Noise in the German Government Bond Market

Home , Government bond

Carl Joel Hartman∗

Simen Spjut†

MASTER THESIS

M.Sc. Advanced Economics and Finance Copenhagen Business School 2016

Supervisor: Sven Klingler Delivery date: 13/5-2016

Number of pages: 51 Number of characters (with spaces): 88 645

∗ [email protected] † [email protected] ABSTRACT

Abstract

For each day from January 2000 through December 2015, we calibrate the Nelson- Siegel-Svensson model using the Differential Evolution algorithm and calculate a noise measure defined as the cross-sectional root mean squared error (RMSE) between quoted market and model implied yields for German government bonds. The noise measure serves as a proxy for liquidity, as in times with high noise arbitrageurs do not have the same possibilities to exploit trading opportunities due to constraints in access to capital. Relating the noise measure to other popular liquidity variables, it becomes evident that the estimated noise measure adds extra value in explaining liquidity in the market. Consistent with previous research on slow-moving capital, our results indicate that returns, a proxy for capital available to arbitrageurs, of certain hedge fund categories have a significant impact on the noise in the following month. The results thus provide further evidence that limits in funding prevents market forces to correct asset mispricings. To test the robustness of the results, the noise measure is calculated using alternative settings in the Differential Evolution algorithm in calibrating the parameters of the Nelson-Siegel-Svensson model. The robustness check stresses the importance of careful calibration, as the series of daily curves end up of having notable worse fits, and the following hedge fund regression analysis overall show weaker relationships.

Keywords: Nelson-Siegel-Svensson model, Diﬀerential Evolution, liquidity, slow- moving capital ACKNOWLEDGEMENTS

Acknowledgements

We would like to thank our supervisor Sven Klingler for his valuable support, insights and comments throughout the thesis work.

Carl Joel Hartman Simen Spjut

Signature Signature

13/5-2016 13/5-2016

Date Date CONTENTS

List of Contents

1 Introduction 1

2 Literature Review 3

3 Bonds 6 3.1 Price and Yield ...... 6 3.2 Duration ...... 8 3.3 Accrued Interest ...... 8

4 Data 10 4.1 The German Government Bond Market ...... 10 4.2 Data Collection ...... 12 4.3 Issuance data ...... 13

5 Yield Curve Estimation 14 5.1 The Nelson-Siegel-Svensson model ...... 14 5.2 Calibrating the Nelson-Siegel-Svensson model ...... 16 5.2.1 The Objective Function ...... 16 5.2.2 The Calibration Problem ...... 17 5.2.3 The Optimization Algorithm ...... 17

6 Methodology 22 6.1 Curve Fitting ...... 22 6.2 Noise Measure ...... 22 6.3 Optimization Algorithm - Diﬀerential Evolution ...... 24

7 Results 26 7.1 Parameters and Curves ...... 26 7.2 Noise and Events ...... 32 7.3 Noise and Market Variables ...... 36 7.3.1 Curve Variables ...... 36 7.3.2 Funding Liquidity Variables ...... 36 7.3.3 Equity Volatility Variables ...... 37 7.3.4 Equity Indices ...... 38 7.3.5 Regression Analysis ...... 38 7.4 Hedge Fund Returns ...... 43 7.4.1 Hedge Fund Regression Analysis ...... 45 7.4.2 Regressing Changes in US Noise on Hedge Fund Returns . . 46 7.5 Sensitivity Analysis ...... 48 CONTENTS

8 Conclusions 51

A Graphs 57

B Tables 62 CONTENTS

List of Figures

1 Example of issuance ...... 13 2 Factor loadings example ...... 15 3 Parameter distributions for 2005-09-20 ...... 19 4 Parameter distributions for 2008-05-12 ...... 20 5 Yield error example ...... 24 6 Estimated term structure ...... 27 7 Market yields and estimated par curves for selected days ...... 28 8 Pricing errors and bid-ask spreads on a day with relatively low noise 29 9 Pricing errors and bid-ask spreads on a day with relatively high noise 29 10 Distributions of Bunds and Bobls yield errors ...... 30 11 Example of the model-market yield diﬀerence over time for two bonds 31 12 German noise measure for the whole sample period ...... 32 13 German noise during the European sovereign debt crisis ...... 35 14 Noise measure compared to other market variables ...... 39 15 Comparison of noise with diﬀerent DE settings ...... 48 16 Linear regressions on US noise ...... 49 17 Issuance of Federal securities ...... 57 18 Parameter estimates ...... 58 19 Example of pricing errors around year-ends ...... 59 20 Noise measure for the period 2008-2011 ...... 60 21 Noise measure for the period 2012-2015 ...... 61 CONTENTS

List of Tables

2 Statistics of empirical parameter distributions ...... 21 3 Monthly spot rate statistics ...... 27 4 Descriptive statistics for market variables ...... 40 5 Pairwise correlations with selected market variables ...... 40 6 Monthly changes in noise regressed on other market variables . . . . 41 7 Summary statistics ...... 62 8 Summary statistics for Schäthe ...... 63 9 Summary statistics for Bobl ...... 63 10 Summary statistics for Bund ...... 63 11 Main hedge fund indices regressions with no lag ...... 64 12 Main hedge fund indices regressions with one lag ...... 65 13 Relative value hedge fund sub-indices regressions with no lag and German noise ...... 66 14 Relative value hedge fund sub-indices regressions with one lag and German noise ...... 67 15 Relative value hedge fund sub-indices regressions with no lag and US noise ...... 68 16 Relative value hedge fund sub-indices regressions with one lag and US noise ...... 69 17 Main hedge fund indices regressions with one lag and German noise estimated with alternative DE settings ...... 70 18 Summary statistics for ﬁltered noise ...... 70 19 Relative value hedge fund sub-indices regressions with one lag and German noise estimated with alternative DE settings ...... 71 20 Short descriptions of selected events ...... 72 CONTENTS

List of Equations

1 Bond price ...... 6 2 Discrete discount factor ...... 6 3 Yield to maturity ...... 6 4 Continuous yield to maturity ...... 6 5 Continuous discount factor ...... 7 6 Discrete forward rate ...... 7 7 Instantaneous forward rate ...... 7 8 Continuous forward rate ...... 7 9 Continuous spot rate ...... 7 10 Par rate ...... 7 11 Macaulay duration ...... 8 12 Duration weights ...... 8 13 Accrued interest ...... 8 14 Svensson forward function ...... 14 15 Svensson spot function ...... 14 16 Price as function of parameters ...... 16 17 Yield as function of parameters ...... 16 LIST OF SYMBOLS

List of Symbols

p Dirty price of a bond p˜ Model implied dirty price of a bond y Yield to maturity of a bond y˜ Model implied yield to maturity of a bond

cti Coupon payment i that occurs at time ti R Redemption value of a bond a Accrued interest of a bond d(·) Discrete discount factor function δ(·) Continuous discount factor function r(·) Discrete forward rate function f(·) Continuous forward rate function s(·) Discrete spot rate function z(·) Continuous spot rate function D Macaulay duration wi Weights in the Macaulay duration formula m Years to maturity from trade date t0 B Number of eligible bonds in the estimation for a given day t B* Number of eligible bonds for calculating the noise measure for a given day t C Cashﬂow matrix for a given day t with B bonds and corresponding cashﬂows β Vector of the NSS parameters β0, β1, β2, β3, τ1 and τ2 P (·) NSS bond price function Y (·) NSS bond yield-to-maturity function ωj Weight for bond j in the objective function for obtaining parameter estimates 1 INTRODUCTION

1 Introduction

The standard models of asset pricing assume an agent who, without costs, par- ticipates in all markets. In these theoretical markets, securities are always traded at their fundamental values. The latest financial crisis clearly demonstrated how, during periods of financial distress, orderly trading can become disrupted, making asset prices deviate from their fundamental values for extended periods of time. A variety of explanations for these persistent mispricings have been suggested, including market liquidity and funding liquidity, which we generally refer to as frictions. Accepting that frictions are intrinsic characteristics of financial markets, we explore mispricings on German government bonds, a safe haven market with one of the highest credit ratings and most favorable liquidity conditions in the world. As a measure of mispricings and general market liquidity, we use the noise measure of Hu et al. (2013). We construct the noise measure by first calibrating the Svensson (1994) model by minimizing duration-weighted differences between market observed and model implied prices using the Differential Evolution algorithm of Storn & Price (1997). We then calculate the noise as the root mean squared error between market observed yields and yields implied by the Svensson (1994) model. The underlying premise of the noise measure is that in highly liquid and creditworthy government bond markets, increased price errors reflects a market-wide shortage of arbitrage capital and deteriorating liquidity conditions. By examining the micro foundations of the noise we show that bid-ask spreads create a lower bound of the noise as the noise is measured in mid yields and arbitrageurs typically have to trade at the provided market quotes, i.e. buying at the ask and selling at the bid. Qualitatively investigating the relationship between the evolution of the noise together with events and economic data releases, we only find a handful events that led to immediate reactions in the noise. As an example, we cannot relate the sharply increased noise at the year-end of 2010 to a significant event or data release. Rather, we deduce the increased noise to reduced trading activity following a period of surging bond yields, decreasing the capital of bond traders. These findings are consistent with the theoretical foundations of the noise as a measure of liquidity. Normally, arbitrageurs have abundant capital which they can deploy to supply liquidity and trade mispricings back to fundamental values. However, during periods of distress and worsened funding conditions, it becomes more difficult for arbitrageurs to finance their positions. Formally exploring the relationship between the noise and popular proxies of funding liquidity, we find that tougher funding conditions are associated with increases in the noise. Our results correspond well with Shleifer & Vishny (1997),

1 1 INTRODUCTION

Gromb & Vayanos (2002), Brunnermeier & Pedersen (2009), Ashcraft et al. (2011), who show that mispricings may persist due to to funding frictions. Following the studies by Mitchell et al. (2007), Duffie (2010), Fleckenstein (2013), and Flecken- stein et al. (2014), we test an implication of the theory of slow-moving capital. We develop a hypothesis where shocks to the assets of arbitrageurs lead to changes in the capital available to exploit mispricings, which leads to subsequent changes in the noise. More specifically, we expect returns of hedge funds specialized in fixed income relative value arbitrage to forecast subsequent changes in the noise measure. Regressing changes in the noise on lagged returns of HFRX indices, we show that, in general, the returns of hedge funds specializing in fixed income relative value arbitrage predict changes in the noise while other hedge fund specializations do not. We also conduct the hedge fund regressions on the US noise measure by Hu et al. (2013) and find the relationship between hedge fund returns and subsequent changes in the noise to be stronger. In the literature on persistent mispricings, our results provide support to the notion that capital availability to specific types of arbitrageurs matter. We outline the rest of the paper as follows. In Section 2 we present a selection of the background literature within term structure modeling, mispiricings and liquidity. In Section 3 we cover basic fixed income calculations employed throughout the paper. Section 4 provides a brief introduction to the German government bond market and the data used in this study. In Section 5 we describe the Svensson (1994) model and the issues associated with calibrating it. In Section 6 we describe our approach in calibrating the Svensson (1994) model and constructing the noise measure. Section 7 presents the resulting time-series of the calibrated spot curves and the noise measure. Further, we investigate the noise series by relating it to events, regressing it on other popular liquidity proxies, and hedge fund returns. In the final part of Section 7, we conduct a sensitivity analysis of our results. Section 8 briefly summarizes our main findings.

2 2 LITERATURE REVIEW

2 Literature Review

In this section we present a selection of the background literature in spot curve modeling, liquidity in asset pricing, and noise as a measure of liquidity. Zero-coupon bonds are securities with one payment at maturity and a term structure is a function that relates a financial variable to its maturity. It follows that the term structure of zero-coupon rates describe the relationship between zero-coupon rates and time to maturity. The limited maturity spectrum and low market liquidity of observed zero-coupon bonds make it necessary to estimate the spot rate curve using observed coupon-bearing bond prices. Over the years several approaches on how to estimate the spot curve have been developed. We begin by dividing spot curve model approaches into a theoretical and an empirical category. While theoretical spot rate models are derived from equilibrium conditions or no- arbitrage conditions, empirical models derive spot rates by matching model implied prices to market observed prices. The theoretical models can be derived back to Vasicek (1977) and Cox et al. (1985) and include e.g. Ho & Lee (1986) and Hull & White (1990). The empirical models can be further be divided into spline-based and function- based models. McCulloch (1971, 1975) models the spot curve by fitting the discount curve using quadratic respectively cubic splines. The discount function and spline-based methodology was improved by Vasicek & Fong (1982) who use exponential splines to fit the discount curve. The aforementioned models attempt to derive spot rates through the discount curve, however this approach have been criticized for having undesirable economic properties.1 Instead of estimating the discount function Fama & Bliss (1987) bootstrap discrete forward rates, which can be smoothed in order to obtain a continuous curve. Nelson & Siegel (1987) fit a smooth forward curve trough a four parameter parsimonious function, later extended by Svensson (1994). A large and continuously growing literature studies the effects of market frictions on asset prices. Such frictions include market liquidity and funding liquidity. Market liquidity refers to the ability transact to large quantities of a security quickly and at low cost, funding liquidity refers to the amount of capital available to investors and their ability to raise cash on demand. These frictions play an important role for arbitrageurs who identify arbitrage opportunities and trade securities back to fundamental values. While earlier work, for instance by Amihud & Mendelson (1986) and Amihud (2002), empirically show a connection between asset returns and liquidity, Acharya & Pedersen (2005) present a theoretical framework for liquidity in asset pricing, formally showing that liquidity is a priced risk factor. In other words, depending on their liquidity conditions, otherwise identi-

1See for instance Diebold & Rudebusch (2013, p. 4).

3 2 LITERATURE REVIEW

cal securities may be priced differently. Brunnermeier & Pedersen (2009) develop a model that links market liquidity and funding liquidity. Arbitrageurs provide market liquidity and their ability to do so depends on their ability to obtain funding. Conversely, arbitrageurs’ funding, i.e. their capital and margin requirements, depends on market liquidity. Consequently, mispricings may persist due to a lack of available arbitrage capital. Arbitrageurs may obtain funding through equity capital supplied by themselves or investors, and leverage through debt financing and/or buying securities on margin. Shleifer & Vishny (1997) further show that actual or possible investor redemptions may cause arbitrageurs to reduce positions as mispricings and ex ante arbitrage profits are increasing. The model of Brunnermeier & Pedersen (2009) demonstrate that the result of Shleifer & Vishny (1997) is further exacerbated when arbitrageurs are facing margin constraints. On that topic, Ashcraft et al. (2011) argue that investors’ leverage is mainly constrained due to margins that prevail in the market. Garleanu & Pedersen (2011) also find that different margins may lead to price discrepancies between assets with identical cash flows. Frazz- ini & Pedersen (2014) study how margin constraints coerce investors to choose non-optimal portfolios. Previous empirical research by Chordia et al. (2005), Mitchell et al. (2007), Fleckenstein (2013), and Fleckenstein et al. (2014) among others analyze how liquidity depend on fund flows and capital available to hedge funds. Chordia et al. (2005) show that capital flows into both bond and equity funds are related to bond market liquidity. Fleckenstein et al. (2014) find that mispricings between nominal and inflation-linked bonds decrease with increased flows into hedge funds. Mitchell et al. (2007) show that large capital outflows from hedge funds specializing in convertible bond arbitrage led to massive bond sales and increased mispricings on convertible bonds. While funds specializing in convertible bond arbitrage were capital constrained, other hedge funds failed to take advantage of the increased mispricings. Similarly, Fleckenstein (2013) find that capital available to specific hedge fund categories narrow inflation-linked bond (IFL) mispricings. More specifically, an increase in available capital to hedge funds more likely to engage in IFL arbitrage reduce IFL mispricings. It follows that not all arbitrageurs have the knowledge or skills in identifying and carrying out every type of arbitrage strategy.2 Thus, the funding conditions of specific arbitrageurs may affect mispricings. Hu et al. (2013) propose a market wide liquidity measure, noise, that is constructed from the root mean squared error (RMSE) between observed and theoret- ically deducted bond prices. Brunnermeier & Pedersen (2009, p. 2202) definition of market liquidity is similar to the Hu et al. (2013) noise measure, as market

2This theory is in line with Duarte et al. (2007), who ﬁnd that arbitrage strategies that require "more human capital" produce signiﬁcant alpha.

4 2 LITERATURE REVIEW

liquidity is considered to make up the difference between the transaction price and the fundamental asset value. The basic premise of their noise measure is that in highly liquid and creditworthy government bond markets, increased price errors reflects a market wide shortage of arbitrage capital and deteriorating liquidity conditions. Other research that use the noise measure to proxy liquidity include Malkhozov et al. (2015), who construct the noise measure for six developed markets and find both global and local components of illiquidity. Schuster & Uhrig-Homburg (2015) computes a German noise measure, as part of three measures of liquidity measures and test the Brunnermeier & Pedersen (2009) model. Musto et al. (2015) construct a security-level noise measure which they use as a measure of relative mispricing and relate these to bond characteristics. The findings include that older, smaller, more-stripped, lower volume and higher-spread bonds trade at discounts, especially when market-wide liquidity is low.

5 3 BONDS

3 Bonds

3.1 Price and Yield

The price p of a bond with N periodic coupon payments cti at times ti = t1, . . . , tN and redemption value R equals the principal, is the present value of its future cash ﬂows

N X p = cti d(t0, ti) + Rd(t0, tN ) (1) i=1

where the i’th cashﬂow occurs at time ti and the last cashﬂow at maturity tN . The discrete discount function from today to time t1 is

1 d(t0, t1) = t −t (2) (1 + s(t0, t1)) 1 0

where the difference t1 − t0 is expressed in years and s(t0, t1) is the discrete spot interest rate for maturity t1. The spot rate is defined as the interest rate of a government zero-coupon bond that pays EUR 1 at maturity. The yield-to- maturity y of a bond is the single interest rate that discount the future cashflows to the present price such as

N X ct R p = i + (3) (1 + y)ti−t0 (1 + y)tN −t0 i=1

The price p can hence be derived by either discount future cashﬂows with the single yield-to-maturity y or by discounting each of the individual cashﬂows by the corresponding spot rates. When modeling the term structure, it is convenient to use continuously compounded interest rates. With continuous compounding we have that Equation (3) becomes

N X −(ti−t0)y (tN −t0)y p = cti e + Re (4) i=1

Let t0 be today and t1 and t2 future dates such that t0 < t1 < t2. Then z(t0, t1) is the the continuous spot rate between today and t1 and the continuously compounded discount rate for the same period is

6 3 BONDS 3.1 Price and Yield

(t1−t0)z(t0,t1) δ(t0, t1) = e (5)

A closely related concept is forward rates. The forward rate is the interest rate contracted at t0 to be paid between t1 and t2. To illustrate the relationship between spot and forward rates, the simply compounded forward rate between time t1 and t2 is

(t2 − t0) s (t0, t2) − (t1 − t0) s (t0, t1) r(t0, t1, t2) = (6) t2 − t1

The instantaneous forward rate f is when the investment period become inﬁnites- imal such that

f(t0, t1) = lim f(t0, t1, t2) (7) t2→t1

and the forward rate with t2 > t1 will be the average of instantaneous forward rates, or equivalently one-period forward rates in continuous time

1 Z t2 f(t0, t1, t2) = f(t0, λ) dλ (8) t2 − t1 λ=t1 Thus, the instantaneous forward rate can be seen as the marginal increase in the total return from a marginal increase in the length of the investment. By taking the average of instantaneous forward rates between t0 and t2, the spot rate z(t0, t2) can be obtained as

1 Z t2 z(t0, t2) = f(t0, λ) dλ (9) t2 − t0 λ=t0 which emphasizes the concept of thinking of spot rates as averages and forward rates as marginals. The par yield of a bond is deﬁned as the coupon rate that given its yield-to- maturity y causes the price of the bond to equal its par value

N X −(ti−t0)y (tN −t0)y p = cti e + Re (10) i=1 where once again R represents the redemption value. (Hull, 2012, p. 81)

7 3 BONDS 3.2 Duration

3.2 Duration The Macaulay duration is deﬁned as a weighted average maturity of cash ﬂows. With continuous compounding the duration D follows as

N PN −y(ti−t0) (ti − t0)ct e X D = i=1 i = w (t − t ) (11) p ti i 0 i=1

where ti − t0 is the time diﬀerence in years between today and the future time of

the i’th cashflow. The weight wi for cashflow cti is defined as

−y(ti−t0) cti e wi = N (12) P −y(ti−t0) i=1 cti e

For a zero-coupon bond, the duration will simplify into the remaining years to maturity. On the contrary, a bond that do pay coupons will have a duration less than its time to maturity. (Hull, 2012, p. 89)

3.3 Accrued Interest The price p has been deﬁned as what is called the dirty price, meaning that potential accrued interest is included. Bonds are however typically quoted in clean price, which is obtained by subtracting potential accrued interest. Thus, accrued interest is the fraction of the next coupon payment that has been accumulated since the last coupon payment (or in the ﬁrst coupon period from the accrual date) to the settlement date. For German government bonds, the settlement occurs two business days after the trade date, conventionally denoted as T + 2. To arrive at the dirty price paid by the buyer accrued interest is added to the clean price, meaning that Clean Price = p − a where the accrued interest a follows as

Days in the accrued coupon period a = c × (13) Days in the coupon period

and is thus a function of the constant coupon payment c, the number of days since the last payment and the number of days in a coupon period. (Hull, 2012, p. 974) Following the preceding, another fundamental aspect of bond pricing becomes evident. Time intervals as years, months and days are not always of the same length, meaning there are not always 30 days in one month or 365 days in a year.

8 3 BONDS 3.3 Accrued Interest

Therefore, day count conventions are used to determine the length of a given time interval, for instance days in a coupon period. The day count conventions generally differ in their use of fixed or actual calendar days and can essentially be categorized into three types. The first convention type use a fixed number of days number of calendar days in a month and in a year and is referred to as e.g. 30/360. The second type use the actual number of calendar days in a month but a fixed number of days per year, this convention is referred as e.g. ACT/360. The third convention type use the actual number of calendar days in a month and the actual number calendar days in a year and is referred to as ACT/ACT.

9 4 DATA

4 Data

4.1 The German Government Bond Market Federal bonds (Bundesanleihen or Bunds) are issued at 10 and 30 year maturities, and make up the primary funding source of the German government. Federal notes (Bundesobligationen or Bobls) are issued at 5 years to maturity. Federal Treasury notes (Bundesschatzanweisungen or Shätze) have an initial maturity of 2 year. The aforementioned bond types have annual coupon frequency, ACT/ACT day count basis and no ex-dividend period. (Fabozzi & Choudhry, 2004) Finally, the discount paper (Unverzinsliche Schatzanweisungen or Bubills) are normally issued at 6 and 12 months maturities. The Bubills are quoted on a discount yield basis and follows the ACT/360 day count convention. Moreover, for instance so called special funding tools may also be used. In 2005 and 2009 USD denominated bonds were issued and in 2013 a Bund-Laender-Anleihe issue was made. (Deutsche Finanzagentur, 2016b) More than 90% of the funding needs are obtained by one off issues in the primary market. Issuance goes through an auction process where members of the Bund Issues Auction Group are eligible to participate. (Deutsche Finanzagentur, 2016b) The auction process is the issuance tool for Federal bonds, Federal notes, Federal Treasury notes, Treasury discount papers and inflation-linked notes and bonds.3 (Deutsche Finanzagentur, 2016d) Unlike many other markets, there is no primary dealer system with predefined obligations for a defined group of banks. The Ministry of Finance is the issuer and debt manager while the German Finance Agency is the central debt management services manager. Issuance is carried out trough the Bundesbank which provides bank services to the issuer. In each auction, a portion of the issue is withheld for market management purposes.4 The market management portion is subsequently placed in the market. Smoothing purchases and sales are conducted in order to ensure liquidity and avoid price fluctuations. Bonds are issued repeatedly due to two reasons. First, in order to build up the outstanding size over time through regular tap issues, supporting market liquidity. The second alternatively reason might be to increase the size which is used for market management operations (IMF, 2001, p. 142). Figure 1 in Section 4.3 illustrates how bonds’ volume may be built up over time.

3Henceforth, Federal bonds/Bunds, Federal notes/Bobls, Federal Treasury notes/Schätze and Treasury discount papers/Bubills will alternately be denoted by their respective Bloomberg ticker abbreviations DBR, OBL, BKO and BUBILL. 4Figure 17 shows the yearly issuance over the past decade. As one can see, the issuance levels are fairly stable over the period, with the exception of 2009-2010 when higher amounts of treasury discount papers were issued.

10 4 DATA 4.1 The German Government Bond Market

The secondary market for German government bonds is one of the largest and most liquid markets for government bonds in Europe (Bundesbank, 2007, p. 57). Trading takes place on stock exchanges and over-the-counter (OTC). The volume of exchange trading is relatively small compared with OTC transactions. According to Bundesbank’s calculations, the OTC trade volume made up 98% of all secondary trading in 2006 (Bundesbank, 2007, p. 53). In exchange trading the Bundesbank is the most important parter for institutional investors and banks. In 2006, the Bundesbank’s market management activities accounted for EUR 81.6 billion out of a total stock exchange turnover of EUR 311.6 billion. For government bonds, the main indices Deutsche Renteindex (REX) and REX- performance index (REXP) were introduced in 1991. (Fabozzi & Choudhry, 2004, p. 20) The REX index is constructed based on a theoretical approach, where the index is a weighted portfolio of 30 synthetic bonds within the maturity range of 1 to 10 years, where the maturities are constant integers. The weighting is determined based on a pre-deﬁned market share, and is reviewed annually. Further, the bonds in the index portfolio are divided into the three coupon levels at 6%, 7.5% and 9%. (Deutsche Börse Group, 2016) Futures contracts are traded on the Eurex platform. The existence of future contracts is supporting liquidity in the underlying government bonds. German government bonds in the 2, 5, 10 and 30 year maturity segments are deliverable in the future contracts. The most liquid segment is in the 10 year point, both in the future contract and in the underlying bond. For instance, the trading volume for 2015 in the 10 year Bund future contract amounted to EUR 27 trillion. (Deutsche Finanzagentur, 2016c) Stripping of government bonds in Germany was introduced by the Bundesbank in 1997 for certain 10- and 30-year Bunds (Bundesbank, 1997).5 As of today, the strippable bonds are Bunds having coupon payments on the 4th of January or 4th of July. Stripped bonds with the same maturity are traded under the same ISIN, however it is not possible to combine strips from diﬀerent types of bonds. (Deutsche Finanzagentur, 2016a) Anti-stripping is also possible, a process where coupon bonds are reconstructed out of stripped bonds. (Batten et al., 2004, p. 259f)

5Stripping is a procedure where zero coupon bonds are created out of coupon bearing bonds. The coupons are "stripped", and thereafter sold separately. Thus, stripping enables direct deriva- tion of zero-coupon bonds. (Batten et al., 2004, p. 259f)

11 4 DATA 4.2 Data Collection

4.2 Data Collection The sample consists of daily observations from the beginning of year 2000 to the end of 2015. Bond data for German government bonds were downloaded from Bloomberg. In the Bloomberg SRCH function, the selection was made on including bonds that have matured or are active6, with fixed coupon, annual payment frequency, bullet maturity type and denominated in either EUR or DEM. Moreover, treasury discount papers (BUBILLs) and inflation-linked bonds were excluded. Further inclusion criterias were put in place, such as only including the tickers BKO, OBL and DBR and only bonds with the main day count conventions used for German bonds ACT/ACT. Finally, we put in the restrictions to include only bonds that have a positive amount outstanding. This initial sample consists of 83 Bunds (DBR), 60 Bobls (OBL) and 72 Schätze (BKO). Furthermore, we do not exclude on-the-run bonds as Ejsing & Sihvonen (2009) do not find any substantial on-the-run liquidity premium on German bonds. We begin by downloading static characteristics, for instance the fields for ISIN, ticker, issue date, initial accrual date, coupon rate, maturity date and day count convention. We then proceed by, for each bond, downloading time series data of bid and ask prices using different pricing sources. Following for example Ejs- ing et al. (2012), quoted dirty prices are downloaded. As we need to consider the outstanding volume for each bond at estimation, we construct series of daily outstanding volumes, further explained in Section 4.3. Bloomberg provides a variety of pricing sources, where our first hand option is to use the CBBT source. The use of CBBT is motivated by Corradin & Rodriguez- Moreno (2014), for instance it puts requirements the time frame, prices have to be within five minutes to be eligible for the weighted average. Following the pecking order defined by Corradin & Rodriguez-Moreno (2014), whenever CBBT is not available we use the BGN source (which is a weighted average of quotes by a minimum of five market participants), and as a final resort, the BVAL source is applied. BVAL is a pricing source which gives a theoretical price of the bond. For the regressions conducted in Section 7, data is downloaded from Bloomberg unless stated otherwise. Daily hedge fund index data are obtained from HFR Research Inc.7 The yield levels on the indices used in constructing the default spread variable have been downloaded from Datastream.8 In Section 7, economic interpretations of the movements in the noise measure are made. As a tool for this, various sources are used. Mainly, the historical

6We put a criteria excluding bonds that matured before the estimation period begins. 7Available at https://www.hedgefundresearch.com/family-indices/hfrx. 8The used indices are the IBOXX Euro Sovereign all Maturities and IBOXX EuroCorp. BBB (All Maturities), with the Datastream ticker names BSEUAL and IBC3BAL, respectively.

12 4 DATA 4.3 Issuance data

archive in Financial Times and a Bloomberg terminal (where news for certain days are ﬁltered out based on dates and keywords) were used for this purpose.

4.3 Issuance data The outstanding volume of bonds may increase over its lifetime due to tap issues and re-openings, thus making a bond initially not eligible for estimation eligible later on. In order to get daily series of outstanding nominal volumes for each bond during the estimation period, we download issue statistics from Deutsche Finanzagentur and Bundesbank. This eases the ﬁltration in the estimation, as we can set a lower acceptance level of outstanding volume for each bond each day. Figure 1 clearly illustrates this with two bonds, with the acceptance level set to EUR 5 billion. Thus, for each bond the issuances are added up creating series of outstanding volumes for each day in the estimation period.

DE0001135481 DE0001102382 EUR billion EUR billion 25 25 20 20 15 15 10 10 5 5 0 0

Jan12 Jan13 Jan14 Jan15 Jul15 Sep15 Nov15 Jan16

Figure 1: Example of issuance. Illustration of the impact of tap issues and re-openings for two selected bonds. The red dotted line is the lower acceptance level later used in daily estimations. By using this acceptance level, bonds are included only the dates their outstanding volume size is equal to or greater than EUR 5 billion.

13 5 YIELD CURVE ESTIMATION

5 Yield Curve Estimation

Suppose there existed tradable riskless zero-coupon bonds for a wide range of maturities. Obtaining the spot rate curve would then be a matter of fitting the observed market rates using standard econometric methods. In reality, zero-coupon bonds issued by the German government have a maximum maturity of 12 months at issuance. Recall from Equations (1), (2) and (5) that the price of a bond can be obtained by discounting its cashflows by the corresponding spot rates. Conse- quently, even though spot rates are not observed in the market, coupon-bearing bonds can be used to extract the spot rates. Due to its popularity among market participants and central banks, we choose to fit a smooth spot curve using the function-based Svensson (1994) model.

5.1 The Nelson-Siegel-Svensson model The Nelson & Siegel (1987) (NS) model ﬁts the forward curve using a Laguerre function plus a constant.9 Svensson (1994) increases the ﬂexibility of the NS model by adding a forth component. The forth component essentially allows for a second "hump" in the spot curve. To simplify, let f(m) be the instantaneous forward rate f(t0, t0 + m), where m is the years to maturity from today t0, then

−m m −m −m −m τ τ τ f(m) = β0 + β1e 1 + β2 e 1 + β3 e 2 (14) τ1 τ2

Recall from Equation (8) that the corresponding spot rate function is obtained by integrating the forward rate function

1 − e−mτ1 1 − e−mτ1 z(m) = β + β + β − e−mτ1 0 1 mτ 2 mτ 1 2 (15) 1 − e−mτ2 −mτ2 + β3 − e mτ2

Consistent with financial theory, Equation (15) implies that the discount curve illustrated in Equation (5) satisfies limm→0 δ(m) = 1 and limm→∞ δ(m) = 0. More- over, limm→0 z(m) = β0 +β1 and limm→∞ z(m) = β0. Hence, β0 can be interpreted as the long-maturity rate and β0 + β1 as the short-maturity rate. Corollary, β1 is the yield difference between the short and long end of the spot curve. β2 and τ1 9Laguerre functions consist of polynomials multiplied by exponential decay terms and are mathematical approximation functions on the domain [0, ∞).

14 5 YIELD CURVE ESTIMATION 5.1 The Nelson-Siegel-Svensson model

control the potential presence of a hump in the spot curve. Specifically, τ1 control the position of the hump and β2 control the the magnitude and direction of the hump. The four components, or factor loadings, of the NSS model have economic interpretations. As illustrated in Figure 2, each of them has a specific contribution to the curve and can be interpreted as short, medium-short, medium-long and long- term components. The long-term component is the loading on β0 and is constant at 1 for all maturities. The short-term component is the loading on β1. For m = 0 the short-term component is equal to one, but exponentially decays to zero as m increase. The loading on β2 is the medium-short component. For m = 0 the medium-term component is equal to zero, it increases with m and reaches its maximum at medium-short maturities and then decays to zero again for longer maturities. τ1 determines at which maturity the medium component reaches its maximum as well as the rate of decay of the β1 loading. A higher value of τ1 means that maximum of the β2 component is reached at earlier maturities and the β1 loading approach zero at lower maturities. τ2 and β3 respectively determine the position and magnitude of a second hump. The forth component increases with m up until medium-long maturities and then stays relatively constant, hence it is referred to as the medium-long component. A higher τ2 means that the maximum of the forth component is reached at a lower maturity. The four components of the NSS model allows the spot curve to assume a variety of shapes: flat, increasing, decreasing, U-shaped and S-shaped. The parsimonious approximation also guards against overfitting, promotes a tractable estimation as well as a continuity of the term structure of forward rates. 1.0 β0 loading

0.8 β1 loading β2 loading 0.6 β3 loading Loading 0.4 0.2 0.0 0 2 4 6 8 10 12 14 Years to maturity

Figure 2: Factor loadings example. Factor loading values of the NSS model for diﬀerent years to maturity m. The parameter values of τ1 and τ2 are set to 0.731 and 0.186, respectively.

15 5 YIELD CURVE ESTIMATION 5.2 Calibrating the Nelson-Siegel-Svensson model

5.2 Calibrating the Nelson-Siegel-Svensson model Recall from Equations (1), (2) and (5) that the price of a bond may be defined as a function of the spot rates and the underlying cash flows of the bond. From Equa- tion (15), we know that the NSS model implies that these spot rates are functions of known maturities, m, and unknown parameters β = [β0, β1, β2, β3, τ1, τ2]. For a given day t, let C denote the cashflow matrix that contains B bonds and their corresponding cashflows. For bond j, Cj is then a vector of its cashflows. Hence, the fitted price p˜j is a function of its known cash flows Cj and NSS parameters β

p˜j = P (Cj, β) (16)

where P is the bond price function. From Equations (3) and (4), we know that this implies that the yield-to-maturity also is a function of β

y˜j = Y (Cj, β) (17)

The parameters of the NSS model are calibrated such that the resulting spot rates discount bond cash flows in C such that the discounted cash flows (NSS model implied prices) best matches market prices. The determination of the parameter values is achieved by a procedure where an objective function is specified and subsequently minimized using an optimization algorithm.

5.2.1 The Objective Function In the objective function either yield or price errors can be minimized. The choice of which difference to minimize is not trivial. Svensson (1994) argue that the NSS model should be calibrated by minimizing the yield differences as minimization of price differences may result in large yield errors for bonds with short time to maturity. This heteroscedasticity problem arises because long-maturity bonds have higher duration, i.e. the yield of long-maturity bonds is more price sensitive than the yield of short-maturity bonds. An optimization procedure that minimizes price differences will consequently result in a yield curve that is underfitted in the short-end, and overfitted in the long-end. The argument above would suggest minimizing the yield differences. However, there exists a practical disadvantage of minimizing yield differences. When bonds are quoted in prices the yield cannot be solved analytically (see Equation (1)). Instead, each yield calculation requires a numerical approximation procedure. Hence, with constrained computational resources it makes sense to minimize price errors. The problem of inappropriate fitting can be mitigated by weighting each price error by a measure related to the inverse of its duration. Following the majority of central banks, we minimize duration weighted price errors (Bank of International Settlements, 2005).

16 5 YIELD CURVE ESTIMATION 5.2 Calibrating the Nelson-Siegel-Svensson model

5.2.2 The Calibration Problem Once the objective function is defined an optimization algorithm is applied to calibrate the NSS parameters. Calibrating consistent NSS parameter estimates is known to be difficult.10 Gilli et al. (2010) argue that the underlying reason of the NSS calibration problem is twofold. First, the optimization problem is non-convex with multiple local minima. The existence of multiple local minima make the algorithm solution sensitive to starting values as the algorithm may stop at a local minimum. In order to be certain that the global minimum has been reached the optimization procedure would have to be run with virtually all possible sets of starting values. Second, the specification of the NSS model in Equation (15) makes it prone to a multicollinearity problem. As an illustrative example, suppose τ1 = τ2, then β2 and β3 have the same factor loading and are perfectly collinear. Equation (15) reduces to the NS model, but with the third factor equal to (β2 + β3). In other words, the estimated coefficient will be the sum of β2 and β3. Hence, the optimization algorithm will not estimate the individual parameters efficiently. A second multicollinearity problem arrises when τ1 approach zero, as the factor loadings on β0 and β1 becomes equal.

5.2.3 The Optimization Algorithm The optimization algorithm can be separated into two parts, a global and a local search component. The global part is used to find the appropriate region over the domain of the the objective function. This can be achieved by a full estimation (coarse grid search) or by partial estimation, i.e. dividing the estimation of the β’s and τ’s, keeping one group fixed while estimating the other.11 The local search algorithm finds the solution from each set of starting values. Manousopoulos & Michalopoulos (2009), as well as Gilli et al. (2010), test different optimization algorithms for the NSS optimization problem. Manousopoulos & Michalopoulos (2009) test direct search, gradient based, and global optimization algorithms and conclude that direct search or global optimization algorithms are preferable to gradient based algorithms for the NSS optimization problem.12 Gilli et al. (2010) compare a gradient-based algorithm to Differential Evolution

10See Gürkaynak et al. (2007) for an example on US data, Bolder & Stréliski (1999) on Cana- dian data, and Gimeno & Nave (2009) on Spanish data. 11This approach is used by e.g. Nelson & Siegel (1987), Diebold & Li (2006), Bolder & Stréliski (1999). 12More speciﬁcally, the following algorithms are tested: Nelder-Mead, Powell’s direct search, Broyden-Fletcher-Goldfarb-Shanno (Judd, 1998, p. 114-115), generalized reduced gradient algorithm (Lasdon et al., 1978), Simulated annealing (Judd, 1998, p. 299-301).

17 5 YIELD CURVE ESTIMATION 5.2 Calibrating the Nelson-Siegel-Svensson model

(DE) (Storn & Price, 1997), an optimization heuristic, and ﬁnd DE to be more appropriate than the gradient-based algorithm.13 We illustrate the calibration problem by running the standard gradient-based optimization function nlminb in R and DE 500 times each on the same optimization problem for two given dates, 2005-09-20 and 2008-05-12. The two dates are selected such that one day is randomly drawn from a period of relatively low market volatility, while the other day is randomly drawn from a period of relatively high market volatility. Both the DE and the nlminb are stochastic algorithms, hence the solution of each iteration is expected to be diﬀerent from the last.14 However, with an increasing number of iterations we will observe a distribution of parameter estimates that is converging to the true distribution of parameter estimates produced by the algorithm. The ideal algorithm is the one that produce objective function values at the true minimum and parameter distribution with low variance. The parameter distributions are presented in Figure 3 and Figure 4. In Table 2 the mean and variance together with the corresponding parameter estimates from the best and worst objective function values are presented. Given the multicollinearity problem associated with the loadings on β2 and β3, it is not surprising that these parameter estimates exhibit the largest variance. It is clear that DE produces more consistent parameter estimations and objective function values. Since we want to avoid spikes in the noise to be driven by random outcomes of a particular estimation, this makes DE a preferable algorithm for our purposes.

13The gradient-based algorithm is the quasi-Newton method nlminb from the stats package in R. 14In nlminb starting values are randomly determined. For DE, the algorithm itself incorporates a stochastic element, and we also set starting values randomly.

18 5 YIELD CURVE ESTIMATION 5.2 Calibrating the Nelson-Siegel-Svensson model

350 320

280 256

210 192

140 128

70 64

0 0 0 5 10 15 -10 0 10 20 30 β0 β1

140 140

112 112

84 84

56 56

28 28

0 0 -20 0 20 -20 0 20 β2 β3

150 170

120 136

90 102

60 68

30 34

0 0 0 10 20 30 0 10 20 30 τ1 τ2

Diﬀerential Evolution nlminb

Figure 3: Parameter distributions for 2005-09-20. Empirical distributions for NSS parameters during 2005-09-20, with the number of iterations set to 500. The DE settings are set to nP = 50, nG = 200, F = 0.50 and CR = 0.90.

19 5 YIELD CURVE ESTIMATION 5.2 Calibrating the Nelson-Siegel-Svensson model

350 320

280 256

210 192

140 128

70 64

0 0 0 5 10 15 -10 0 10 20 30 β0 β1

180 180

144 144

108 108

72 72

36 36

0 0 -20 0 20 -20 0 20 β2 β3

190 200

152 160

114 120

76 80

38 40

0 0 0 5 10 15 0 10 20 30 τ1 τ2

Diﬀerential Evolution nlminb

Figure 4: Parameter distributions for 2008-05-12. Empirical distributions for NSS parameters during 2008-05-12, with the number of iterations set to 500. The DE settings are set to nP = 50, nG = 200, F = 0.50 and CR = 0.90.

20 5 YIELD CURVE ESTIMATION 5.2 Calibrating the Nelson-Siegel-Svensson model

Table 2: Statistics of empirical parameter distributions. µ denotes the mean, σ2 the variance and Best is the value of the parameter than belongs to the set of parameters that minimizes the objective function out of 500 iterations. Obj. is the objective function outcome, where Best simply represents the lowest achieved result. The DE settings are set to nP = 50, nG = 200, F = 0.50 and CR = 0.90.

2005-09-20 2008-05-12 DE nlminb DE nlminb µ 2.762 2.226 1.554 3.417 σ2 2.817 17.963 6.273 28.167 β 0 Best 4.136 4.139 4.369 4.254 Worst 6.166 2.600 4.974 3.872 µ −0.750 −0.070 2.513 1.263 σ2 2.819 49.182 6.289 61.146 β 1 Best −2.166 −2.168 −0.997 −0.866 Worst −4.176 29.701 −0.923 −14.322 µ 0.498 2.885 1.788 4.439 σ2 145.541 471.533 149.644 484.627 β 2 Best −2.471 −2.455 −2.162 30.000 Worst 6.186 −12.757 −6.605 −29.706 µ 2.322 −2.054 2.358 −7.273 σ2 132.054 485.978 167.593 500.880 β 3 Best 0.528 0.517 2.761 −29.814 Worst −5.494 −30.000 4.879 −29.999 µ 6.533 5.937 5.886 4.723 σ2 5.767 16.044 5.243 9.684 τ 1 Best 2.322 2.332 1.312 0.637 Worst 13.785 0.000 4.514 0.000 µ 7.021 11.226 8.772 11.939 σ2 5.898 93.206 48.134 128.983 τ 2 Best 0.907 0.898 0.411 0.692 Worst 10.762 0.008 5.819 0.000 µ 0.006 0.068 0.181 0.194 σ2 0.000 0.135 0.000 0.008 Obj. Best 0.004 0.004 0.085 0.077 Worst 0.008 2.385 0.185 0.453

21 6 METHODOLOGY

6 Methodology

6.1 Curve Fitting For all traded bonds on a given day, bonds with an outstanding nominal volume less than EUR 5 billion, bonds with time to maturity less than 3 months and longer than 10 years, and finally bonds with time since issuance less than 1 month are excluded from the sample. With the remaining bonds we construct the cashflow matrix C, earlier introduced in Section 5.2, for each day t. We construct C such that each column contains all the future cash flows of a bond. Many bonds have stub periods, i.e. a longer or shorter time interval over which interest accrues than the usual interval between coupons. In order to obtain proper cash flows, and correctly price each bond, potential stub periods are adjusted for.15 The procedure then works as follows: Given a set of parameter values β determined by the optimization algorithm, the NSS model Equation (15) returns spot rates which are subsequently transformed into discount factors using Equa- tion (2). Theoretical bond prices are obtained by multiplying the resulting vector of discount factors with the cashflow matrix. The difference between observed market prices p and the model implied prices p˜ are calculated. Thus, for a given day with B eligible bonds, the estimates for β are obtained by minimizing the objective function defined in Equation (18).

B X 2 argmin [(˜pj (β) − pj) ωj] (18) β j=1 To mitigate the heterogeneity problem caused by the lower duration of bonds closer to maturity, the price diﬀerence for each bond is multiplied by a weight ωj deﬁned as

1 ω = Dj (19) j PB 1 i Di which is an approach proposed by for instance Bolder & Stréliski (1999, p. 11).

6.2 Noise Measure The idea of the noise measure is to capture the general level of mispricings on a given day. That is, we want to avoid situations where one or a few individual

15Among the selected bonds, 99 have long ﬁrst coupon period and the remaining 116 have normal periods.

22 6 METHODOLOGY 6.2 Noise Measure

bonds are driving the noise, perhaps because of data errors. We implement a two step filtration process. As a first step in our filtration process we remove or adjust bond observations that are obvious errors.16 As a second step we conduct an initial calibration of the NSS model using the procedure outlined in Section 6.1. By calculating market and model yields, we consider individual bonds with absolute yield errors exceeding four standard deviations as outliers and remove those bonds. After the filtration process is finished, the calibration procedure in Section 6.1 is now performed again without outliers. The solution with parameters β resulting in the lowest objective function value is then used to calculate the daily noise in Equation (20). In the curve fitting procedure we used observed prices as inputs to calibrate the NSS spot curve. The noise measure in Equation (20) is defined in the yield space. Therefore, market and model dirty prices are converted into yields. For a given day t, the noise measure is calculated as

v u B* u 1 X 2 Noise = t [y − y˜ (β)] (20) B* j j j=1

where B* denotes the number of eligible bonds in the construction of the noise measure. While bonds with time to maturity ranging from 3 months to 10 years was used in calibrating the NSS model, the maturity space used in computing the noise measure is reduced to 1 to 10 years. Consequently, for all days B* is less than B. The average number of bonds used in constructing the noise measure is 34.6.17 On average, 7.7 bonds that were previously included in the NSS curve calibration are excluded in the noise calculations. Bonds in the short-end of the curve are excluded because this part of the curve exhibits more noise than other parts of the curve. Hu et al. (2013, p. 2350) argue that the higher noise in the the short-end of the spot curve is a result of temporary demand and supply ﬂuctuations in the short-end maturity segment of the market. The resulting time-series of the noise measure tends to contain abnormal spikes around year-ends and on one occasion at the turn of the half year. Excluding daily abnormal noise spikes, the average monthly noise rarely exceeds 5 while on days close to year-ends it can suddenly reach values between 30 and 60. In Figure 5, the

16An obvious mispricing might be for example DE0001137073 on the day of 2005-02-22. Recall our pecking order in pricing sources described in Section 4.2, where we at ﬁrst hand choose CBBT, then BGN and ﬁnally BVAL. The CBBT source reports a bid dirty price of 146.173, which equals a yield of -19.208%, whereas the BGN pricing source reports a bid dirty price of 101.157, equaling the yield of 2.437%. Adjusting for these errors in the input estimation data is thus necessary. 17The maximum number of bonds was 41 (2001-02-14), and the minimum was 28 (2008-10-07).

23 6 METHODOLOGY 6.3 Optimization Algorithm - Diﬀerential Evolution

pricing errors of the three last trading days of 2003 are plotted. The aggregated noise measure spikes at 61.48 on 2003-12-29 but is stable at 2.19 and 2.60 on the preceding and succeeding trading days. It is not the case that one or two mispriced bonds are driving the noise on this day. Rather, a wide range of the bonds deviate by a large extent (additional examples are given in Figure 19 in Appendix A). The pre-estimation filter is designed to exclude one or two outliers, not this kind of across the curve outliers. To avoid extreme spikes in the noise to affect our later regression analysis, we drop days where the noise measure spikes abnormally. The filtered noise series is presented in Figure 12, and in Table 18 in Appendix B, the yearly summary statistics of the noise measure is presented together with statistics for the full estimation period 2000-2015. 5 4

3 2003-12-23 2003-12-29 2 2003-12-30 1 0 Market - Model (bps) -1 -2

0 1 2 3 4 5 6 7 8 9 10

Years to maturity

Figure 5: Yield error example. The pricing error in yield space and basis points under three following business days, illustrating the reasoning of why certain days have been removed from the ﬁnal series of noise.

6.3 Optimization Algorithm - Diﬀerential Evolution The DE algorithm is implemented through the NMOF package in R. A population matrix nP = 200 of initial solutions (sets of the six NSS parameters) are produced by randomly drawing from a uniform distribution of the parameter ranges in Equation (6.3). These ranges are also enforced as parameter constraints through a penalty function that is directly added to the objective function for solutions where the parameter constraints are violated. The DE algorithm provided in the NMOF package further requires us to specify the step size F and the probability crossover CR. A smaller step size mean that new candidate solutions makes smaller steps when moving through the solution space.18 A high CR results in new solutions

18With a small F , we evolve the solutions by adding small changes at several dimensions of the solution. In a sense then we have a population of local searches, or at least of slowly-moving individuals.

24 6 METHODOLOGY 6.3 Optimization Algorithm - Diﬀerential Evolution

being changed along many dimensions, i.e. several parameters are changed, while a low CR indicate that a new solution only affects a few parameters. Gilli et al. (2011, p. 477) find that a rather small F value (between 0.3 and 0.5) and high CR value (around 0.9) are good parameter settings that increases the speed of convergence when calibrating the NSS model.19 We set F = 0.50 and CR = 0.99 in our NSS calibration. The stopping criterion is simply set to a fixed number of evolutions nG × nP , where nG = 600 is the number of generations. The chosen parameter constraints are shown in Equation (6.3)

0 ≤ β0 ≤ 15, −15 ≤ β1 ≤ 30, −30 ≤ β2 ≤ 30, −30 ≤ β3 ≤ 30,

0 ≤ τ1 ≤ 30, 0 ≤ τ2 ≤ 30, (21)

0 ≤ β0 + β1

In Appendix A, Figure 18 illustrates the estimated parameters over the estimation sample period. Parameters tend to occasionally take on limit values. Recall however, that as the parameters have economic interpretation it makes sense to restrict them to only be allowed to take on values within certain limits.

19The performance of DE is generally improved when setting F to a low value while diﬀerent choices of CR has less inﬂuence (Gilli & Schumann, 2009, p. 17).

25 7 RESULTS

7 Results

We begin by presenting the time-series of our calibrated NSS curves and the estimated German noise. We further describe the micro foundation of the noise measure, giving cross-sectional and time-series examples of mispricings on individual bonds. We then move on to investigate what cause the noise to change and its qualities as a market-wide measure of liquidity. We begin by qualitatively investigate the relationship between noise and events and then explore its relation to other popular measures of liquidity. We then investigate the relationship between changes in the noise and hedge fund returns. Using the noise as a measure of mispricings, we test the implications of the theory of slow-moving capital. Finally, we present a sensitivity analysis where we test the robustness of our results.

7.1 Parameters and Curves Figure 18 in Appendix A shows the evolution of the daily estimated NSS parameters β during the estimation period. Parameter estimates are notably volatile and often at their allowed boundaries. One example is β0, which represents the long-term rate. Economic intuition would lead us to expect the long term rate to be rather stable. Contrary to economic intuition, the estimates of this parameter make notable shifts of several percentage points from one day to another and is often at the limits of its allowed range. Given our discussion and findings of the calibration problem of the NSS model in Section 5.2.2, it is not surprising that β2 and β3 exhibit the most volatile estimates. Figure 6 illustrates the estimated spot curve from January 2000 to December 2015. Descriptive statistics of spot rates at different maturities is presented in Table 3. Figure 6 and Table 3 are based on a monthly frequency of estimated spot rates. The spot rates are calculated for fixed maturity points at 6 months, 1 year, 2 years, 4 years, 6 years, 8 years and 10 years. The spot rates are calculated using the daily estimated NSS parameters β.

26 7 RESULTS 7.1 Parameters and Curves

6.0%

5.0%

4.0%

Jan 3.0% 2000 Jan 2.0% 2002 Jan 1.0% 2005 0.0% Jan 2008 10Y 8Y Jan 2011 6Y 4Y Jan 2Y 2014 1Y 6M

Jan 2016

Figure 6: Estimated term structure. 193 monthly spot curves based on calibrated NSS parameters β. Monthly spot rates are calculated as a mean of daily spot rates.

The shape of the spot curve moves between diﬀerent shapes, and as expected in general spot rates increases with maturity.20

Table 3: Monthly spot rate statistics. Descriptive statistics of monthly spot rates. Monthly spot rates are calculated as an average of daily estimated NSS spot rates.

Time to maturity 6m 1y 2y 5y 7y 10y Mean 1.869 1.960 2.129 2.662 2.974 3.334 St. Dev. 1.652 1.670 1.672 1.612 1.537 1.406

In Figure 7, we plot six examples of par-curves and the market-observed bond yields. The left panel of Figure 7 illustrates the par-curve on three days with relatively low noise. The market observed bond yields follow the par-curves quite closely. The right panel of Figure 7 shows three days of relatively high noise. As can be seen, and as expected, the dispersions around the par-curves are notably larger on the high-noise days. Whether mispricings, or noise, are traded away can be argued to depend on bid-ask spreads. To explore the relation between bid-ask spreads and misspricings,

20The main theories behind the shape of the yield curve are the Pure Expectations theory, the Market Segmentation theory and the Liquidity Premium theory.

27 7 RESULTS 7.1 Parameters and Curves

as an illustration we plotted quoted bid-ask spreads and security-level mispricings seen in Figure 8 and 9. As traders typically have to buy (sell) at the oﬀer side (bid side) of the bid-ask spread, the noise has to be larger than the bid-ask spread. Such potential trading opportunity would only arise when there is a positive distance between the outer borders of the market bid-ask and the model bid-ask, emphasized in the left panel in Figure 8. To clarify, as the noise measure is calculated as diﬀerences between mid market yields y and model implied yields y˜, these deviations may not be enough for arbitrageurs facing the cost of crossing the bid-ask spread. Thus, naturally assuming that market players trade on the bids and asks, the bid-ask spread of individual bonds will inevitably contribute to the noise being larger than zero. 4.00 4.00 3.00 3.00

2005-01-04

2.00 2009-12-01 2.00 2015-05-05 Yield (%) Yield (%) 1.00 1.00 2008-04-04 2008-12-19

0.00 0.00 2010-06-25

0 2 4 6 8 10 0 2 4 6 8 10

Years to maturity Years to maturity Figure 7: Market yields and estimated par curves for selected days. Diﬀerent days are separated by diﬀerent colors and symbols. The left panel plots three days with relatively low noise. The right panel outlines three days with relative high noise. Black lines are par curves for each day, which has been smoothed using cubic splines. The noise on the low-noise days are 0.94 for 2005-01-04, 2.69 for 2009-12-01 and 0.83 for 2015-05-05. The noise on the high-noise days are 6.21 for 2008-04-04, 7.80 for 2008-12-19 and 5.58 for 2010-06-25.

The left panel of Figure 8 and 9 illustrate the bid-ask spreads for a random selection of bonds in the two to ﬁve year maturity segment. Model implied bid-ask spreads are computed by adding the quoted bid-ask spread to the model implied mid-yield. The right panel of Figure 8 and 9 illustrate the basis point diﬀerence between market yields and model implied yields. The main takeaway from Figure 8 is that the noise observed on this day is not mainly due to arbitrage opportunities and could not be traded away by arbitrageurs, even if the prices immediately adjusted to the model implied prices.

28 7 RESULTS 7.1 Parameters and Curves 3.0 Market Bid-Ask 2 Model Bid-Ask 1 2.9 0 2.8 -1 Yield (%) 2.7 Yield Error (bps) -2

2.6 BKO DBR -3 OBL 2.5

2.0 2.5 3.0 3.5 4.0 4.5 5.0 0 2 4 6 8 10

Years to maturity Years to maturity Figure 8: Pricing errors and bid-ask spreads on a day with relatively low noise. The left graph shows the bid-ask spread measured in yields within the 2 to 5 year maturity segement on the date 2005-01-04. The shaded area highlights a bond whose model and quoted bid-ask spreads are non-overlapping. Model implied bid-ask spreads are computed by adding the quoted bid-ask spread to the model implied mid-yield. In the bid-ask figure to the left a few bond observations have been removed in order to make the visualization more clear. The yield error figure to the left illustrates mid-price differences between market quotes and model implied prices, measured in yields, for the maturity range 0 to 10 years. The highlighted bond is DE0001141422, a 3.00% coupon Bobl with 3.26 years to maturity and EUR 14 billion in nominal outstanding volume at the given date. The estimated noise on this day is 1.09.

Compared to Figure 8, Figure 9 instead shows a day with high noise. The scale on the horizontal axis in the left graph have been adjusted to the maturity segment between 2 and 5 year. In comparison with Figure 9, the distance between bid-ask boundaries for market and model yields are high across the curve, implying that there are arbitrage trading opportunities on this day.

5.2 Market Bid-Ask 10 Model Bid-Ask 5.0 5 0 4.8 Yield (%) -5 4.6 Yield Error (bps)

BKO -10 4.4 DBR OBL

2.0 2.5 3.0 3.5 4.0 4.5 5.0 0 2 4 6 8 10

Years to maturity Years to maturity Figure 9: Pricing errors and bid-ask spreads on a day with relatively high noise. Illustrates the same concept as in Figure 8, but for the date 2000-02-02. The left panel is scaled to illustrate the 2 to 5 year segment of the curve. The estimated noise during this day is 5.04

29 7 RESULTS 7.1 Parameters and Curves

Observing the right panel in Figure 9, another interesting pattern emerges. As the graph is showing market - model yields, it seems as Bunds (DBR) are above the no deviation line (horizontal black dotted line at 0), whereas Bobls (OBL) are below. To further examine this, for each day we take the series of yield diﬀerences for Bunds and for Bobls, and plot their distributions throughout the estimation period.

0.30

0.25 DBR

OBL

0.20

0.15 Density

0.10

0.05

0.00 -15 -10 -5 0 5 10 15

Market - model (bps)

Figure 10: Distributions of Bunds and Bobls yield errors. The distributions of yield diﬀerences between mid yields quoted by market and mid yield implied by the NSS model. The distributions are computed using all Bunds and Bobls that are used in estimations over the whole estimation period and that are eligible in calculating the noise. Thus, bond observations considered as outliers removed in the estimation are thus excluded and only bonds within the maturity range 1 to 10 years a given day are included.

Despite the fact that the amount of observations differ for each category (as there might be more bond type observations of a kind during low or high noise periods), Figure 10 do illustrate the difference in means, which for Bunds is 0.37 bp, whereas the mean for Bobls is -0.97.21 21The phenomena of different pricing of medium and long term bonds has been documented on the US market by Musto et al. (2015).

30 7 RESULTS 7.1 Parameters and Curves

Bund (DE0001102325) 2.00% 15/08/2023 Bobl (DE0001141703) 0.25% 11/10/2019 4.0 2.0 3.0 1.5 2.0 1.0 0.5 1.0 0.0 0.0 Abs. diﬀ. in model and market yield (bps) Abs. diﬀ. in model and market yield (bps) Jan2014 Apr2014 Jul2014 Okt2014 Jan2015 Okt2014 Jan2015 Apr2015 Jul2015 Okt2015

Figure 11: Example of the model-market yield difference over time for two bonds. Illustration of how the difference between the model yield y˜ and market yield y varies over time for one bund (left panel) and one Bobl (right panel). The red dotted lines are the mean difference over the chosen time intervals. Note that the scale of the vertical axes differ between the graphs, as the Bobl exhibits larger deviations.

As the graphs in Figure 11 outlines different periods of time, a direct comparison is cumbersome, although the illustration implies that mispricings in different bonds may be determined by bond specific characteristics, as for instance the size or the degree of seasoning. Another interesting aspect is that the individual noise for the bonds in Figure 11 are oscillating around a deviation around one basis point. For the Bund in the left panel, the mean of absolute yield basis point difference is 0.768, whereas for the Bobl in the right graph the mean is 1.135. This is, however, not surprising, as bid-ask spreads deter potential arbitrageurs to trade down the noise closer to zero, as seen in Figure 8.

31 7 RESULTS 7.2 Noise and Events

7.2 Noise and Events Figure 12 shows the estimated noise for the whole period. As one can see, the period with the highest noise is during the US subprime crisis.

7.0

6.0

5.0

4.0

3.0

Noise measure (bps) 2.0

1.0

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016

Figure 12: German noise measure for the whole sample period. The noise measure calculated as the RMSE of model yields versus market yields for eligible German sovereign bonds during the period 2000 to 2015.

Two-year time series plots of the noise are presented in Figure 20 and 21 in Ap- pendix A. These figures together with Table 20, in Appendix B, illustrate and provide a brief description of selected events during our sample period. This sub- chapter describes the evolution of the noise and a selection of these events. Special attention is given to the periods with the highest noise, the US subprime crisis of 2008 and the first years of the European sovereign debt crisis that became evident in 2010.22 The noise is generally declining in the early 2000s. While the 9/11 terrorist attack had major and long-lasting effects on the U.S. market (Hu et al., 2013, p. 2353), it does not seem to have been a driving factor of our estimated German noise measure. During 2005 - June 2006 the noise remains low and stable. The average and standard deviation of the noise during this period was 0.841 and 0.196. The same statistics for the whole estimation period are 2.159 and 1.084, respectively. By the end of June 2007, signs of a of a US subprime mortgage crisis is starting to unfold. Problems in the US subprime mortgage market started to become

22The description of events has to a large degree been based on news articles from Bloomberg and The Financial Times Historical Archive.

32 7 RESULTS 7.2 Noise and Events

apparent in the beginning of 2007 as several subprime lenders are forced to shut down. The Federal Home Loan Mortgage Corporation (Freddie Mac) announce that they will no longer buy the most risky subprime mortgages and mortgage- related securities (2007-02-27). In April New Century Financial Corporation, a leading subprime mortgage lender files for bankruptcy (2007-04-02). During June 2007 Countrywide Financial, the biggest US mortgage lender, reveals that foreclo- sures are doubled since 2006. The rating agencies S&P and Moody’s downgrade hundreds of bonds backed by subprime mortgages. However, the noise remains low and relatively stable during the first half of 2007 as macroeconomic factors and European financial markets remain strong. During January through June 2007, mean noise is about 0.822 basis points. In July, funds are seized and sold at fire-sale prices after being seized from two failed Bear Stearns hedge funds (2007-07-19). In August American Home Mortgage Investment Corp files for bankruptcy (2007-08-06), BNP Paribas halts redemptions of three of its funds (2007-08-09), Lehman Brothers announce that their subprime mortgage unit, BNC Mortgage would be shut down (2007-08-23). In the beginning of August, the subprime contagion hits Europe as the euro interbank market dries up23, prompting the ECB to inject EUR 95 billion into the banking system on August 9. On February 17 2008, the UK bank Northern Rock is nationalized. One month later on March 14 2008, the Fed approves the financing of JP Morgan’s purchase of Bear Stearns, which had failed to obtain short-term funding to survive on its own. Contrary to the US noise estimated by Hu et al. (2013), there is a considerable drop in the German noise at the end of May. Also, while the German noise have a similar spike as the US noise, it does not continue to increase at the same rate during the rest of the fall and at the end of 2008. At the end of 2009, it is apparent that Greece is experiencing financial problems as the three rating agencies Moody’s, S&P and Fitch downgrade Greece’s long- term credit rating. Following on February 23 2010, Fitch downgrade Greece’s four major banks, from this point until Greece’s first bailout deal is sealed at the end of April the estimated German noise remains volatile. The highest noise during the first part of the Greek debt crisis is observed on the days leading up to the first bailout. The next peak in the noise is observed on July 27, following a bank stress test conducted by The Committee of European Banking Supervisors (CEBS)24. During the fall of 2010, it has become apparent that the 2008 banking crisis has left the Irish government finances in a bad shape. On September 30 it is announced

23See e.g. Cassola et al. (2013). 24The CEBS conclude that 7 out of 91 banks would need to raise more capital in order keep their Tier 1 ratios above the required level of 6%. (Committee of European Banking Supervisors, 2010)

33 7 RESULTS 7.2 Noise and Events

that the cost of bailing out Irish banks may cost up to EUR 50 billion.25 While the noise had been falling during the summer and fall of 2010, yields of government bonds were rapidly increasing. In one month, from November 8 to December 8, the 10 year German benchmark yield increased from 2.392% to 3.032%. US and Japanese bond yields also experienced similar increases while UK bond yields increased 18%. As illustrated in Figure 13 the noise peaks around the year-end of 2010. In an article on 9 December, a Financial Times headline states "Eurozone bond markets face testing December". The authors point out that December is traditionally a month with declining market activity as financial institutions typically adjust their balance sheets and close their books for the year-end. In combination with reports of thin trading volumes, we can infer that general market liquidity was most likely low during the turn of the year. End- of-year financial tensions and the demand to hold on to liquid assets is further demonstrated by the ECBs failed sterilization on 28 December. March 15 to April 6 2011 is a turbulent period in Portugal. During the short time period Portugal’s government collapses, its credit rating is cut six times, and finally a bailout request is made on April 6.26 In the summer of 2011, EU leaders approve a draft of Greece’s second bailout (2011-07-21). In addition to the EUR 109 billion bailout, other EU countries promise to fund Greece indefinitely until Greece is able to return to the market and issue bonds on its own again. The final bailout is later signed on 1 Mars 2012. Later in the fall Greek bond holders are forced to take a 50% haircut (2011-10-27). Also, during the fall of 2011, S&P downgraded the US credit rating below AAA (2011-08-05). On July 26 2012, the somewhat iconic "whatever it takes" speech by Draghi takes place, and the noise is estimated to about 1.854. At the end of 2012, S&P upgrade the Greek credit rating to B- (CCC). In 2013 the Greek situation becomes somewhat more calm. Figure 13 visualizes the noise and chosen markets related events from early 2010 to late 2011.

25The GDP of Ireland 2010 was EUR 156 billion, making the bail out cost roughly equal to one third of Ireland’s GDP (Central Statistics Oﬃce, 2011, Table A). 26Moody’s, S&P and Fitch cut Portugal’s long-term credit rating two times each.

34 7 RESULTS 7.2 Noise and Events

Noise

9 07dec11: 15 EU 28maj10: Spain loses AAA rating members risk 14jun10: Moody cut Greece downgrade 8 14jan11: Fitch downgrades 27okt11: Haircut Greece on Greece bonds 7 23feb10: Greece 23jul10: Bank stress banks downgraded test results

6 15mar11: Portugal downgraded 30sep10: Cost of bailing 23mar11: Portuguese government collapse 09apr10: Irish banks higher than exp. 06apr11: Portugal asks for bailout

5 Greece rating cut 16maj11: Port. bailout approved 4 3 2 27apr10: Greece and Port. rating cut 25mar10: 28apr10: Spain downgraded

1 05jul11: Portuguese rating cut Bailout for 30apr10: Greece bailout 1 21nov10: Ireland apply for bailout 10maj10: EU agrees on 750bn EUR bailout fund 28nov10: EU approves Irish rescue 12jul11: Irish rating cut Greece 21jul11: Greece bailout 2

Apr10 Jul10 Okt10 Jan11 Apr11 Jul11 Okt11

Figure 13: German noise during the European sovereign debt crisis. Estimated Ger- man noise during a chosen period of time, from early 2010 to late 2011. During this period, a sequence of market related events occurred, that in the graphs has been highlighted and marked out.

We also investigate the reaction of the German noise to main macroeconomic releases for the Eurozone and the US.27 In general, the noise does not seem to be driven by specific events or releases of economic data.28 For example, the increase in the noise around the year-end of 2010 does not seem to be related to any particular event. Rather the increase in the noise occur during a period normally known to have low liquidity, after a surge in bond yields that made market participants more reluctant than usual to participate in trading. Although not formally tested, our findings suggest that something else but events is driving the noise. As theory suggests, that mispricings are driven by lack of arbitrage capital and this seems to be consistent with our findings thus far, we proceed by formally examining the relationship between the noise and various funding and liquidity variables.

27For instance, CPI ﬁgures, US Non-farm Payroll, and changes in German unemployment. 28The exceptions include a few events related to the European sovereign debt crisis and the bank stress results on 23 July 2010.

35 7 RESULTS 7.3 Noise and Market Variables

7.3 Noise and Market Variables Hu et al. (2013) argue that the US noise measure is designed to capture overall market liquidity conditions. In order to determine if the German noise measure has similar properties, we conduct OLS regressions to assess how the German noise is related to I) changes in the spot curve II) other measures of liquidity III) equity markets. We group twelve variable into four groups and report the summary statistics of the regressions in Table 6. The ﬁrst group is related to the shape of the spot curve and is referred to as curve variables. These include the level, slope and volatility. The second group consists of proxies of funding liquidity and is referred to as funding variables. These include the TED spread, the 3 month EURIBOR EONIA spread and the yield spread between AAA and BBB bond indices. The third group consists of three equity indices representing diﬀerent regions. Finally, the fourth group of variables include the implied volatility indices of the equity indices in group three. All variables are regressed both individually and in multivariate form within their corresponding group.

7.3.1 Curve Variables The level of interest rates (Level) is represented by the 3 month spot rate implied by our estimated NSS function. As as short-term rates as well as liquidity tend to decrease during recessions we expect a negative relation between the noise and the 3-month spot rate. The term spread (Curve) of the spot curve is defined as as the spread between the 10 year and 1 year spot rates implied by our estimated NSS curve. Changes in the slope occurs either because of changes in the short-end of the curve or the long-end of the curve. As long-term rates are typically more stable and short-term rates tend to decrease during recessions we expect a positive coefficient. Bond return volatility (Bond V.) is defined as the annualized 21 day rolling standard deviation of the return of REX, the German sovereign bond index introduced in Section 4.1. As previous studies have shown commonality between liquidity and volatility we expect a negative coefficient.29

7.3.2 Funding Liquidity Variables The theoretical model by Brunnermeier & Pedersen (2009) formally link market liquidity to funding liquidity. When funding is, or might become tight, investors do not want to take on leveraged positions and as investors reduce their activity in the market, market liquidity declines. Schuster & Uhrig-Homburg (2015) provide empirical support to the Brunnermeier & Pedersen (2009) model on German data.

29See e.g. Chordia et al. (2005).

36 7 RESULTS 7.3 Noise and Market Variables

Accordingly, we explore how the German noise measure is related to three common proxies of funding liquidity. The TED30 spread (TED) generally refers to the difference of the rate of interbank loans and the rate of government securities, in our case the basis point difference between the 3 month EURIBOR and 3 month spot rate implied by our NSS model (Hull, 2012, p. 809). During times of financial distress or uncertainty the LIBOR rate increase. Thus we expect the relationship between the noise and the TED spread to be positive. The EURIBOR EONIA spread (EE) represent the spread between unse- cured interbank borrowing rates (EURIBOR) and overnight-indexed swap (EO- NIA) rates. Our EURIBOR EONIA spread is the basis point difference between the 3 month EURIBOR and the 3 month EONIA rate. A bank can borrow at the overnight rate and lend at the 3 month EURIBOR rate and hedge its interest risk with and 3 month EONIA swap. The credit risk in a EONIA swap is lower than in EURIBOR lending as no principal is exchanged in the swap. As the perceived risk of interbank lending increase the EURIBOR EONIA spread increase. As an increased spread indicate tighter funding conditions we expect a positive relation to the noise measure. The Default spread (Default) is the difference in basis points between the yields of the iBoxx indices: Euro Corporates BBB Rated All Maturities and Euro Sovereign all Maturities. In periods of uncertainty the rate charged for risky corporate bonds increase relative to risk free government bonds. Consequently, we expect a positive relationship between the default spread.

7.3.3 Equity Volatility Variables The empirical relationship between liquidity and volatility is well known.31 The model of Brunnermeier & Pedersen (2009) predict that market liquidity is related to volatility and suggest the CBOE VIX index as a measure of volatility. A liquidity shock leads to price volatility, which raises expectation about future volatility, which leads to increased margins, and decreased liquidity. We test regressing changes in the noise on three different regional versions of the VIX index, representing the German, European and US markets. We expect the coefficients of all volatility indices to be positive. We proceed this section by first present the three indices and then comment on the regression results. The CBOE VIX index (VIX) represents the implied 30 day volatility of the S&P 500 index expressed in annualized percentage points. The VIX index is constructed using a weighted average of S&P 500 index option prices such that it

30TED is an acronym for its initial deﬁnition as the diﬀerence between 3 month Treasury Bill (T) and Eurodollar futures contracts (ED). 31See e.g. Hu et al. (2013, p. 2343).

37 7 RESULTS 7.3 Noise and Market Variables

represents the implied volatility over the next 30 days. The underlying S&P 500 index consists of the 500 large firms listed on the NYSE or NASDAQ exchanges and captures approximately 80% of the market capitalization. (S&P Dow Jones Indices, 2015) The VDAX index (VDAX) is the German analogue of the VIX index. The VDAX index represent the implied 45 day volatility of the DAX index, a German equity index. The underlying stock index DAX comprises the 30 largest firms, representing more than 80% of the traded German equity on exchanges. Thus, even though the number of firms included in the underlying DAX index is small compared to S&P 500, the DAX index is still a rather good proxy for the German equity market. The VSTOXX index (VSTOXX) is the European version of the VIX index. It measures the implied volatility of the EURO STOXX 50 Index over the next 30 days. Analogue to the S&P 500 and DAX indices the underlying EURO STOXX 50 Index consists of large firms in the Eurozone. The index represents approximately 57% of the eurozone total free-float market capitalization.

7.3.4 Equity Indices We also include the indices DAX, S&P500 and STOXX, that respectively are equity indices for Germany, the US and Europe. As Duarte et al. (2007, p. 797) point out, a body of literature have earlier emphasized the importance of considering equity markets in fixed income dedicated research, as there are common risk factors having impact on both equity and bond returns. The findings of Duarte et al. (2007) also support the previously documented commonality in equity and bond returns. Consequently, by including the mentioned indices, we can obtain some understanding to what extent the German noise measure is related to equities. In the light of previous research, we thus expect a significant relationship.

7.3.5 Regression Analysis Figure 14 illustrates the noise measure (black line and on the left hand side axis) compared to a selection of the previously discussed market variables. Notable is that the middle two graphs, the EURIBOR EONIA spread and the TED spread exhibited low volatility before the US subprime crisis. For instance the default spread and the VDAX appear to show somewhat more resemblance with the noise measure in the early years.

38 7 RESULTS 7.3 Noise and Market Variables

bps % bps bps

VDAX (rhs) 70 10y1y (rhs) 7 7 Noise (lhs) Noise (lhs) 6 6 200 5 5 50 4 4 3 3 100 30 2 2 1 1 0 10 2000 2002 2004 2006 2008 2010 2012 2014 2016 2000 2002 2004 2006 2008 2010 2012 2014 2016

bps bps bps bps

EURIBOR-EONIA (rhs) TED (rhs) 7 7 Noise (lhs) Noise (lhs) 6 6 150 300 5 5 4 4 75 3 3 150 2 2 1 1 0 0

2000 2002 2004 2006 2008 2010 2012 2014 2016 2000 2002 2004 2006 2008 2010 2012 2014 2016 % bps bps bps

Volatility in REX (rhs) Default Spread (rhs) 7 7 Noise (lhs) Noise (lhs) 3.0 6 6 400 5 5 4 4 2.0 3 3 200 2 2 1.0 1 1 0

2000 2002 2004 2006 2008 2010 2012 2014 2016 2000 2002 2004 2006 2008 2010 2012 2014 2016

Figure 14: Noise measure compared to other market variables. VDAX is the German equivalent of the US CBOE VIX index. It measures the implied volatility of the DAX index. 10y1y is the the spread in basis points between the 10 year and 1 year spot rates. EURIBOR- EONIA is the spread between the 3 month EURIBOR and 3 month EONIA. The TED spread is the rate between 3 month EURIBOR and the 3 month spot rate. Spot rates are calibrated through the NSS model. Volatility in REX is the 21-day rolling standard deviation of annualized returns in the REX index. Default is the yield spread in basis points between two iBoxx indices, Euro Corporates BBB Rated All Maturities and Euro Sovereign all Maturities.

Table 4 outlines the descriptive statistics for the chosen market variables, where all but the equity indices are in monthly changes based on month end values. The equity indices are in monthly returns, nonetheless also based on month end values.

39 7 RESULTS 7.3 Noise and Market Variables

Table 4: Descriptive statistics for market variables. Statistics for market variables used to regress the calculated German noise measure. DAX, STOXX and SP500 are in monthly log returns, other variables in monthly changes based on month end values. First observation is February 2000 and last December 2015. ∆Level is the 3 month German zero coupon rate based on estimated NSS parameters β. ∆Curve is the spread between 10 year and 1 year on the curve, also based on estimated β. ∆TED is the spread in basis points between 3 month EURIBOR and estimated 3 month German zero coupon rate. ∆Bond V. is the 21-day rolling standard deviation of annualized returns of the German sovereign bond index REX. ∆Default is the yield spread in basis points between two iBoxx indices, Euro Corporates BBB Rated All Maturities and Euro Sovereign all Maturities. ∆VDAX, ∆VSTOXX and ∆VIX represents the monthly change in equity volatility indices.

# months Mean Median St. Dev. Min Max ∆Level 191 −0.0191 0.0105 0.1870 −1.32 0.36 ∆Curve 191 −0.3538 −2.4894 20.1090 −51.22 94.70 ∆Bond V. 191 0.0000 −0.0133 0.3813 −2.55 1.54 ∆TED 191 0.0146 −1.0712 18.3356 −61.62 163.02 ∆Euribor-EONIA 3m 191 0.0215 −0.0900 11.1319 −41.40 68.10 ∆Default 191 −0.0196 −1.2000 28.1424 −97.09 176.34 DAX 191 0.4268 0.9579 6.2584 −25.42 21.38 STOXX 191 0.2684 0.8090 4.3649 −16.94 10.77 SP500 191 −0.0611 0.5822 5.4403 −18.64 14.69 ∆VDAX 191 −0.0575 −0.4100 4.8076 −13.62 23.76 ∆VSTOXX 191 −0.0543 −0.5977 5.4766 −16.47 20.29 ∆VIX 191 −0.0464 −0.4300 4.5704 −13.00 20.50

Following, in Table 5, the correlations between the chosen market variables are shown. As in Table 4, the correlations are also based on monthly changes in month end values, expect for equity indices that are in monthly returns. Of main interest is clearly how the noise relates to the market variables. Indicated by Figure 14, which although is in levels, the default spread shows relatively high correlation with the noise, being about 23%. The VDAX volatility index also has a correlation with the noise of 23%. The VSTOXX volatility index exhibits the third strongest correlation coeﬃcient with the noise at 21%.

Table 5: Pairwise correlations with selected market variables. Correlations in percentages in monthly changes based on month end values for chosen market variables.

(2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (1) ∆Noise −12 10 15 11 12 23 −9 −11 −4 23 21 16 (2) ∆Level −58 −18 −58 −41 −35 10 13 11 −30 −30 −29 (3) ∆Curve 29 30 28 15 −11 −13 −11 24 21 14 (4) ∆BondV 36 28 16 −3 −9 −9 25 22 25 (5) ∆TED 79 33 −4 −10 −6 27 27 33 (6) ∆EE3m 47 −6 −12 −7 33 28 36 (7) ∆Def. −18 −22 −13 55 50 51 (8) DAX 74 94 −56 −58 −46 (9) STOXX 78 −59 −61 −62 (10) SP500 −56 −60 −49 (11) ∆VDAX 96 83 (12) ∆VSTOXX 84 (13) ∆VIX

The regression results using the monthly change in noise as dependent variable are presented in below Table 6, where the regressions have been divided into panels in accordance with the earlier deﬁned four categories.

40 7 RESULTS 7.3 Noise and Market Variables

Table 6: Monthly changes in noise regressed on other market variables. Variables are in monthly changes based on the last observation of each month. Reported p-values within brackets are based on Newey-West standard errors. Panel (a) illustrates the noise measure’s relationship to variables related to the spot curve. ∆Level is the three month spot rate, representing the level of the spot curve. ∆Curve is the 10 year rate subtracted by the 1 year rate and represent the spot curve slope. The spot rates are calculated using our estimated NSS model. ∆Bond V. is the 21-day rolling standard deviation of the German sovereign bond index REX. Panel (b) shows the regressions based on related the fear variables ∆TED, ∆EE and ∆VDAX. ∆EE denotes the monthly change in the 3 month EURIBOR-EONIA spread.

(a) Curve Variables (b) Funding Variables

Dependent variable: Dependent variable: ∆Noise ∆Noise (1) (2) (3) (4) (1) (2) (3) (4) ∆Level -0.3458∗∗ -0.2603 ∆TED 0.0032 0.0019 (-1.9839) (-1.0694) (1.1989) (0.4417)

∆Curve 0.0027 0.0003 ∆EE 0.0057∗ -0.0021 (1.0923) (0.0739) (1.9303) (-0.3069)

∆Bond V. 0.2123∗∗∗ 0.1849 ∆Default 0.0045∗∗∗ 0.0045∗∗∗ (1.8063) (1.1025) (4.3760) (3.1686)

Obs 192 192 192 192 Obs 192 192 192 192 R2 (%) 1.377 0.950 2.147 3.003 R2 (%) 1.124 1.339 5.274 5.437 Adj R2 (%) 1.447 Adj R2 (%) 3.920

Note: ∗p < 0.01; ∗∗p < 0.05; ∗p < 0.10 Note: ∗p < 0.01; ∗∗p < 0.05; ∗p < 0.10

Dependent variable: Dependent variable: ∆Noise ∆Noise (1) (2) (3) (4) (1) (2) (3) (4) DAX -1.2919∗∗ -1.1922 ∆VDAX 0.0259∗∗∗ 0.0356 (-2.2071) (-0.5253) (3.6701) (1.5617)

STOXX -0.0253∗∗∗ -3.5247∗∗ ∆VSTOXX 0.0211∗∗∗ -0.0028 (-2.7932) (-2.4129) (3.1233) (-0.1263)

SP500 -1.3467∗∗ 2.2906 ∆VIX 0.0198∗∗ -0.0085 (-2.0583) (0.8650) (2.0010) (-0.4756)

Obs 192 192 192 192 Obs 192 192 192 192 R2 (%) 2.16 4.01 1.77 4.57 R2 (%) 5.104 4.411 2.685 5.285 Adj R2 (%) 3.03 Adj R2 (%) 3.765

Note: ∗p < 0.01; ∗∗p < 0.05; ∗p < 0.10 Note: ∗p < 0.01; ∗∗p < 0.05; ∗p < 0.10

Starting with Panel (a) in Table 6, for the 3 month spot rate ∆Level we find a negative and statistically significant relation. However, it only explains about 1.38% of the monthly variation in the noise. For ∆Curve, we find a positive but not significant relation with the noise. As with the short-term rate variable ∆Level, the explanatory power of ∆Curve is low, under 1%. We do find a positive and on the 1% level significant relationship between the monthly changes in noise and bond return volatility ∆Bond. V. Accordingly, liquidity, proxied by the noise measure, is related to fundamental volatility in the bond market. The monthly changes in bond volatility explain approximately 2.15% of the variation in the noise. The low explanatory power of the of the curve variables indicate that the NSS model

41 7 RESULTS 7.3 Noise and Market Variables

manages to capture different shapes and levels of the spot curve and that the noise is to a very large extent driven by factors other than movements of the German sovereign bond curve. In column (4), the multiple regression including all three curve variables is presented. None of the curve variables stays significant however. In Panel (b), the regression result for the TED spread shows a positive, but not significant relationship. On the contrary, and in line with expectations, our results show a positive and significant relationship between the EURIBOR EONIA spread and the noise measure. Further, the results confirm that changes in the default spread are significantly related to changes in the noise measure. Even though changes in the default spread exhibit the highest coefficient of determination of the funding variables, it does not explain more than about 5.27% of the variation in the noise measure. For the equity indices in Panel (c), among the single regressions the STOXX index is the most significant regressor, being significant on the 1% level and having an R2 value of 4.41%. The STOXX index also stays significant in the multiple regression in column (4). All three indices are however significant in their single OLS regressions, DAX and S&P500 on the 5% level, and having coefficient estimates taking negative values. Among the single coefficient estimates in Panel (c), the S&P500 is the lowest with -1.3467, followed by the DAX with -1.2919. Con- sequently, it appears as increased returns in equity indices thus imply decreased noise. Note however that the multiple regression of the equity variables suffers from strong multicollinearity, as one could anticipate from observing Table 5. Thus, the individual coefficient estimates are not reliable. Finally, as expected all volatility indices in Panel (d) have a significant positive relation with the noise measure. As for equity variables, one can suspect issues with multicollinearity in the multiple regression in column (4). As seen in the lower right corner of Table 5, for instance the correlation between ∆VDAX and ∆VSTOXX is 96%. Nonetheless, as expected, all single estimated coefficients take on positive values, implying that positive changes in the equity volatility variables results in positive changes in German noise. While Hu et al. (2013) obtain higher explanatory power on similar variables, our main results show a high degree of resemblance. Liquidity variables such as the risk variables and the equity volatility variables have high explanatory power, whereas the curve variables do not have strong explanatory power for the monthly changes in the noise measure.

42 7 RESULTS 7.4 Hedge Fund Returns

7.4 Hedge Fund Returns Motivated by studies by Gromb & Vayanos (2002), Mitchell et al. (2007), Brunner- meier & Pedersen (2009), Duffie (2010), Ashcraft et al. (2011), and in particular Fleckenstein (2013), we explore if the theory of slow-moving capital may help explain some of the noise on the German government bond market. We continue this section by first presenting the intuition behind the theory of slow-moving capital as an explanation for observed mispricings. We then translate this intuition into a testable hypothesis and present our results. Textbook arbitrage is often defined as a trading strategy that is riskless and does not require any capital. Real world arbitrage on the other hand, is almost always risky and requires capital.32 For example, if a hedge fund buys a bond, it can use the bond as collateral and borrow against it. However, the collateral value of the bond will not equal its cash price. The difference between the two values, i.e. the margin, must be financed by the hedge fund itself. Similarly, short selling also requires a margin. If the market moves against the position of a hedge fund, it has to post additional capital. If the hedge fund lacks the sufficient capital to do this, it is forced to close its position at a loss. Consequently, arbitrageurs may not be able to exploit arbitrage opportunities if they are capital constrained. The total capital of a hedge fund consists of equity capital provided by its investors, and possible debt financing. Thus, the amount of capital available to a hedge fund may be affected by withdraws by investors and general funding liquidity, including margin requirements. Shleifer & Vishny (1997) show how mispricings may persist due to the principal- agent problem that arises between hedge fund managers and investors. Suppose that a hedge fund identifies an arbitrage opportunity and takes a position. Unfor- tunately, due to the idiosyncratic nature of asset prices, the market moves against the position of the hedge fund. Further suppose that investors infer past hedge fund returns as indicators of future performance. In this case, rational investors reallocate their capital from the hedge fund that is loosing money. Apart from the reduction in capital due to the loss in the position, the hedge fund is now loosing capital due to investor redemptions. If the mispricing continue to grow, and/or investors withdraw their capital, the hedge fund eventually reaches its capital con- straint and is forced to liquidate its position. In effect, as mispricings grow and ex ante profits increase, hedge funds may be too constrained to trade prices back to fundamental values. The very risk that funding constraints become binding makes the hedge fund want to have a "safety buffer". This limits hedge funds provision of market liquidity. Moreover, available capital to specialized arbitrageurs matter. Mitchell et al.

32For examples on common ﬁxed income arbitrage strategies see e.g. Duarte et al. (2007).

43 7 RESULTS 7.4 Hedge Fund Returns

(2007) argue that an initial shock to hedge fund wealth caused persistent mispricings on the convertible bond market. An initial negative shock to the net asset value of convertible bond hedge funds caused investor redemptions and subsequent fire sale of assets. Hedge funds specialized in convertible arbitrage became capital constrained and were not able to exploit increased mispricings. Other hedge funds that were not specialized in convertible arbitrage and not capital constrained did not exploit the increased mispircings at an early stage or in a large aggregate scale. Fleckenstein (2013) who study mispricings between Treasury bonds and inflation-swapped Treasury-Protected Securities also obtain similar results. Re- turns of hedge fund categories considered more likely to take advantage of inflation- linked bond mispricings forecast subsequent changes in mispricings, while returns of hedge funds considered not likely to take advantage of inflation-linked bond mispricings did not forecast subsequent changes in mispricings. Our hypothesis is founded on the literature and intuition above, which we generally refer to as the theory of slow-moving capital; changes in the capital of arbitrageurs should forecast subsequent changes in the noise. More specifically, a negative shock to the return of fixed income arbitrageurs at t − 1 constrain fixed income arbitrageurs, and mispricings measured as noise at time t widens. Conversely, a positive shock to the return of fixed income hedge funds at time t − 1 increase their available capital available to exploit mispricings, consequently lowering the noise at time t. It is worth emphasizing that we do not expect results on the same magnitude as Mitchell et al. (2007) or Fleckenstein (2013). Our measure of mispricings, the noise, consists of nominal German government bonds, a safe haven market, with one of the highest credit ratings and best liquidity in the world. Moreover, the noise measure excludes the most illiquid maturity segments, and it is cleaned from the largest pricing errors. To empirically test the slow-moving capital predictions, we regress month-end noise on one-month lagged monthly returns on the HFRX Hedge Fund indices.33 In order to determine if available arbitrage capital to specific types of arbitrageurs matter, it is important that the hedge fund index as best as possible represent a specific subset of hedge fund strategies. According to Hedge Fund Research Inc., the HFRX methodology "have the highest statistical likelihood of producing a return series that is most representative of the reference universe of strategies".34 HFR classifies hedge fund strategies into four main categories: Equity Hedge, Event Driven, Macro, and Relative Value. HFRX Equity Hedge Index maintain long and short positions in primarily equity and equity derivatives. HFRX Event

33We also perform the regressions using changes in mean and median monthly noise and obtain similar results. 34We also conduct the regressions with the alternative HFRI indices, the main results are similar.

44 7 RESULTS 7.4 Hedge Fund Returns

Driven Index maintain positions in companies currently or prospectively involved in corporate transactions including, but not limited to, mergers, restructurings, financial distress, tender offers, shareholder buybacks, debt exchanges, security issuance or other capital structure adjustments. HFRX Macro/CTA Index consist of a broad range of strategies, in which the investment process is predicated on movements in underlying economic variables, and the impact these have on equity, fixed income, hard currency and commodity markets. HFRX Relative Value Arbitrage Index consist of investment strategies where the investment thesis is predicated on realization of a value discrepancy in the relationship between multiple securities. These four main categories are further divided into more specialized sub-strategies. As pointed out by Hu et al. (2013, p. 2342), fixed income hedge funds focused on relative value trading are the most likely agents to arbitrage away dispersions along the yield curve. Consequently, from the descriptions of the main HFRX indices, we expect the Relative Value index to have the most significant effect on subsequent changes in the noise. The HFRX Relative Value index is further subdivided into nine sub-indices: Energy Infrastructure, FI - Asset Backed Securities, FI - Convertible Arbitrage, FI - Corporate, FI - Sovereign, Real Estate, Multi Strategy, Volatility, Alternative.35 Out of the Relative Value sub-indices, we expect the fixed income indices to have the most significant effect on subsequent changes in the noise.

7.4.1 Hedge Fund Regression Analysis The results of the regressions using the estimated German noise and one-month lagged returns of the four main HFRX indices are shown in Panel (a) in Table 12 in Appendix B. The results are in line with our expectations. The Equity Hedge, Event Driven and Relative Value coefficients are negative, indicating that a decrease in hedge fund capital is followed by a subsequent increase in the noise measure. Conversely, an increase in hedge fund capital is generally associated with a decrease in the noise measure. The only significant hedge fund index is the HFRX Relative Value index. This indicates that changes in capital to this type of hedge funds are particularly important in explaining subsequent changes in the noise measure. However, both the coefficient and explanatory power of the HFRX Relative Value index are low. An 1% increase in capital available to Relative Value hedge funds is expected to decrease the noise by 0.062 basis points the following month. Considering the noise monthly standard deviation of 1.079, the impact of hedge fund returns on the noise is limited. We further investigate the impact of the one-month lagged returns of the sub- indices within the Relative Value index. The results are presented in Table 16.

35Index descriptions are available at https://www.hedgefundresearch.com/ hfrx-indices-index-descriptions.

45 7 RESULTS 7.4 Hedge Fund Returns

In the individual regressions, the coefficients of all Relative Value sub-indices are negative and the HFRX FI - Convertible Arbitrage index is significant on a 1% level. In the multiple regression on the one-month lagged returns of the Relative Value sub-indices, the FI - Corporate Bond index is significant on a 1% level and the FI Convertible Arbitrage index is significant on a 5% level. As with the main HFRX indices, the coefficients and explanatory powers are low. This indicates that the fixed income indices within the Relative Value category do drive changes in the noise, but to a small degree. In summary, changes in hedge fund capital predict subsequent changes in the German noise measure and our results confirm the the predictions of the theory of slow-moving capital. We also regress changes in the noise on contemporaneous hedge fund returns. In the contemporaneous regressions, all main category indices are significant and the R2 values are generally higher compared to the one-month lagged regressions. In particular, for the Relative Value index, the R2 value increases from 1.47% to 6.05%. Within the Relative Value subcategory all the fixed income indices are significant on a 1% level. In the multiple regressions the Macro index becomes significant. We show that, in general, changes in the capital hedge funds specializing in fixed income arbitrage predict changes in the noise while other hedge fund specializations do not. Our results provide support to the slow-moving capital theory. It is interesting that the German government bond market, a safe haven market with one of the highest credit ratings and liquidity in the world, is affected by capital changes of global hedge fund indices.

7.4.2 Regressing Changes in US Noise on Hedge Fund Returns We download the US noise measure by Hu et al. (2013) from 2000 through 2014 and conduct the same hedge fund regressions as on the German noise measure.36 Since we conduct the German regression on a longer sample he comparison between the regression results on the German and US noise should not be interpreted as formal comparisons, but rather as an indication of diﬀerences.37 The summary statistics from regressing end-of-month changes in the US noise on the one-month lagged and contemporaneous hedge fund returns are presented in Panel (b) in Table 11 and 12, respectively. Regressing the one-month lagged hedge fund returns on changes in the US noise notably increase the coeﬃcients and t-

36The US noise estimated by Hu et al. (2013) is available at http://www.mit.edu/ ~junpan/. 37Shortening the sample period on the German regressions result in inconsequential diﬀerences in the regression results. Rather than shortening our German sample period to end in 2014, we decide present the longer sample period that runs until the end of 2015. This give us a longer sample period for our main interest, the German market.

46 7 RESULTS 7.4 Hedge Fund Returns

values of hedge fund coefficients compared to the regressions on the German noise. Among the main indices the coefficient on the HFRX Relative Value index increase from -0.031 (t-value of -2.00) on the German noise to -0.122 (t-value of -6.86) on the US noise. Among the Relative-Value sub-indices the FI Convertible Arbitrage, FI Corporate and FI Sovereign indices all have t-values exceeding 2. The coefficient and t-value of the FI Corporate index increase from -0.007 and -0.27 (on German noise) to -0.132 and -3.17 (on US noise) respectively. The individual one-month lagged hedge fund returns explain between 1 to 8 percent of the monthly changes in the US noise, and up to 0.04 to 3% on the German noise. The coefficients and explanatory power of the contemporaneous regressions are substantially higher on the US noise. Together, the monthly returns of the Relative Value sub-indices explain 60% of the variation in the monthly changes of the US noise, compared to only 13% on the German noise. In particular the coefficients on the fixed income Relative Value sub-indices are notably higher. The regressions on the US noise give further confirm our hypothesis that hedge fund returns forecast subsequent changes in the noise.

47 7 RESULTS 7.5 Sensitivity Analysis

7.5 Sensitivity Analysis To assess the robustness of the regression results we run the optimization process and estimate the noise once again, this time with the DE settings of nP = 50 and 38 nG = 200. In this section we will refer to the original DE settings as "refined" and the new settings as "simple". The left panel in Figure 15 show the resulting noise series when running our optimization procedure with the simple and refined DE settings. The right panel of Figure 15 illustrates the basis point difference between the noise produced by the simple settings and the noise based on the refined DE settings. The differences in the resulting series are notable. The refined DE settings on average produce a noise measure 0.798 basis points (standard deviation 0.864) lower than the simple settings, the maximum difference is 6.864 basis points. The simple DE settings produce a lower noise measure, and lower objective function value, on only 3.87% of the total 4028 estimated days. Given that the simple settings produce a lower noise on a given day, the difference between the noise produced by the refined and simple algorithm settings is small, on average 0.072 basis points.

Simple settings

Reﬁned settings 6 8 4 6 2 4 Noise (bps) Diﬀerence in Noise (bps) 0 2

2000 2005 2010 2015 2000 2005 2010 2015

Figure 15: Comparison of noise with different DE settings. The left panel plots the noise measure estimated based on different DE settings. The right panel illustrates the difference between the simple and refined noise series. The refined DE settings are nP = 1000 and nG = 600. whereas the simple settings are nP = 50 and nG = 200. For both estimations, F = 0.30 and CR = 0.99.

To investigate if the diﬀerence in the noise series have implications on our

38 The settings used in our original calibration are nP = 1000 and nG = 600.

48 7 RESULTS 7.5 Sensitivity Analysis

regression results, we regress the noise produced by the simple algorithm settings on the one-month lagged hedge fund returns. The results from the regression on the main and the Relative Value HFRX indices are presented in Appendix A, Table 17 and Table 19 respectively. Comparing the results of the regressions on the noise produced by the simple settings to the regression results on the refined settings in Panel (a) in Table 12 and 14, we note that the results are rather different. Among the main indices, the Relative Value Index is no longer significant. With the exception of the Equity Hedge index, all R2 values on the individual regressions decrease notably. Also in the regressions on the one-month lagged Relative Value sub-indices the R2 values decrease. Moreover, the FI - Arbitrage index goes from being insignificant to significant. Using the two sets of German noise, we compare the relationship of them to the US noise estimated by Hu et al. (2013). Figure 16 shows the scatterplot for each case.

nP = 50 and nG = 200 nP = 1000 and nG = 600 10 10 8 8 6 6 4 4 German Noise (bps) German Noise (bps) 2 2 0 0

0 5 10 15 20 0 5 10 15 20

HPW US Noise (bps) HPW US Noise (bps)

Figure 16: Linear regressions on US noise. Linear regressions on the daily US noise estimated by Hu et al. (2013). The left graph is versus the daily estimated German noise with nP = 50, nG = 200, CR = 0.90 and F = 0.50. The right graph is against the daily estimated German noise with nP = 1000, nG = 600, CR = 0.99 and F = 0.30. HPW is an abbreviation for Hu, Pan and Wang.

Comparing the (untabulated) regression results visualized in Figure 16, the refined noise measure exhibits a stronger relationship to the US noise estimated by Hu et al. (2013). The correlation coefficients ρ are 0.55 and 0.69, or equivalently in single OLS regressions the R2 values are 30.69% and 47.17%. The slope coefficients are in both cases significant on the 1% level, but the estimate goes from

49 7 RESULTS 7.5 Sensitivity Analysis

1.034 to 1.672. This sensitivity assessment stresses the importance of thoughtful calibration, to avoid the noise being too "noisy". Finally, recall the results in Table 6 in Section 7. In Panel (a), the noise is regressed on curve variables. Running the same (untabulated) regressions with the noise estimated with the simple DE settings, the variables ∆Level and ∆Bond V. are no longer significant. On the contrary, the variable ∆Curve is significant on the 5% level with a positive coefficient estimate, implying as the curve increases the noise does too. This may be interpreted as when the NSS parameters β are not carefully enough calibrated, shifts in the yield curve will not sufficiently be captured, thus amplifying the noise due to poor curve fitting. This, once again, strengthens the validity of our calibration outcome for the noise with improved DE estimation settings. In Panel (b), the default spread, which was the single most significant variable in Table 6, is now insignificant with a t-statistic equal to 0.44, and the correlation coefficient ρ drops from 23% to 2%.

50 8 CONCLUSIONS

8 Conclusions

By calibrating the term structure using the Nelson-Siegel-Svensson (NSS) model, we constructed a daily noise measure using 16 years of market data for German government bonds. The parameters of the Nelson-Siegel-Svensson were calibrated by minimizing duration-weighted differences in observed market and implied model bond prices using the Differential Evolution (DE) algorithm. The noise measure was constructed as a root mean square error (RMSE) of daily cross-sectional differences between market yields and model implied yields. The estimated series of noise serve as a proxy for market liquidity, as when markets become distressed the capital available to arbitrageurs becomes scarce. Consequently, the dispersions between market and model yields are increasing, as mispricings are not traded away to the same extent as in normal market conditions. By taking monthly changes in the estimated noise, and examining its relationship with variables related to funding liquidity, we find a positive and significant relation with the EURIBOR EONIA spread and the spread between a sovereign bond index and BBB corporate bond index. Overall, the results imply that the noise measure is not only noise, but rather contains valuable information about the liquidity condition of the market. Furthering strengthening this finding, the results show significant relationship with equity volatility variables, including not only on the German market but also European and US equity markets. Inspired by the preceding literature of slow-moving capital, we explored how the estimated noise relates to lagged hedge fund returns, a proxy for capital available to arbitrageurs, expected to exploit potential dispersions in asset prices. The obtained results clearly indicate a relationship, supporting the theory of slow- moving capital, and thus in line with previously documented findings. Accordingly, in this study we do not only confirm that noise, measured as dispersions between market and model yields, provides insights to liquidity conditions, but we further empirically establish the importance of market frictions within the field of asset pricing.

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56 APPENDIX A Graphs

A Graphs

2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016

175 15

82 150

125 Bunds30y

60 Bunds10y Bobls5y 100

53 Schaetze2y Bubill12m 75 23 47 75 74 41 72 72 Bubill9m 69 64 Bubill6m 60 60 60 59 58 56 57 12 56 54 54 54 52 52 53 53 50 50 50 51 51 Bubill3m 48 32 46 46 45 44 41 8 41 39 20 39 36 37 32 33 33 33 33 25 20

12 20 10 10 11 10 16 8 9 6 7 6 8 8 0

Figure 17: Issuance of Federal securities. Issuance of Federal securities excluding inﬂation- linked and USD denominated issues. Source: Deutsche Finanzagentur (2016).

57 APPENDIX A Graphs

β0 β1 30 15 20 10 10 5 0 -10 0

2000 2002 2004 2006 2008 2010 2012 2014 2016 2000 2002 2004 2006 2008 2010 2012 2014 2016

β2 β3 30 30 20 20 10 10 0 0 -10 -10 -20 -20 -30 -30

2000 2002 2004 2006 2008 2010 2012 2014 2016 2000 2002 2004 2006 2008 2010 2012 2014 2016

τ1 τ2 30 30 25 25 20 20 15 15 10 10 5 5 0 0

2000 2002 2004 2006 2008 2010 2012 2014 2016 2000 2002 2004 2006 2008 2010 2012 2014 2016

Figure 18: Parameter estimates. Evolution of the Nelson-Siegel-Svensson parameters during the estimation range. Red horizontal lines represents the lower and upper limits used is the estimations. The settings used in the DE estimation are nP = 1000, nG = 600, F = 0.3 and CR = 0.99.

58 APPENDIX A Graphs

Date: 2002-12-30 Date: 2014-12-29 6 BKO BKO DBR DBR OBL OBL 4 4 2 2 0 0 Market - Model (Price) Market - Model (Price) -2 -2 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 5 5 0 0 -5 -5 -10 -10 Market - Model (bps) Market - Model (bps) -15

0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 Years to maturity Years to maturity

Figure 19: Example of pricing errors around year-ends. Example of pricing errors around year-ends. Illustrated are the prices and yields reported by the market minus the prices and yields implied by the estimated NSS parameters. The bond types are emphasized in the figures by different symbols and colors. For instance, in the lower panel outlining the difference in yield space, bonds above the dotted line could be considered "cheap" and bonds below as "expensive", without considering other factors such as funding costs, investor preferences etcetera.

59 APPENDIX A Graphs

EU US Other 2000 - 2001

5.0 B01 4.0 3.0 2.0 1.0 A00 A01 0.0

15Jan00 15Mar00 15Maj00 15Jul00 15Sep00 15Nov00 15Jan01 15Mar01 15Maj01 15Jul01 15Sep01 15Nov01

2002 - 2003 4.0 B02 B03 3.0 2.0 1.0

A02 C02 A03 0.0

15Jan02 15Mar02 15Maj02 15Jul02 15Sep02 15Nov02 15Jan03 15Mar03 15Maj03 15Jul03 15Sep03 15Nov03 0

2004 - 2005 3.0 B05 2.0 1.0

A04 A05 0.0

15Jan04 15Mar04 15Maj04 15Jul04 15Sep04 15Nov04 15Jan05 15Mar05 15Maj05 15Jul05 15Sep05 15Nov05

2006 - 2007 3.0 B07 2.0 1.0

A06 A07 0.0

15Jan06 15Mar06 15Maj06 15Jul06 15Sep06 15Nov06 15Jan07 15Mar07 15Maj07 15Jul07 15Sep07 15Nov07

Figure 20: Noise measure for the period 2008-2011. Noise measure for the years 2008- 2011 with selected events. Events are described in Table 20. Red highlighting means US related event, blue EU event and green an event elsewhere.

60 APPENDIX A Graphs

EU US Other 2008 - 2009

8.0 B08 D08 F08 7.0 6.0 5.0 4.0 3.0 2.0 1.0 A08 C08 E08 G08 A09 0.0

15Jan08 15Mar08 15Maj08 15Jul08 15Sep08 15Nov08 15Jan09 15Mar09 15Maj09 15Jul09 15Sep09 15Nov09

2010 - 2011

6.0 B10 D10 F10 H10 J10 B11 D11 F11 H11 5.0 4.0 3.0 2.0 1.0 A10 C10 E10 G10 I10 A11 C11 E11 G11 I11 0.0

15Jan10 15Mar10 15Maj10 15Jul10 15Sep10 15Nov10 15Jan11 15Mar11 15Maj11 15Jul11 15Sep11 15Nov11 0

2012 - 2013 3.0 B12 B13 D13 2.0 1.0

A12 A13 C13 0.0

15Jan12 15Mar12 15Maj12 15Jul12 15Sep12 15Nov12 15Jan13 15Mar13 15Maj13 15Jul13 15Sep13 15Nov13

2014 - 2015 3.0 B15 D15 F15 2.0 1.0

A14 A15 C15 E15 G15 0.0

15Jan14 15Mar14 15Maj14 15Jul14 15Sep14 15Nov14 15Jan15 15Mar15 15Maj15 15Jul15 15Sep15 15Nov15

Figure 21: Noise measure for the period 2012-2015. Noise measure for the years period 2012-2015 with selected events. Events are described in Table 20. Red highlighting means US related event, blue EU event and green an event elsewhere.

61 APPENDIX B Tables

B Tables

Table 7: Summary statistics. Statistics for German bond characteristics are based on the 1Y-10Y sample, used in calculating the noise measure.

Period # Bonds (3M - 10Y) # Bonds (1Y - 10Y) Coupon Size Maturity Age Duration (%) (EUR bn) (Years) (Years) (Years)

Mean

2000 45.8 39.3 5.45 9.15 4.47 8.06 3.89 2001 45.3 39.5 5.33 10.30 4.21 8.00 3.69 2002 45.0 36.3 5.10 12.18 4.19 7.91 3.68 2003 43.2 36.1 4.92 14.13 4.09 7.93 3.63 2004 44.3 34.4 4.53 16.13 4.15 7.90 3.70 2005 41.5 32.2 4.18 17.92 4.24 7.82 3.80 2006 40.1 32.3 4.00 18.55 4.15 7.82 3.73 2007 40.1 31.1 4.02 19.17 4.19 7.85 3.76 2008 38.6 30.1 4.08 19.78 4.27 7.79 3.83 2009 38.9 29.6 3.81 20.09 4.33 7.74 3.90 2010 40.4 31.0 3.39 20.25 4.36 7.64 3.97 2011 39.3 32.9 3.08 19.74 4.48 7.65 4.11 2012 40.8 34.8 2.72 19.12 4.53 7.66 4.21 2013 42.8 36.8 2.41 18.65 4.51 7.64 4.24 2014 45.6 38.9 2.20 18.00 4.66 8.26 4.38 2015 45.7 39.0 2.03 17.87 4.61 8.32 4.38 Total 42.3 34.6 3.85 16.72 4.35 7.89 3.94

Median

2000 5.25 7.67 4.03 9.96 3.59 2001 5.00 8.18 3.74 9.98 3.44 2002 5.00 11.00 3.56 9.98 3.25 2003 5.00 14.00 3.55 9.98 3.28 2004 4.50 14.90 3.74 9.98 3.51 2005 4.25 18.00 3.66 9.98 3.45 2006 4.00 18.00 3.45 9.98 3.22 2007 3.75 20.00 3.58 10.00 3.35 2008 4.00 20.00 3.72 10.00 3.48 2009 4.00 20.00 3.78 10.00 3.57 2010 3.75 20.00 3.88 10.00 3.69 2011 3.50 19.00 3.93 10.04 3.76 2012 2.75 19.00 3.93 10.09 3.76 2013 2.50 18.00 3.92 10.03 3.77 2014 2.00 18.00 4.11 9.99 3.95 2015 1.75 18.00 4.13 9.99 3.98 Total 4.00 17.00 3.80 9.98 3.57

Standard deviation

2000 1.34 4.04 2.49 2.57 1.96 2001 1.19 4.87 2.50 2.73 2.00 2002 1.12 5.52 2.48 2.89 1.98 2003 1.19 5.86 2.48 2.89 2.00 2004 1.12 5.38 2.52 2.92 2.06 2005 1.02 4.76 2.52 2.99 2.08 2006 0.90 4.58 2.57 3.00 2.14 2007 0.70 4.37 2.61 3.02 2.17 2008 0.69 3.86 2.62 3.06 2.17 2009 0.92 3.77 2.61 3.07 2.18 2010 1.13 3.52 2.61 3.07 2.22 2011 1.08 3.25 2.67 3.04 2.30 2012 1.18 3.29 2.68 3.02 2.38 2013 1.22 3.24 2.68 3.00 2.42 2014 1.42 3.35 2.73 4.59 2.46 2015 1.45 3.28 2.64 4.57 2.44 Total 1.59 5.55 2.59 3.24 2.21

62 APPENDIX B Tables

Table 8: Summary statistics for Schäthe. Summary statistics for used Schäthe bonds.

N Coupon (%) Volume (EUR bln) Duration Year Mean Max Min Mean Max Min Mean Max Min Mean Max Min 2000 3.1 4.0 2.0 3.449 4.500 3.000 6.2 7.0 5.1 0.88 1.86 0.25 2001 3.5 5.0 2.0 4.356 4.750 3.000 8.2 10.0 6.0 1.22 1.86 0.25 2002 5.9 7.0 4.0 4.055 4.750 3.250 10.2 12.0 8.0 1.11 1.87 0.25 2003 6.6 7.0 6.0 3.371 4.250 2.000 11.9 14.0 7.0 1.07 1.88 0.25 2004 6.6 7.0 6.0 2.577 4.000 2.000 12.9 17.0 7.0 1.06 1.89 0.25 2005 6.6 7.0 6.0 2.399 2.750 2.000 14.4 17.0 7.0 1.07 1.90 0.25 2006 6.7 7.0 6.0 2.544 3.500 2.000 14.7 17.0 7.0 1.06 1.88 0.25 2007 6.7 7.0 6.0 3.334 4.500 2.000 14.6 16.0 7.0 1.05 1.86 0.25 2008 6.6 7.0 6.0 3.903 4.750 3.000 14.3 15.0 7.0 1.05 1.87 0.25 2009 6.7 7.0 6.0 3.141 4.750 1.250 14.6 16.0 7.0 1.07 1.96 0.25 2010 6.9 7.0 6.0 1.533 4.750 0.500 16.2 19.0 11.0 1.11 1.99 0.25 2011 6.9 7.0 6.0 1.047 1.750 0.250 17.6 19.0 7.0 1.11 1.98 0.25 2012 6.9 7.0 6.0 0.802 1.750 0.000 16.3 19.0 9.0 1.10 1.98 0.25 2013 6.9 7.0 6.0 0.195 1.750 0.000 14.9 17.0 10.0 1.11 1.98 0.25 2014 6.9 7.0 6.0 0.119 0.250 0.000 14.1 15.0 9.0 1.11 2.00 0.25 2015 6.9 7.0 6.0 0.086 0.250 0.000 13.4 15.0 8.0 1.11 2.00 0.25 2016 7.0 7.0 7.0 0.036 0.250 0.000 13.2 14.0 8.0 1.15 1.94 0.35

Table 9: Summary statistics for Bobl. Summary statistics for used Bobl bonds.

N Coupon (%) Volume (EUR bln) Duration Year Mean Max Min Mean Max Min Mean Max Min Mean Max Min 2000 15.9 17.0 15.0 4.497 5.880 3.250 6.9 15.0 5.1 2.39 4.59 0.25 2001 15.1 16.0 14.0 4.419 5.000 3.250 7.9 15.0 6.0 2.22 4.57 0.25 2002 13.2 14.0 12.0 4.339 5.000 3.250 9.3 20.0 6.0 2.18 4.60 0.25 2003 11.7 13.0 11.0 4.259 5.000 3.000 11.7 20.0 6.0 2.37 4.61 0.25 2004 11.6 12.0 11.0 4.207 5.000 3.000 13.5 20.0 6.0 2.37 4.69 0.25 2005 10.2 11.0 10.0 3.908 5.000 2.500 15.8 20.0 6.0 2.44 4.71 0.25 2006 10.1 11.0 10.0 3.555 4.500 2.500 16.6 20.0 8.0 2.41 4.62 0.25 2007 10.1 11.0 10.0 3.449 4.500 2.500 16.5 20.0 6.0 2.33 4.56 0.25 2008 9.4 10.0 9.0 3.482 4.250 2.500 16.9 19.0 7.0 2.46 4.61 0.25 2009 9.4 10.0 9.0 3.442 4.250 2.250 17.0 19.0 7.0 2.46 4.72 0.25 2010 10.2 11.0 9.0 3.204 4.250 1.750 17.0 19.0 6.0 2.65 4.78 0.25 2011 11.2 12.0 10.0 2.904 4.250 1.250 16.8 19.0 6.0 2.73 4.82 0.25 2012 12.2 13.0 11.0 2.304 4.250 0.500 16.8 19.0 8.0 2.74 4.94 0.25 2013 13.2 14.0 12.0 1.660 4.000 0.250 16.8 19.0 9.0 2.69 4.95 0.25 2014 14.1 15.0 13.0 1.297 2.750 0.250 16.7 19.0 9.0 2.57 4.97 0.25 2015 13.7 14.0 13.0 0.882 2.750 0.000 16.6 20.0 9.0 2.54 5.15 0.25 2016 12.1 13.0 12.0 0.572 2.750 0.000 17.0 20.0 16.0 2.54 4.75 0.25

Table 10: Summary statistics for Bund. Summary statistics for used Bund bonds.

N Coupon (%) Volume (EUR bln) Duration Year Mean Max Min Mean Max Min Mean Max Min Mean Max Min 2000 26.9 28.0 26.0 6.332 9.000 3.750 10.1 20.0 5.1 4.33 8.03 0.25 2001 26.1 27.0 25.0 5.999 8.380 3.750 11.4 24.0 5.1 4.26 8.10 0.25 2002 25.9 27.0 25.0 5.788 8.000 3.750 12.5 27.0 5.1 4.04 8.05 0.25 2003 24.1 25.0 23.0 5.518 7.500 3.750 14.5 27.0 5.1 4.16 8.40 0.25 2004 24.1 25.0 23.0 5.279 7.500 3.750 16.2 27.0 5.1 4.06 8.52 0.25 2005 22.8 24.0 21.0 4.899 6.880 3.250 18.0 27.0 6.1 4.17 8.70 0.25 2006 21.3 22.0 21.0 4.570 6.250 3.250 19.8 27.0 7.2 4.35 8.62 0.25 2007 21.5 22.0 21.0 4.399 6.000 3.250 20.3 27.0 7.0 4.23 8.51 0.25 2008 20.8 22.0 20.0 4.327 5.380 3.250 21.2 27.0 7.0 4.30 8.58 0.25 2009 19.8 21.0 19.0 4.286 5.380 3.250 22.3 27.0 6.0 4.50 8.69 0.25 2010 19.8 21.0 19.0 4.062 5.250 2.250 22.6 27.0 6.0 4.65 9.00 0.25 2011 20.8 22.0 20.0 3.754 5.000 2.250 21.9 27.0 6.0 4.85 9.01 0.25 2012 21.7 22.0 21.0 3.433 5.000 1.500 21.2 27.0 6.0 5.03 9.27 0.25 2013 22.6 24.0 22.0 3.124 6.250 1.500 20.7 25.0 10.0 5.14 9.33 0.25 2014 24.6 25.0 24.0 2.992 6.250 1.000 19.7 25.0 10.0 5.30 9.42 0.25 2015 25.2 26.0 25.0 2.733 6.250 0.500 19.5 24.0 9.0 5.27 9.76 0.26 2016 25.0 25.0 25.0 2.650 6.250 0.500 19.6 24.0 10.2 5.12 9.18 0.42

63 Table 11: Main hedge fund indices regressions with no lag. Regressions are based on Newey-West standard errors. t-values are within brackets. Panel (a) contains regressions for the calculated German noise, in monthly changes based on month end values. Panel (b) contains regressions for the US noise estimated by Hu et al. (2013). The very right column in each panel represent a multiple regression including all the individual indices. HFRXEH is Equity Hedge Index, HFRXED is Event Driven Index, HFRXM is Macro/CTA Index and HFRXRVA is Relative Value Arbitrage Index.

(a) Monthly change in German noise (b) Monthly change in US noise

Dependent variable: Dependent variable: ∆German Noise ∆US Noise (1) (2) (3) (4) (5) (1) (2) (3) (4) (5) ∗∗∗ ∗∗ HFRXEHt −0.054 −0.066 HFRXEHt −0.249 −0.033 (−3.283) (-1.569) (−2.529) (−0.360) ∗∗∗ ∗∗∗ HFRXEDt −0.051 0.052 HFRXEDt −0.309 −0.028 (−2.653) (1.381) (−2.787) (−0.338) ∗ ∗∗∗ HFRXMt 0.027 0.046 HFRXMt −0.036 0.004 (1.828) (2.836) (−1.019) (0.068) ∗∗∗ ∗ ∗∗∗ ∗∗∗ HFRXRVAt −0.062 −0.051 HFRXRVAt −0.295 −0.254 (−4.387) (-1.855) (−6.285) (−3.010)

Constant 0.004 0.008 0.003 0.005 −0.001 Constant −0.0002 0.043 −0.005 0.010 0.013 (0.141) (0.264) (0.102) (0.187) (−0.037) (−0.002) (0.426) (−0.040) (0.135) (0.152)

Observations 132 132 132 132 132 Observations 120 120 120 120 120 R2 (%) 4.18 2.69 0.92 6.05 8.83 R2 (%) 24.83 25.81 0.43 37.73 38.15 Adjusted R2 (%) 5.96 Adjusted R2 (%) 36.00 F Statistic 3.074∗∗ F Statistic 17.736∗∗∗ Degrees of Freedom (df =1; 130) (df =1; 130) (df =1; 130) (df =1; 130) (df =4; 127) Degrees of Freedom (df =1; 118) (df =1; 118) (df =1; 118) (df =1; 118) (df =4; 115) Note: ∗p <0.1; ∗∗p <0.05; ∗∗∗p <0.01 Note: ∗p <0.1; ∗∗p <0.05; ∗∗∗p <0.01 Table 12: Main hedge fund indices regressions with one lag. Regressions are based on Newey-West standard errors. t-values are within brackets. Panel (a) contains regressions for the calculated German noise, in monthly changes based on month end values. Panel (b) contains regressions for the US noise estimated by Hu et al. (2013). The very right column in each panel represent a multiple regression including all the individual indices. HFRXEH is Equity Hedge Index, HFRXED is Event Driven Index, HFRXM is Macro/CTA Index and HFRXRVA is Relative Value Arbitrage Index.

(a) Monthly change in German noise (b) Monthly change in US noise

Dependent variable: Dependent variable: ∆German Noise ∆US Noise (1) (2) (3) (4) (5) (1) (2) (3) (4) (5) HFRXEHt−1 −0.010 0.035 HFRXEHt−1 −0.096 −0.011 (−0.518) (0.802) (−1.071) (−0.270)

HFRXEDt−1 −0.032 −0.048 HFRXEDt−1 −0.134 −0.053 (−1.452) (−1.093) (−1.229) (−0.492)

HFRXMt−1 0.037 0.035 HFRXMt−1 0.037 0.059 (1.339) (1.025) (0.495) (1.171) ∗∗ ∗∗∗ ∗ HFRXRVAt−1 −0.031 −0.028 HFRXRVAt−1 −0.122 −0.086 (−2.000) (−1.168) (−6.864) (−1.828)

Constant 0.006 0.009 0.004 0.007 0.010 Constant 0.005 0.025 0.000 0.011 0.015 (0.155) (0.264) (0.123) (0.211) (0.288) (0.044) (0.225) (0.002) (0.320) (0.215)

Observations 131 131 131 131 131 Observations 120 120 120 120 120 R2 (%) 0.15 1.07 1.69 1.47 4.02 R2 (%) 3.74 4.90 0.45 6.48 7.65 Adjusted R2 (%) 0.97 Adjusted R2 (%) 4.41 F Statistic 1.318 F Statistic 17.736∗∗∗ Degrees of Freedom (df =1; 129) (df =1; 129) (df =1; 129) (df =1; 129) (df =4; 126) Degrees of Freedom (df =1; 118) (df =1; 118) (df =1; 118) (df =1; 118) (df =4; 115) Note: ∗p <0.1; ∗∗p <0.05; ∗∗∗p <0.01 Note: ∗p <0.1; ∗∗p <0.05; ∗∗∗p <0.01 Table 13: Relative value hedge fund sub-indices regressions with no lag and German noise. Regressions are based on Newey-West standard errors. t-values are within brackets. The very right column represent a multiple regression including all the individual indices. Explanatory variables are sub-indices of the HFRX Relative Value Arbitrage Index (HFRXRVA). HFRXEINF is Energy Infrastructure Index, HFRXFAB is FI-Asset Backed Index, HFRXCA is Convertible Arbitrage Index, HFRXFCO is FI- Corporate Index, HFRXFSV is Sovereign Index, HFRXREAL is Real Estate Index, HFRXRVMS is Multi-Strategy Index, HFRXVOL is Volatility Index and HFRXYA is Yield Alternative Index.

Dependent variable: ∆ German Noise (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) ∗∗ ∗ HFRXEINFt −0.029 0.060 (−2.147) (1.818) ∗∗∗ ∗∗∗ HFRXFABt −0.165 −0.124 (−4.746) (−2.654) ∗∗∗ HFRXCAt −0.031 −0.002 (−6.897) (−0.101) ∗∗∗ HFRXFCOt −0.083 −0.005 (−4.931) (−0.075) ∗∗∗ HFRXFSVt −0.041 0.024 (−3.441) (0.973) ∗∗ HFRXREALt −0.040 0.047 (−2.380) (1.408) ∗∗∗ ∗∗ HFRXRVMSt −0.089 −0.080 (−5.251) (−1.991)

HFRXVOLt −0.030 −0.008 (−0.786) (−0.199) ∗∗ ∗ HFRXYAt −0.045 −0.094 (−2.500) (−1.695)

Constant 0.020 0.156∗∗∗ −0.005 0.032 0.015 0.012 0.042 0.013 0.025 0.150∗∗∗ (0.591) (4.386) (−0.254) (1.149) (0.454) (0.394) (1.346) (0.330) (0.737) (3.130)

Observations 132 132 132 132 132 132 132 132 132 132 R2 (%) 2.98 7.47 5.59 6.52 2.45 2.07 6.72 0.54 4.93 12.72 Adjusted R2 (%) 6.28 F Statistic 1.976∗∗ Degrees of Freedom (df =1; 130) (df =1; 130) (df =1; 130) (df =1; 130) (df =1; 130) (df =1; 130) (df =1; 130) (df =1; 130) (df =1; 130) (df =9; 122) Note: ∗p <0.1; ∗∗p <0.05; ∗∗∗p <0.01 Table 14: Relative value hedge fund sub-indices regressions with one lag and German noise. Regressions are based on Newey-West standard errors. t-values are within brackets. The very right column represent a multiple regression including all the individual indices. Explanatory variables are sub-indices of the HFRX Relative Value Arbitrage Index (HFRXRVA). HFRXEINF is Energy Infrastructure Index, HFRXFAB is FI-Asset Backed Index, HFRXCA is Convertible Arbitrage Index, HFRXFCO is FI- Corporate Index, HFRXFSV is Sovereign Index, HFRXREAL is Real Estate Index, HFRXRVMS is Multi-Strategy Index, HFRXVOL is Volatility Index and HFRXYA is Yield Alternative Index.

Dependent variable: ∆ German Noise (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) ∗ HFRXEINFt−1 −0.024 0.006 (−1.744) (0.104)

HFRXFABt−1 −0.084 −0.066 (−1.591) (−0.962) ∗∗∗ ∗∗ HFRXCAt−1 −0.011 −0.053 (−2.735) (−2.388) ∗∗∗ HFRXFCOt−1 −0.007 0.193 (−0.271) (3.481)

HFRXFSVt−1 −0.017 0.021 (−1.502) (0.607) ∗ HFRXREALt−1 −0.045 −0.024 (−1.740) (−0.666)

HFRXRVMSt−1 −0.018 −0.018 (−0.906) (−0.329) ∗ HFRXVOLt−1 −0.070 −0.077 (−1.544) (−1.942) ∗ HFRXYAt−1 −0.038 −0.050 (−1.870) (−0.624) Constant 0.021 0.085 0.003 0.008 0.010 0.015 0.014 0.027 0.024 0.038 (0.532) (1.405) (0.072) (0.209) (0.363) (0.407) (0.434) (0.659) (0.784) (0.447)

Observations 131 131 131 131 131 131 131 131 131 131 R2 (%) 1.98 1.95 0.75 0.04 0.40 2.56 0.28 2.93 3.34 12.10 Adjusted R2 (%) 5.57 F Statistic 1.851∗ Degrees of Freedom (df =1; 129) (df =1; 129) (df =1; 129) (df =1; 129) (df =1; 129) (df =1; 129) (df =1; 129) (df =1; 129) (df =1; 129) (df =9; 121) Note: ∗p <0.1; ∗∗p <0.05; ∗∗∗p <0.01 Table 15: Relative value hedge fund sub-indices regressions with no lag and US noise. Regressions are based on Newey- West standard errors. t-values are within brackets. The noise based on US data and estimated by Hu et al. (2013). The very right column represent a multiple regression including all the individual indices. Explanatory variables are sub-indices of the HFRX Relative Value Arbitrage Index (HFRXRVA). HFRXEINF is Energy Infrastructure Index, HFRXFAB is FI-Asset Backed Index, HFRXCA is Convertible Arbitrage Index, HFRXFCO is FI-Corporate Index, HFRXFSV is Sovereign Index, HFRXREAL is Real Estate Index, HFRXRVMS is Multi-Strategy Index, HFRXVOL is Volatility Index and HFRXYA is Yield Alternative Index.

Dependent variable: ∆ US Noise (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) ∗∗∗ HFRXEINFt −0.152 −0.014 (−3.708) (−0.262) ∗∗ ∗ HFRXFABt −0.563 −0.185 (−2.310) (−1.772) ∗∗∗ ∗∗ HFRXCAt −0.174 −0.126 (−27.655) (−2.541) ∗∗∗ HFRXFCOt −0.353 0.222 (−3.804) (1.290) ∗∗∗ ∗∗∗ HFRXFSVt −0.345 −0.211 (−6.818) (−3.247)

HFRXREALt −0.171 0.002 (−1.561) (0.030) ∗∗∗ ∗∗ HFRXRVMSt −0.372 −0.111 (−2.623) (−2.586)

HFRXVOLt −0.091 −0.008 (−0.809) (−0.175) ∗∗∗ HFRXYAt −0.156 0.034 (−3.051) (0.522)

Constant 0.128 0.546∗ −0.061 0.123 0.089 0.033 0.178 0.019 0.094 0.159 (1.429) (1.920) (−1.417) (1.080) (1.126) (0.268) (1.327) (0.117) (1.073) (1.269)

Observations 120 120 120 120 120 120 120 120 120 120 R2 (%) 15.96 23.77 50.20 33.87 49.75 10.41 31.36 1.37 14.46 60.78 Adjusted R2 (%) 57.57 F Statistic 18.943∗∗∗ Degrees of Freedom (df =1; 118) (df =1; 118) (df =1; 118) (df =1; 118) (df =1; 118) (df =1; 118) (df =1; 118) (df =1; 118) (df =1; 118) (df =9; 110) Note: ∗p <0.1; ∗∗p <0.05; ∗∗∗p <0.01 Table 16: Relative value hedge fund sub-indices regressions with one lag and US noise. Regressions are based on Newey- West standard errors. t-values are within brackets. The noise based on US data and estimated by Hu et al. (2013). The very right column represent a multiple regression including all the individual indices. Explanatory variables are sub-indices of the HFRX Relative Value Arbitrage Index (HFRXRVA). HFRXEINF is Energy Infrastructure Index, HFRXFAB is FI-Asset Backed Index, HFRXCA is Convertible Arbitrage Index, HFRXFCO is FI-Corporate Index, HFRXFSV is Sovereign Index, HFRXREAL is Real Estate Index, HFRXRVMS is Multi-Strategy Index, HFRXVOL is Volatility Index and HFRXYA is Yield Alternative Index.

Dependent variable: ∆ US Noise (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) HFRXEINFt−1 −0.090 0.064 (−0.996) (0.603) ∗ HFRXFABt−1 −0.211 −0.095 (−1.674) (−0.840) ∗∗∗ HFRXCAt−1 −0.062 −0.033 (−6.071) (−0.349) ∗∗∗ HFRXFCOt−1 −0.132 0.078 (−3.174) (0.293) ∗∗ HFRXFSVt−1 −0.131 −0.072 (−2.211) (−0.549)

HFRXREALt−1 −0.067 0.097 (−1.332) (1.080) ∗ HFRXRVMSt−1 −0.112 0.013 (−1.800) (0.087)

HFRXVOLt−1 −0.078 −0.033 (−0.589) (−0.589)

HFRXYAt−1 −0.118 −0.188 (−1.266) (−1.278) Constant 0.084 0.210 −0.017 0.052 0.039 0.017 0.060 0.025 0.079 0.121 (0.485) (1.093) (−0.231) (0.707) (0.380) (0.152) (0.595) (0.165) (0.570) (0.677)

Observations 119 119 119 119 119 119 119 119 119 119 R2 (%) 5.96 3.37 6.33 4.72 7.15 1.63 2.78 1.02 8.27 12.99 Adjusted R2 (%) 5.80 F Statistic 1.808∗ Degrees of Freedom (df =1; 117) (df =1; 117) (df =1; 117) (df =1; 117) (df =1; 117) (df =1; 117) (df =1; 117) (df =1; 117) (df =1; 117) (df =9; 109) Note: ∗p <0.1; ∗∗p <0.05; ∗∗∗p <0.01 APPENDIX B Tables

Table 17: Main hedge fund indices regressions with one lag and German noise estimated with alternative DE settings. Regressions are based on Newey-West standard errors. t-values are within brackets. The very right column represent a multiple regression including all the individual indices. HFRXEH is Equity Hedge Index, HFRXED is Event Driven Index, HFRXM is Macro/CTA Index and HFRXRVA is Relative Value Arbitrage Index. The alternative Diﬀerential Evolution (DE) settings used for calibration are nP = 50, nG = 200, CR = 0.90 and F = 0.50.

Dependent variable: ∆German Noise estimated with alternative DE settings (1) (2) (3) (4) (5)

HFRXEHt−1 −0.050 −0.140 (−1.103) (−1.238)

HFRXEDt−1 −0.018 0.152 (−0.350) (1.574)

HFRXMt−1 0.030 0.057 (0.585) (0.979)

HFRXRVAt−1 −0.048 −0.045 (−1.590) (−0.757)

Constant 0.009 0.010 0.007 0.011 −0.005 (0.192) (0.182) (0.150) (0.195) (−0.103)

Observations 131 131 131 131 131 R2 0.67 0.06 0.21 0.66 2.39 Adjusted R2 −0.71 F Statistic 0.771 Degrees of Freedom (df =1; 129) (df =1; 129) (df =1; 129) (df =1; 129) (df =4; 126)

Note: ∗p <0.1; ∗∗p <0.05; ∗∗∗p <0.01

Table 18: Summary statistics for ﬁltered noise. Yearly statistics over ﬁltered German noise measure. Days with abnormal spikes in the noise measure have been removed. The last row contains summary statistics for the full sample period.

Year Mean Median St. Deviation 2000 3.458 3.377 0.598 2001 2.729 2.702 0.317 2002 2.576 2.531 0.300 2003 1.826 1.817 0.229 2004 1.217 1.087 0.326 2005 0.810 0.789 0.203 2006 0.960 0.950 0.164 2007 1.245 1.128 0.548 2008 3.865 3.809 1.060 2009 3.234 2.991 0.818 2010 3.185 3.018 0.787 2011 2.589 2.600 0.588 2012 1.829 1.767 0.308 2013 1.505 1.507 0.326 2014 1.446 1.483 0.324 2015 1.012 0.961 0.277 Total 2.093 1.870 1.092

70 Table 19: Relative value hedge fund sub-indices regressions with one lag and German noise estimated with alternative DE settings. Regressions are based on Newey-West standard errors. t-values are within brackets. The very right column represent a multiple regression including all the individual indices. Explanatory variables are sub-indices of the HFRX Relative Value Arbitrage Index (HFRXRVA). HFRXEINF is Energy Infrastructure Index, HFRXFAB is FI-Asset Backed Index, HFRXCA is Convertible Arbitrage Index, HFRXFCO is FI-Corporate Index, HFRXFSV is Sovereign Index, HFRXREAL is Real Estate Index, HFRXRVMS is Multi-Strategy Index, HFRXVOL is Volatility Index and HFRXYA is Yield Alternative Index. The alternative Diﬀerential Evolution (DE) settings used for calibration are nP = 50, nG = 200, CR = 0.90 and F = 0.50.

Dependent variable: ∆ German Noise estimated with alternative DE settings (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) HFRXEINFt−1 0.016 0.129 (0.379) (1.575) ∗∗ ∗∗ HFRXFABt−1 −0.185 −0.244 (−2.533) (−2.355) ∗∗∗ HFRXCAt−1 −0.019 −0.027 (−2.761) (−0.661) ∗ HFRXFCOt−1 −0.021 0.213 (−0.514) (1.795)

HFRXFSVt−1 −0.030 −0.003 (−1.347) (−0.062)

HFRXREALt−1 −0.018 0.050 (−0.451) (0.713) ∗∗ HFRXRVMSt−1 −0.056 −0.187 (−1.011) (−2.142)

HFRXVOLt−1 −0.086 −0.101 (−1.290) (−1.608)

HFRXYAt−1 −0.002 −0.116 (−0.040) (−1.506)

Constant −0.001 0.181∗∗ 0.003 0.016 0.017 0.012 0.034 0.035 0.009 0.238∗ (−0.021) (2.261) (0.062) (0.262) (0.296) (0.195) (0.553) (0.534) (0.160) (1.792)

Observations 131 131 131 131 131 131 131 131 131 131 R2 0.16 1.76 0.39 0.08 0.24 0.08 0.49 0.84 0.00 5.85 Adjusted R2 −1.15 F Statistic 0.836 Degrees of Freedom (df =1; 129) (df =1; 129) (df =1; 129) (df =1; 129) (df =1; 129) (df =1; 129) (df =1; 129) (df =1; 129) (df =1; 129) (df =9; 121) Note: ∗p <0.1; ∗∗p <0.05; ∗∗∗p <0.01 APPENDIX B Tables

Table 20: Short descriptions of selected events. A selection of events that occurred during the full estimation period. The category column shows whether the event is mainly thought as an EU, US or other region event. The ID column shows an abbreviation used in Figure 20 and Figure 21, found in Appendix A.

Date Category ID Description 2000-12-06 OTHER A00 IMF emergency loan to Turkey 2001-09-11 US A01 Terrorist attack 9/11 2001-11-14 EU B01 Volatility in German bond market amid bond yield decline 2002-03-06 EU A02 Volatility in German bond market amid high euro growth expectations and speculation on ECB 2002-05-07 EU B02 Bond yields fall on unexpectedly high German factory orders and high global growth 2002-11-27 EU C02 Stock market recovery and speculation on ECB rate cut 2003-03-19 US A03 US coalittion invasion of Iraq 2003-07-15 US B03 FED chef Alan Greenspan speach 2004-03-03 EUR A04 Volatility prior to ECB policy meeting on 4th Mars 2005-04-16 EU A05 Bundesbank President resigns after dispute with government 2005-04-21 EU B05 New Bundesbank President appointed 2006-02-07 EU A06 Talks of Euronext tie-up with Deutche Borse stall, ECB decides not to publish expected rate 2007-07-19 US A07 Firesale in CDOs as creditors sell USD 1 bn of assets siezed from Bear Stearn funds 2007-08-09 EU B07 Euro interbank dires up. ECB inject EUR 95 bn into the banking system 2008-02-17 EU A08 Nothern Rock is taken into government ownership 2008-03-11 US B08 The Federal Reserve Board announces the creation of the Term Securities Lending Facility 2008-03-14 US C08 The Federal Reserve Board approves the financing arrangement announced by JP Morgan Chase 2008-04-04 US D08 Banks borrowing from the FEDs discount window surges 2008-09-15 US E08 Lehman bankruptcy 2008-09-16 US F08 The Federal Reserve Board authorizes the Federal Reserve Bank of New York to lend up to $85 2008-10-06 OTHER G08 Global equity sell-off, "worst falls since Black Monday 1987" 2009-02-17 US A09 American Recovery and Reinvestment Act of 2009 2010-02-23 EU A10 Fitch downgrade Greece’s four major banks to triple-b 2010-03-25 EU B10 Euro-zone and IMF agrees on how to bail out Greece 2010-04-27 EU C10 S&P downgrade Greek debt to junk (BB+) and Portugese by two notches (from A+ to A-), day after Spain is downgraded by S&P with neg. outlook (from AA+ to AA) 2010-05-02 EU D10 Greece: Bailout #1. EUR 110 bn. A week after EU agrees on EUR 750 bn bailout plan for failing euro-zone economies. 2010-05-28 EU E10 Spain lose its AAA rating after S&P downgrade to AA+ 2010-06-14 EU F10 Moody’s becomes the second rating agency to cut Greece to jun cut to Ba1 (junk) 2010-07-23 EU G10 Results of bank stress test released. 7 out of 91 banks would need to raise more capital 2010-09-30 EU H10 Cost of bailing out Irish banks may cost up to EUR 50 bn, more than 1/3 of 2009 national income 2010-11-21 EU I10 Ireland becomes second country to apply for EU/IMF bailout 2010-11-28 EU J10 EU approves EUR 85 bn Irish resque and outlines permanent system to resolve debt crisis in Europe 2011-01-12 EU A11 Merkel: "Germany will do whatever is necessary so that the euro remains stable". Two days after Fitch cut Greece to BB+. Now all three major credit rating institutions have junk rating on Greece. 2011-03-23 EU B11 The Portuguese government collapses. From Mars 15 to April 5 Portugal is downgraded 6 times by Moody’s, S&P and Fitch. 2011-04-06 EU C11 Portgual becomes the third country to ask for bailout 2011-05-16 EU D11 Portgual’s bailout is approved 2011-07-05 EU E11 Portgual’s credit rating is cut to Ba2 (junk) by Moody’s. A week after Ireland’s credit rating is cut to Ba1 (junk) by Moody’s 2011-07-21 EU F11 Greece: Draft of Bailout #2. EUR 109 bn. Other EU cuntries promise to fund Greece until it is able to issue bonds again 2011-08-05 EU G11 US loses AAA rating 2011-10-27 EU H11 50% haircut on Greek bonds 2011-12-07 EU I11 S&P warn 15 Eurozone members of possible downgrades 2012-02-21 EU A12 Greece: Final agreement of bailout #2 2012-07-26 EU B12 Draghi’s "whatever it takes" speech 2013-05-12 US A13 Taptering: FOMC minutes and Berkanke speak to congress about tapering 2013-06-19 US B13 Tapering: FOMC meeting and press conference 2013-10-01 US C13 US government shutdown 2013-12-18 US D13 Tapering: FED scaling down asset purchases from USD 85 bn to 75 bn 2014-06-05 EU A14 ECB cuts rates, introduces TLTRO and "intensifies work" on the upcoming ABS purchase program (ABSPP) 2015-01-15 EU A15 SNB remove EUR cap 2015-03-09 EU B15 ECB QE of EUR 60 bn monthly purchases about to begin 2015-03-25 EU C15 Cypruss bailout. EUR 10 bn 2015-06-11 EU D15 IMF states that no progress in discussions with Greece. Negotiators withdrawn due to major differences in key areas 2015-06-30 EU E15 Greece misses deadline pf IMF payment 2015-07-13 EU F15 Greek bailout #3 2015-08-11 OTHER G15 China devalues the CNY