The Advanced Fixed Income and Derivatives Management Guide

By Saied Simozar

A highly-detailed, practical analysis of fixed income management

The Advanced Fixed Income and Derivatives Management Guide provides a completely novel framework for analysis of fixed income securities and portfolio management, with over 700 useful equations. The most detailed analysis of inflation linked and corporate securities and bond options analysis available;, this book features numerous practical examples that can be used for creating alpha transfer to any fixed income portfolio. With a framework that unifies back office operations, such as risk management and portfolio management in a consistent way, readers will be able to better manage all sectors of fixed income, including bonds, mortgages, credits, and currencies, and their respective derivatives, including bond and interest rate futures and options, callable bonds, credit default swaps, interest rate swaps, swaptions and inflation swaps. Coverage includes never-before-seen detail on topics including recovery value, partial yields, arbitrage, and more, and the companion website features downloadable worksheets that can be used for measuring the risks of securities based on the term structure models.

Many theoretical models of the Term Structure of Interest Rates (TSIR) lack the accuracy to be used by market practitioners, and the most popular models are not mathematically stable. This book helps readers develop stable and accurate TSIR for all fundamental rates, enabling analysis of even the most complex securities or cash flow structure. The components of the TSIR are almost identical to the modes of fluctuations of interest rates and represent the language with which the markets speak.

• Examine unique arbitrage, risk measurement, performance attribution, and replication of bond futures
• Learn to estimate recovery value from market data, and the impact of recovery value on risks
• Gain deeper insight into partial yields, product design, and portfolio construction
• Discover the proof that corporate bonds cannot follow efficient market hypothesis

This useful guide provides a framework for systematic and consistent management of all global fixed income assets based on the term structure of rates. Practitioners seeking a more thorough management system will find solutions in The Advanced Fixed Income and Derivatives Management Guide.

• Language: English
• Publisher: Wiley
• Release date: Apr 23, 2015
• ISBN: 9781119014171

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This edition first published 2015

© 2015 Saied Simozar

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Library of Congress Cataloging-in-Publication Data

ISBN 978-1-119-01414-0 (hardback); ISBN 978-1-119-01416-4 (ebk);

ISBN 978-1-119-01417-1 (ebk)

Cover Design: Wiley

Top Image: ©Shutterstock.com/bluebay

Bottom Image: ©Shutterstock.com/wongwean

List of Tables

1.1 Yield and duration of a portfolio

1.2 Key rate duration of a portfolio

2.1 US historical term structure components

2.2 US historical volatility of term structure components

3.1 Weights of principal components, 1992–2012

3.2 Historical half-life (mean reversion) of US treasury term structure components

3.3t-test of half-life of US treasury term structure components

3.4 Average value of US treasury term structure components

3.5 Annualized absolute volatility of US treasury term structure components

4.1 Duration components of zero coupon bonds

4.2 Curve exposure of portfolios of zero coupon bonds

4.3 Curve exposure of eurodollar futures contracts

4.4 Conventional yield and duration of portfolios of securities

4.5 Duration components of key rate securities

4.6 Transposed and scaled duration components of key rate securities

4.7 Duration components and yield of an equal weighted treasury index

4.8 Average duration components of an equal weighted treasury index

4.9 Duration components of global treasuries, January 3, 2013

5.1 Index performance attribution using coupon bonds for the TSIR

5.2 Index performance attribution using coupon Strips

5.3 Decay coefficient and contribution to performance, 1992–2012

5.4 Decay coefficient and volatility of performance, 1992–2012

5.5 Comparison of aggregated daily performance by basis function, 1992–2012

5.6 Comparison of annualized volatility by basis function

6.1 Selected term structure of swaps, July 30, 2012

6.2 Selected adjustment table for TSLR, July 30, 2012

6.3 Swap valuation table, July 30, 2012

7.1 Selected treasury bonds, 2012

7.2 Analysis of EUR term structure components

7.3 EUR swap trade, April 22, 2008

7.4 USD swap trade data, November 26, 2007

7.5 USD swap trade performance, November 26, 2007

7.6 USD swap trade data, June 28, 2004

7.7 USD swap trade performance, November 26, 2007

7.8 Durations of streams of cash flows

7.9 Summary of trade result, December 18, 2012

8.1 Performance of index replicating portfolio using five components, 1992–2012

8.2 Performance of index replicating portfolio using three components, 1992–2012

8.3 Performance of hedging methods, 1998–2012

9.1 Correlations of historical components of TSLV, 2000–2012

9.2 Principal components of historical components of TSLV, 2008–2012

9.3 Adjustment table for US swap volatility, June 30, 2012

9.4 Market, fair, and model volatilities, June 30, 2012

10.1 Components of the TSIR

10.2 Return attribution of coupon Strips 2/15/2027, 1997–2012

10.3 Eurodollar futures contracts, July 30, 2012

10.4 Euribor futures contracts, July 30, 2012

11.1 Timeline for cash flow analysis of inflation linked bonds

11.2 Price and spreads for selected IL bonds, July 30, 2012

11.3 Yield and interest rate durations for selected IL bonds, July 30, 2012

11.4 Real and credit durations for selected IL bonds, July 30, 2012

11.5 Sample US headline inflation index

11.6 Seasonal factors for US CPI

11.7 Yield of short maturity Tips, July 31, 2012

11.8 Risks of selected inflation swaps, July 31, 2012

12.1 Comparison of duration components of credit securities, July 30, 2012

12.2 Term structure of Brazil, May 25, 2012

12.3 Term structure of European credit spreads, May 25, 2012

12.4 Analytics for selected credit securities, July 31, 2012

12.5 Emerging markets portfolio report

12.6 Performance contribution example

12.7 Performance contribution example

12.8 Partial yields of selected securities, July 31, 2012

13.1 Selected analytics with recovery or guarantee, July 31, 2012

13.2 Partial yield and TSCS, July 31, 2012

14.1 Futures options analytics, July 31, 2012

14.2 Futures valuations analytics, July 31, 2012

14.3 Futures risk analytics, July 31, 2012

14.4 Replicating futures risks, July 31, 2012

14.5 Bond futures backtest results, July 31, 2012

14.6 Bond futures backtest underperformers, July 31, 2012

15.1 Bond option premiums, July 8, 2011

15.2 Early exercise of American call option, July 8, 2011

15.3 Bond option Greeks, July 8, 2011

15.4 Bond option durations, July 8, 2011

15.5 Bond option TSLV sensitivities, July 8, 2011

15.6 Bond option beta sensitivities, July 8, 2011

15.7 Call values of credit bonds, July 8, 2011

15.8 Option values for varying correlation parameters, July 8, 2011

15.9 Call risks of credit bonds, July 8, 2011

16.1 Long/short currency trades

18.1 Valuation of mortgage bonds, settlement August 3, 2012

18.2 Risk measures of mortgage bonds, July 31, 2012

18.3 Principal components of mortgage volatility, July 31, 2012

18.4 Principal components of swaption volatility, July 31, 2012

18.5 Hedging volatility of a mortgage

19.1 Sample portfolio analyzer output

19.2 Sample linear optimization constraints

19.3 Sample linear optimization trades, July 31, 2012

19.4 Sample portfolio preview

21.1 Practical discount yields

21.2 Practical floating discount benchmarks

21.3 Types of cash flow

21.4 Matrix of methods of risk calculation

List of Figures

2.1 Chebyshev term structure components in τ space

2.2 Chebyshev term structure components in time space

2.3 Forward rate components in τ space

2.4 Forward rate components in time space

2.5 US term structure of interest rates for September 30, 2010

2.6 Components of US yield curve for September 30, 2010

2.7 Level of yield curve shifted by 50 bps.

2.8 Slope of yield curve shifted by 50 bps.

2.9 Bend of yield curve shifted by 50 bps.

2.10 Yield curve on December 11, 2008

2.11 Comparison of ISM manufacturing index and bend of the TSIR

2.12 Implied historical decay coefficient

2.13 Implied historical decay coefficient from treasury market

3.1 Orthogonal term structure components in τ space

3.2 Orthogonal term structure and principal components in τ space, 1992–2012

3.3 Term structure and volatility adjusted principal components in τ space, 1992–2012

3.4 Historical bend of the Chebyshev basis function

4.1 Eurodollar futures contracts VBP

4.2 Key rate contribution to duration, time space

6.1 Term structure of swap curve, May 25, 2012

6.2 Spread of repo and Libor over treasury bills

7.1 Historical term structures of euro swaps

7.2 Historical term structures of USD swaps

7.3 AUD and NZD swap curves, May 24, 2012

7.4 AUD and NZD instantaneous forward swap curves, May, 24, 2012

7.5 AUD and NZD swap curves, December, 18, 2012

8.1 Portfolio optimization example

9.1 Selected cross-sections of relative Libor volatility, June 30, 2012

9.2 Selected cross-sections of absolute Libor volatility, June 30, 2012

10.1 Convexity adjusted yield curve, May 28, 1999

10.2 Yield curve without convexity adjustment, May 28, 1999

10.3 Convexity adjusted long zero curves

10.4 Treasury and swap curves for calculations of EDFC, July 30, 2012

11.1 Spot real (Rts) and nominal (Tsy) rates, July 30, 2012

11.2 Term structure of inflation expectations, July 30, 2012

11.3 Average monthly inflation rates

11.4 Standard deviation of monthly inflation in the US

11.5 Cumulative seasonal inflation adjustment for US

11.6 Implied and market inflation rates, July 31, 2012

12.1 Credit spread of Brazil, May 25, 2012

12.2 Term structures of rates in France and Germany, July 31, 2012

12.3 Contribution to partial yield

13.1 TSCS and TSDP for Ford Motor Co., July 31, 2012

15.1 European at-the-money call swaption, July 8, 2011

15.2 Log-normal probability distribution

15.3 American at-the-money call swaption, July 8, 2011

15.4 American at-the-money put swaption, July 8, 2011

15.5 Correlation functions

17.1 Fraction of homes sold per year

17.2 Natural log of mortgage factor due to incentive

18.1 Conventional 30-year mortgage rates

18.2 Calculation error for 30-year conventional mortgages

18.3 Conventional 15-year mortgage rates

20.1 Newton's optimization method

21.1 Propagation from bucket j to bucket k

Abbreviations

Notation

For notational convenience most variable names have been limited to a single character. Subscripts have been used to differentiate related variables. Subscripts i, j, and k have been used exclusively as running integers and are interchangeable. Other subscript letters are used to differentiate closely related names. For example, pm and pc are used for the market price and calculated price of a security, respectively. When these subscripts are mixed with running subscripts, a comma is inserted between them (e.g. pm,i or pc,k).

SUBSCRIPTS

VARIABLE NAMES

Preface

Fixed income management has become significantly more quantitative and competitive over the last 20 years or so, and the days where fund managers could make very large duration bets are mostly over. Most clients prefer portfolios with diversified sources of alpha and duration targets that are comparable to the risk profiles of their liabilities or their intended risk/return expectations. Developments of strategies that are quantifiable and repeatable are essential for the success of fixed income business.

Understanding the factors that contribute to risk and return are essential, in order to structure a sound portfolio. Risk management and return attribution require the quantification of sources of risk and return and thus are math intensive. A portfolio manager who is familiar with linear programming can structure an optimum portfolio based on analysts' recommendations, portfolios policies and guidelines as well as his own views of the markets that is likely to have a superior return than another portfolio of similar weights and risk profiles.

This book provides a comprehensive framework for the management of fixed income, both horizontally and vertically. It covers in detail all sectors of fixed income, including treasuries, mortgages, international bonds, swaps, inflation linked securities, credits and currencies and their respective derivatives. We develop a methodology for decomposing valuation metrics and risks into common components that can easily be understood and managed. Valuation, risk measurement and management, performance attribution, hedging and cheap/rich analysis are the natural byproducts of the framework.

Nearly all the concepts in the book were developed out of necessity over more than 20 years as a fund manager at DuPont Capital Management, Putnam Investments, Banc of America Capital Management and Nuveen Investments. Even though the book is rich in theory and mathematical derivations, the primary focus is alpha generation, understanding valuations and exploiting market opportunities.

The intended audience of the book includes the following:

Portfolio managers – Throughout the book there are numerous strategies and valuation formulas to help portfolio managers structure optimal portfolios and identify value opportunities without changing their intended risk profile.

Analysts – Estimation of default probability and recovery value from market prices of securities as well as recovery adjusted yield and duration can help analysts compare securities on a level playing field.

Traders – Throughout the book there are numerous examples of cheap/rich analysis of securities to help traders identify trading opportunities. Synthetic securities can be constructed when a security that provides the necessary exposure does not exist or is not available for trading.

Hedge funds – There is coverage for nearly all liquid fixed income derivatives together with methods for the identification of value and hedging the risks of derivatives. Several backtests demonstrate the efficacy of value identification and provide systematic approaches to long/short and leveraged strategies.

Proprietary trading desks – There is broad coverage of risk decomposition and hedging for all securities and their derivatives, including credit securities and credit default swaps.

Risk measurement/management – The risks of all securities are decomposed into components that can be separately measured or hedged by both the back office and portfolio managers.

Performance attribution – Performance attribution and contribution at the security and portfolio levels for all asset classes and derivatives is performed using the same methodology. The performance of a treasury portfolio can be measured to within 1 basis point on an annual basis, with similar accuracy for other sectors.

Central bankers – The analysis of default probability and recovery for sovereign countries based on the traded price of their securities and precise calculations of the term structure of inflation expectations provide methods for the measurements of systemic risk in global markets.

Academics – There are a few concepts covered in the book that have not been published elsewhere, including:

proof that long term yields cannot change;

structural problems of swaps and why they are subject to arbitrage;

why corporate bonds violate the efficient market hypothesis;

real rates cannot have log-normal distribution.

Finance and financial engineering textbook – This book can serve as an advanced book for graduate students in finance or financial engineering.

Many of the mathematical derivations are followed by practical examples or backtests to show how the analysis can be used to uncover value or measure risks in fixed income portfolios.

This book assumes that the reader is familiar with basic fixed income securities and their analysis. Knowledge of calculus, linear algebra and matrix operations is necessary to follow many of the quantitative aspects of the book. Some of the math concepts that are not covered in calculus can be easily found in online sources such as Wikipedia, including Chebyshev polynomials, the gamma function, principal components analysis, and eigenvalues and eigenvectors.

Most of the derivations in the book are original and therefore only a few external references have been mentioned. For some areas that have been extensively studied in the market, we provide comprehensive coverage within our framework, including:

Mortgage valuations – We provide very detailed measurements of sensitivity to the term structure of volatility and rates by matching volatility across its surface precisely and using a method similar to a closed form solution. We show that hedging the volatility of mortgages requires multiple swaptions.

Corporate bonds – We estimate the recovery value from the market price of securities and calculate the recovery adjusted spread and credit and interest rate durations. We show that option adjusted spread is not the best measure of value for corporate bonds.

Bond futures – A self-consistent probability weighted method for the valuation and risk measurement is developed. The valuation result is used in backtests for long/short strategies that produce very respectable information ratios.

Inflation linked – The decomposition of risks of inflation linked bonds and inflation swaps into the respective components of real and nominal along with seasonal adjustments provides very accurate hedging and valuations.

Bond options – It is argued that Black-76 model is not arbitrage-free for bond options and we develop a model for pricing American bond options with the accuracy of a closed form solution, if one existed. In the options chapter we show that the most widely used platform to value American bond options is sometimes off by a factor of more than 2 at the time of this analysis.

The backbone of our framework is the term structure of rates, including interest rates, real rates, swap rates (Libor), credit rates and volatility. Through principal components analysis we show that the market's own modes of fluctuations of interest rates are nearly identical to the components of our term structure of interest rates. Essentially, our term structure model speaks the language of the markets. Thus, the model requires the minimum number of components to explain all changes in interest rates. Five components can price all zero coupon treasuries within 2 basis points (bps) of market rates. More importantly, a different number of components can be used for risk management than for valuation without loss of generality. Exact pricing of all interest rate swaps that is provided by our methodology can be used for valuation of swap transactions.

The components of the term structure model represent weakly correlated sectors of the yield curve and can be used for structuring and risk measurement of portfolios. The first component, level, is associated with the duration of the portfolio. The second component, slope, is associated with the flattening/steepening structure and can be used to structure a barbell trade. The third component, bend, represents the exposure of a portfolio at the long and short ends relative to the middle of the curve and is used to structure a butterfly trade.

Valuation metrics along with the term structure durations for the identification of sources of alpha and risk are provided for all asset classes. We introduce the concept of partial yields as a way to decompose the contribution of different sectors to the yield of a portfolio. It is not reasonable to aggregate the yield of a security that has a high probability of default in a portfolio, since the resulting portfolio yield is not likely to be realized. Partial yield addresses this issue, by calculating the default probability and decomposing the yield into components that can be used to aggregate a portfolio's yield.

The valuation metrics and term structure durations along with linear programming provide tools for portfolio construction at the security level. This is also known as the bottom-up approach to portfolio construction and is useful for daily maintenance of a portfolio. Sector allocations and analysis of the portfolio's mix of assets and durations and correlation among different asset classes are the subject of the top-down method of portfolio construction in fixed income. The two methods are complementary to each other; however, top-down is usually analyzed on a monthly or quarterly basis.

There is a step-by-step outline of building a spreadsheet based tool for designing new products or maintaining an existing portfolio. This tool provides the tracking error, marginal contribution to risk, and can be used for what-if analysis or to see how the portfolio would have performed during prior financial crises or how additions of new asset classes or sectors alter the risk profile of the portfolio. There is also a method to identify the structure of the competitive universe and design a product that could compete in that space.

We have provided detailed steps and formulation for the implementation of the framework that is outlined in the book. Many of the components can be built in spreadsheets; however, reliable and efficient analytics require the development of the necessary tools as separate programs. The benefits of such a framework and the potential performance improvements significantly outweigh its development costs.

Acknowledgement

You might think that following some of the seven hundred or so formulas in the book is not a trivial task, let alone deriving them. Kris Kowal, Managing Director and Chief Investment Officer of DuPont Capital Management, Fixed Income Division, offered to review the manuscript and re-derive nearly all the formulas in the book. Kris provided numerous helpful suggestions and comments that were instrumental in reshaping the book into its present form. In many cases, following Kris's recommendations additional steps were added to the derivations to make it easier for the reader to follow. Thanks Kris.

Foreword

In 1998, shortly after arriving at Putnam Investments, Saied Simozar began work on a model for the term structure of interest rates that was to become a cornerstone of an entire complex of portfolio management tools and infrastructure. It was fortuitous timing because that rate model had the dual benefits of being derived through current market pricing structure (rather than historical regressions) and the flexibility to quickly incorporate new security types.

The late 1990s marked something of a sea change in the fixed income markets. The years leading up to that period had been defined by big global themes and trends like receding global inflation rates and the development of out of benchmark sectors like high yield corporate bonds and emerging market debt, as well as global interest rate convergence under the nascent stages of European Monetary Union. Under these broad trends, return opportunities, portfolio positioning, and risk could easily be characterized in terms of duration and sector allocation percentages.

Much of that changed in 1998 when the combination of increasingly complex security types, rapid globalization of financial markets, and large mobile pools of capital set the stage for a series of rolling financial crises that rocked global financial markets and eventually led to the collapse of one of the most sophisticated hedge funds of that era – Long Term Capital Management. In the aftermath, it became clear that traditional methods of monitoring portfolio positioning and risk were insufficient to manage all the moving parts in modern fixed income portfolios.

Fortuitously, that term model (and the portfolio management tools built around it) allowed Putnam to effectively navigate through that financial storm. Perhaps more importantly, it provided the basis for an infrastructure that could easily adapt and change with the ever evolving fixed income landscape. Today, while many of the original components of that infrastructure have been augmented and updated, the basic tenants of the philosophical approach remains in place.

In his book, Saied lays out a blueprint for a set of integrated tools that can be used in all aspects of fixed income portfolio management from term structure positioning, analysis of spread product, security valuation, risk measurement, and performance attribution. While the work is firmly grounded in mathematical theory, it is conceptually intuitive and imminently practical to implement. Whether you are currently involved in the management of fixed income portfolios or are looking to get a better understanding of all the inherent complexities, you won't find a more comprehensive and flexible approach.

D. William Kohli

Co-Head of Fixed Income

Putnam Investments

About the Author

Saied Simozar, PhD has spent almost 30 years in fixed income portfolio management, fixed income analytics, scientific software development and consulting. He is a principal at Fipmar, Inc., an investment management consulting firm in Beverly Hills, CA. Prior to that, Saied was a Managing Director at Nuveen Investments, with responsibilities for all global fixed income investments. He has also been a Managing Director at Bank of America Capital Management responsible for all global and emerging markets portfolios of the fixed income division. Prior to that, he was a senior portfolio manager at Putnam Investments and DuPont Pension Fund Investments.

Introduction

One of the keys to managing investment portfolios is identification and measurement of sources of risk and return. In fixed income, the most important source is the movement of interest rates. Even though changes in interest rates at different maturities are not perfectly correlated, diversifying a portfolio across the maturity spectrum will not lead to interest rate risk reduction. In general, a portfolio of one security that matches the duration of a benchmark tends to have a lower tracking error with the benchmark than a well-diversified portfolio that ignores duration.

Historically, portfolio managers have used Macaulay or modified duration to measure the sensitivity of a portfolio to changes in interest rates. With the increased efficiency of the markets and clients' demands for better risk measurement and management, several approaches for modeling the movements of the term structure of interest rates (TSIR) have been introduced.

A few TSIR models are based on theoretical considerations and have focused on the time evolution or stochastic nature of interest rates. These models have traditionally been used for building interest rate trees and for pricing contingent claims. For a review of these models, see Boero and Torricelli [1].

Another class of TSIR models is based on parametric variables, which may or may not have a theoretical basis, and their primary emphasis is to explain the shape of the TSIR. An analytical solution of the theoretical models would also lead to a parametric solution of the TSIR; see Ferguson and Raymar for a review [2]. Parametric models can be easily used for risk management and they almost always lead to an improvement over the traditional duration measurement. Willner [3] has applied the term structure model proposed by Nelson and Siegel [4] to measure level, slope and curvature durations of securities.

Key rate duration (KRD) proposed by Ho [5] is another attempt to account for non-parallel movements of the TSIR. A major shortcoming of KRD is that the optimum number and maturity of key rates are not known, and often on-the-run treasuries are used for this purpose. Additionally, key rates tend to have very high correlations with one another, especially at long maturities, and it is difficult to attach much significance to individual KRDs. The most important feature of KRD is that the duration contribution of a key rate represents the correct hedge for that part of the curve.

Another approach that has recently received some attention for risk management is the principal components analysis (PCA) developed by Litterman and Scheinkman [6]. In PCA, the most significant components of the yield curve movements are calculated through the statistical analysis of historical yields at various maturities. A very attractive feature of principal components, as far as risk management