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- Herausgeber: John Wiley & Sons
- Kategorie: Fachliteratur
- Serie: Bloomberg Professional
- Sprache: Englisch
As western governments issue increasing amounts of debt, the fixed income markets have never been more important. Yet the methods for analyzing these markets have failed to keep pace with recent developments, including the deterioration in the credit quality of many sovereign issuers. In Fixed Income Relative Value Analysis, Doug Huggins and Christian Schaller address this gap with a set of analytic tools for assessing value in the markets for government bonds, interest rate swaps, and related basis swaps, as well as associated futures and options.
Taking a practitioner’s point of view, the book presents the theory behind market analysis in connection with tools for finding and expressing trade ideas. The extensive use of actual market examples illustrates the ways these analytic tools can be applied in practice.
The book covers:
- Statistical models for quantitative market analysis, in particular mean reversion models and principal component analysis.
- An in-depth approach to understanding swap spreads in theory and in practice.
- A comprehensive discussion of the various basis swaps and their combinations.
- The incorporation of credit default swaps in yield curve analysis.
- A classification of option trades, with appropriate analysis tools for each category.
- Fitted curve techniques for identifying relative value among different bonds.
- A multi-factor delivery option model for bond future contracts.
Fixed Income Relative Value Analysis provides an insightful presentation of the relevant statistical and financial theories, a detailed set of statistical and financial tools derived from these theories, and a multitude of actual trades resulting from the application of these tools to the fixed income markets. As such, it’s an indispensable guide for relative value analysts, relative value traders, and portfolio managers for whom security selection and hedging are part of the investment process.
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Contents
Foreword
Chapter 1: Relative Value
The Concept of Relative Value
The Sources of Relative Value Opportunities
The Insights from Relative Value Analysis
The Applications of Relative Value Analysis
The Craft of Relative Value Analysis
Summary of Contents
Part I: Statistical Models
Chapter 2: Mean Reversion
What Is Mean Reversion and How Does It Help Us?
Diagnostics for Model Selection
Model Estimation
Calculating Conditional Expectations and Probability Densities
Calculating Conditional, Ex Ante Risk-Adjusted Returns
First Passage Times
A Practical Example Incorporating all the Ideas
Conclusion
Chapter 3: Principal Component Analysis
Introduction: Goal and Method
An Intuitive Approach toward PCA
Factor Models: General Structure and Definitions
PCA: Mathematics
PCA as Factor Model
Insight into Market Mechanisms through Interpretation of the Eigenvectors
Applying Eigenvector Interpretation in Different Markets
Decomposing Markets into Uncorrelated Factors
Embedding PCA in Trade Ideas
Appropriate Hedging
Analyzing the Exposure of Trading Positions and Investment Portfolios
Market Reconstruction and Forecasting
A Yield Curve Model Based on PCA
PCA as a Tool for Screening the Market for Trade Ideas
PCA as a Tool for Asset Selection
Example of a PCA-based Trade Idea
Problems and Pitfalls of PCA 1: Correlation between Factors during Subperiods
Problems and Pitfalls of PCA 2: Instability of Eigenvectors over Time
PCA as a Tool to Construct New Types of Trades
Part II: Financial Models
Chapter 4: Some Comments on Yield, Duration, and Convexity
Introduction
Some Brief Comments on the Yield of a Coupon-Paying Bond
A Brief Comment on Duration
A Common Misapplication of Convexity
Chapter 5: Bond Futures Contracts
Futures Price and Delivery Option
One-Factor Delivery Option Models
The Need for Multi-Factor Delivery Option Models
A Flexible Multi-Factor Delivery Option Model
Chapter 6: LIBOR, OIS Rates, and Repo Rates
Introduction
Brief Definitions
Differences between LIBOR and OIS Rates
Repo Rates in Greater Detail
Capital Treatment for Secured and Unsecured Loans
A Model Relating LIBOR and Repo Rates
Conclusions
Chapter 7: Intra-Currency Basis Swaps
Definition
Factors that Determine Pricing
Role as Building Blocks
Conclusion
Chapter 8: Theoretical Determinants of Swap Spreads
Old Approach: Looking at Default Risk of Swap Counterparties
The Modern Approach
Viewing the Swap as a Derivative
Insurance Properties of LIBOR–Repo Spreads
Other Practical Modeling Issues
The Subprime Crisis as a Brief Case Study
Conclusions
Chapter 9: Swap Spreads from an Empirical Perspective
Empirical Analysis of Swap Spreads
Conclusions
Chapter 10: Swap Spreads as Relative Value Indicators for Government Bonds
Introduction
Typical Use of Swap Spreads as a Relative Value Indicator for Government Bonds
Problems with the Use of Swap Spreads as Relative Value Indicators for Government Bonds
Attempts to Solve these Problems
Conclusions
Chapter 11: Fitted Bond Curves
Introduction
Framework of Analysis
Specifying a Function for Discount Factors
Weights
Example: Fitting the German Bund Curve
Statistical Analysis of Rich/Cheap Figures
Applications
Conclusions
Chapter 12: A Brief Comment on Interpolated Swap Spreads
Chapter 13: Cross-Currency Basis Swaps
Introduction
Definitions
Economic Functions
Pricing the CCBS
Conclusion
Chapter 14: Relative Values of Bonds Denominated in Different Currencies
Introduction
Calculating USD LIBOR Swap Spreads for Foreign Bonds
Rich/Cheap Analysis through Fitted Curves for Bonds Denominated in Different Currencies
Separate Yield Curves in Respective Currencies
The Equilibrium between ASW and CCBS Markets
The Equilibrium for Bunds (Low-Risk Bonds)
The ASW Model Revisited
The Equilibrium in Case of JGBs (Risky Bonds)
Conclusion
Chapter 15: Credit Default Swaps
Introduction
Structure of a CDS
Two Different Pricing Approaches toward CDS: Pricing of CDS versus Other CDS or versus Bonds
A PCA on the CDS Curve
A PCA on the EUR Sovereign CDS Universe
A PCA on CDS-Adjusted Bond Yields
Pitfalls
Conclusion
Chapter 16: USD Asset Swap Spreads versus Credit Default Swaps
Introduction
General Concept of No-Arbitrage Models
Credit Information in the CDS
Credit Information in the (USD) ASW of Bonds
The No-Arbitrage Relationship in Practice
The ASW Model Revisited Once More
Applying the General Concept to an EMU Model
Conclusions
Chapter 17: Options
Introduction
A Brief Review of Option Pricing Theory
Classification of Option Trades
Option Trade Type 1: Single Underlying
Option Trade Type 1: Two or More Underlyings
Option Trade Type 2: Single Underlying
Option Trade Type 2: Two or More Underlyings
Option Trade Type 3: Factor Model for the Vega Sector
Pitfalls of Option Trades of Type 3
Conclusion: Summary of Option Trade Types and Their Different Exposure
Chapter 18: Relative Value in a Broader Perspective
Introduction
The Macroeconomic Role of Relative Value Analysis and Trading
Arbitrageurs and Politicians
The Misrepresentation of Arbitrage by Politicians
Conclusion: Political Implications of Relative Value
Bibliography
Index
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Library of Congress Cataloging-in-Publication Data
Huggins, Doug, 1965–
Fixed income relative value analysis : a practitioner’s guide to the theory, tools, and trades / Doug Huggins and Christian Schaller.
1 online resource.
Includes bibliographical references and index.
Description based on print version record and CIP data provided by publisher; resource not viewed.
ISBN 978-1-118-47720-5 (hbk) — ISBN 978-1-118-47721-2 (ebk)
ISBN 978-1-118-47722-9 (ebk) — ISBN 978-1-118-47719-9 (ebk)
1. Fixed-income securities. 2. Securities—Valuation. I. Schaller, Christian, 1971– II. Title.
HG4650
332.63′2044–dc23
2013009914
A catalogue record for this book is available from the British Library.
ISBN 978-1-118-47719-9 (hbk) ISBN 978-1-118-47720-5 (ebk) ISBN 978-1-118-74197-9 (ebk)
ISBN 978-1-118-47722-9 (ebk) ISBN 978-1-118-47721-2 (ebk)
Relative Value: a Practitioner’s Guide
I remember very clearly the beginnings of the Relative Value Group at Deutsche Bank. The year was 1995. I was one of a small group of Research and Sales professionals who had recently arrived at Deutsche Bank. We became convinced that significant opportunities existed to apply relative value concepts to fixed income instruments in a way that was highly interesting for sophisticated clients. We realized that by analyzing separately the opportunities and risks of certain fixed income products, we could help our clients achieve the performance they aimed for while mitigating credit, market and liquidity risk. Our goal was simple: to help our clients achieve the best possible risk–reward equation.
We soon realized that we could apply these principles more widely across our client base in fixed income. For example, some of our clients held clear beliefs on areas of value in the market, but were seeking new ways to invest which reflected those beliefs. These were clients who provided important liquidity to the markets in which they operated, and this provided us with another important insight: as relative value addressed irrational differences between the prices of related instruments, we saw markets become more transparent, more liquid, and more efficient. Unquestionably, the science of relative value, and the transparency it brings to relationships between the prices of different instruments, has contributed to the growth of derivatives and other financial products which reduce market risk.
These were years in which Deutsche Bank was building up a world-leading markets platform, and as a comparable new entrant in many areas, we needed to innovate to prosper. Relative value disciplines formed a core part of our intellectual capital. Relative value provided us with a way to reduce risk and spot opportunities between different instruments, both within and across asset classes, and thus to help our clients perform better for their investors. Relative value gave us a systematic way to address the fundamental question: what’s expensive and what’s cheap? That discipline contributed greatly as we advised clients on asset allocation in their portfolios, and gave us valuable insights about how to deploy our own resources: capital, technology and people. Already, relative value at Deutsche Bank had evolved far beyond its origins as a method of identifying pricing inefficiencies in fixed income instruments. It gave us a framework for a much wider range of portfolio and business decisions.
The financial crisis of 2008 and early 2009 was a defining period in the development of relative value analysis. Under conditions of extreme market stress and acute shortages of liquidity, we saw the ‘conventional’ relationships between the prices of related securities break down. Put simply: the normal rules ceased to apply. The risk of sovereign default, suddenly much more apparent, profoundly impacted the prices of government debt and the derivatives related to it. This posed a major challenge for our clients and for the sound functioning of financial markets on which the global economy depends. But this extremely difficult period also brought us fundamental insights. Our experience with Long-Term Capital Management, and the Russian and Asian crises, warned us that at times of significant market stress, the conventional ‘rules’ governing relationships between assets cease to function and for us, this was a clear signal to reduce our balance sheet and risk exposures. Perhaps most significantly of all: as market conditions stabilized and liquidity returned, well-funded investors were able to invest in good-quality assets at very favorable prices. Relative value analysis was able to guide us toward many opportunities for us to create value for our clients.
In retrospect, the relative value perspective was invaluable to us throughout the most difficult months of 2008 and 2009. This perspective helped us protect ourselves better during the crisis, and take advantage of opportunities faster as stability returned. For Deutsche Bank in 2009, the ability to identify pricing anomalies enabled us to spot investment opportunities for our clients right across the fixed income spectrum. Relative value brings clarity to complex products, and helps us understand market behavior. We know now how important that is. There is no doubt that a deeper understanding and use of relative value disciplines across the world’s markets would have helped the financial industry navigate through the financial crisis and contribute to the stabilization of financial markets which was the first step toward global economic recovery.
It’s therefore no surprise that relative value has become increasingly important in post-crisis markets. It has evolved into a way of comparing prices and valuing the different risk elements across a broad range of asset classes. It gives market participants a method for arriving at a deeper understanding of new instruments as they appear. This added visibility contributes to market transparency, which in turn gives investors confidence. This encourages liquidity across a wide range of market areas, and makes pricing more efficient – which benefits issuers and investors alike. In emerging or immature markets, relative value disciplines can contribute to a ‘virtuous circle’ of improved transparency, liquidity and pricing efficiency. This lays the ground for healthy and sustainable growth in a wide diversity of markets – from the funding of governments and large corporations, to the pricing of a wide range of essential commodities. In other words, relative value disciplines make a host of real-world decisions clearer and easier to make. Deep, diverse, well-functioning and well-trusted markets have never been more important for the real economy than they are today.
Relative value thinking has played a significant role in my professional life. At its outset a way to serve sophisticated clients in fixed income markets, relative value gave me a methodology for identifying investment opportunities across a wide range of asset classes and added to my understanding of how markets behave. We can turn that knowledge to the advantage of our clients, and help financial markets do their job for the wider economy. At Deutsche Bank, relative value disciplines have contributed significantly to the development of our markets platform, as we allocate resources, mitigate risks, and find new ways of helping our clients perform. In my career, I have witnessed the evolution of relative value from a specialist methodology to an essential part of the market practitioner’s toolbox. This Practitioner’s Guide is both timely and relevant.
Enjoy the book.
Henry Ritchotte, Chief Operating Officer and member of the management board, Deutsche Bank
Chapter 1
Relative Value
The Concept of Relative Value
Relative value is a quantitative analytical approach toward financial markets based on two fundamental notions of modern financial economics.
Violation of this principle would result in the existence of an arbitrage opportunity, which is inconsistent with equilibrium in financial markets.
This proposition seems relatively straightforward now, but this wasn’t always the case. In fact, Kenneth Arrow and Gérard Debreu won Nobel prizes in economics in 1972 and 1983 in part for their work establishing this result. And Myron Scholes and Robert Merton later won Nobel prizes in economics in 1997 for applying this proposition to the valuation of options. In particular, along with Fischer Black, they identified a self-financing portfolio that could dynamically replicate the payoff of an option, and they were able to determine the value of this underlying option by valuing this replicating portfolio.
Most of the financial models discussed in this book are based on the application of this proposition in various contexts.
This result may appear intuitive, but it’s somewhat more difficult to establish than the first result. Of particular interest for our purposes is that the result can be established via the Arbitrage Pricing Theory, which assumes the existence of unobservable, linear factors that drive returns.
In this case, it’s possible to combine securities into portfolios that expose investors to any one of the risk factors without involving exposure to any of the other risk factors. In the limit, as the number of securities in the portfolio increases, the security-specific risks can be diversified away. And in this case, any security-specific risk that offered a non-zero expected return would present investors with an arbitrage opportunity, at least in the limit, as the remaining risk factors could be immunized by creating an appropriate portfolio of tradable securities.
For our purposes, this is a powerful result, as it allows us to analyze historical data for the existence of linear factors and to construct portfolios that expose us either to these specific factors or to security-specific risks, at our discretion. In fact, principal component analysis (PCA) can be applied directly in this framework, and we’ll rely heavily on PCA as one of the two main statistical models we discuss in this book.
The Sources of Relative Value Opportunities
From these two propositions, it’s clear that the absence of arbitrage is the assumption that drives many of the models we use as relative value analysts. This should come as no surprise, since one of the main roles of a relative value analyst is to search for arbitrage opportunities.
But for some people, this state of affairs presents a bit of a paradox. If our modeling assumptions are correct about the absence of free lunches, why do analysts and traders search so hard for them?
This apparent paradox can be resolved with two observations. The first is the recognition that arbitrage opportunities are rare precisely because hard-working analysts invest considerable effort trying to find them. If these opportunities could never be found, or if they never generated any profits for those who found them, analysts would stop searching for them. But in this case, opportunities would reappear, and analysts would renew their search for them as reports of their existence circulated.
The second observation that helps resolve this paradox is that even seemingly riskless arbitrage opportunities carry some risk when pursued in practice. For example, one of the simpler arbitrages in fixed income markets is the relation between bond prices, repo rates, and bond futures prices. If a bond futures contract is too rich, a trader can sell the futures contract, buy the bond, and borrow the purchase price of the bond in the repo market, with the bond being used as collateral for the loan. At the expiration of the contract, the bond will be returned to the trader by his repo counterparty, and the trader can deliver the bond into the futures contract. In theory, this would allow the trader to make a riskless arbitrage profit. But in practice, there are risks to this strategy.
For example, the repo counterparty may fail to deliver the bonds to the trader promptly at the end of the repo transaction, in which case the trader may have difficulty delivering the bonds into the futures contract. Failure to deliver carries significant penalties in some cases, and the risk of incurring these penalties needs to be incorporated into the evaluation of this seemingly riskless arbitrage opportunity.
These perspectives help us reconcile the existence of arbitrage opportunities in practice with the theoretical assumptions behind the valuation models we use. But they don’t explain the sources of these arbitrage or relative value opportunities, and we’ll discuss a few of the more important sources here.
Demand for Immediacy
In many cases, relative value opportunities will appear when some trader experiences an unusually urgent need to transact, particularly in large size. Such a trader will transact his initial business at a price that reflects typical liquidity in the market. But if the trader then needs to transact additional trades in the same security, he may have to entice other market participants to provide the necessary liquidity by agreeing to transact at a more attractive price. For example, he may have to agree to sell at a lower price or to buy at a higher price than would be typical for that security. In so doing, this trader is signaling a demand for immediacy in trading, and he’s offering a premium to other traders who can satisfy this demand.
The relative value trader searches for opportunities in which he can be paid attractive premiums for satisfying these demands for immediacy. He uses his capital to satisfy these demands, warehousing the securities until he can liquidate them at more typical prices, being careful to hedge the risks of the transactions in a cost-effective and prudent manner.
Because these markets are so competitive, the premiums paid for immediacy are often small relative to the sizes of the positions. As a result, the typical relative value fund will be run with leverage that is higher than the leverage of, say, a global macro fund. Consequently, it’s important to pay attention to small details and to hedge risks carefully.
Misspecified Models
It sometimes happens that market participants overlook relevant issues when modeling security prices, and the use of misspecified models can result in attractive relative value opportunities for those who spot these errors early.
For example, until the mid-1990s, most analysts failed to incorporate the convexity bias when assessing the relative valuations of Eurodollar futures contracts and forward rate agreements. As market participants came to realize the importance of this adjustment, the relative valuations of these two instruments changed over time, resulting in attractive profits for those who identified this issue relatively early.
As another example, until the late 1990s, most academics and market participants believed vanilla swap rates exceeded the yields of default-free government bonds as a result of the credit risk of the two swap counterparties. Due in part to our work in this area, this paradigm has been shown to be flawed. In particular, the difference over time between LIBOR and repo rates now is considered to be a more important factor in the relative valuations between swaps and government bonds.
In recent years, as credit concerns have increased for many governments, it has become increasingly important to reflect sovereign credit risk as an explicit factor in swap spread valuation models, and we discuss this issue in considerable detail in this book.
Regulatory Arbitrage
The fixed income markets are populated by market participants of many types across many different regulatory jurisdictions, and the regulatory differences between them can produce relative value opportunities for some.
For example, when thinking about the relative valuations of unsecured short-term loans and loans secured by government bonds in the repo market, traders at European banks will consider the fact that the unsecured loan will attract a greater regulatory charge under the Basel accords. On the other hand, traders working for money market funds in the US won’t be subject to the Basel accords and are likely to focus instead on the relative credit risks of the two short-term deposits. The difference in regulatory treatment may result in relative valuations that leave the European bank indifferent between the two alternatives but that present a relative value opportunity for the US money market fund.
The Insights from Relative Value Analysis
In some sense, relative value analysis can be defined as the process of gaining insights into the relationships between different market instruments and the external forces driving their pricing. These insights facilitate arbitrage trading, but they also allow us more generally to develop an understanding of the market mechanisms that drive valuations and of the ways seemingly different markets are interconnected.
As a consequence, relative value analysis, which originated in arbitrage trading, has a much broader scope of applications. It can reveal the origins of certain market relations, the reasons a security is priced a certain way, and the relative value of this pricing in relation to the prices of other securities. And in the event that a security is found to be misvalued, relative value analysis suggests ways in which the mispricing can be exploited through specific trading positions. In brief, relative value analysis is a prism through which we view the machinery driving market pricing amidst a multitude of changing market prices.
As an example, consider the divergence of swap spreads for German Bunds and US Treasuries in recent quarters, which might appear inextricable without considering the effects of cross-currency basis swaps (CCBS), intra-currency basis swaps (ICBS), and credit default swaps (CDS).
In this case, CCBS spreads widened as a result of the difficulties that European banks experienced in raising USD liabilities against their USD assets. On the other hand, arbitrage between Bunds, swapped into USD, and Treasuries prevented an excessive cheapening of Bunds versus USD LIBOR. As a consequence, Bunds richened significantly against EURIBOR (see Chapter 14 for more details).
However, given the relationship between European banks and sovereigns, the difficulties of European banks were also reflected in a widening of European sovereign CDS levels. Hence, Bunds richened versus EURIBOR at the same time as German CDS levels increased.
An analyst who fails to consider these interconnected valuation relations may find the combination of richening Bunds and increasing German CDS opaque and puzzling. But a well-equipped relative value analyst can disentangle these valuation relations explicitly to identify the factors that are driving valuations in these markets. And armed with this knowledge, the analyst can apply these insights to other instruments, potentially uncovering additional relative value opportunities.
The Applications of Relative Value Analysis
Relative value analysis has a number of applications.
Trading
One of the most important applications of relative value analysis is relative value trading, in which various securities are bought and others sold with the goal of enhancing the risk-adjusted expected return of a trading book.
Identifying relatively rich and relatively cheap securities is an important skill for a relative value trader, but additional skills are required to be successful as a relative value trader. For example, rich securities can and often do become richer, while cheap securities can and often do become cheaper. A successful relative value trader needs to be able to identify some of the reasons that securities are rich or cheap in order to form realistic expectations about the likelihood of future richening or cheapening. We discuss this and other important skills throughout this book.
Hedging and Immunization
Relative value analysis is also an important consideration when hedging or otherwise immunizing positions against various risks. For example, consider a flow trader who is sold a position in ten-year (10Y) French government bonds by a customer. This trader faces a number of alternatives for hedging this risk.
He could try to sell the French bond to another client or to an interdealer broker. He could sell another French bond with a similar maturity. He could sell Bund futures contracts or German Bunds with similar maturities. He could pay fixed in a plain vanilla interest rate swap or perhaps a euro overnight index average (EONIA) swap. He could buy payer swaptions or sell receiver swaptions with various strikes. He could sell liquid supranational or agency bonds issued by entities such as the European Investment Bank. Depending on his expectations, he might even sell bonds denominated in other currencies, such as US Treasuries or UK Gilts. Or he might choose to implement a combination of these hedging strategies.
In devising a hedging strategy, a skilled trader will consider the relative valuations of the various securities that can be used as hedging instruments. If he expects Bunds to cheapen relative to the alternatives, he may choose to sell German Bunds as a hedge. And if he believes Bund futures are likely to cheapen relative to cash Bunds, he may choose to implement this hedge via futures contracts rather than in the cash market.
By considering the relative value implications of these hedging alternatives, a skilled flow trader can enhance the risk-adjusted expected return of his book. In this way, the value of the book reflects not only the franchise value of the customer flow but also the relative value opportunities in the market and the analytical skills of the trader managing the book.
Given the increasing competitiveness of running a fixed income flow business, firms that incorporate relative value analysis as part of their business can expect to increase their marginal revenues, allowing them to generate higher profits and/or to offer liquidity to customers at more competitive rates.
Security Selection
In many respects, a long-only investment manager faces many of the same issues as the flow trader in the previous example. Just as a flow trader can expect to enhance the risk-adjusted performance of his book by incorporating relative value analysis into his hedging choices, a long-only investment manager can expect to enhance the risk-adjusted performance of his portfolio by incorporating relative value analysis into his security selection process.
For example, an investment manager who wants to increase his exposure to the 10Y sector of the EUR debt market could buy government bonds issued by France, Germany, Italy, Spain, the Netherlands, or any of the other EMU member states. Or he could buy Bund futures or receive fixed in a EURIBOR or EONIA interest rate swap. Or he might buy a US Treasury in conjunction with a cross-currency basis swap, thereby synthetically creating a US government bond denominated in euros.
An investment manager who incorporates relative value analysis as part of his investment process is likely to increase his alpha and therefore over time to outperform an otherwise similar manager with the same beta who doesn’t incorporate relative value analysis.
The Craft of Relative Value Analysis
Relative value analysis is neither a science nor an art. Rather, it’s a craft, with elements of both science and art. For a practitioner to complete the journey from apprentice to master craftsman, he needs to learn to use the tools of the trade, and in this book we introduce these tools along with their foundations in the mathematical science of statistics and in the social science of financial economics.
We also do our best to explain the practical benefits and potential pitfalls of applying these tools in practice. In the development of an apprentice, there is no substitute for repeated use of the tools of the trade in the presence of a master craftsman. But we make every effort in this book to convey the benefit of our experience over many years of applying these tools.
Since financial and statistical models are the tools of the trade for a relative value analyst, it’s important that the analyst choose these tools carefully, with an eye toward usefulness, analytical scope, and parsimony.
Usefulness
In our view, models are neither right nor wrong. Pure mathematicians may be impressed by truth and beauty, but the craftsman is concerned with usefulness. To us, various models have varying degrees of usefulness, depending on the context in which they’re applied.
As Milton Friedman reminds us in his 1966 essay “The Methodology of Positive Economics”, models are appropriately judged by their implications. The usefulness of a particular model is not a function of the realism of its assumptions but rather of the quality of its predictions.
For relative value analysts, models are useful if they allow us to identify relative misvaluations between and among securities, and if they improve the quality of the predictions we make about the future richening and cheapening of these securities.
For example, we agree with critics who note that the Black–Scholes model is wrong, in the sense that it makes predictions about option prices that are in some ways systematically inconsistent with the prices of options as repeatedly observed in various markets. However, we’ve found the Black–Scholes model to be useful in many contexts, as have a large number of analysts and traders. It’s important to be familiar with its problems and pitfalls, and like most tools it can do damage if used improperly. But we recommend it as a tool of the trade that is quite useful in a number of contexts.
Analytical Scope (Applicability)
For our purposes, it’s also useful for a model to have a broad scope, with applicability to a wide range of situations. For example, principal component analysis (PCA) has proven to be useful in a large number of applications, including interest rates, swap spreads, implied volatilities, and the prices of equities, grains, metals, energy, and other commodities. As with any powerful model, there is a cost to implementing PCA, but the applicability of the model once it has been built means that the benefits of the implementation tend to be well worth the costs.
Other statistical models with broad applicability are those that characterize the mean-reverting properties of various financial variables. Over considerable periods of time, persistent mean reversion has been observed in quite a large number of financial variables, including interest rates, curve slopes, butterfly spreads, term premiums, and implied volatilities. And in the commodity markets, mean reversion has been found in quite a number of spreads, such as those between gold and silver, corn and wheat, crack spreads in the energy complexes, and crush spreads in the soybean complex.
The ubiquity of mean-reverting behavior in financial markets means that mean reversion models have a tremendous applicability. As a result, we consider them some of the more useful tools of a well-equipped relative value analyst, and we discuss them in some detail in this book.
Parsimony
From our perspective, it’s also useful for a model to be parsimonious. As Einstein articulated in his 1933 lecture “On the Method of Theoretical Physics”, “It can scarcely be denied that the supreme goal of all theory is to make irreducible basic elements as simple and as few as possible without having to surrender the adequate representation of a single datum of experience”.
In our context, it’s important to note the relative nature of the word “adequate”. In most circumstances, there is an inevitable trade-off between the parsimony of a model and its ability to represent experience. The goal of people developing models is to improve this tradeoff in various contexts. The goal of people using models is to select those models that offer the best tradeoff between costs and benefits in specific applications. And it’s in that sense that we characterize the models in this book as being useful in the context of relative value analysis.
Summary of Contents
Relative value analysis models can be divided into two categories: statistical and financial. Statistical models require no specific knowledge about the instrument that is being modeled and are hence universally applicable. For example, a mean reversion model only needs to know the time series, not whether the time series represents yields, swap spreads, or volatilities, nor what drives that time series.
Financial models, on the other hand, give insight into the specific driving forces and relationships of a particular instrument (and are therefore different for each instrument). For example, the specific knowledge that swap spreads are a function of the cost of equity of LIBOR panel banks can explain why their time series exhibits a certain statistical behavior.
While we present the models in two separate categories, comprehensive relative value analysis combines both. The successful relative value trader described above might first use statistical models to identify which instruments are rich and cheap relative to each other, and then apply financial models in order to gain insights into the reasons for that richness and cheapness, on which basis he can assess the likelihood for the richness and cheapness to correct. If he sees a sufficient probability for the spread position to be an attractive trade, he can then use statistical models again to calculate, among others, the appropriate hedge ratios and the expected holding horizon.
Statistical Models
The two types of statistical models presented here are designed to capture two of the most useful statistical properties frequently observed in the fixed income markets: the tendency for many spreads to revert toward their longer-run means over time and the tendency for many variables to increase and decrease together. Chapter 2 and Chapter 3 are largely independent and therefore do not need to be read sequentially. However, Chapter 3 does refer to the application of mean-reverting models to the estimated factors and to specific residuals, so a reader with no preference would do well to read the chapter on mean reversion first.
Mean Reversion
Many financial spreads exhibit a persistent tendency to revert toward their means, providing a potential source of return predictability. In this chapter, we discuss stochastic processes that are useful in modeling this mean reversion, and we present ways in which data can be used to estimate the parameters of these processes. Once the parameters have been estimated, we can calculate the half-life of a process and make probabilistic statements about the value of the spread at various points in the future.
We also present the concept of a first passage time and show ways to calculate probabilities for first passage times. Once we have these first passage time densities, we can provide probabilistic answers to some of the more perplexing questions that are typical on a trading desk. Over what time period should I expect this trade to perform? What sort of return target is reasonable over the next month? How likely am I to hit a stop-loss if placed at this level? First passage time densities can provide probabilistic answers to these questions, and we discuss practical ways in which they can be implemented in a trading environment.
Principal Component Analysis
Many large data sets in finance appear to be driven by a smaller number of factors, and the ability to reduce the dimensionality of these data sets by projecting them onto these factors is a very useful method for analyzing and identifying relative value opportunities. In this chapter we discuss PCA in some detail. We address not only the mathematics of the approach but also the practicalities involved in applying PCA in real-world applications, including trading the underlying factors and hedging the factor risk when trading specific securities.
Financial Models
The financial models in this section are relative value models in that they value one security in relation to one or more other securities. To some extent, the chapters build on one another, with the material for one chapter serving as a starting point for the material in another chapter. For example, the chapter comparing risky bonds denominated in multiple currencies synthesizes the material on OIS–repo spreads, ICBS, cross-currency basis swaps, swap spreads, and CDS. Not every chapter needs to be read sequentially, but readers should be alert to the dependencies that exist between the various chapters, which we do our best to highlight in the subsequent previews.
Some Comments on Yield, Duration, and Convexity
A working knowledge of bond and interest rate mathematics is a prerequisite for this book. But we believe some of the basic bond math taught to practitioners is simply wrong, or at the very least misleading. For example, the basis point value of a bond is fundamentally a different concept from the value of a basis point for a swap, yet many practitioners are unclear about this difference. As another example, the Macaulay duration of a bond is often referred to as the weighted average time to maturity of a bond, but this is only true when all the zero-coupon bonds that constitute the coupon-paying bond have the same yield, a condition that is almost never observed in practice. We also discuss the frequent misuse of bond convexity and suggest a more practical interpretation of the concept.
Bond Futures Contracts
A simple no-arbitrage relation applies to the relative values of a cash bond, the repo rate for the bond, and the forward price of the bond. But government bond futures contracts typically contain embedded delivery options, which complicate the analysis. We present a multi-factor model for valuing the embedded delivery option, which can be implemented in a spreadsheet using basic stochastic simulation.
LIBOR, OIS Rates, and Repo Rates
Overnight index swaps (OIS) are based on unsecured overnight lending rates, whereas repo transactions are secured with collateral. In addition to the difference in credit risks, the two transactions will be subject to different treatment with regard to regulatory capital. We present a simple model for OIS rates that incorporates repo rates, the default probability, the presumed recovery rate, the risk-weighting of the transaction, the amount of regulatory capital required for the transaction, and the cost of the regulatory capital.
Intra-currency Basis Swaps
For this purpose, an ICBS is one in which the legs of the swap reference floating rates are in the same currency but with different maturities. For example, one party might agree to pay three-month EURIBOR for five years in exchange for receiving six-month EURIBOR less a spread for five years. We present a simple model for valuing these swaps based on the concepts presented in the OIS–repo model of the preceding section.
Theoretical Determinants of Swap Spreads
Up until the mid-1990s, it was widely believed that swap rates tended to be greater than government bond yields because of the credit risk of the two swap counterparties. Now, swap spreads are seen to be a function of the spreads between the LIBOR and repo rates over the life of the bond being swapped. We present this model in detail, incorporating the results of the OIS–repo model and the ICBS model of the preceding sections.
Swap Spreads from an Empirical Perspective
While it’s critical to consider the theoretical determinants of swap spreads, it’s also important to consider swap spreads from an empirical perspective. In particular, we examine the crucial link between swap spreads and LIBOR–repo spreads and find considerable empirical support for our conceptual framework. We also consider the role of credit quality in the valuation of sovereign debt relative to swaps in the aftermath of the subprime and European debt crises of recent years.
Swap Spreads as Relative Value Indicators for Government Bonds
Swap spreads often have been used to assess the relative valuations between different bonds along an issuer’s yield curve. We discuss the different ways this can be done and chronicle the numerous pitfalls that accompany these approaches. We conclude that none of these approaches is particularly good for assessing relative valuations among bonds, and we suggest using fitted bond curves as an alternative approach.
Fitted Bond Curves
There are many functional forms that are candidates for fitting yield curves, discount curves, and forward curves. In our experience, the particular functional form chosen is less important than the careful selection of the bonds used to fit the curve and the weighting methods used in the calibrations. In this chapter, we use a basic but widely used functional form to illustrate the important considerations that should apply when fitting bond curves. We then discuss the way in which the results can be used to identify relatively rich and cheap bonds within particular sectors.
A Brief Comment on Interpolated Swap Spreads
The most popular structure for trading bonds against swaps is the interpolated swap spread, with the end date of the swap set equal to the maturity date of the bond. While this structure has advantages relative to alternative structures, it can subject a trader to curve steepening or flattening positions, an issue we discuss in the context of an example.
Cross-Currency Basis Swaps
For our purposes, a CCBS is one in which the two legs are floating rates denominated in different currencies. For example, one party might agree to pay three-month EURIBOR for five years in exchange for receiving three-month USD LIBOR plus a spread for five years. If the tenor of the swap is less than one year, we typically refer to this as an FX swap, and there are no intermediate interest payments. Because the counterparties exchange principal at the beginning and end of the swap, these swaps have been in considerable demand in recent years. We discuss the valuation issues in this chapter.
Relative Values of Bonds Denominated in Different Currencies
A fundamental proposition of international financial economics is that in open and integrated capital markets securities should have the same risk-adjusted expected real return regardless of the currency of denomination. One implication of this is that two otherwise identical bonds, denominated in different currencies, should have identical yields once one is combined with the relevant interest rate swap and relevant basis swaps. We apply this notion in the context of global asset selection, by incorporating CCBS in the techniques for fitting bond curves.
Credit Default Swaps
The time has long since passed that we could assume the existence of default-free sovereign debt. CDS can play a role in assessing and adjusting for these credit implications, and in this chapter we review the salient features of these instruments.
USD Asset Swap Spreads versus Credit Default Swaps
The swap spread model developed in the preceding section assumed the sovereign bond had no default risk. That assumption has become increasingly less tenable in the current environment, and we discuss ways in which CDS can be used to reflect the default risk of specific issuers.
Options
We address the analysis and trading of options in a relative value context by discussing three broad categories of option trades. In the first, the trader simply buys or sells an option with a view that the underlying will finish in-the-money or out-of-the-money, with no dynamic trading. In the second, the trader attempts to capitalize on the difference between the implied volatility of the option and the actual volatility that the trader anticipates for the underlying instrument, by trading the option against a dynamic position in the underlying. In the third, the trader positions for a change in the implied volatility of the option, irrespective of the actual volatility of the underlying instrument.
Relative Value in a Broader Perspective
We conclude our sometimes rather technical description of relative value analysis by taking a broader perspective on its macroeconomic functions. At a time when professionals in the financial services industry increasingly need to justify their role in society, we present a few thoughts about the benefits of arbitrage for society.
Throughout the book, we offer pieces of general advice – words of wisdom that we’ve gleaned over time. We’ve been mentored by some of the best in the business over the years, with particular thanks to our managers and colleagues in Anshu Jain’s Global Relative Value Group at Deutsche Morgan Grenfell, and especially to David Knott, Pam Moulton, and Henry Ritchotte. They were good enough to impart their wisdom to us, and we’re happy to pass along this treasure trove of useful advice, hopefully with a few additional pearls of insight and experience that we’ve been able to add over the years.1
Please visit the website accompanying this book to gain access to additional material www.wiley.com/go/fixedincome
1When reviewing this book, Christian Carrillo, Martin Hohensee, Antti Ilmanen and Kaare Simonsen have provided valuable feedback, enhancing our product.
Part I
Statistical Models
Chapter 2
Mean Reversion
What Is Mean Reversion and How Does It Help Us?
Mean reversion is one of the most fundamental concepts underpinning relative value analysis. But while mean reversion is widely understood at an intuitive level, surprisingly few analysts are familiar with the specific tools available for characterizing mean-reverting processes.
In this chapter, we discuss some of the key characteristics of mean-reverting processes and the mean reversion tools that can be used to identify attractive trading opportunities. In particular, we address:
model selection;
model estimation;
calculating conditional expectations and probabilities;
calculating ex ante risk-adjusted returns, particularly Sharpe ratios;
calculating first passage times, also known as stopping times.
For each concept, we start with a verbal and intuitive description of the concept, followed by a mathematical definition of the concept, and finish with an example application of the concept to market data.
A variable is said to exhibit mean reversion if it shows a tendency to return to its long-term average over time. Mathematicians will object that this definition is simply an exercise in replacing the words “exhibit”, “mean”, and “reversion” with the synonyms “shows”, “long-term average”, and “return”. To address such objections, we’ll provide a more mathematical definition shortly. But first we’ll attempt to establish some further intuition about mean-reverting processes. To some extent, Justice Stewart’s famous maxim on pornography, “I know it when I see it”, applies to mean reversion. With that in mind, let’s take a look at some processes that exhibit mean reversion and a few that don’t.
Figure 2.1 and Figure 2.2 show two simulated time series. Both have an initial value of zero, and both have identical volatilities. But one is constructed to be a simple random walk, with zero drift, while the other is constructed to have a tendency to return toward its long-run mean, constructed to be zero in this example. In fact, the two series were constructed with identical normal random variates. In the case of the random walk, the mean of each observation was the value of the previous observation, so that the process was a martingale. In the case of the mean-reverting process, the mean of each observation was set to reflect the tendency for the process to return to the mean. At this point, we’d hope most readers would identify Figure 2.2 as the one with the mean-reverting variable. If we observe both figures closely, we can see that the mean-reverting process is in some sense a transformation of the random walk in Figure 2.1.
FIGURE 2.1 Simulated random walk.
Source: Authors.
FIGURE 2.2 Simulated mean-reverting process.
Source: Authors.
The speed with which a variable tends to revert toward its mean can vary. For example, Figure 2.3 and Figure 2.4 show time series that were simulated using the same random normal variates that generated the mean-reverting variable in Figure 2.2 but with an important difference. The variable in Figure 2.3 was constructed to have a faster speed of mean reversion than the variable in Figure 2.2, while the variable in Figure 2.4 was constructed to have a still faster speed of mean reversion.
FIGURE 2.3 Simulated mean-reverting process: Faster mean reversion.
Source: Authors.
FIGURE 2.4 Simulated mean-reverting process: Even faster mean reversion.
Source: Authors.
While it’s well and good to consider variables simulated via known equations by a computer, traders and analysts have to make judgments about real-world data, which are almost always messier in some respects than simulated data. So it’s also useful to consider a few real-world examples.
Figure 2.5 shows the spot price of gold in US dollars since January 1975. In our view, the strong upward drift exhibited in this series makes it a poor candidate to be modeled by a mean-reverting process.
FIGURE 2.5 Spot price of gold in US dollars since January 1975.
Source: Bloomberg.
Figure 2.6 shows the realized volatility of the ten-year (10Y) US Treasury yield since January 1962. Given that this series has repeatedly returned to a long-run mean in the past, it appears to be a relatively good candidate for modeling with a mean-reverting process.
FIGURE 2.6 Realized volatility of 10Y US Treasury bond yield (bp/year).
Source: Bloomberg.
As another example, Figure 2.7 shows the 2Y/5Y/10Y butterfly spread along the USD swap curve since 1998. Given the number of times during the sample that this spread returns to its long-run mean, we consider it another good candidate for modeling with a mean-reverting process.
FIGURE 2.7 2/5/10 butterfly spread along USD swap curve since 1988.
Source: Bloomberg.
Mathematical Definitions
Having provided verbal and graphical intuition regarding mean reversion, it’s time to attempt to provide a few useful mathematical definitions.
Stochastic Differential Equation
First, we’ll provide a brief definition of a stochastic differential equation (SDE). In practice, this term is fairly simple to define, as most of the definition is contained within the name. In other words, it’s an equation that characterizes the random behavior of a variable over an infinitesimal period of time. As a result, it gives us the data-generating mechanism for the variable. For example, the equation would allow us to simulate the variable over time using a computer.
For example, dxt = k(μ−xt)dt + σdWt is the SDE for an Ornstein–Uhlenbeck (OU) process, the continuous-time limit of a first-order autoregressive process. The OU process is a popular SDE for modeling mean-reverting variables, as it has moments and densities that can be expressed analytically. In this equation, dxt is the change in the value of the random variable x at time t, over the infinitesimal interval dt. The speed of mean reversion is given by the parameter k and the long-run mean of the variable is given by μ. The instantaneous volatility of the variable is given by σ, and the term dWt is the change in the value of Wt over the instantaneous time interval dt. In fact, Wt is ultimately the source of randomness that drives the process in this equation. In particular, Wt is a pure random walk, often referred to as Gaussian white noise. Wt is also referred to as a Wiener process, after the American mathematician Norbert Wiener.
In general, SDEs take the form
The term f(xt) is the drift coefficient of the equation, and it defines the mean of the process. The term g(xt) is the diffusion coefficient of the equation, and it defines the volatility of the process.
Conditional Density
Next, we’ll define the conditional density of a process, also referred to as a transition density. In particular, the conditional density gives us the probability density for the future value of a random variable conditional on knowing some other information about the variable. In the case of a time series process, the conditioning information is usually some earlier value of the variable. For example, in the case of the OU process, the transition density of xt+τ for τ > 0 is a normal density with mean given by μ + (xt − μ)e−kτ and with a variance given by .
Unconditional Density
The unconditional density of a process is the probability density for the future value of a random variable without being able to condition the density on any additional information. You could think of the unconditional density as the histogram that would result from simulating the process over an infinitely long period. More precisely, it’s the limit of the conditional density p(xt+τ) as τ goes to infinity. So in the case of the OU process, the unconditional density is a normal density with mean given by μ and with variance given by .
Stationary Densities and Mean-Reverting Processes
In some cases, a variable will have a conditional density, but it won’t have an unconditional density. In other words, the limit of the conditional density p(xt) won’t converge to a limiting density.
A simple example of this would be a random walk with drift, given by the SDE dxt = ρdt + σdWt. The transition density or unconditional density for this process is normal with mean xt + ρτ and variance given by σ2τ. In this case, neither the mean nor the variance has a limit as τ → ∞, and the limit of the conditional density doesn’t exist.
Even in the case of a random walk with no drift, given by the SDE dxt = σdWt, there is no unconditional density. In this case, the mean for all future transition densities is simply the current value of the variable xt, but since there is no limit to the variance of the process, there is no limit to the conditional density, and the unconditional density doesn’t exist.
However, in many cases the limit of the conditional density will exist, and the random variable is said to be mean reverting. We also say that stationary density exists and that the process is stationary.
Return Predictability and Alpha
Having provided some intuition and some mathematical definitions of mean-reverting processes, it’s helpful to take a step back and consider the usefulness of mean-reverting models for investors and traders.
Return predictability is a necessary, though not sufficient, condition for generating alpha, defined here as an atypically high, risk-adjusted return. If we identify a financial variable that exhibits return predictability, then either the risks of that variable are predictable or the risk-adjusted returns are predictable.
Mean reversion is a form of return predictability. If a financial variable exhibits mean reversion, then we can use that information to improve our predictions for the future value of the variable. In our view, more often than not, mean reversion in a financial variable is an indication that risk-adjusted returns of the variable are predictable. Of course, in some cases, some or all of the mean reversion in a variable will be the result of risks that exhibit mean reversion rather than the result of mean reversion in the risk-adjusted returns. But in our view, the more typical result is that the risk-adjusted returns are predictable. In this case, mean reversion can be used to generate alpha for traders and investors.
Diagnostics for Model Selection
The key for modeling any mean reversion in the variable x is to select a functional form for the drift coefficient, f(x), that is useful in depicting the tendency of x to decline toward its long-run mean when it’s above the mean and to increase toward its long-run mean when it’s below the mean. So at a minimum, we need a function f(x) that satisfies three conditions:
The value of
f
(
x
) is negative when
x
is above the long-run mean.
The value of
f
(
x
) is positive when
x
is below the long-run mean.
The value of
f
(
x
) is zero when
x
is equal to the long-run mean.
Of course, one simple function that satisfies these properties is a line, in which f(x) could be parameterized as f(x) = k(μ − x). In this case, μ is the long-run mean of the process and k is the strength with which the variable x is “pulled” toward the long-run mean. An equivalent parameterization of f(x) is f(x) = a + bx, in which case a = kμ, and b = k.
As it happens, the simplicity of this linear parameterization simplifies the estimation of parameters from historical data, as the likelihood function will often have a closed-form representation in this case, depending on the specification of the diffusion coefficient, g(x). The linear specification for f(x) also simplifies the calculation of transition densities and first passage time densities for x.
But of course a line is not the only function that could be used to represent the mean-reverting tendencies of x, and we may be willing to sacrifice some simplicity in exchange for a functional form for the drift coefficient that is more useful in capturing the actual mean-reverting tendencies exhibited in the data.
For example, a more flexible functional form that has been used in a variety of applications is f(x) = a + bx + cx2 + dx3, a third-order polynomial in x. In particular, this nonlinear specification allows for the variable x to exhibit increasingly strong mean reversion as it moves further away from the long-run mean. A similar function form that has been used successfully in a variety of applications is , as the term allows for the drift coefficient to become increasingly strong as x approaches zero, allowing zero to act as a reflecting barrier for the process. Examples of these three functional forms appear in Figure 2.8.
FIGURE 2.8 Examples of drift coefficients.
Source: Authors.
In this case, we want to restrict the value of d to be non-positive, to avoid the drift coefficient going to infinity as x increases, in which case x would be an explosive process rather than a stationary, mean-reverting process. For a similar reason, we’d like to restrict the value of a to be non-negative.
Strictly speaking, the drift coefficient, f(x) also needs to satisfy other, rather technical, mathematical conditions in order to ensure that this SDE has a solution. In practice, we tend to assume that these conditions are satisfied (perhaps more often than we should), as the processes typically studied in financial applications tend to be well behaved.
At this point, we should stress that the drift coefficient, f(x), can assume an unlimited variety of functional forms, including nonparametric forms, and the particular form used in practical applications must be specified by the analyst.
If the analyst is open to sacrificing some analytical tractability in an attempt to more usefully model certain aspects of the process, he would benefit from some diagnostic guidance as to the functional forms for f(x) that are likely to be most useful.
To our knowledge, the most useful diagnostic is one suggested by Richard Stanton in a Stanford research paper in the mid-1990s. The basic idea behind this diagnostic tool is to create a nonparametric, empirical approximation of the drift coefficient using historical data. An illustration of this is provided in Figure 2.9.
FIGURE 2.9 Diagnostic tool for drift coefficient.
Source: Authors.
Each point in the graph represents the estimated strength of mean reversion when the variable assumes values in the neighborhood of that point. Perhaps the most expedient way to explain this concept is to start by listing the steps involved in creating the graph.
