between the two bonds differing only in their liquidity. Another strain of literature has tried to construct measures of market liquidity focussing on trading or transaction costs; or on demand and supply equilibria thereby establishing the claim of ‘commonality in liquidity’ (Domowitz and Wang 2002). For example, Chordia, Roll and Subrahmanyam (2001) exploit aggregate daily order imbalances by averaging each individual stock’s quoted spreads over all daily transactions, and then value-weighting the average daily spreads across stocks – effectively deriving overall market liquidity from an equally-weighted average of individual asset market liquidity. Pastor and Stambaugh (2003) using market microstructure data focus on market liquidity associated with temporary price fluctuations induced by order flow, using the idea that order flow induces greater risk reversals when liquidity is lower. In this regard, Hu, Pan and Wang (2013) document that the measure of market liquidity that Pastor and Stambaugh (2003) deploy actually has a statistically significant relation with their own noise measure, thereby providing evidence for the linkage between market liquidity and funding liquidity evoked by Brunnermeier and Pedersen (2008).

2.2 The German stock market

The subject of investigation of this thesis is the behaviour of German stocks’ returns. Therefore the peculiarities of the German stock market should be explained, before proceeding and giving further details regarding the selection of data. Understanding the German stock market necessitates knowledge about the different stock exchanges that exist, the different market segments, classes of stocks and also taxation. The following issues that I

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illustrate have been addressed by Stehle and Schmidt (2015) in their paper ‘Returns on German Stocks 1954 to 2013’.

Historically, Germany used to have several stock exchanges of which Frankfurt is the oldest one with roots going back to the period of medieval fairs. Existing since before 1870, when Germany was made up of independent kingdoms, 21 stock exchanges where known when the Third Reich government decided to close down 12 of these. Besides Frankfurt being the centre of banking industry, Düsseldorf, Munich and Hamburg were important trading venues at that time. Today, there are seven stock exchanges in Germany, one foreign currency exchange, and one exchange each for stock and commodity futures. Of the seven stock exchanges, the Frankfurt Stock Exchange run by Deutsche Börse AG is presently the most important one with 90% of all earnings generated at German stock exchanges. Investors can choose between the fully electronic «Xetra» trading platform or (for institutional and private investors) the «Börse Frankfurt» trading venue where less liquid shares are traded. The remaining six stock exchanges are often labelled as «regional stock exchanges» and are located in Stuttgart, Munich, Düsseldorf, Berlin, Hamburg, and Hannover. They are often specialized in specific industry sectors and «niches» targeting smaller, private investors. Stocks can be traded simultaneously at all these stock exchanges; however, the prices from the Frankfurt Stock Exchange are used as reference prices.

The stock market in Germany is further segmented vertically along different admission requirements, together with certain transparency standards. The «Regulated Market» (German: Regulierter Markt) is an organised market in the sense of the Securities Trading Act meaning admission conditions and follow-up duties are set out in law; whereas the «Open Market» (German: Freiverkehr) is regulated by the stock exchanges themselves. At the Frankfurt Stock Exchange the «Regulated Market» is since 2003 sub-divided into the General Standard and the Prime Standard differing in their level of required transparency, e.g.

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the frequency of disclosures or availability of information in English. In the «Open Market» a firm’s stock can be listed in the Scale or on the Quotation Board, being more attractive to smaller and medium-sized companies due to the lower transparency requirements (Deutsche Börse 2017). Before the current market segmentation came in place, traditionally three different segments existed: the top segment or «Official Market» (German: Amtlicher Markt), the middle segment (German: Geregelter Freiverkehr, Geregelter Markt) and the lowest segment (German: Ungeregelter Freiverkehr, Freiverkehr). Hereby, only the «Official Market» was regulated by the national stock exchange act. An additional segment was created in Frankfurt in 1997 called «New Market» (German: Neuer Markt) designed to attract technology firms usually listed in the middle segment. At its peak in the year 2000, the segment contained 283 companies (year-end). After a huge initial success the respective NEMAX All Shares index plummeted dramatically with the bursting of the dot-com bubble and only 198 companies were left at year-end 2002. The segment was closed in June 2003 as a consequence. Figure 1 visualizes the development of the number of quoted companies and the different segments that existed over time at the Frankfurt Stock Exchange. No data was available for the period after 2014.

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In principle, the dataset used includes stocks from all segments; however, due to further sampling requirements set (please see chapter 3.1), many stocks listed in the Open Market (also Entry Standard and Other) will be excluded.

The two most relevant types of stocks in Germany are common stock (German: Stammaktie) and preferred stock (German: Vorzugsaktie). Thus both stock classes will be included in the data selection. Contrary to other countries, preferred stock in Germany is very similar to common stock and should therefore have comparable risk-return characteristics. The main difference is the inferior voting power of preferred stock which is usually compensated by a small dividend advantage and a set minimum cumulative dividend which has to be paid out to shareholders. In case a company has not paid the minimum dividend in any year, it must be paid in the subsequent years and common stockholders may not receive any dividend payments before the preferred stockholders are compensated. Preferred stock can be converted into common stocks if decided so by the majority of all stockholders and consensus given by preferred stockholders.

Stehle and Schmidt (2015) note that the tax regime for dividends changed over time, and highlight the fact that between 1977 and 2000 double taxation of dividends (at corporate and personal level) was suspended. During this period, German investors received a credit voucher in the amount of the corporate income tax they had paid on their dividends to pay their personal income taxes or claim a tax refund. For the period 1977 to 2000 Stehle and Schmidt (2015) estimate an average return advantage of approx. 1.4% per annum for German investors benefitting from these tax refunds. The data collected and used for this thesis does not take into account these corporate tax effects. However, considering that the time series used here (January 2000 until July 2017) includes only the last twelve months (or 5.7% of the total observation period) in which this dividend advantage was in existence, the impact on the overall results are expected to be small.

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Chapter 3 Data and methodology

3.1 Data selection

For a stock to be included in my dataset it has to fulfil a number of criteria: firstly, it has to be traded on the Frankfurt Stock Exchange; secondly, it needs to belong to a German company, i.e. the country code DE must appear in the ISIN; and lastly, it has to be either a common or a preferred stock. Observations with a stock price below EUR 1, so-called «penny stocks» are excluded from the data because of the potential bias they cause due to the considerably higher standard deviation of their yield. The Federal Financial Supervisory Authority of Germany has warned about price manipulations that can easily occur in the penny stocks market (Bundesanstalt für Finanzdienstleistungsaufsicht 2013, 2015). Furthermore the stock has to have at least 30 months with at least 10 days of observations per month over the period of January 1999 to July 2017 to be included. While the regressions will be performed for the period from January 2000 until July 2017, the year 1999 is included in order to be able to construct the portfolio for 2000 based on previous year figures.

The data on stocks is entirely obtained from Datastream which reports the figures from the Frankfurt Stock Exchange. This encompasses the daily individual stock closing prices (in EUR), the daily individual stock trading volumes (in thousands), the daily individual stocks market values (in millions) used to approximate the market capitalization of a company; the

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daily individual stock book-to-market price ratios; and the daily individual stock total return index. The daily individual stock returns are calculated from the daily individual stock total return index.

The market return is approximated by the returns of the performance index Composite DAX (CDAX), which is a value-weighted German stock market index calculated by Deutsche Börse and composed of all German stocks traded on the Frankfurt Stock Exchange that are listed in the General Standard or Prime Standard market segments. Daily closing prices for the CDAX are obtained from Wallstreet-online (2017) and monthly returns calculated on that basis. The funding liquidity innovations are approximated by mispricings in German Government Listed Securities and represent changes in market-wide systemic funding liquidity risk. Input data is obtained from the Deutsche Bundesbank and used for calculations (see 3.2). The risk free rate is approximated by the one-month EURIBOR, which is obtained from the website of the Deutsche Bundesbank (2017b). The reported nominal annual rates are converted to monthly rates.

The monthly measure for market liquidity of a stock is defined according to Amihud (2002): 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑖𝑖,𝑡𝑡 =_𝑁𝑁1

𝑖𝑖𝑖𝑖∑

�𝑟𝑟𝑖𝑖,𝑑𝑑,𝑖𝑖�

𝑣𝑣𝑣𝑣𝑣𝑣𝑖𝑖,𝑑𝑑,𝑖𝑖

𝑑𝑑 ∗ 106 (1)

where �𝑟𝑟_{𝑖𝑖,𝑑𝑑,𝑡𝑡}� is the absolute value of return of stock 𝑖𝑖 on day 𝑑𝑑 in period 𝑡𝑡 (which is monthly), 𝑣𝑣𝑣𝑣𝑣𝑣_{𝑖𝑖,𝑑𝑑,𝑡𝑡} is the trading volume in Euros of stock 𝑖𝑖 on day 𝑡𝑡 which is obtained by multiplying the number of shares traded on day 𝑑𝑑 by the closing price on day 𝑑𝑑. This measure for market liquidity is the daily ratio of absolute stock return to its respective monetary value equivalent in Euros, averaged over a period of one month. It has the intuitive interpretation of the daily price response to one Euro of trading volume, serving as an approximation of price impact. The higher the value of this measure, the more illiquid the respective stock is. While better measures of market liquidity of stocks exist, they require more microstructure data,

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which is not readily available on a daily basis and for a larger set of stocks and time period. Stange and Kaserer (2009) discuss a well-chosen selection of models that can be used for the determination of an asset’s market liquidity, ranging from models based on bid-ask spread data over models based on volume or transaction data to models based on limit order book data.

For the monthly measure of stock return volatility, the standard deviation is obtained by calculating the arithmetic mean for each month, subtracting this arithmetic mean from the daily return observations, squaring these differences and summing them up to average them thereby forming the variance. The standard deviation is the square root of this variance.

3.2 Construction of the variable funding liquidity

In order to construct the measure for funding liquidity, the intention is to capture the ‘noise’ in the price, i.e. the mispricings of German Federal Listed Securities and exploit its informativeness as a measure of market-wide funding liquidity. As stated before, the price deviations in German Federal Listed Securities can be used as an approximation for funding liquidity of financial markets participants, since the price deviations are likely to occur due to shortage in arbitrage capital instead of market frictions because of the profound market depth and high market liquidity of these assets. The three following types of German Federal Listed Securities are relevant for constructing the zero-coupon yield curve and calculating the model-implied theoretical prices and yields: Federal Government Bonds (Bundesanleihen) with maturities of ten or 30 years, Federal Debt Obligations (Bundesobligationen) with maturities of five years and Federal Treasury Notes (Bundesschatzanweisungen) with maturities of two years. All of these securities are traded on stock exchanges all over Europe, various international electronic trading platforms and in the OTC market. They are quoted continuously by market makers on a voluntary basis which ensures that no artificial liquidity is created which could give rise to misconceptions about the depth of the market among

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investors. German Government securities are traded at the tightest bid-ask spread among the euro-denominated sovereign debt instruments and their market is one of the world's largest and most liquid markets for sovereign debt. In 2016, the average nominal volume in circulation of around EUR 1.1 trillion (excluding the Federal Government's own holdings) was turned over almost 4 times. This corresponds to a daily trading volume of more than EUR 17 billion. This high market liquidity is primarily due to the futures contracts traded on the German futures and options exchange «Eurex» whose pricing is oriented to German Government securities, which are the only securities accepted for delivery. The main turnover on the futures market is in the contract for the ten-year Federal Government Bond future in which about 186 million contracts and a volume of more than EUR 30 trillion (market value) were traded in 2016. Accordingly, the ten-year Federal Government Bond, which serves as this future's underlying, also accounts for the lion’s share of the turnover in the international secondary market for German Government securities (Bundesrepublik Deutschland Finanzagentur GmbH 2017).

I construct the monthly noise measure by following the work done by Hu, Pan and Wang (2013) who calculate the root mean squared distance between the observed yields of bonds and the computed yields of bonds derived from the term structure of interest rates thereby creating the monthly time-series of the new variable ‘funding liquidity’ (𝐹𝐹𝐼𝐼):

𝐹𝐹𝐼𝐼𝑡𝑡= �_𝑁𝑁1_𝑖𝑖∑ �𝑦𝑦𝑁𝑁𝑖𝑖=1𝑖𝑖 𝑡𝑡𝑖𝑖 − 𝑦𝑦𝑖𝑖(𝑏𝑏𝑡𝑡)�2 (2)

where for each of the 𝑁𝑁_𝑡𝑡 bonds, obligations and notes which are traded in each month 𝑡𝑡, 𝑦𝑦_𝑡𝑡𝑖𝑖 denotes its market observed yield and 𝑦𝑦𝑖𝑖(𝑏𝑏_𝑡𝑡) denotes its model-implied yield, where 𝑏𝑏_𝑡𝑡 is the vector of model parameters. The observed market yields 𝑦𝑦_𝑡𝑡𝑖𝑖 were kindly provided by the Deutsche Bundesbank upon request. The Deutsche Bundesbank performs the estimation of the zero-coupon yield curve for German Federal Listed Securities on a daily basis using the

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Nelson-Siegel (1987) approach extended by Svensson (1994) which is a parametric curve-fitting technique that assumes a certain function for forward rates using six parameters to specify the functional form of the curve (Schich 1997:15). Svensson (1994) assumes that the instantaneous forward curve is given by

𝑓𝑓(𝑏𝑏𝑡𝑡) = 𝛽𝛽0 + 𝛽𝛽1exp �−_𝜏𝜏𝑚𝑚₁� + 𝛽𝛽2𝑚𝑚_𝜏𝜏₁exp �−𝑚𝑚_𝜏𝜏₁� + 𝛽𝛽3_𝜏𝜏𝑚𝑚₂exp (−𝑚𝑚_𝜏𝜏₂). (3)

By integrating the forward curve, I derive the zero-coupon spot curve:

𝑧𝑧̃(𝑏𝑏𝑡𝑡) = 𝛽𝛽0,𝑡𝑡+ 𝛽𝛽1,𝑡𝑡 1 − exp �− 𝑚𝑚_𝜏𝜏 1� �𝑚𝑚_𝜏𝜏 1� + 𝛽𝛽2,𝑡𝑡� 1 − exp �− 𝑚𝑚_𝜏𝜏 1� �𝑚𝑚_𝜏𝜏 1� − exp �−𝑚𝑚_𝜏𝜏 1�� +𝛽𝛽3,𝑡𝑡� 1−exp�−_𝜏𝜏2𝑚𝑚� 𝑚𝑚 𝜏𝜏2 − exp (− 𝑚𝑚 𝜏𝜏2)� (4)

The vector of model parameters 𝑏𝑏_𝑡𝑡 including 𝛽𝛽₀, 𝛽𝛽₁, 𝛽𝛽₂, 𝛽𝛽₃, 𝜏𝜏₁, 𝜏𝜏₂ can be obtained from the website of the Deutsche Bundesbank (2017a). I use these parameters to first calculate the theoretical interest rates 𝑧𝑧̃_{𝑡𝑡,𝐼𝐼−𝑚𝑚} (according to (7)) needed for the discounted cash flow calculation and then run this discounted cash flow calculation for each single security with its remaining maturity at each end-of-month for the chosen observation period of January 1999 to July 2017 to compute the theoretical prices for all the relevant securities outstanding. Upon request, the Deutsche Bundesbank kindly provided the necessary input variables, i.e. coupon and time to maturity for all relevant outstanding securities.

The model-implied price of each bond is the «dirty price» hence; accrued interest needs to be deducted: 𝑝𝑝𝑖𝑖_(𝑏𝑏 𝑡𝑡) =_{�1+𝑧𝑧�} 𝑐𝑐 𝑖𝑖,𝑀𝑀−1�𝑀𝑀−1+ 𝑐𝑐 �1+𝑧𝑧�𝑖𝑖,𝑀𝑀−2�𝑀𝑀−2+ ⋯ + 𝑐𝑐 �1+𝑧𝑧�𝑖𝑖,𝑀𝑀−𝑚𝑚�𝑀𝑀−𝑚𝑚+ 𝑐𝑐+𝑛𝑛 �1+𝑧𝑧�𝑖𝑖,𝑀𝑀�𝑀𝑀− (𝑐𝑐 ∗ 𝑠𝑠) = ∑ 𝑐𝑐 �1+𝑧𝑧�𝑖𝑖,𝑀𝑀−𝑚𝑚�𝑀𝑀−𝑚𝑚+ (𝑐𝑐+𝑛𝑛) �1+𝑧𝑧�𝑖𝑖,𝑀𝑀�𝑀𝑀 𝐼𝐼 𝑚𝑚=1 − (𝑐𝑐 ∗ 𝑠𝑠) (5)

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where 𝑝𝑝𝑖𝑖(𝑏𝑏_𝑡𝑡) is the model-implied price of bond 𝑖𝑖, 𝑐𝑐 is the coupon payment, 𝑛𝑛 is the principal amount, 𝑧𝑧̃_{𝑡𝑡,𝐼𝐼} is the zero-coupon spot rate for time to maturity 𝐼𝐼 where 𝑚𝑚 approaches 𝐼𝐼 up to the last integer before 𝐼𝐼; and 𝑐𝑐 ∗ 𝑠𝑠 is the accrued interest where:

𝑠𝑠 =𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 𝑏𝑏𝑏𝑏𝑡𝑡𝑏𝑏𝑏𝑏𝑏𝑏𝑛𝑛 𝐷𝐷𝑏𝑏𝑡𝑡𝑡𝑡𝑣𝑣𝑏𝑏𝑚𝑚𝑏𝑏𝑛𝑛𝑡𝑡 𝐷𝐷𝑛𝑛𝑑𝑑 𝑣𝑣𝐷𝐷𝐷𝐷𝑡𝑡 𝑐𝑐𝑣𝑣𝑐𝑐𝑐𝑐𝑛𝑛 𝑐𝑐𝐷𝐷𝐷𝐷𝑚𝑚𝑏𝑏𝑛𝑛𝑡𝑡_{𝑀𝑀𝑣𝑣𝑡𝑡𝐷𝐷𝑣𝑣 𝑑𝑑𝐷𝐷𝐷𝐷𝐷𝐷 𝑖𝑖𝑛𝑛 𝑐𝑐𝑣𝑣𝑐𝑐𝑐𝑐𝑣𝑣𝑛𝑛 𝑐𝑐𝐷𝐷𝐷𝐷𝑚𝑚𝑏𝑏𝑛𝑛𝑡𝑡 𝑐𝑐𝑏𝑏𝑟𝑟𝑖𝑖𝑣𝑣𝑑𝑑} . (6)

The bonds in scope all have annual coupon payments, hence Total days in coupon payment period is equal to the number of days in a year, i.e. 365 or 366. On the basis of the obtained model-implied prices the respective model-implied yields 𝑦𝑦𝑖𝑖(𝑏𝑏_𝑡𝑡) can be calculated by solving the following equation for 𝑦𝑦𝑖𝑖(𝑏𝑏_𝑡𝑡):

𝑝𝑝𝑖𝑖_(𝑏𝑏 𝑡𝑡) =_{�1+𝐷𝐷}_𝑖𝑖_(𝑏𝑏𝑐𝑐 𝑖𝑖)𝑀𝑀−1�𝑀𝑀−1+ 𝑐𝑐 �1+𝐷𝐷𝑖𝑖_(𝑏𝑏_𝑖𝑖₎_𝑀𝑀−2_�𝑀𝑀−2+ ⋯ + 𝑐𝑐 �1+𝐷𝐷𝑖𝑖_(𝑏𝑏_𝑖𝑖₎_{𝑀𝑀−𝑚𝑚}_�𝑀𝑀−𝑚𝑚+ 𝑐𝑐+𝑛𝑛 �1+𝐷𝐷𝑖𝑖_(𝑏𝑏_𝑖𝑖₎_{𝑀𝑀−𝑚𝑚}_�𝑀𝑀−𝑚𝑚= ∑ 𝑐𝑐 �1+𝐷𝐷𝑖𝑖_(𝑏𝑏_𝑖𝑖₎_{𝑀𝑀−𝑚𝑚}_�𝑀𝑀−𝑚𝑚+ (𝑐𝑐+𝑛𝑛) �1+𝐷𝐷𝑖𝑖_(𝑏𝑏_𝑖𝑖₎_{𝑀𝑀−𝑚𝑚}_�𝑀𝑀 𝐼𝐼 𝑚𝑚=1 (7)

Securities that were included are all the outstanding Federal Government Bonds, Federal Debt Obligations and Federal Treasury Notes with a time to maturity of at least 6 months but no more than 30 years. I exclude securities with remaining maturities below 6 months and above 30 years because on the one hand side the short end of the yield curve is usually more noisy and also less likely to be subject to arbitrage; and on the other hand side the Nelson-Siegel-Svensson modelled yield curve comes with some loss of accuracy at the very short and very long end (Choundry 2012: 315). To avoid that pricing errors drive the noise measure, I put in place a filter effectively excluding observations where the difference between the observed yield and the theoretical yield is larger than four standard deviations away from the mean. This leads to an exclusion of only 1% of the observations and is therefore a reasonably mild filter. The monthly time-series variation of the constructed noise measure calculated

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according to formula (6) for the period of January 1999 throughout July 2017 is shown in Figure 2.1

Figure 2: Yield spreads in German Government bonds over the period of January 1999 to July 2017

During normal times the noise measure fluctuates around its time-series average of 3.89 basis points with a standard deviation of 2.47 basis points, and is highly persistent with a monthly autocorrelation of 92.62%.2 The average month included 46 bonds. The two most distinguishable peaks are the dotcom crises in 2002/2003 and the financial crises in 2008/2009. The highest spike in the time-series occurred in November 2002 attaining 19.68 basis points; while the other extreme is the lowest level in November 2005 at 1.10 basis points. It is noticeable that the dotcom crises had a relatively larger impact onto the prices and yields of German Government bonds compared to the financial crises. Possible explanations could be that the overall market liquidity in 2002/2003 as such was lower (compared to 2008/2009 or even today) due to less sophisticated trading systems and platforms; and that Germany was particularly affected by the dotcom crises because its real economy consisted of those new technology firms. This latter point is evidenced by the

1

See Annex 1 for the complete time series of the variable FL.

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amount of companies that were listed in the newly created segment «Neuer Market» at the Frankfurt Stock Exchange, which was aimed to attract technology firms (see chapter 2.3). The obtained results resemble those of Hu, Pan and Wang (2013) who do the same for the U.S. Government bond market and found a time-series average of 3.61 basis points with a standard deviation of 2.17 basis points and a monthly autocorrelation of 90.75%.

Their noise measure peaks at 20.47 basis points in December 2008 three months after the default of Lehman Brothers when concerns over the financial crises deepened. In that respect the authors emphasize that “[…] our noise measure comes from the U.S. Treasuries bond market – the one with the highest credit and liquidity quality and is the number one safe haven during crises, and yet it was able to reflect liquidity crises of varying origins and magnitudes. In this respect, what is captured in our noise measure is not the liquidity concerns specific to the Treasury market, but liquidity conditions across the overall financial market.” (Hu, Pan and Wang 2013: 3)

In order to try to explicate their funding liquidity noise measure, Hu, Pan and Wang (2013) regress their measure onto various other liquidity measures such as the liquidity premia in government-backed bonds and in «on-the-run » bonds; the Pastor and Stambaugh (2003) liquidity factor (see chapter 2.1); the VIX index, which is a popular measure of the implied volatility of S&P 500 index options calculated and published by the Chicago Board Options Exchange; and default spreads, measured as the difference in yield between Baa and Aaa rated bonds. Their results show that collectively, these variables can only explain about 43.7% of the monthly variation in the noise measure. Consequently, they take this as evidence that more than 50% of the variation remains unexplained and claim that this unexplained component is additional informativeness about overall liquidity conditions across financial markets.

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3.3 Descriptive statistics

Over the period of January 2000 until July 2017, i.e. 211 months, on average 435 companies are included in the dataset each month and represent on average EUR 1,035 billion. Figure 3 shows the two largest dips in market values around 2002/2003 and 2008/2009, the gradual increase after the dotcom crises leading up to the financial crises, and the increase in market values of German stock market companies again since 2012. What is noticeable is the decrease in the number of stock companies since 2012, while the total market capitalisation continues to increase.

Figure 3: Evolution of total market capitalisation and number of stock companies included in the dataset

Hence, this evolution seems to be driven by the rising prices of shares included in the dataset. This claim is supported by the evolution of month-end figures of the CDAX index, which can be seen below in Figures 4 and 5. Since the end of 2012 the CDAX performance index has almost doubled and reached a first peak in November 2017 at 1,250 points.

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Figure 4: Evolution of month-end CDAX performance index over the period January 2000 to July 2017

Figure 5: Evolution of month-end CDAX performance index over the period January 2013 to July 2017

As can be seen in Figure 6, the book-to-market price ratio graph indicates current valuation ratios attaining the same and even higher heights as during the financial crisis, but still below those of the dotcom bubble.

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Figure 6: Evolution of total market capitalisation and median book-to-market-price ratio across stock companies included in the dataset

The variable market excess return is calculated as the difference between the monthly returns of the CDAX performance index and the risk free rate approximated by the one-month EURIBOR; both are graphed below in Figures 7 and 8.

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Figure 8: Evolution of the EURIBOR over the period January 2000 to July 2017

As the graph in Figure 7 already suggests, market return is revealed to be stationary when performing a test for a unit root. The variable risk free rate depicted in Figure 8, however, shows clear signs of a trend which is corroborated by a unit root test, which cannot reject the null hypothesis for any of the selected lags. Nevertheless, since the relevant variable for the estimations eventually is the market excess return variable, which again turns out to be stationary, no further transformations of variables are necessary.3

The variables 𝑆𝑆𝐼𝐼𝑆𝑆𝑆𝑆 and 𝑉𝑉𝑉𝑉𝐼𝐼𝑉𝑉𝑆𝑆 in the following equations represent the Fama-French factors SMB (Small minus Big) and HML (High minus Low) respectively. All stocks are categorized depending on whether their market capitalisation (𝑆𝑆𝐼𝐼𝑆𝑆𝑆𝑆) is small (“S”), i.e. below the median or big (“B”), i.e. on or above the median; and depending on whether their book-to-market-price (𝑉𝑉𝑉𝑉𝐼𝐼𝑉𝑉𝑆𝑆) is high (“H”), i.e. on or above the 70th percentile, medium (“M”), i.e. on or above the 30th percentile, or low (“L”), i.e. below the 30th percentile. The variables are consequently calculated as follows:

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𝑆𝑆𝐼𝐼𝑆𝑆𝑆𝑆𝑡𝑡 =�𝑟𝑟𝑖𝑖 𝑆𝑆/𝐻𝐻_−𝑟𝑟 𝑖𝑖𝐵𝐵/𝐻𝐻�+�𝑟𝑟𝑖𝑖𝑆𝑆/𝑀𝑀−𝑟𝑟𝑖𝑖𝐵𝐵/𝑀𝑀�+(𝑟𝑟𝑖𝑖𝑆𝑆/𝐿𝐿−𝑟𝑟𝑖𝑖𝐵𝐵/𝐿𝐿) 3 (8) 𝑉𝑉𝑉𝑉𝐼𝐼𝑉𝑉𝑆𝑆𝑡𝑡 =�𝑟𝑟𝑖𝑖 𝑆𝑆/𝐻𝐻_−𝑟𝑟 𝑖𝑖𝑆𝑆/𝐿𝐿�+�𝑟𝑟𝑖𝑖𝐵𝐵/𝐻𝐻−𝑟𝑟𝑖𝑖𝐵𝐵/𝐿𝐿� 2 (9)

where 𝑟𝑟_𝑡𝑡 stands for the aggregated monthly value-weighted return of the stocks in the respective portfolios. For example, 𝑟𝑟_𝑡𝑡𝑆𝑆/𝐻𝐻 is the monthly value-weighted return of all stocks with a small market capitalisation and a high book-to-market-price ratio. The formula for 𝑆𝑆𝐼𝐼𝑆𝑆𝑆𝑆 eliminates the book-to-market-price effect, while capturing the differences in monthly returns for stocks with larger market capitalisation. The formula for 𝑉𝑉𝑉𝑉𝐼𝐼𝑉𝑉𝑆𝑆 does the opposite and captures the differences in monthly returns of stocks with high book-to-market-price compared to those with low book-to-market-book-to-market-price and eliminates the size effect. Figure 9 shows both time series on 𝑉𝑉𝑉𝑉𝐼𝐼𝑉𝑉𝑆𝑆 and 𝑆𝑆𝐼𝐼𝑆𝑆𝑆𝑆.

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Unit root tests for both variables provided evidence in favour of the variables being stationary.4 The variable 𝐼𝐼𝐼𝐼𝐼𝐼 (Illiquid-Minus-Liquid) is constructed following the work of Amihud et al. (2015) who calculate it as the differential return between the average monthly returns of a portfolio of the 20% most illiquid stocks and the portfolio of the 20% most liquid stocks, as measured by ILLIQ. Also for this variable, the test for a unit root indicates stationarity.5 Figure 10 depicts the evolution of the variable 𝐼𝐼𝐼𝐼𝐼𝐼 over time

Figure 10: Evolution of the variable IML in the period from January 1999 to July 2017

4

See Annex D for unit root tests of the variables SIZE and VALUE.

5

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3.4 Factor pricing models

Being interested in how prices of assets change as a function of a certain risk inevitably brings one to linear factor models, since the basic consumption-based pricing model faces serious empirical difficulties. Factor pricing models replace the consumption-based expression6 for marginal utility growth with a linear model of the form:

mt+1= a + b′ft+1 (10)

where 𝑎𝑎 and 𝑏𝑏 are free parameters, m_t+1 is the stochastic discount factor, i.e. really the marginal rate substitution and f_t+1 are the factors in the sense of proxies for marginal utility that “describe whether investors are happy or unhappy” as Cochrane (2005:45) puts it. The most famous and widely used linear factor model is the Capital Asset Pricing Model (CAPM) developed by Sharpe (1964) and Lintner (1965a, b) based on Markowitz’ Modern Portfolio Theory (Markowitz 1952). It is a single-factor model which links the discount factor 𝑚𝑚 only and entirely to the return of a so-called ‘wealth portfolio’ also referred to as the ‘market portfolio’:

mt+1= a + b𝑅𝑅t+1𝑚𝑚 . (11)

According to Modern Portfolio Theory, only the non-diversifiable risk of an asset is priced. This non-diversifiable risk is entirely represented by the risk of the market portfolio. The market portfolio has the highest attainable Sharpe Ratio, which is the ratio that describes how much excess return an investor should be receiving to be compensated for the extra volatility (i.e. the potential deviation from the average portfolio return) he endures for adding a riskier asset to his portfolio. The CAPM hence establishes that only systematic market risk plays a

6

The most important formula in asset pricing gives the first-order conditions for optimal consumption and portfolio formation by linking the market price 𝑝𝑝_𝑡𝑡, that is to be expected, to the payoff 𝑥𝑥_𝑡𝑡+1 and the investor’s consumption choice 𝑐𝑐_𝑡𝑡 and 𝑐𝑐_𝑡𝑡+1: 𝑝𝑝_𝑡𝑡= 𝑆𝑆_𝑡𝑡�𝛽𝛽𝑐𝑐′(𝑐𝑐𝑖𝑖+1)

𝑐𝑐′(𝑐𝑐𝑖𝑖) 𝑥𝑥𝑡𝑡+1�. The stochastic discount factor 𝑚𝑚 captures the

marginal rate of substitution 𝛽𝛽𝑐𝑐′(𝑐𝑐_{𝑐𝑐′(𝑐𝑐}𝑖𝑖+1)

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role in the pricing of assets thereby disregarding idiosyncratic risks. In the expected return-beta notation the extent to which an asset’s excess return co-moves with the market portfolio’s excess return is shown by beta (𝛽𝛽_𝑖𝑖):

𝑆𝑆(𝑟𝑟𝑖𝑖) = 𝑟𝑟𝑓𝑓+ 𝛽𝛽𝑖𝑖[𝑆𝑆(𝑟𝑟𝑚𝑚) − 𝑟𝑟𝑓𝑓]. (12)

Over the past decades the CAPM has become the benchmark model for asset pricing; however, it does rely on a set of simplifying assumptions, such as that: “(1) Investors are risk-averse individuals who maximize the expected utility of their wealth; (2) Investors are price takers and have homogeneous expectations about asset returns that have a joint normal distributions; (3) There exists a risk-free asset such that investors may borrow or lend unlimited amounts at a risk-free rate; (4) The quantities of assets are fixed, and all assets are marketable and perfectly divisible; (5) Asset markets are frictionless, and information is costless and simultaneously available to all investors; and (6) There are no market imperfections such as taxes, regulations, or restrictions on short selling” (Copeland, et al. 2005). Since these critical assumptions considerably limit the applicability of the basic CAPM model, plenty of research has tried to further develop the model by relaxing assumptions and testing empirical implications. Shih et al (2013) provide a comprehensive summary and overview of the evolution of the CAPM. One of the model extensions they discuss and that should be mentioned here is the liquidity-adjusted model by Acharya and Pedersen (2005). They show that the beta (𝛽𝛽_𝑖𝑖) in the CAPM model can be decomposed into the standard market excess return beta plus three additional betas representing each a different form of liquidity risk: (1) commonality in liquidity referring to “investors requiring a return premium for a security that is illiquid when the market as a whole is illiquid”; (2) return sensitivity to market liquidity referring to “investors’ preference for securities with high returns when the market is illiquid”; and (3) liquidity sensitivity to market returns referring to investors’ willingness to “pay a premium for a security that is liquid when the

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return is low”. Acharya and Pedersen (2005) note that the last source seems to be the most significant source of liquidity risk even though it has not yet received much attention in the literature and empirical studies.

A second strand of linear factor models is based on the Arbitrage Pricing Theory (APT) developed by Ross (1976) and has borne a multiple-factor model of which the most prominent one is the Fama-French three-factor model. Contrary to the CAPM, which is an economic model that tries to explain returns by finding the actual risks which justify higher returns, the APT starts from the statistical characterization of the common components or ‘factors’ explaining average returns. Cochrane (2005:182) states that “[t]he biggest difference between APT and ICAPM7 for empirical work is the inspiration for factors. The

APT suggests that one starts with a statistical analysis of the covariance matrix of returns and finds portfolios that characterize common movement. The ICAPM suggests that one starts by thinking about state variables that describe the conditional distributions of future asset returns.” He furthermore points out that the APT is a relative pricing model which prices one security given the prices of others, while an absolute pricing model would price any one security by reference to its fundamental sources of risk. The notation of a multiple-factor model can look as follows:

𝑆𝑆(𝑟𝑟𝑡𝑡) = 𝑟𝑟𝑡𝑡𝑓𝑓+ 𝛽𝛽1�𝑟𝑟1𝑡𝑡 − 𝑟𝑟𝑡𝑡𝑓𝑓� + 𝛽𝛽2�𝑟𝑟2𝑡𝑡− 𝑟𝑟𝑡𝑡𝑓𝑓� + ⋯ + 𝛽𝛽𝑛𝑛�𝑟𝑟𝑛𝑛𝑡𝑡− 𝑟𝑟𝑡𝑡𝑓𝑓� (13)

where 𝑆𝑆(𝑟𝑟_𝑡𝑡) is the expected return of the portfolio at time 𝑡𝑡, 𝑟𝑟_𝑡𝑡𝑓𝑓 is the risk-free rate, 𝛽𝛽_𝑛𝑛 is the sensitivity of the portfolio to factor 𝑛𝑛, and �𝑟𝑟_{𝑛𝑛𝑡𝑡}− 𝑟𝑟_𝑡𝑡𝑓𝑓� is the risk premium of the portfolio associated to the respective factor 𝑛𝑛. Fama-French’s model states that one can explain stock returns by their covariances - or betas - with market risk-return, size and book-to-market value.

7

ICAPM refers to the Intertemporal Capital Asset Pricing Model that employs stochastic investment opportunities in a continuous-time model.

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The time series regression equation consequently would look like:

𝑅𝑅𝑡𝑡 = 𝛼𝛼 + 𝛽𝛽1𝑅𝑅𝑡𝑡𝑚𝑚+ 𝛽𝛽2𝑆𝑆𝐼𝐼𝑆𝑆𝑡𝑡+ 𝛽𝛽3𝐻𝐻𝐼𝐼𝐼𝐼𝑡𝑡+ 𝜀𝜀𝑡𝑡 (14)

where 𝑅𝑅_𝑡𝑡 is excess return, 𝛼𝛼 is the intercept, 𝛽𝛽𝑛𝑛are the covariances of the asset’s excess return with the factors (also called factor loadings), 𝑅𝑅_𝑡𝑡𝑚𝑚 is the excess return on the market, 𝑆𝑆𝐼𝐼𝑆𝑆𝑡𝑡 is the factor capturing size (e.g. market capitalization of a firm), 𝐻𝐻𝐼𝐼𝐼𝐼𝑡𝑡 is the book-to-value ratio and 𝜀𝜀_𝑡𝑡 is the error term. The aim of their time series regression is to establish the cross-sectional relation between average return and betas and to see if the alpha is low, i.e. to see if high average returns are associated with high betas on the factors excess market return, size, and book-to-market value (Cochrane, 2017).

Bearing in mind their conceptual differences, I will apply both the basic CAPM model as well as the Fama-French three-factor model, since the two models can both deliver reliable results depending on the portfolio formation.

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Chapter 4 Empirical results

4.1 Panel data regressions

4.1.1 Cross-sectional regression of stock returns onto market liquidity

The first cross-sectional estimation that I will perform follows the course of investigation of Amihud et al. (2015) and regresses the monthly return of an individual stock onto the stock’s market liquidity (𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼3), controlling for other stock characteristics such as market capitalisation (𝐼𝐼𝑀𝑀𝑉𝑉𝑀𝑀), book-to-market ratio (𝑆𝑆𝑡𝑡𝐼𝐼) and return volatility (𝑉𝑉𝑉𝑉𝐼𝐼𝑉𝑉). The monthly regressions cover the period from January 2000 to July 2017. The dependent variable is the monthly return (𝑟𝑟_{𝑖𝑖𝑡𝑡}) of an individual stock in a given month; the independent main explanatory variable is an individual stock’s 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼3 measure, calculated as the stock’s average ILLIQ of the three directly preceding months, given a specific month. The independent control variable 𝑉𝑉𝑉𝑉𝐼𝐼𝑉𝑉3 is the average return volatility over the directly preceding three months, given a specific month. The independent control variables 𝐼𝐼1. 𝐼𝐼𝑀𝑀𝑉𝑉𝑀𝑀 and 𝐼𝐼1. 𝑆𝑆𝑡𝑡𝐼𝐼 capture respectively the first lag of the natural logarithm of market capitalisation and the first lag of the book-to-market ratio of an individual stock.

𝑟𝑟𝑖𝑖𝑡𝑡 = 𝛽𝛽𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼3𝑖𝑖𝑡𝑡 + 𝛽𝛽𝐼𝐼𝑀𝑀𝑉𝑉𝑀𝑀L1. MCAP𝑖𝑖𝑡𝑡+ 𝛽𝛽𝐵𝐵𝑡𝑡𝐼𝐼𝐼𝐼1. 𝑆𝑆𝑡𝑡𝐼𝐼𝑖𝑖𝑡𝑡 +

𝛽𝛽𝑉𝑉𝑉𝑉𝐼𝐼𝑉𝑉_{𝑉𝑉𝑉𝑉𝐼𝐼𝑉𝑉3}

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Following the specification of this fixed effects regression model, the coefficient on 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼3 captures the marginal effect of a stock’s market liquidity on its excess return after controlling for different firm characteristics. As in the paper of Amihud et al. (2015), I standardize the variable 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼3, so that the monthly average in each month is always 1.0. This mean-adjustment enables me to interpret e.g. 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼3 = 2 as twice the average 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼3 across all individual stocks in a given month. Robust standard errors are reported in parenthesis and statistically significant coefficients are labelled with *** if p < 0.01, with ** if p < 0.05 and with * if p < 0.10.

Table 1: Fixed effects cross-section regression results on stocks’ monthly returns

The results of this cross-sectional regression shown in Table 1 indicate that on an individual stock level, a stock’s market capitalisation, return volatility and market liquidity influence the stock’s return in a statistically significant way. All three variables are statistically significant at the 1% critical value. The coefficient on market liquidity is at 0.0206, which means that a stock whose average market liquidity is half the average market liquidity (or double the average market illiquidity) has a higher monthly return of 0.0206%, equal to approx. 0.25% per year. This result is similar to the findings of Amihud et al. (2015: 70), however the impact they find is somewhat higher; approx. 1.27% per year higher returns for stocks with double the average market illiquidity. Higher return volatility in turn is associated with lower

Return 0.0262*** (0.0013) 0.0000 (0.0000) 0.0206*** (0.0015) -0.2516*** (0.0612) -0.1484*** (0.0077) 79,759 820 2.01% 𝛽𝛽𝐼𝐼𝑀𝑀𝑉𝑉𝑀𝑀 𝛽𝛽𝐵𝐵𝑡𝑡𝐼𝐼 𝛽𝛽𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼3 𝛽𝛽𝑉𝑉𝑉𝑉𝐼𝐼𝑉𝑉3 𝑅𝑅2 𝑉𝑉𝑏𝑏𝑠𝑠𝑒𝑟𝑟𝑣𝑣𝑎𝑎𝑡𝑡𝑖𝑖𝑣𝑣𝑛𝑛𝑠𝑠 𝑁𝑁𝑢𝑚𝑚𝑏𝑏𝑒𝑟𝑟 of IDs 𝑀𝑀𝑣𝑣𝑛𝑛𝑠𝑠𝑡𝑡𝑎𝑎𝑛𝑛𝑡𝑡

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average monthly returns, which contradicts the standard theory of a positive risk-return relation, however this finding is consistent with Amihud et al. (2015) and other earlier studies. Also typically for Germany is the lack of statistical significance of the book-to-market ratio indicating a missing premium for value stocks compared to growth stocks.

4.1.2 Cross-sectional regression of stocks’ market liquidity onto funding liquidity

The second cross-sectional estimation aims at finding a significant relationship between stocks’ market liquidity and market-wide funding liquidity. For this purpose, I introduce the macroeconomic time-series on funding liquidity into the panel data with stock-specific characteristics and regress stocks’ monthly market liquidity (𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼) onto the first lags of their book-to-market ratios (𝐼𝐼1. 𝑆𝑆𝑡𝑡𝐼𝐼), their market capitalisation in natural logarithms (𝐼𝐼1. 𝐼𝐼𝑀𝑀𝑉𝑉𝑀𝑀), and their return volatility (𝐼𝐼1. 𝑉𝑉𝑉𝑉𝐼𝐼𝑉𝑉); as well as onto the first lag of funding liquidity (𝐼𝐼1. 𝐹𝐹𝐼𝐼).

𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑖𝑖𝑡𝑡 = 𝛽𝛽𝐹𝐹𝐼𝐼𝐼𝐼1. 𝐹𝐹𝐼𝐼𝑖𝑖𝑡𝑡+ 𝛽𝛽𝐼𝐼𝑀𝑀𝑉𝑉𝑀𝑀L1. MCAP𝑖𝑖𝑡𝑡+ 𝛽𝛽𝐵𝐵𝑡𝑡𝐼𝐼𝐼𝐼1. 𝑆𝑆𝑡𝑡𝐼𝐼𝑖𝑖𝑡𝑡

+𝛽𝛽𝑉𝑉𝑉𝑉𝐼𝐼𝑉𝑉_{𝑉𝑉𝑉𝑉𝐼𝐼𝑉𝑉}

𝑖𝑖𝑡𝑡+ 𝛼𝛼𝑖𝑖 + 𝜗𝜗𝑖𝑖 (16)

Same as in the previous regressions, I standardize the independent variables 𝐼𝐼1. 𝑆𝑆𝑡𝑡𝐼𝐼, 𝐼𝐼1. 𝐼𝐼𝑀𝑀𝑉𝑉𝑀𝑀, and 𝐼𝐼1. 𝑉𝑉𝑉𝑉𝐼𝐼𝑉𝑉 so that the monthly averages in each month for each of these variables is always 1.0; as well as 𝐼𝐼1. 𝐹𝐹𝐼𝐼 so that its average throughout time is 1.0. This mean-adjustment enables me to interpret e.g. 𝐼𝐼1. 𝐹𝐹𝐼𝐼 = 2 as twice the average 𝐼𝐼1. 𝐹𝐹𝐼𝐼 across time. As I suspect that especially the effect of funding liquidity onto stocks’ market liquidity will vary depending on the market liquidity itself, I run fixed effects regressions grouped by percentiles of 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 into ten portfolios, where portfolio 1 contains the most liquid stocks (i.e. with the lowest ILLIQ value) and portfolio 10 contains the least liquid stocks (i.e. with the highest ILLIQ value). Percentiles are calculated separately for each month so that for every month equal amounts of stocks are allocated to the 10 percentiles; this leads to the

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unbalanced amounts of observations within each of the ten portfolios, but avoids the otherwise created time effects. The regressions cover the period from January 2000 to July 2017 and include all stocks in the dataset. Standard errors are reported in parenthesis and statistically significant coefficients are labelled with *** if p < 0.01, with ** if p < 0.05 and with * if p < 0.10.

Table 2: Fixed effects cross-section regression results on stocks’ monthly market liquidity

The results of the second cross-sectional regression depicted in Table2 indicate that on an individual stock level, a stock’s market capitalisation and return volatility influence the stock’s market liquidity in a statistically significant way, as both variables are statistically significant at the 1% critical value. The book-to-market value is statistically significant at the 1% critical value in half of the cases, notably for the relatively more liquid portfolios 1 to 5. Funding liquidity is statistically significant at the 1% level for all 10 portfolios and the coefficient on this variable increases exponentially as portfolios become more illiquid. For the portfolio 1 containing only the most liquid stocks throughout the observation period, the coefficient on funding liquidity is at 2.33 which means that when average market-wide funding liquidity conditions worsen and only half of the funding liquidity that pertains during “ordinary” times is available, the market illiquidity of these stocks increases by 2.33%. However, for the portfolio 10 containing only the most illiquid stocks throughout the observation period, the coefficient on funding liquidity is at 4,143.14 which means that when

Liquid 2 3 4 5 6 7 8 9 Illiquid 2.3321*** 17.3424*** 47.9983*** 119.8790*** 254.1587*** 454.1866*** 778.4591*** 1,284.0327*** 2,201.5554*** 4,143.1476*** (0.2855) (1.0719) (2.1279) (3.7578) (6.5142) (10.3636) (17.6164) (30.2354) (52.9315) (328.6963) 7.7126*** 30.2668*** 46.7626*** 16.3911 -161.9650*** -553.3742*** -1,240.8778*** -2,651.4326*** -5,213.3200*** -33,250.3387*** (1.2877) (4.1620) (8.0218) (13.6809) (23.5917) (38.2870) (64.9642) (114.6913) (210.2155) (1,291.9270) 3.5022*** 12.6547*** 30.1471*** 38.4513*** 44.9397*** 35.6700*** 123.2051*** 126.2348*** 585.5039*** 3,503.5234*** (0.5233) (1.6659) (3.5040) (5.9131) (9.2537) (11.8015) (20.5994) (29.2516) (57.8698) (243.3694) -159.3514*** -305.8637*** -383.1531*** -417.7220*** -494.2089*** -248.3877* -34.8981 9.8178 10.5321 -34.8850 (10.1280) (31.9931) (53.3232) (70.9751) (96.5582) (133.8030) (48.9466) (24.2896) (51.7489) (170.4021) 19.9692*** 129.6890*** 320.4428*** 629.2739*** 1,085.7811*** 1,649.6017*** 2,395.5045*** 3,300.3502*** 4,608.7405*** -10,196.5296*** (2.2306) (3.4416) (4.1876) (4.5641) (5.8997) (13.1119) (31.5508) (73.5507) (172.1439) (1,439.3813) 9,160 9,006 8,926 8,882 8,778 8,668 8,441 8,315 8,025 7,489 3.96% 4.88% 7.92% 13.06% 19.93% 25.17% 27.94% 28.30% 29.89% 15.97% 𝑉𝑉𝑏𝑏𝑠𝑠𝑒𝑟𝑟𝑣𝑣𝑎𝑎𝑡𝑡𝑖𝑖𝑣𝑣𝑛𝑛𝑠𝑠 𝑅𝑅2 𝑀𝑀𝑣𝑣𝑛𝑛𝑠𝑠𝑡𝑡𝑎𝑎𝑛𝑛𝑡𝑡 𝛽𝛽𝐹𝐹𝐼𝐼 𝛽𝛽𝑆𝑆𝐼𝐼𝑆𝑆𝑆𝑆 𝛽𝛽𝑉𝑉𝑉𝑉𝐼𝐼𝑉𝑉 𝛽𝛽𝑉𝑉𝑉𝑉𝐼𝐼𝑉𝑉𝑆𝑆

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average market-wide funding liquidity conditions worsen and only half of the funding liquidity that pertains during “ordinary” times is available, the market illiquidity of these stocks increases by 4,143.14%.

4.2 Time series regressions of stock returns onto funding liquidity

4.2.1 Portfolio formation

For the purposes of this analysis, time series regressions will be run for a number of sorted monthly portfolios. The portfolio’s average monthly returns (𝑅𝑅_{𝑖𝑖𝑡𝑡}) will be calculated using the value-weighted method, thereby weighting stock returns by their capitalization at the end of the preceding month. This weighting method is chosen because also the market return used in the regressions is approximated by a value-weighted index. Stocks will be sorted in three different ways: (1) only by market liquidity into ten decile portfolios, (2) only by return volatility into ten decile portfolios and (3) by market liquidity controlling for return volatility into 15 portfolios. In this latter, the sorting of stock by market liquidity into quintiles is conditioned on a pre-sorting by return volatility into terciles, hence creating 15 (3x5) portfolios which are sorted by market liquidity within each return volatility tercile. The 3x5 matrix which produces the 15 portfolios, is constructed as follows:

Table 3: Portfolio matrix sorting by return volatility and market liquidity

where the numbers in the fields correspond to the number given to the respective portfolio. The first volatility quintile refers to the least volatile stocks and the first illiquidity quintile represents the least illiquid stocks, i.e. the most liquid stocks. Consequently, the portfolios 1

1st quintile 2nd quintile 3rd quintile 4th quintile 5th quintile

1st tercile 1 2 3 4 5 2nd tercile 6 7 8 9 10 3rd tercile 11 12 13 14 15 Illiquidity V o la tility

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consists of the least volatile and most liquid stocks, whereas the portfolio 15 contains the most volatile and illiquid stocks. Liquidity of stocks will be measured by Amihud’s measure of illiquidity as described in 3.1, where also the measure for volatility is explained. The sorting of stock will be done according to the previous year’s December illiquidity or volatility measure. This implies that re-calibration of the portfolios is done annually and is kept constant for 12 months (with the exception of the year 2017, since the observation period ends in July 2017). The same procedure is being followed for the sorting by volatility. Return volatility is measured as the monthly standard deviation of all daily returns within a month. For reasons of comparison, I also create decile portfolios sorted by market capitalisation and book-to-market price respectively. Figure 11 shows the average market liquidity (as measured by ILLIQ) and the average stock return volatility (as measured by its standard deviation) over all individual equally-weighted stocks in the period from January 1999 to July 2017. Clearly, average market liquidity and average return volatility move together; regressing one time-series onto the other reveals a statistically significant correlation coefficient. This correlation between average market illiquidity and average return volatility is the motivation behind the above third portfolio formation sorting by market illiquidity while controlling for return volatility thereby creating 15 portfolios.

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Figure 11: Evolution of average market liquidity and return volatility of stocks from January 1999 to July 2017

4.2.2 Augmented CAPM single-factor model

The notation is amended by replacing a portfolio’s excess return (𝑟𝑟_𝑡𝑡− 𝑟𝑟_𝑡𝑡𝑓𝑓) by 𝑅𝑅_𝑡𝑡 and market excess return by (𝑟𝑟_𝑡𝑡𝑚𝑚− 𝑟𝑟_𝑡𝑡𝑓𝑓) by 𝐼𝐼𝑀𝑀𝑀𝑀_𝑡𝑡. Consequently, the time-series regression of portfolio excess return 𝑅𝑅_{𝑖𝑖𝑡𝑡} on funding liquidity changes ∆𝐹𝐹𝐼𝐼_𝑡𝑡 and market excess returns 𝐼𝐼𝑀𝑀𝑀𝑀𝑡𝑡 which has to be fitted is as follows:

𝑅𝑅𝑡𝑡 = 𝛼𝛼 + 𝛽𝛽∆𝐹𝐹𝐼𝐼∆𝐹𝐹𝐼𝐼𝑡𝑡+ 𝛽𝛽𝐼𝐼𝑀𝑀𝑀𝑀𝐼𝐼𝑀𝑀𝑀𝑀𝑡𝑡+ 𝜀𝜀𝑡𝑡 (17)

where 𝑡𝑡 refers to the respective month in the period from January 2000 to July 2017. The regressions are estimated of each of the 10 portfolio, use heteroscedasticity-robust standard errors and are run each separately to allow for greater flexibility of betas which are portfolio-specific. The results of the time-series regressions on the ten portfolios sorted by different characteristics can be seen in Tables 4 - 8. Robust standard errors are reported in parenthesis and statistically significant coefficients are labelled with *** if p < 0.01, with ** if p < 0.05 and with * if p < 0.10.

The role of market and funding liquidity innovations in German stock market returns

Name

Elsa Feldmann

Date

15

December 2017

MSc. Thesis

Abstract

Statement of Originality

Contents

Table of Figures

Table of Tables

Chapter 1

Introduction

Chapter 2

Literature review

Chapter 3

Data and methodology

Chapter 4

Empirical results