Table Of ContentLecture Notes in Economics
and Mathematical Systems 605
FoundingEditors:
M.Beckmann
H.P.Künzi
ManagingEditors:
Prof.Dr.G.Fandel
FachbereichWirtschaftswissenschaften
FernuniversitätHagen
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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
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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. J´anos
Sz´az 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 fianc´ee 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
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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
