Fixed Income Securities - Bruce Tuckman - ebook

Fixed Income Securities ebook

Bruce Tuckman

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Fixed income practitioners need to understand the conceptualframeworks of their field; to master its quantitative tool-kit; andto be well-versed in its cash-flow and pricing conventions.Fixed Income Securities, Third Edition by Bruce Tuckman andAngel Serrat is designed to balance these three objectives. Thebook presents theory without unnecessary abstraction; quantitativetechniques with a minimum of mathematics; and conventions at auseful level of detail. The book begins with an overview of global fixed income marketsand continues with the fundamentals, namely, arbitrage pricing,interest rates, risk metrics, and term structure models to pricecontingent claims. Subsequent chapters cover individual markets andsecurities: repo, rate and bond forwards and futures, interest rateand basis swaps, credit markets, fixed income options, andmortgage-backed-securities. Fixed Income Securities, Third Edition is full ofexamples, applications, and case studies. Practically everyquantitative concept is illustrated through real market data. Thispractice-oriented approach makes the book particularly useful forthe working professional. This third edition is a considerable revision and expansion ofthe second. Most examples have been updated. The chapters on fixedincome options and mortgage-backed securities have beenconsiderably expanded to include a broader range of securities andvaluation methodologies. Also, three new chapters have been added:the global overview of fixed income markets; a chapter on corporatebonds and credit default swaps; and a chapter on discounting withbases, which is the foundation for the relatively recent practiceof discounting swap cash flows with curves based on money marketrates. [FOR THE UNIVERSITY EDITION] This university edition includes problems which students can useto test and enhance their understanding of the text.

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Contents

Cover

Endorsement

Series

Title Page

Copyright

Preface to the Third Edition

Acknowledgments

An Overview of Global Fixed Income Markets

A SNAPSHOT OF GLOBAL FIXED INCOME MARKETS

FIXED INCOME MARKETS IN THE UNITED STATES

FIXED INCOME MARKETS IN EUROPE

FIXED INCOME MARKETS IN JAPAN

Part One: The Relative Pricing of Securities with Fixed Cash Flows

Chapter 1: Prices, Discount Factors, and Arbitrage

THE CASH FLOWS FROM FIXED-RATE GOVERNMENT COUPON BONDS

DISCOUNT FACTORS

THE LAW OF ONE PRICE

ARBITRAGE AND THE LAW OF ONE PRICE

APPLICATION: STRIPS AND THE IDIOSYNCRATIC PRICING OF U.S. TREASURY NOTES AND BONDS

ACCRUED INTEREST

APPENDIX A: DERIVING REPLICATING PORTFOLIOS

APPENDIX B: THE EQUIVALENCE OF THE DISCOUNTING AND ARBITRAGE PRICING APPROACHES

Chapter 2: Spot, Forward, and Par Rates

SIMPLE INTEREST AND COMPOUNDING

EXTRACTING DISCOUNT FACTORS FROM INTEREST RATE SWAPS

DEFINITIONS OF SPOT, FORWARD, AND PAR RATES

CHARACTERISTICS OF SPOT, FORWARD, AND PAR RATES

TRADING CASE STUDY: TRADING AN ABNORMALLY DOWNWARD-SLOPING 10S-30S EUR FORWARD RATE CURVE IN Q2 2010

APPENDIX A: COMPOUNDING CONVENTIONS

APPENDIX B: CONTINUOUSLY COMPOUNDED SPOT AND FORWARD RATES

APPENDIX C: FLAT SPOT RATES IMPLY FLAT PAR RATES

APPENDIX D: A USEFUL SUMMATION FORMULA

APPENDIX E: THE RELATIONSHIP BETWEEN SPOT AND FORWARD RATES AND THE SLOPE OF THE TERM STRUCTURE

APPENDIX F: THE RELATIONSHIP BETWEEN SPOT AND PAR RATES AND THE SLOPE OF THE TERM STRUCTURE

APPENDIX G: MATURITY, PRESENT VALUE, AND FORWARD RATES

Chapter 3: Returns, Spreads, and Yields

DEFINITIONS

COMPONENTS OF P&L AND RETURN

CARRY-ROLL-DOWN SCENARIOS

APPENDIX A: YIELD ON SETTLEMENT DATES OTHER THAN COUPON PAYMENT DATES

APPENDIX B: P&L DECOMPOSITION ON DATES OTHER THAN COUPON PAYMENT DATES

Part Two: Measures of Interest Rate Risk and Hedging

Chapter 4: One-Factor Risk Metrics and Hedges

DV01

A HEDGING APPLICATION, PART I: HEDGING A FUTURES OPTION

DURATION

CONVEXITY

A HEDGING APPLICATION, PART II: A SHORT CONVEXITY POSITION

ESTIMATING PRICE CHANGES AND RETURNS WITH DV01, DURATION, AND CONVEXITY

CONVEXITY IN THE INVESTMENT AND ASSET-LIABILITY MANAGEMENT CONTEXTS

MEASURING THE PRICE SENSITIVITY OF PORTFOLIOS

YIELD-BASED RISK METRICS

APPLICATION: THE BARBELL VERSUS THE BULLET

Chapter 5: Multi-Factor Risk Metrics and Hedges

KEY-RATE ’01S AND DURATIONS

PARTIAL ’01S AND PV01

FORWARD-BUCKET ’01S

MULTI-FACTOR EXPOSURES AND MEASURING PORTFOLIO VOLATILITY

APPENDIX: SELECTED DETERMINANTS OF FORWARD-BUCKET ’01S

Chapter 6: Empirical Approaches to Risk Metrics and Hedging

SINGLE-VARIABLE REGRESSION-BASED HEDGING

TWO-VARIABLE REGRESSION-BASED HEDGING

LEVEL VERSUS CHANGE REGRESSIONS

PRINCIPAL COMPONENTS ANALYSIS

APPENDIX A: THE LEAST-SQUARES HEDGE MINIMIZES THE VARIANCE OF THE P&L OF THE HEDGED POSITION

APPENDIX B: CONSTRUCTING PRINCIPAL COMPONENTS FROM THREE RATES

Part Three: Term Structure Models

Chapter 7: The Science of Term Structure Models

RATE AND PRICE TREES

ARBITRAGE PRICING OF DERIVATIVES

RISK-NEUTRAL PRICING

ARBITRAGE PRICING IN A MULTI-PERIOD SETTING

EXAMPLE: PRICING A CONSTANT-MATURITY TREASURY SWAP

OPTION-ADJUSTED SPREAD

PROFIT AND LOSS ATTRIBUTION WITH AN OAS

REDUCING THE TIME STEP

FIXED INCOME VERSUS EQUITY DERIVATIVES

Chapter 8: The Evolution of Short Rates and the Shape of the Term Structure

INTRODUCTION

EXPECTATIONS

VOLATILITY AND CONVEXITY

RISK PREMIUM

A MATHEMATICAL DESCRIPTION OF EXPECTATIONS, CONVEXITY, AND RISK PREMIUM

APPLICATION: EXPECTATIONS, CONVEXITY, AND RISK PREMIUM IN USD AND JPY SWAP MARKETS

APPENDIX: PROOF OF EQUATION (8.26)

Chapter 9: The Art of Term Structure Models: Drift

MODEL 1: NORMALLY DISTRIBUTED RATES AND NO DRIFT

MODEL 2: DRIFT AND RISK PREMIUM

THE HO-LEE MODEL: TIME-DEPENDENT DRIFT

DESIRABILITY OF FITTING TO THE TERM STRUCTURE

THE VASICEK MODEL: MEAN REVERSION

Chapter 10: The Art of Term Structure Models: Volatility and Distribution

TIME-DEPENDENT VOLATILITY: MODEL 3

THE COX-INGERSOLL-ROSS AND LOGNORMAL MODELS: VOLATILITY AS A FUNCTION OF THE SHORT RATE

TREE FOR THE ORIGINAL SALOMON BROTHERS MODEL

THE BLACK-KARASINSKI MODEL: A LOGNORMAL MODEL WITH MEAN REVERSION

APPENDIX: CLOSED-FORM SOLUTIONS FOR SPOT RATES

Chapter 11: The Gauss+ and LIBOR Market Models

THE GAUSS+ MODEL

SOLUTION AND ESTIMATION

USD AND EUR SAMPLE RESULTS

MODEL EXTENSIONS

THE LIBOR MARKET MODEL

APPENDIX A: EQUIVALENCE OF THE CASCADE AND REDUCED FORMS OF THE GAUSS+ MODEL

APPENDIX B: THE FUNCTION FOR THE GAUSS+ MODEL

APPENDIX C: ESTIMATING THE PARAMETERS OF THE GAUSS+ MODEL

APPENDIX D: FITTING THE INITIAL TERM STRUCTURE IN THE GAUSS+ MODEL

APPENDIX E: DRAWING RANDOM NUMBERS FROM A MULTIVARIATE NORMAL DISTRIBUTION

Part Four: Selected Securities and Topics

Chapter 12: Repurchase Agreements and Financing

REPURCHASE AGREEMENTS: STRUCTURE AND USES

REPO, LIQUIDITY MANAGEMENT, AND THE FINANCIAL CRISIS OF 2007–2009

GENERAL AND SPECIAL REPO RATES

Chapter 13: Forwards and Futures: Preliminaries

FORWARD CONTRACTS AND FORWARD PRICES

THE FORWARD DROP AND CASH CARRY

FORWARD BOND YIELDS

FORWARD SWAP RATES

INTEREST RATE SENSITIVITY OF FORWARDS

DAILY SETTLEMENT OF FUTURES CONTRACTS

FORWARD AND FUTURES PRICES IN A TERM STRUCTURE MODEL

THE FUTURES-FORWARD DIFFERENCE

FORWARD RATES VERSUS FUTURES ON RATES

TAILS

Chapter 14: Note and Bond Futures

MECHANICS

COST OF DELIVERY AND THE DETERMINATION OF THE FINAL SETTLEMENT PRICE

MOTIVATIONS FOR A DELIVERY BASKET AND CONVERSION FACTORS

IMPERFECTION OF CONVERSION FACTORS AND THE DELIVERY OPTION AT EXPIRATION

GROSS AND NET BASIS

THE QUALITY OPTION BEFORE DELIVERY

SOME NOTES ON PRICING THE QUALITY OPTION IN TERM STRUCTURE MODELS

THE TIMING OPTION

THE END-OF-MONTH OPTION

TRADING CASE STUDY: NOVEMBER ’08 BASIS INTO TYM0 (JUNE 2000)

Chapter 15: Short-Term Rates and Their Derivatives

LIBOR AND LIBOR-RELATED SECURITIES

FED FUNDS AND RELATED SECURITIES

LIBOR-OIS AS AN INDICATOR OF FINANCIAL STRESS

TRADING CASE STUDY: SHORTING THE TED SPREAD OF THE 13/8S OF MARCH 15, 2012

Chapter 16: Swaps

SWAP CASH FLOWS

THE VALUATION OF SWAPS

A NOTE ON THE INTEREST RATE RISK OF SWAPS

ON CREDIT RISK AND INTEREST RATE SWAPS

MAJOR USES OF INTEREST RATE SWAPS

THE REGULATORY AND LEGISLATIVE MANDATES TO CLEAR OVER-THE-COUNTER DERIVATIVES

BASIS SWAPS AND SPREADS

CONSTANT MATURITY SWAPS

APPENDIX: DERIVATION OF CONVEXITY CORRECTION FOR CMS SWAPS

Chapter 17: Arbitrage with Financing and Two-Curve Discounting

BOND TRADING WITH FINANCING

BOND ARBITRAGE WITH FINANCING

SWAP TRADING WITH FINANCING

SWAP ARBITRAGE WITH FINANCING

PRICING A USD LIBOR SWAP WITH FED FUNDS AS THE INVESTABLE AND COLLATERAL RATE

APPENDIX A: ARBITRAGE RELATIONSHIPS ACROSS BONDS AND SWAPS WITH FINANCING

APPENDIX B: PRICING SWAPS WITH THE TWO-CURVE APPROACH

Chapter 18: Fixed Income Options

CAPS AND FLOORS

SWAPTIONS

BOND OPTIONS

EURODOLLAR AND EURIBOR FUTURES OPTIONS

BOND FUTURES OPTIONS

SUMMARY OF APPLYING BS TO FIXED INCOME OPTIONS

SWAPTION SKEW

THEORETICAL FOUNDATIONS FOR APPLYING BLACK-SCHOLES TO SELECTED FIXED INCOME OPTIONS

APPENDIX A: EXPECTATIONS FOR BLACK-SCHOLES-STYLE OPTION PRICING

APPENDIX B: EARLY EXERCISE OF AMERICAN-STYLE FUTURES OPTIONS

APPENDIX C: FUTURES PRICES ARE MARTINGALES WITH THE MONEY MARKET ACCOUNT AS A NUMERAIRE

Chapter 19: Corporate Bonds and Credit Default Swaps

CORPORATE SECURITIES

RATINGS, DEFAULT, AND RECOVERY

CREDIT SPREADS

CREDIT SPREADS AND DEFAULT RATES

CREDIT DEFAULT SWAPS

APPENDIX A: CUMULATIVE DEFAULT RATES

APPENDIX B: CDS-BOND BASIS AS THE DIFFERENCE BETWEEN THE CDS SPREAD AND THE PAR ASSET SWAP SPREAD

Chapter 20: Mortgages and Mortgage-Backed Securities

MORTGAGE LOANS

MORTGAGE-BACKED SECURITIES

PREPAYMENT MODELING

MBS VALUATION AND TRADING

PRICE-RATE BEHAVIOR OF MBS

HEDGING REQUIREMENTS OF SELECTED MORTGAGE MARKET PARTICIPANTS

Chapter 21: Curve Construction

INTRODUCTION

FLAT FORWARDS

FLAT FORWARDS FOR A USD LIBOR CURVE

SMOOTH FORWARDS BY PIECEWISE QUADRATICS

LOCALITY PROPERTIES AND HEDGING

References

Index

Additional Praise forFixed Income Securities: Tools for Today's Markets, 3rd Edition

“The coverage of fixed income markets and instruments is even better than in previous editions while the book retains the same clarity of exposition via extensive, carefully worked examples. An outstanding textbook that is extensively used by practitioners is something special. This is indeed the standout text on fixed income.”

Stephen M. Schaefer, Professor of Finance, London Business School

“This is a terrific reference text that combines a strong conceptual framework with real-world pricing and hedging applications. It is a must-read for any serious investor in fixed income markets.”

Terry Belton, Global Head of Fixed Income Research, JPMorgan

“This outstanding book achieves the perfect balance between presenting the foundational principles of fixed income markets and providing interesting and insightful practical applications. This classic is required reading for anyone interested in understanding fixed income markets.”

Francis Longstaff, Allstate Professor of Insurance and Finance, The Anderson School at UCLA

“This is a great book. It covers the most current issues in fixed income and reflects the authors' deep understanding of the markets grounded in the theory of finance and many years of practical experience.”

Ardavan Nozari, Treasurer, Citigroup Global Markets Holding Inc.

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For a list of available titles, visit our Web site at www.WileyFinance.com.

Copyright © 2012 by Bruce Tuckman and Angel Serrat. All rights reserved.

Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada.

No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 646-8600, or on the Web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at www.wiley.com/go/permissions.

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Library of Congress Cataloging-in-Publication Data:

Tuckman, Bruce. Fixed income securities : tools for today's markets / Bruce Tuckman, Angel Serrat. – 3rd ed. p. cm. – (Wiley finance series) Includes index. ISBN 978-0-470-89169-8 (hardback) ISBN 978-0-470-90403-9 (paperback); ISBN 978-1-118-13394-1 (ebk); ISBN 978-1-118-13395-8 (ebk); ISBN 978-1-118-13396-5 (ebk) 1. Fixed-income securities. I. Serrat, Angel. II. Title. HG4650.T83 2012 332.63′2044–dc23 2011037178

Preface to the Third Edition

The goal of this book is to present conceptual frameworks for pricing and hedging a broad range of fixed income securities in an intuitive, mathematically simple, and applied manner. Conceptual frameworks are necessary so as to connect ideas across products and to learn new material more easily. An intuitive and mathematically simple approach is certainly useful to students and practitioners without very advanced mathematical training, but it is also really a good way for everyone to learn new material. Finally, an applied approach is crucial for several reasons. First, examples go a long way in solidifying conceptual understanding. The introduction of practically every concept in this book is followed by an example taken from the markets or, at the very least, by an appropriately calibrated example. Second, important details emerge from applications. Third, only by working through real or realistic examples can orders of magnitude be learned and appreciated. For example, a study of DV01 is not complete without having absorbed that the sensitivity of a 10-year bond is about 8 cents per 100 face amount per basis point, as opposed to 0.8 cents, 80 cents, or 8 dollars.

The book begins with an Overview of global fixed income markets. This section provides institutional descriptions of securities and market participants along with data designed to illustrate absolute and relative sizes of markets and players. A well-informed fixed income market professional has some idea about how central banks around the world have reacted to the financial crisis of 2007–2009 and can say whether the size of the mortgage market in the United States is one-tenth the size of GDP, about equal to GDP, or 10 times GDP.

For securities with fixed cash flows, Part One of the book presents the relationships across prices, spot rates, forward rates, returns, and yields. The fundamental notion of arbitrage pricing is introduced and is central to the analysis. Part Two describes how to measure and hedge interest rate risk, covering one-factor metrics, namely, DV01, duration, and convexity (in both their general and yield-based forms); two-factor metrics like key-rate ’01s, partial PV01s, and forward bucket ’01s; and empirical methods like regression and principal component analysis.

Part Three turns to the arbitrage pricing of contingent claims, i.e., of securities with cash flows that depend on interest rates, like options. The science of arbitrage pricing in this context is followed by a framework in which to think about the shape of the term structure of interest rates in terms of expectations, risk premium, and convexity. One-factor term structure models are then described, to be used both in their own right, when appropriate, and as building blocks toward more sophisticated models. Chapter 11, the last chapter in Part Three, has two parts. First, it presents a multi-factor model for use in relative value applications, along with suggestions for estimating its parameters empirically. Second, it introduces the LIBOR Market Model, an extremely popular model for pricing exotic derivatives, in a particularly accessible manner.

Finally, Part Four applies the knowledge gained in the previous three parts to present and analyze a broad and extensive range of fixed income topics and products including repo, bond and note futures, rate futures, swaps, options, corporate bonds and credit default swaps (CDS), and mortgage-backed securities.

This edition substantially revises and expands the second. The only parts of the book that have remained essentially unchanged are Chapters 7 through 10 on pricing contingent claims with one-factor term structure models. The rest of the material that was in the second edition has been updated and, with the exception of a couple of particularly interesting case studies, the numerical illustrations, examples, and applications are all new. In addition, several chapters in this third edition are completely new and others significantly expanded. New chapters include the Overview, Chapter 17 on how the realities of financing have changed the practice of discounting cash flows, and Chapter 19 on corporate bond and CDS markets. Significantly expanded chapters include Chapter 6 on empirical hedging, which now includes principal component analysis; Chapter 11, which was discussed above; Chapter 18 on volatility and fixed income options, which now covers a very broad range of products, Black-Scholes pricing, and a mathematically simple introduction to martingale pricing; and Chapter 20, on mortgages and mortgage-backed securities, which takes a much more market-oriented approach and adds material on pool characteristics, TBAs, and dollar rolls.

Acknowledgments

The authors would like to thank the following people for helpful comments and suggestions: Amitabh Arora, Larry Bernstein, John Feraca, Lawrence Goodman, Jean-Baptiste Homé, Dick Kazarian, Peyron Law, Marco Naldi, Chris Striesow, and Doug Whang. We would like to thank Helen Edersheim for carefully reviewing the manuscript and sparing readers from phrases like “options wroth about $2” and the like. Finally, we would like to thank Bill Falloon, Meg Freeborn, and Natasha Andrews-Noel at Wiley for their patience and support throughout the process of putting this book together.

An Overview of Global Fixed Income Markets

This overview begins with a snapshot of fixed income markets across the globe and continues with concise reviews of fixed income markets in the United States, Europe, and Japan. These reviews have three goals: one, to describe how households and institutions achieve their borrowing and investing objectives through fixed income markets; two, to highlight the magnitude of the amounts of securities outstanding and of the balance sheets of market participants; and three, to emphasize the themes that are particularly relevant and significant for understanding today’s markets.

A SNAPSHOT OF GLOBAL FIXED INCOME MARKETS

While fixed income markets are truly global, the vast majority of securities originate in the United States, Europe, and Japan. Figure 0.1 shows the notional amounts outstanding of debt securities by residence of issuer, arranged in order of decreasing size. The largest markets by far are in the United States, the Eurozone, Japan, and the United Kingdom. (The Eurozone includes countries that are part of the European Union and also use the Euro as their currency.) The amounts outstanding for many Eurozone countries are shown individually in the graph, and indicated with asterisks, because several of these markets rank among the largest in the world on their own.

Figure 0.1 Debt Securities by Residence of Issuer as of March 2010

Source: Bank for International Settlements.

Derivative securities do not have an issuer in the same sense as do debt securities, but the distribution of the notional amounts of over-the-counter (OTC) interest rate derivatives across currencies tells a story similar to that of Figure 0.1. According to Figure 0.2, which shows amounts outstanding of single-currency, OTC interest rate derivatives, markets are dominated by contracts denominated in EUR (Euro), USD (United States dollar), JPY (Japanese Yen), and GPB (British Pound).1 And with respect to exchange-traded derivatives, Table 0.1 shows that Europe and North America comprise almost all of the outstanding notional amount.

Figure 0.2 Amounts Outstanding of OTC Single-Currency Interest Rate Derivatives as of December 2009

Source: Bank for International Settlements.

Table 0.1 Exchange-Traded Interest Rate Derivatives as of March 2010, in Billions of U.S. Dollars

Source: Bank for International Settlements.

RegionNotionalEurope27,807North America22,604Asia and Pacific10Other934

It is worthwhile noting that Figures 0.1, 0.2, and Table 0.1 report the place of origination of fixed income securities rather than the place of residence of the ultimate owners or counterparties. So, to take one of the more significant examples, China’s ownership of nearly $850 billion of U.S. Treasury securities does not appear anywhere in Figure 0.1. Nevertheless, even accounting for such instances, the data presented here justify this book’s focus on fixed income securities and markets in the United States, Europe, and Japan.

As a final note before turning to the three overviews, Figure 0.3 gives a coarse breakdown of the composition of debt securities in the United States, the Eurozone, Japan, and the United Kingdom. (The totals are the same as those reported in Figure 0.1.) While the proportions of debt issued by governments, financial institutions, and corporations are similar in the United States and the Eurozone, debt markets in Japan are dominated by governments while those in the United Kingdom are dominated by the issues of financial institutions.

Figure 0.3 Debt Securities by Residence of Issuer and Sector as of March 2010

Source: Bank for International Settlements.

FIXED INCOME MARKETS IN THE UNITED STATES

Securities and Other Assets

Figure 0.4 shows the major categories of credit market debt in the United States, along with the size of the market for each, as of March 31, 2010.2 Due to the definition of credit market debt in this cut of the data, several assets are not explicitly mentioned here (e.g., deposits, money-market fund shares, security repurchase agreements, insurance and pension reserves, equities), but will be included in the discussions of households and institutions later in this section.

Figure 0.4 Credit Market Debt in the United States as of March 2010

Source: Flow of Funds Accounts of the United States.

Mortgages

The largest single category of debt in the United States is mortgages, at a size of $14.2 trillion. A mortgage is a loan secured by property, so that if a borrower fails to make the payments required by a mortgage, the lender has a claim on the property itself. Exercising this claim, the lender could keep proceeds from the sale of the property up to the amount still owed; or the lender could seize or on the property, sell it, and recover the outstanding loan amount that way. In practice there might be restrictions on the immediate or full exercise of this claim, like bankruptcy and other borrower protections or any tax liens on the same property. Finally, depending on the laws of the relevant state, the lender might or might not have to the borrower’s other assets to collect any remaining amount owed after the sale of the property.

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