## Bond Portfolio Optimization1 The tools of modern portfolio theory are in general use in the equity markets, either in the form of portfolio optimization software or as an accepted frame- 2 work in which the asset managers think about stock selection. In the ?xed income market on the other hand, these tools seem irrelevant or inapplicable. Bond portfolios are nowadays mainly managed by a comparison of portfolio 3 4 risk measures vis ¶a vis a benchmark. The portfolio manager’s views about the future evolution of the term structure of interest rates translate th- selves directly into a positioning relative to his benchmark, taking the risks of these deviations from the benchmark into account only in a very crude 5 fashion, i.e. without really quantifying them probabilistically. This is quite surprising since sophisticated models for the evolution of interest rates are commonly used for interest rate derivatives pricing and the derivation of ?xed 6 income risk measures. Wilhelm (1992) explains the absence of modern portfolio tools in the ?xed 7 income markets with two factors: historically relatively stable interest rates and systematic di?erences between stocks and bonds that make an application of modern portfolio theory di–cult. These systematic di?erences relate mainly to the ?xed maturity of bonds. Whereas possible future stock prices become more dispersed as the time horizon widens, the bond price at maturity is 8 ?xed. This implies that the probabilistic models for stocks and bonds have 1 Starting with the seminal work of Markowitz (1952). |

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### Contents

2 | |

6 | |

7 | |

9 | |

24 Estimating the Term Structure of Interest Rates | 10 |

25 Classical Theories of the Term Structure of Interest Rates | 11 |

26 ArbitrageFree Term Structure Theories | 12 |

Term Structure Modeling in Continuous Time | 13 |

433 Passive Bond Portfolio Selection Strategies | 77 |

434 Summary and Conclusion | 82 |

Dynamic Bond Portfolio Optimization in Continuous Time | 85 |

52 Bond Portfolio Selection Problem in a HJM Framework | 87 |

522 The HamiltonJacobiBellman Equation | 89 |

523 Derivation of Optimum Portfolio Weights | 91 |

524 The Value Function for CRRA Utility Functions | 94 |

53 Special Cases | 96 |

32 Interest Rate Modeling Approaches | 14 |

33 HeathJarrowMorton 1992 | 17 |

332 Dynamics of Traded Securities | 18 |

333 ArbitrageFree Pricing | 19 |

The HJM Drift Condition | 20 |

335 The Short Rate of Interest | 21 |

336 Special Cases | 22 |

34 Vasicek 1977 | 23 |

343 Properties | 26 |

35 HullWhite 1994 | 30 |

352 Derivation of ZeroCoupon Bond Prices | 31 |

353 Properties | 35 |

36 Summary and Conclusion | 39 |

Static Bond Portfolio Optimization | 41 |

422 Application to Bond Portfolios | 43 |

423 Obtaining the Parameters | 48 |

424 OneFactor Vasicek 1977 Model | 51 |

425 TwoFactor HullWhite 1994 Model | 60 |

43 Static Bond Portfolio Selection in Practice | 66 |

432 Active Bond Portfolio Selection Strategies | 67 |

532 TwoFactor HullWhite 1994 Model | 100 |

54 International Bond Investing | 105 |

542 Model Setup | 106 |

543 Derivation of the Optimum Portfolio Weights | 109 |

544 Interpretation of the Optimum Portfolio Weights | 111 |

545 Numerical Example | 112 |

55 Summary and Conclusion | 113 |

Summary and Conclusion | 114 |

HeathJarrowMorton 1992 | 119 |

A2 ArbitrageFree Pricing | 120 |

A3 HJM Drift Condition | 121 |

HullWhite 1994 | 122 |

Dynamic Bond Portfolio Optimization | 123 |

Dynamic Bond Portfolio Optimization | 124 |

C3 International Bond Portfolio Selection | 126 |

127 | |

List of Tables | 134 |

136 | |

### Common terms and phrases

analyze arbitrage arbitrage-free assets assume bond market Bond Portfolio Optimization bond portfolio selection Brownian motions Chapter continuous-time correlation coupon bonds covariance matrix different maturities duration strategy dynamic term structure Equation expected values Fabozzi fixed income forward rate curve Heath/Jarrow/Morton 1992 hedge portfolio hence HJM framework Hull/White HW2 model instantaneous forward rates interest rate models interest rate risk introduced investor Itô's lemma Korn/Kraft Macaulay durations market prices Markowitz Martellini/Priaulet/Priaulet 2003 martingale mean-variance efficient portfolios minimum-variance portfolio modern portfolio theory money market account normally distributed numerical example optimum portfolio weights parameter values portfolio selection problem prices of interest short rate models short-sale constrained portfolios solution spot interest rate spot rate stochastic discount factor structure of interest Table term structure model terminal wealth two-factor variables variance Vasicek model volatility structure Wilhelm yield curve yield curve strategies zero zero-coupon bond prices Zero-coupon bond weights

### Popular passages

Page 127 - Crane, DB, 1972. A dynamic model for bond portfolio management.

Page 132 - Dynamic Asset Allocation and Fixed Income Management." Journal of Financial and Quantitative Analysis 34: 513-531. Vasicek, O. (1977) "An Equilibrium Characterization of the Term Structure.

Page 7 - It is equal to the dirty price minus accrued interest. The accrued interest is equal to the amount of the next coupon payment multiplied by the proportion of the current inter-coupon period so far elapsed, ie the buyer of the bond "compensates...

Page 7 - The dirty price is the actual amount in return for the right to the full amount of each future coupon payment and the redemption proceeds.