Table Of Content

Lecture Notes in Economics and Mathematical Systems 605 FoundingEditors: M.Beckmann H.P.Künzi ManagingEditors: Prof.Dr.G.Fandel FachbereichWirtschaftswissenschaften FernuniversitätHagen Feithstr.140/AVZII,58084Hagen,Germany Prof.Dr.W.Trockel InstitutfürMathematischeWirtschaftsforschung(IMW) UniversitätBielefeld Universitätsstr.25,33615Bielefeld,Germany EditorialBoard: A.Basile,A.Drexl,H.Dawid,K.Inderfurth,W.Kürsten Michael Puhle Bond Portfolio Optimization 123 Dr.MichaelPuhle AllianzGlobalInvestorsKapitalanlagegesellschaftmbH NymphenburgerStraße112-116 80636Munich Germany [email protected] Doctoralthesis,UniversityofPassau,2007 ISBN978-3-540-76592-9 e-ISBN978-3-540-76593-6 DOI10.1007/978-3-540-76593-6 LectureNotesinEconomicsandMathematicalSystemsISSN0075-8442 LibraryofCongressControlNumber:2007938963 ©2008 Springer-VerlagBerlinHeidelberg Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerial is concerned, specificallythe rights of translation, reprinting, reuseof illustrations, recitation, broadcasting,reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplication ofthispublicationorpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyright LawofSeptember9,1965,initscurrentversion,andpermissionforusemustalwaysbeobtained fromSpringer.ViolationsareliabletoprosecutionundertheGermanCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoes notimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Production:LE-TEXJelonek,Schmidt&VöcklerGbR,Leipzig Coverdesign:WMXDesignGmbH,Heidelberg Printedonacid-freepaper 987654321 springer.com Acknowledgements Iwouldliketoexpressmygratitudetoanumberofpeople.Firstofall,Ithank my thesis advisor Prof. Dr. Jochen Wilhelm for supervising and guiding me during my years at the chair of finance at Passau University. Prof. Dr. János Száz gave me the opportunity to present previous versions at seminars in Budapest. He also provided valuable feedback and agreed to be the second referee. Prof. Dr. Bernhard Nietert took the time to discuss several parts of this thesis with me. Dipl.-Kffr. Marion Trautbeck-Kim and Dipl.-Kfm. Andreas Kremer read a draft of this thesis on short notice. Furthermore, I would like to thank Dr. Wolfram Peters for giving me the opportunity to complete this thesis during my first months at Allianz Global Investors. My parents and my brother supported this project from the beginning and provided warm encouragement during difficult days. Last but not least, I would like to thank my fiancée Veronika for her support especially during the final stages. Contents Acknowledgements ............................................ V Abbreviations ................................................. XI Commonly Used Symbols......................................XIII 1 Introduction............................................... 1 2 Bond Market Terminology................................. 5 2.1 Characteristics of Bonds.................................. 5 2.2 Interest Rates........................................... 6 2.3 Term Structure of Interest Rates .......................... 8 2.4 Estimating the Term Structure of Interest Rates............. 9 2.5 Classical Theories of the Term Structure of Interest Rates .... 10 2.6 Arbitrage-Free Term Structure Theories .................... 11 2.7 Empirical Properties of the Term Structure of Interest Rates.. 11 3 Term Structure Modeling in Continuous Time............. 13 3.1 Introduction ............................................ 13 3.2 Interest Rate Modeling Approaches ........................ 14 3.3 Heath/Jarrow/Morton (1992) ............................. 17 3.3.1 Introduction ...................................... 17 3.3.2 Dynamics of Traded Securities ...................... 18 3.3.3 Arbitrage-Free Pricing ............................. 19 3.3.4 Excursus: The HJM Drift Condition ................. 20 3.3.5 The Short Rate of Interest.......................... 21 3.3.6 Special Cases ..................................... 22 3.4 Vasicek (1977) .......................................... 23 3.4.1 Introduction ...................................... 23 3.4.2 Derivation of Zero-Coupon Bond Prices .............. 23 3.4.3 Properties ........................................ 26 VIII Contents 3.5 Hull/White (1994)....................................... 30 3.5.1 Introduction ...................................... 30 3.5.2 Derivation of Zero-Coupon Bond Prices .............. 31 3.5.3 Properties ........................................ 35 3.6 Summary and Conclusion ................................ 39 4 Static Bond Portfolio Optimization ........................ 41 4.1 Introduction ............................................ 41 4.2 Static Bond Portfolio Selection in Theory................... 41 4.2.1 A Short Review of Modern Portfolio Theory .......... 41 4.2.2 Application to Bond Portfolios ...................... 43 4.2.3 Obtaining the Parameters .......................... 48 4.2.4 One-Factor Vasicek (1977) Model.................... 51 4.2.5 Two-Factor Hull/White (1994) Model................ 60 4.3 Static Bond Portfolio Selection in Practice.................. 66 4.3.1 Introduction ...................................... 66 4.3.2 Active Bond Portfolio Selection Strategies ............ 67 4.3.3 Passive Bond Portfolio Selection Strategies ........... 77 4.3.4 Summary and Conclusion .......................... 82 5 Dynamic Bond Portfolio Optimization in Continuous Time 85 5.1 Introduction ............................................ 85 5.2 Bond Portfolio Selection Problem in a HJM Framework ...... 87 5.2.1 Dynamics of Prices and Wealth ..................... 87 5.2.2 The Hamilton/Jacobi/Bellman Equation ............. 89 5.2.3 Derivation of Optimum Portfolio Weights............. 91 5.2.4 The Value Function for CRRA Utility Functions ...... 94 5.3 Special Cases ........................................... 96 5.3.1 One-Factor Vasicek (1977) Model.................... 96 5.3.2 Two-Factor Hull/White (1994) Model................100 5.4 International Bond Investing..............................105 5.4.1 Introduction ......................................105 5.4.2 Model Setup......................................106 5.4.3 Derivation of the Optimum Portfolio Weights .........109 5.4.4 Interpretation of the Optimum Portfolio Weights ......111 5.4.5 Numerical Example................................112 5.5 Summary and Conclusion ................................113 6 Summary and Conclusion..................................115 A Heath/Jarrow/Morton (1992) .............................119 A.1 Dynamics of Zero-Coupon Bonds ..........................119 A.2 Arbitrage-Free Pricing ...................................120 A.3 HJM Drift Condition ....................................121 A.4 Special Case: Hull/White (1994) ..........................122 Contents IX B Dynamic Bond Portfolio Optimization .....................123 C Dynamic Bond Portfolio Optimization .....................125 C.1 Vasicek (1977) ..........................................125 C.2 Hull/White (1994).......................................125 C.3 International Bond Portfolio Selection......................126 References.....................................................127 List of Tables ..................................................135 List of Figures.................................................137 Abbreviations CIR Cox/Ingersoll/Ross (1985) CRRA Constant relative risk aversion EUR Currency code for the Euro HJB Hamilton-Jacobi-Bellman equation HJM Heath/Jarrow/Morton (1992) HW2 Hull/White (1994) InvG Investmentgesetz (German Investment Act) MA Martingale approach MMA Money market account ODE Ordinary differential equation PDE Partial differential equation RRA Relative risk aversion SCA Stochastic control approach SDE Stochastic differential equation USD Currency code US Dollar e.g. Exempli gratia, for example ff. And following pages i.e. Id est, that is max Maximize min Minimize p. Page pp. Pages s.t. Subject to std. Standard deviation w.r.t. With respect to Commonly Used Symbols t Time T Maturity date or investment horizon R(t,T) Continuously-compounded spot interest rate from t to T P(t,T) Price at time t for a zero-coupon bond with maturity date T µ Drift of the zero-coupon bond price σ i-th volatility of the zero-coupon bond price, 1≤i≤d i λ i-th market price of interest rate risk, 1≤i≤d i r(t) Short rate of interest, r(t)=f(t,t) α Drift of the short rate σ Volatility of the short rate r R(T,τ) Forward interest rate set at time t for a loan that starts at time t T and is to be repaid at time τ f(t,T) Instantaneous forward rate set at time t for a loan that begins at time T and is to be repaid an instant later m Drift of the instantaneous forward rates s i-th volatility of the instantaneous forward rates, 1≤i≤d i dt Instantaneous time period ∆t Short time period dz Vector of Brownian motion increments (d×1) d Number of Brownian motions B(t) Value at time t of a money market account with B(0)=1 E[x] Expectation of x std(x) Standard deviation of x var(x) Variance of x cov(x,y) Covariance between x and y corr(x,y) Correlation between x and y ζ Stochastic discount factor a Drift of the stochastic discount factor b i-th volatility of the stochastic discount factor, 1≤i≤d i N Holdings vector XIV Commonly Used Symbols Nˆ Holdings vector of risky zero-coupon bonds W Wealth at time t t n Number of assets τ Maximum maturity of zero-coupon bonds Pˆ Vector of current prices of risky zero-coupon bonds 0 C Covariance matrix ε Second factor in HW2 model ρ Correlation between r and ε in HW2 model θ Mean reversion level κ Mean reversion speed of short rate in Vasicek and HW2 model r κ Mean reversion speed of ε in HW2 model ε σ Volatility of ε in HW2 model ε u(W) Utility of wealth function γ Risk aversion parameter in CRRA utility function x Vector of state variables (d×1) d Number of state variables α Drift vector of state variables β Volatility matrix of state variables σ Matrix of volatilities of bond price returns (n×d) t w Portfolio weights vector w∗ Optimum portfolio weights vector J Optimal value function J Partial derivative of J with respect to x x I Identity matrix