Trading Strategies In Bond Markets

Contents

Abbreviations Symbols... iii

1 Preface... 1

2 An Overview of Bonds... 3

2.1 Bond Prices and The Yield Curve . . . 5

2.2 Forces Driving Bond Prices . . . 8

3 Arbitrage-free Yield Modelling 12

3.1 General Concepts . . . 12

3.2 The Heath-Jarrow-Morton Model . . . 13

3.3 The Time Discrete Heath-Jarrow-Morton Model . . . 16

3.4 Empirical Quality of the Time Discrete HJM Model . . . 25

4 Parametric Yield Curve Estimation ... 32

4.1 The Dynamic Nelson-Siegel Method . . . 32

4.2 Shortcomings of the Dynamic Nelson-Siegel Method . . . 34

4.3 Empirical Quality of the DNS Method . . . 41

5 Portfolio Optimization... 50

5.1 Optimal Portfolio with Regards to Conditional Value at Risk . . . 50

5.2 Efficient Implementation . . . 56

5.3 Example of Optimal Portfolios . . . 60

6 Conclusion and Outlook... 63

A Appendix ...64

A.1 Coupon Reinvestment at Yield to Maturity . . . 64

A.2 Information on Yield Data and Data Sources . . . 65

A.3 Source Code for Discrete HJM . . . 68

A.4 Statements for Chapter 4 . . . 71

A.5 Statements for Chapter 5 . . . 72

1 Preface

Bonds are important instruments of funding, investment and hedging. Bonds are commonly used for conservative investment and in order to control the risk of a portfolio. However, bonds are actively traded which makes it important to find strategies on how to trade them.

Essential for understanding bond prices and their development is the yield curve. The yield curve itself and effects of other quantities on the yield curve are thoroughly studied topics in financial economics with a vast literature field concerned with it. Among introductory literature are the books Choudhry (2010) and Choudhry (2001). Also mathematics is concerned with the yield curve. The main focus of mathematics literature is on how to describe the yield curve mathematically. Comprehensive books on this matter are Brigo and Mercurio (2006) and Filipovic (2009) which explain a variety of yield curve models and estimation methods. Amidst are two popular approaches: The Heath-Jarrow-Morton (HJM) Model developed in Heath et al. (1992), and the parametric Nelson-Siegel (NS) Method established in the work of Nelson and Siegel (1987).

The Heath-Jarrow-Morton Model is a sophisticated term structure model which directly models the entire forward rate curve. A variety of other term-structure models are special cases of the general Heath-Jarrow-Morton Model. Teichmann and Wuethrich (2013) recently presented a time-discrete version of the Heath-Jarrow-Morton Model. Within the here presented timediscrete framework consistent yield curve prediction is possible.

The parametric Nelson-Siegel Method is used as the basis for yield curve estimation by many central banks. The German Bundesbank is one of them as explained in Schich (1997). The flaws of the Nelson-Siegel Method are criticized by Bjoerk and Christensen (1997) and Filipovic (1999). Among the criticism is that arbitrage is not suspended directly. Emanating from the Nelson-Siegel Method is the Dynamic-Nelson-Siegel (DNS) Method formulated by Diebold et al. (2005) describing the development of Nelson-Siegel curves in time. This method is augmented in Christensen et al. (2007) to an arbitrage-free version by fitting DNS into the class of exponentialaffine term structure. The class of exponential-affine term structure was firstly describe by Duffie and Kan (1996). Despite the criticism the Dynamic Nelson-Siegel method has the reputation of being reliable in yield forecasting as assessed by Diebold and Rudebusch (2013), Coroneo et al. (2008), and many others.

With the HJM model or the DNS method forming expectations about the future yield curve, the theory of portfolio optimization can be consulted to establish trading strategies. Portfolio Optimization dates back to Modern Portfolio Theory by Markowitz (1952) where the variance is used as a measure of risk. The more recent approach by Rockafellar and Uryasev (2000) suggests instead conditional value at risk as a measure of risk. Including various constraints for realistic portfolios is extensively studied in Krokhmal et al. (2002). In the work of Alexander et al. (2006) and Iyengar and Ma (2013) methods of efficient implementation of the conditional value at risk portfolio selection problem are assessed. The latter is based on an iterative algorithm developed in Nesterov (2005).

This work contributes to the research by theoretically and empirically investigating the accuracy of the time-discrete Heath-Jarrow-Morton and the Dynamic Nelson-Siegel yield expectations in view of bond trading. Both approaches are compared to each other with regards to how the bond market works. Additionally, the HJM and DNS yield expectations are incorporated into the portfolio optimization with respect to conditional value at risk. The resulting portfolio selection problem is also assessed empirically. The empirical part includes observations of the US Treasury yield curve from January 2009 to November 2013.

This text requires some essential mathematical statements which are used here without further explanation. Among these are the Ito-Doeblin Formula, the Fundamental Theorems of Asset Pricing, and the Karush-Kuhn-Tucker Theorem. Standard notation is used if not indicated differently. Matrix entries equal to zero are left blank for notational convenience, and the expression almost safe is spared for equalities.

The remainder of this work is organized as follows. Chapter two describes the working of the bond market. Different basic variables related to bonds are reviewed. The yield curve is discussed in greater detail. Moreover, the forces which drive bond prices are examined closer, examining how recent financial, economic and political events are taken into account. The relationship between bond prices and economic indicators is assessed. Uncertainties that surround any investment action are discussed specifically.

Chapter three focuses on arbitrage-free modelling of bond prices. The findings of the preceding chapter are given a mathematical framework. After that, the Heath-Jarrow-Morton yield model is introduced. A time-discrete version of the Heath-Jarrow-Morton model is presented in order to predict future yields. Empirical performance of the time-discrete HJM model is further investigated.

In the fourth chapter the Dynamic Nelson-Siegel method is considered to fit and forecast the yield curve. The shortcomings of such a parametric model are mentioned, and a possible resolution is provided by extending it to an arbitrage-free version. It is further empirically assessed how well the Dynamic Nelson-Siegel method fits to observed yields of US Treasuries. In addition, the results are compared to those of the HJM-based approach.

The fifth chapter demonstrates how a portfolio of bonds can be optimized if the expectations on return are determined by one of the methods introduced in the two earlier chapters. As a measure of risk the conditional value at risk is chosen. Methods to efficiently solve the resulting optimization problem are provided. Lastly, an extensive example is given in which the arbitragefree model is run against the parametric method in order to optimize a portfolio consisting of US Treasuries.

Chapter six concludes the work. In the appendix a common misconception regarding coupon reinvestment at yield to maturity is addressed. Also information on yield data is given. Essential parts of C++ source code for implementation of the time-discrete HJM model are appended, as well. At the end, technical statements are provided without proof but with literature references to proofs. Generally, proofs are given only of statements which are essential for practical matter or understanding.

2 An Overview of Bonds

This chapter provides a general overview of the bond market and explains important parameters. At first, the question what are bonds is briefly discussed. Then two sections follow in this chapter. In the first section bond prices and related factors are explained. Special attention is aimed towards the yield curve which is identified to be a crucial feature in bond trading. Throughout the second section the forces which influence the yield curve and their connection to risk are examined closer. The main sources for this chapter are Term-Structure Models, A Graduate Course by Filipovic (2009), The Bond and Money Markets: Strategy, Trading, Analysis by Choudhry (2010), and The Bond and Money Markets: Trading, Strategy, Analysis by Choudhry (2001).

A bond is a tradable financial instrument of debt. In order to borrow money, bonds are issued and sold. Bonds are different from loans because bonds can be traded at the stock exchange, and so creditor is the general public. Whoever buys a bond and holds it, is the creditor and called the holder of the bond. The initial price of the bond is the amount of money lent by the holder to the issuer. The issuer of the bond redeems his debt to the holder of the bond by making regular payments. These payments and its dates are defined at the issuance of the bond. It can be differentiated between coupon payments and the payment of the face value. [1] Coupon payments are payments made repeatedly after a certain time period which is called coupon period. With the last coupon the face value is paid. At this time the bond expires because the debt has been redeemed appropriately. The lifetime of the bond from issuance to redemption date is called maturity. Figure 2.1 visualizes a bond’s cash flow. The bond is sold to the holder for the bond price; that is a cash flow from the holder to the issuer at the amount of the bond price. Throughout the bond’s lifetime, the holder receives regular coupon payments. That are cash flows from the issuer to the holder at the amount of the coupon. Finally, the face value is paid at maturity which is a cash flow from the issuer to the holder at the amount of the face value.

[Figures and tables are omitted from this preview.]

Figure 2.1: Example of a bond’s cash flow from the perspective of the holder. The arrows in red (i.e. from left to right) and blue (i.e. from right to left) symbolize the outgoing and incoming payments, respectively. The green arrow (i.e. pointing downwards) depicts the time order in which the cash flows take place. The cash flows are characterized beneath.

The coupon payments are commonly given as a fixed percentage of the face value. Coupon payment dates occur usually semi-annually or annually. If the amount of the coupon payment is the same at the end of each coupon period like in the example of Figure 2.1 then the bond is called a fixed-coupon bond. If the coupon can vary over time instead, the bond is called floating coupon bond or index-linked coupon bond. Such bonds are linked to underlyings and move accordingly with them. Floating coupons are linked to a reference interest rate like the London Interbank Offered Rate (LIBOR). Index-linked bonds have an economic indicator as underlying. This underlying can be for example the Consumer Price Index (CPI). Floating coupon bonds and index-linked coupon bonds are supposed to protect the holder of price depreciation through interest rate changes and inflation, respectively. Therefore, floats and linkers are popular in environments with unstable price changes like in emerging countries. [2] For so-called amortised bonds the payment of the face value is split into more than one payment. Bonds can have embedded options which leave the possibility to either holder or issuer to demand back or to redeem the debt at an earlier time than maturity. A bond with a call option allows the issuer to redeem the bond early. Such an option protects the issuer of paying higher interest on his debt than elsewhere in the market.

A put option, in contrast to the call, gives the holder the possibility to demand an early redemption. There are also convertible bonds. Those include the option to convert the bond into a specified amount of shares. Bonds can also be issued with a warrant which gives the holder the possibility to buy more of the bond at a later time. Asset-backed securities are bundled loans which are then resold as bonds. Also bonds without any maturity exist. Those undated bonds are never redeemed. Finally, bonds without any coupon payments just pay the face value at maturity. Such bonds are called zero coupon bonds. They are rarely found in the market but they are an important theoretical instrument of analysis. This will become clear in the later chapters.

Bonds are issued by governments in order to finance their debt. After the issuance the bonds are sold in an auction on the primary market to banks. The banks again offer the bonds on the secondary market to the general public. The secondary market is where bonds are traded. That is mostly through over the counter (OTC) transactions but also on stock exchanges. Also agencies issue bonds on request of the government. For instance on request of the United States government the Government National Mortgage Association (GNMA), the Federal National Mortgage Association (FNMA) and the Federal Home Loan Mortgage Corporation (FHLMC) issue bonds. Other issuers of bonds are institutions like the World Bank, or even a private person can issue bonds.

Companies issue bonds to raise capital without selling more shares. Holding a bond of a company means lending money to this company which is opposed to holding shares of a company since holding shares means owning a percentage of the company. Correspondingly, in case of the company’s bankruptcy bond holders are compensated before shareholders. Thus, bonds bear a generally lower default risk than shares. [3] Also government bonds bear a default risk. Nevertheless, the default risk on government bonds is considered the lowest in the market since the central bank can usually print money to redeem any debt. After the bond is sold from the issuer to the holder, both have opposing interest in terms of the investment horizon. The issuer has a long term view because redeeming his debt after a short time period comes with a risk - the risk to miss out on investment opportunities.

On the other hand, the holder has a short term view as she is exposed to illiquidity during the lending period. Both parties are further put into jeopardy through changing interest rates. If interest rates rise but the coupon payments do not rise in the same manner then the holder might be better off investing in other assets. Yet, the issuer of a fixed coupon bond benefits from rising interests since he pays only the relative low coupon instead of paying high interest which would have been payable on a normal loan. Additional risk comes with inflation. Depreciating monetary value drives down the bond’s real rate of return. In other words, because of rising prices the purchasing power of the coupons and the face value decreases. This effect is advantageous for the bond issuer has he has to give up less purchasing power in order to pay the holder. And it is disadvantageous for the latter as she receives payments with a lower purchasing power than initially expected. Another uncertainty for the investor of corporate bonds is the possibility of credit rating upgrades or downgrades. An upgrade in credit rating of the issuing company enables the holder to sell the bond to a higher price than the original purchase price. Despite, the holder is not able to sell the bond price even for the original purchase price if the company’s credit rating is downgraded. Sometimes there also exists the risk of a thin market. That means that the bond holder has problems selling her issue of the bond for a reasonable price due to a low demand for that bond. Instead the bond can only be sold for a price much lower or it even cannot be sold at all. For most government bonds this kind of liquidity risk is not an issue. The following section engages in the calculation of a bond’s fair price. Besides that, the zerocoupon yield curve is explained.

2.1 Bond Prices and The Yield Curve

Bond prices and the yield curve are closely related to each other. In this section an overview of the terms bond price, yield and yield curve is provided.

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[1] The terms maturity value, nominal value, principle, redemption value, par value or par are often used synonymously.

[2] "Brazil, India, Turkey and Thailand together have issued $12.7 billion in inflation-linked bonds this year." Trivedi (2013).

[3] Some companies issue bond-like instruments whose holders are last creditors to be compensated in case of bankruptcy. A recent example is the default of participation certificates issued by Prokon Regenerative Energien GmbH.