Sustainable asset accumulation and dynamic portfolio decisions

Dynamic Modeling and Econometrics in
Economics and Finance 18

Carl Chiarella
Willi Semmler
Chih-Ying Hsiao
Lebogang Mateane

Sustainable Asset
Accumulation and
Dynamic Portfolio
Decisions


Dynamic Modeling and Econometrics
in Economics and Finance

Volume 18

Editors
Stefan Mittnik
Ludwig Maximillian University Munich
Munich, Germany
Willi Semmler
Bielefeld University
Bielefeld, Germany
and
New School for Social Research
New York, USA

More information about this series at />

Carl Chiarella • Willi Semmler • Chih-Ying Hsiao •
Lebogang Mateane

Sustainable Asset
Accumulation and Dynamic
Portfolio Decisions

123


Carl Chiarella
School of Finance and Economics
University of Technology
Sydney, New South Wales, Australia

Willi Semmler
Henry Arnhold Professor of Economics
Department of Economics
New School for Social Research
New York, NY, USA
and
Bielefeld University
Bielefeld, Germany

Chih-Ying Hsiao
School of Finance and Economics
University of Technology
Sydney, New South Wales, Australia

Lebogang Mateane
Department of Economics
New School for Social Research
New York, NY, USA

ISSN 1566-0419
ISSN 2363-8370 (electronic)
Dynamic Modeling and Econometrics in Economics and Finance
ISBN 978-3-662-49228-4
ISBN 978-3-662-49229-1 (eBook)
DOI 10.1007/978-3-662-49229-1
Library of Congress Control Number: 2016942551
© Springer-Verlag Berlin Heidelberg 2016
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Preface

The global economy experienced a worldwide meltdown of asset markets in the
years 2007–2009. This posed great challenges for asset and portfolio managers.
Many funds such as university endowments, sovereign wealth funds, and pension
funds were overexposed to risky returns and suffered considerable losses. On the
other hand, the long-run upswing in the stock market since 2010, induced by a
monetary policy of quantitative easing in the USA, and later in Europe and Asia,
led to asset price booms and new wealth formation. In both cases quite significant
differences in asset management and wealth accumulation were visible. Our book
aims at dealing with sustainable wealth formation and dynamic decision making.
We have three perspectives in mind.
A first perspective is how wealth formation and the proper management of
financial funds can help to buffer income risk sufficiently and to obtain adequate
risk-free income at a later stage of life. This is an important concern in the current
public debate on asset accumulation and wealth disparity. In whatever institutional
form saving takes place, in mutual funds, public pension funds, corporate pension
funds, or private saving accounts, the generic issue is how much to save and invest
and how to make proper asset allocation decisions.
A second important issue for sustainable wealth accumulation is that many
agents and institutions in financial markets tend to put some constraints on the
accumulation and allocation of assets—following some rules, guidelines and restrictions concerning risk-taking, safeness of investments, as well as social, ethical,
environmental, and climate change aspects. Thus investments are often restricted
to certain risk classes, classes of assets or particular assets. Much investment and
asset allocation decisions are therefore made following behavioral and institutional
rules, responding to some given constraints and guidelines, without necessarily
being optimal in the narrow sense.
A third perspective of sustainable wealth formation is that we want to move more
toward dynamic decision making and dynamic re-balancing of portfolios. Portfolio

decisions are frequently modeled as static decisions problems. Yet, how should the
investors respond to expected future returns, changing return differentials, global or
idiosyncratic risk, change of inflation rates, affecting the real value of their assets,
v


vi

Preface

and so on? In standard literature, the modeling of savings and wealth accumulation
are often separated from asset allocation decisions. We pursue a simultaneous and
dynamic treatment of both savings behavior and portfolio decision making, taking
into account expected returns. Expected returns are evaluated here, using a new
method—harmonic estimations of returns.
In order to solve such dynamic decision problems in portfolio theory and
portfolio practice—solving saving as well as asset accumulation problems
simultaneously—we put forward dynamic programming as a procedure for dynamic
decision making that allows to integrate sustainable wealth accumulation as well as
asset allocation decisions. Although some shortcomings of this procedure exist, a
careful use of it can help to not only undertake dynamic modeling but also aid online
decision making once some pattern of expected returns of different asset classes,
for example estimated through using harmonic estimations, has been recognized.
The book is written in a way that it can be used by researchers and in graduate
classes on financial economics, asset pricing and portfolio theory, finance and
macro, portfolio theory and practice, pension fund theory and management, socially
responsible investment decisions, financial market and wealth disparities, methodology of dynamic portfolio theory, intertemporal asset allocation and households’
saving, and applied dynamic programming.
Parts of the book are based on lectures delivered at the University of Bielefeld,
Germany, the University of Technology, Sydney, Australia, The New School for

Social Research, New York City, USA, and University of Economics, Vienna,
Austria, as well as conferences and workshops at the ZEW, Mannheim, Germany.
We are very grateful to our colleagues at those institutions as well as to several
generations of students who took our classes in this area and gave comments on
these lectures in their formative stages. We are also grateful for discussions with
Hans Amman, Lucas Bernard, Raphaele Chappe, Peter Flaschel, Lars Gruene,
Stefan Mittnik, Unra Nyambuu, Eckhard Platen, and James Ramsey.
Individually, many of the chapters of the book have been presented at conferences, workshops, and seminars throughout the United States, Europe, and
Australia. Many chapters of this book are also based on previous article by the
authors, published with a variety of different coauthors. Each chapter acknowledges
the particular coauthors involved, and a general acknowledgment can be found
below.
In preparing this manuscript, we in particular relied on the help of Tony Bonen
and Uwe Koeller whom we want to thank for extensive assistance in editing this
volume. Willi Semmler wants to thank the Fulbright Foundation for a Fulbright
Professorship at the University of Economics, Vienna, in the Winter Term 2011, as
well as the German Research Foundation for financial support.
Sydney, NSW, Australia
New York, NY, USA
Sydney, NSW, Australia
New York, NY, USA
December 10, 2015

Carl Chiarella
Willi Semmler
Chi-Ying Hsiao
Lebogang Mateane


Acknowledgements

The following material and journal articles have been used as foundations for
various chapters of the book. The chapters have to some extent been reworked in
the light of new developments in their subject areas and are not necessarily identical
in their titles to the ones of the original papers. We thank the editors and publishers
of this material for the permission of reusing it in this book.
• Chap. 4: Semmler, W., L. Gruene and L. Oehrlein (2009), Dynamic Consumption
and Portfolio Decisions with Time Varying Asset Returns, Journal of Wealth
Management, vol. 12, no. 2.
• Chap. 5: Semmler, W. and C-Y. Hsiao (2009), Dynamic Consumption and
Portfolio Decisions with Low Frequency Movements of Asset Returns, Journal
of Wealth Management, vol 14, no 2: 101-111.
• Chap. 6: Semmler, W., ”Dynamic Consumption and Portfolio Decisions with
Estimated Low Frequency Movements of Asset Returns and Labor Income”,
Journal of Wealth Management, vol. 14, no 2:101-111, 2012.
• Chap. 7: Some of this material has been published in the edited volume
“Financial Econometrics Modeling – derivatives pricing, hedge fund and term
structure models”, MacMillan.

vii



Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.1 Institutions, Models and Empirics.. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.2 Dynamic Programming as Solution Method .. . . . . .. . . . . . . . . . . . . . . . . . . .

1.3 Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.4 Outline and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

1
2
4
5
7

2 Forecasting and Low Frequency Movements of Asset Returns .. . . . . . . .
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.2 Limits on Forecasting Asset Returns .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.3 The Use of Periodic Returns. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

9
9
9
14
17

3 Portfolio Modeling with Sustainability Constraints ... . . . . . . . . . . . . . . . . . . .
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.2 Mean-Variance Portfolio Models .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.3 Description of Statistical Properties of Returns Data . . . . . . . . . . . . . . . . .
3.3.1 Computing Expected Real Returns on Risky Assets . . . . . . . . . .
3.3.2 Variance-Covariance and Correlation Matrices
and Volatility of Real Returns . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.3.3 Eigenvalue and Eigenvector Properties
of the Empirical Covariance and Correlation Matrix . . . . . . . . .

3.4 Estimation Results of the Portfolio Models .. . . . . . .. . . . . . . . . . . . . . . . . . . .
3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

19
19
21
27
27

33
37
47
48

4 Dynamic Saving and Portfolio Decisions-Theory . . . . .. . . . . . . . . . . . . . . . . . . .
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
4.2 The Model with One Asset and Constant Returns.. . . . . . . . . . . . . . . . . . . .
4.2.1 Numerical Results for the Benchmark Model .. . . . . . . . . . . . . . . .
4.2.2 Variation of Risk Aversion, Returns and Discount Rate . . . . . .

53
53
53
55
57

30

ix

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Contents

4.3 Dynamic Consumption and Portfolio Decisions: Two
Assets and Time Varying Returns . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
4.3.1 The Model with Time Varying Returns .. . .. . . . . . . . . . . . . . . . . . . .
4.3.2 Numerical Results on a Benchmark Case .. . . . . . . . . . . . . . . . . . . .
4.3.3 Variation of Risk Aversion . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
4.3.4 Variation of Returns . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
4.3.5 Variation of Time Horizon . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
4.4 A Stochastic Model with Mean Reversion in Returns.. . . . . . . . . . . . . . . .
4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

61
62
64
66
67
69
73
76
77

5 Asset Accumulation with Estimated Low Frequency
Movements of Asset Returns.. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

5.2 The Literature and Results . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.3 The Dynamic Programming Solution .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.4 Varying Risk Aversion Across Investors .. . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.5 Varying Time Horizon Across Investors . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.6 Some Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

81
81
82
85
86
90
95

6 Asset Accumulation and Portfolio Decisions with Time
Varying Asset Returns and Labor Income . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.2 Literature and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.3 Business Cycles, Asset Returns and Labor Income . . . . . . . . . . . . . . . . . . .
6.4 Dynamic Decisions on Asset Accumulation .. . . . . .. . . . . . . . . . . . . . . . . . . .
6.5 Wealth Disparities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

97
97
100
103
106
112
113

7 Continuous and Discrete Time Modeling . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
7.2 Literature and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
7.3 Discrete-Time Approximation .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
7.3.1 Euler Method.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
7.3.2 Milstein Method . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
7.3.3 New Local Linearization Method .. . . . . . . . .. . . . . . . . . . . . . . . . . . . .
7.3.4 Equivalence of the Euler and NLL Predictors .. . . . . . . . . . . . . . . .
7.4 Empirical Results on Modeling Short Term Interest Rates . . . . . . . . . . .
7.4.1 Specification Test . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
7.4.2 Results of Estimating CKLS Model . . . . . . .. . . . . . . . . . . . . . . . . . . .
7.5 Searching for New Models . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
7.5.1 Improvement in the Continuous-Time Framework.. . . . . . . . . . .
7.5.2 Modeling Autocorrelations in the Estimated Noise .. . . . . . . . . .
7.5.3 Modeling Thick-Tails in the Estimated Noise .. . . . . . . . . . . . . . . .

115
115
116
119
119
120
120
121
122
123
124
126
126

127
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xi

7.5.4 Model Identification . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
7.5.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
7.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

129
130
133
134

8 Asset Accumulation and Portfolio Decisions Under Inflation Risk . . . . .
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
8.2 A New Multi-factor Model for Nominal
and Inflation-Indexed Bonds. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
8.2.1 The Factors .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
8.2.2 The Nominal Bonds . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
8.2.3 The Inflation Indexed Bonds (IIB) .. . . . . . . .. . . . . . . . . . . . . . . . . . . .
8.2.4 The No-Arbitrage Pricing . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
8.3 Intertemporal Asset Accumulation with Inflation Risk . . . . . . . . . . . . . . .
8.3.1 The Intertemporal Asset Allocation Model.. . . . . . . . . . . . . . . . . . .
8.3.2 The Systematic Factors .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
8.3.3 The Investment Opportunity Set . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

8.3.4 Agents’ Action .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
8.3.5 Dynamic Programming Approach . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
8.3.6 Solving for the Intertemporal Portfolio .. . .. . . . . . . . . . . . . . . . . . . .
8.4 Model Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
8.4.1 The Term Structure of Real Yields . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
8.4.2 The Term Structure of Nominal Yields . . . .. . . . . . . . . . . . . . . . . . . .
8.4.3 Estimation of Realized Inflation Dynamics . . . . . . . . . . . . . . . . . . .
8.4.4 Estimation of Stock Return Dynamics .. . . .. . . . . . . . . . . . . . . . . . . .
8.5 Application of Intertemporal Optimal Portfolios ... . . . . . . . . . . . . . . . . . . .
8.5.1 Example 1: Expected Optimal Final Utility
and the Factors .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
8.5.2 Example 2: Asset Allocation and Risk Aversion
Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
8.5.3 Example 3: Asset Allocation and Investment Horizon.. . . . . . .
8.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

139
139
142
142
143
144
145
148
148
149
149
150
151

155
157
157
160
164
166
167
167
169
171
172
173

9 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 179
A Dynamic Programming as Solution Method . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 181
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 183



List of Figures

Fig. 2.1
Fig. 2.2

Original and de-trended real BAA Yields . . . . . . .. . . . . . . . . . . . . . . . . . . .
Harmonic fit of the BAA yields . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

16
16

Fig. 3.1
Fig. 3.2

Distribution of eigenvector components .. . . . . . . .. . . . . . . . . . . . . . . . . . . .
Unconstrained optimization with capital allocation line
(annualized returns).. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Efficient frontiers (annualized returns). (a) Short
selling allowed. (b) No short selling allowed. (c) No
short selling and first constraint. (d) No short selling
and extra constraints .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

36

Welfare in the interval D Œ0; 1 . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
The consumption wealth ratio for the interval D Œ0; 1 . . . . . . . . . .
Trajectories for wealth for different initial levels of wealth . . . . . . . .
Consumption-wealth ratio decreasing with rising
(for r < ı/, D 0:1; D 0:5; D 0:75; D 5 . . . . . . . . . . . . . . . . . .
Welfare falling with rising . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Welfare rising with rising r . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Welfare falling with rising discount rates
0:01; 0:03; 0:06 and 1 .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Consumption wealth ratio rising for ı D 0:01; 0:03; 0:06 . . . . . . . . . .
Consumption-wealth ratio for ı D 1 .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Value function (top), optimal consumption (left) and
consumption wealth ratio (right) for D 0:75 and ı D 0:05 . . . . . .
Vector field and trajectories (top) and optimal trajectory
(bottom) for W0 D .1; 0/ and for D 0:75 .. . . . .. . . . . . . . . . . . . . . . . . . .
Value function for W 0:1 (upper left), D 1 (upper
right) D 2 (bottom) .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

Consumption-wealth ratio for W 0:1; (upper left)
D 1 (upper right) D 2 (bottom) . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

56
56
57

Fig. 3.3

Fig. 4.1
Fig. 4.2
Fig. 4.3
Fig. 4.4
Fig. 4.5
Fig. 4.6
Fig. 4.7
Fig. 4.8
Fig. 4.9
Fig. 4.10
Fig. 4.11
Fig. 4.12
Fig. 4.13

51

51

58
58
59

60
60
61
64
65
66
67

xiii


xiv

List of Figures

Fig. 4.14 Vector fields (left) for W 0:1; 1 and 2 ( from above to
below) and the corresponding trajectories for wealth,
consumption and portfolio weight ˛ (also from above to below).. .
Fig. 4.15 Value function for variation of returns, Re;t , Rf ;t (see
Table 4.2). Variant 1 (upper left), variant 2 (upper
right), variant 3 (below) . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Fig. 4.16 Vector fields and optimal trajectories for the variation
in returns Re;t and Rf ;t . Variant 1 (upper panel), variant
2 (middle panel), variant 3 (lower panel). In each of the
variants the trajectories of wealth, consumption and
portfolio weight, ˛ (right panel from above to below) .. . . . . . . . . . . . .
Fig. 4.17 Value function for discount rates: ı W 0:1 (upper left), 1
(upper right), 2 (lower left) and 11 (lower right).. . . . . . . . . . . . . . . . . . .
Fig. 4.18 Consumption (left) and consumption-wealth ratio
(right) for discount rates ı: 0.1 (upper panel); 0.5

(middle panel); 1 (lower panel) . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Fig. 4.19 Vector fields (left) and trajectories (right) for discount
rates ı: 0.1 (upper panel), 0.5 (middle panel), 1
(lower panel), trajectories of wealth, consumption and
portfolio weight, ˛ (right panel from above to below) .. . . . . . . . . . . . .

68

69

70
71

72

73

Fig. 5.1
Fig. 5.2
Fig. 5.3
Fig. 5.4
Fig. 5.5
Fig. 5.6
Fig. 5.7
Fig. 5.8
Fig. 5.9

Long swings in asset build up for D 0:5 . . . . . .. . . . . . . . . . . . . . . . . . . .
Value function for D 0:5 .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Long swings in asset build-up for D 0:8 . . . . . .. . . . . . . . . . . . . . . . . . . .

Value function for D 0:8 .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Long swings in asset movements for D 5 . . . . .. . . . . . . . . . . . . . . . . . . .
Long swings in asset build up for D 0:8 and ı D 0:05 .. . . . . . . . . .
Value function for D 0:8 and ı D 0:05 . . . . . . .. . . . . . . . . . . . . . . . . . . .
Dissipating wealth for D 0:8 and ı D 0:5 . . . .. . . . . . . . . . . . . . . . . . . .
Value function for D 0:8 and ı D 0:5 . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

87
88
88
89
89
92
93
94
94

Fig. 6.1
Fig. 6.2
Fig. 6.3
Fig. 6.4
Fig. 6.5

Long swings in asset build up for D 0:8 and ı D 0:03 .. . . . . . . . . .
Value function for D 0:8 and ı D 0:03 . . . . . . .. . . . . . . . . . . . . . . . . . . .
Decline in assets for D 0:8 and ı D 0:5 . . . . . .. . . . . . . . . . . . . . . . . . . .
Value function for D 0:8 and ı D 0:5 . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Shrinkage of wealth with no labor income, for D 0:8
and ı D 0:5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Value function with no labor income, for D 0:8

and ı D 0:5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

108
109
110
110

Fig. 6.6
Fig. 7.1
Fig. 7.2
Fig. 7.3

111
111

Interbank rate (Source: OECD Main economic
indicators). (a) Germany; (b) U.K.; (c) U.S. . . . .. . . . . . . . . . . . . . . . . . . . 122
Distribution of estimated white noise (I). (a) Germany;
(b) U.K.; (c) U.S. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 125
Normalized autocorrelation of the estimated noise.
(a) Germany; (b) U.K.; (c) U.S.. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 126


List of Figures

Fig. 7.4
Fig. 7.5
Fig. 7.6
Fig. 7.7

Fig. 8.1
Fig. 8.2
Fig. 8.3
Fig. 8.4
Fig. 8.5
Fig. 8.6
Fig. 8.7
Fig. 8.8
Fig. 8.9
Fig. 8.10
Fig. 8.11
Fig. 8.12
Fig. 8.13

Simulated data for the Ait-Sahalia and Andersen-Lund
model. (a) Ait-Sahalia model. (b) Andersen-Lund model .. . . . . . . . .
Normalized autocorrelation of the estimated noise for
the continuous-time models . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Distribution of estimated white noise (II). (a) Germany;
(b) U.K.; (c) U.S. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Original (left) and simulated (right) short rate.
(a) Germany, original; (b) Germany, simulated; (c)
U.K., original; (d) U.K., simulated; (e) U.S., original;
(f) U.S., simulated . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Real yields from TIPS and estimated real rate . .. . . . . . . . . . . . . . . . . . . .
US nominal bond yields and federal funds rate (FFR) . . . . . . . . . . . . . .
Nominal yields and estimated factors. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Federal fund rate and the estimated instantaneous rate . . . . . . . . . . . . .
Realized and filtered annualized inflation . . . . . . .. . . . . . . . . . . . . . . . . . . .
SP500 index .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

Expected optimal final utility ( I D 0:23 inflation
premia effect) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Expected optimal final utility ( I D 0:85 depreciation effect) .. . . .
Example risk aversion with I D 0:23 (smaller ) . . . . . . . . . . . . . . . . .
Example risk aversion with I D 0:23 (larger ) .. . . . . . . . . . . . . . . . . .
Example risk aversion with I D 0:85 . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Example investment horizons with I D 0:23 ... . . . . . . . . . . . . . . . . . . .
Example investment horizons with I D 0:85 ... . . . . . . . . . . . . . . . . . . .

xv

127
128
131

132
158
160
163
164
165
166
168
168
169
170
170
171
172

List of Tables

Table 3.1

Variance-covariance matrix of annualized expected
real returns (1983:02–2008:06) . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Table 3.2 Descriptive statistics and mean equation of expected
real returns.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Table 3.3 Eigenvalues of empirical variance-covariance matrix .. . . . . . . . . . . .
Table 3.4 Theoretical and actual eigenvalues for empirical
correlation matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Table 3.5 Mean-variance quadratic utility optimal portfolio
weights (Riskfree rate D 4:2 %, RiskAverParam D 2,
Borrowing rate D 8 %) . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Table 3.6 Mean-variance quadratic utility optimal portfolio
weights (Riskfree rate D 4:2 %, RiskAverParam D 3,
Borrowing rate D 8 %) . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Table 3.7 Mean-variance quadratic utility optimal portfolio
weights (Riskfree rate D 4.2 %, RiskAverParam D 4,
Borrowing rate D 8 %). . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Table 3.8 Mean-variance efficient frontier portfolio weights
(short selling allowed/borrowing rate D 8 % . .. . . . . . . . . . . . . . . . . . . .
Table 3.9 Mean-variance efficient frontier portfolio weights (no
short selling allowed/upper and lower bound constraints) . . . . . . . .
Table 3.10 Mean-variance efficient frontier portfolio weights (no
short selling allowed/combination constraints) .. . . . . . . . . . . . . . . . . . .

31

32
33
36

38

38

39
40
41
42

Table 4.1
Table 4.2

Parameter values for the model.. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Parameters for the returns, Re;t ; Rf ;t . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

63
69

Table 5.1

Coefficients of the harmonic fit (real stock return) of Eq. (5.3) .. .

86

Table 6.1

Coefficient estimates of the low frequency
components of labor income . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 106

Table 7.1
Table 7.2

Results of estimation and forecast for Germany .. . . . . . . . . . . . . . . . . . 134
Results of estimation and forecast for the U.K... . . . . . . . . . . . . . . . . . . 135
xvii


xviii

List of Tables

Table 7.3

Results of estimation and forecast for the U.S. .. . . . . . . . . . . . . . . . . . . 135

Table 8.1
Table 8.2
Table 8.3
Table 8.4
Table 8.5
Table 8.6

Estimations of real yields and their statistics . .. . . . . . . . . . . . . . . . . . . .
LR test for H0 W Är D ÄQ r . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
LR test for H0 W Ä D ÄQ . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Estimations of nominal yields and their statistics . . . . . . . . . . . . . . . . .

Estimation results for the CPIU . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Parameter summary .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

159
159
162
162
163
167


Chapter 1

Introduction

The world-wide meltdown of asset markets in the years 2007–2009 posed great challenges for asset and portfolio managers. Many funds such as university endowments,
sovereign wealth funds and pension funds were overexposed to risky returns and
suffered considerable losses. On the other hand, the long run upswing in the stock
market since 2010, induced by a monetary policy of quantitative easing in the US,
led to asset price booms and new wealth formation. In both cases quite significant
differences in wealth accumulation were visible. It is now well recognized that
disparity in wealth accumulation has become more distinct then the disparity in
income. This seems to have emerged in the last few decades or so for the US, as well
as for other countries.1 Our book aims at dealing with sustainable wealth formation
and dynamic decision making. For this we have three perspectives in mind.
A first perspective is how wealth formation and the proper management of
financial funds can help to buffer income risk sufficiently, and to obtain adequate
risk free income at a later stage of life. Preventing excessive meltdowns of portfolios,
while maintaining adequate wealth growth as a buffer against risk so as to have
sufficient risk free income later in life, is an important concern in the current public

debate on asset accumulation and wealth disparity. In whatever institutional form
saving takes place, the generic issue is how much to save and invest, and how
to make proper portfolio decisions. In particular the creation and management of
pension and retirement funds have become a significant public policy issue. The
debate on this appears to have accelerated by the prediction that, in the not-toodistant future, old-age wealth inequality will remerge.2 Thus, the more generic issue

1

See Jacoby (2008), Milanovic (2010) and Piketty (2014). Note that most of the recent literature
on wealth disparity is concerned with a broader notion of wealth, that might include, real estate,
land, resources and assets in retirement funds. We here are mainly concerned with the financial
market, wealth accumulation and disparity.
2
See Ghilarducci (2008).
© Springer-Verlag Berlin Heidelberg 2016
C. Chiarella et al., Sustainable Asset Accumulation and Dynamic Portfolio
Decisions, Dynamic Modeling and Econometrics in Economics and Finance 18,
DOI 10.1007/978-3-662-49229-1_1

1


2

1 Introduction

is: what is the role of financial markets in sustainable asset accumulation and in
reducing wealth disparities?
A second important perspective is that many agents and institutions in financial
markets tend to put some constraints on the accumulation and allocation of

assets—following some rules, guidelines and restrictions concerning risk-taking,
safeness of investments, as well as social, ethical and environmental aspects. Thus
investments are often restricted to certain asset classes or risk classes of assets. Much
investment and asset allocation decisions are therefore made following behavioral
and intuitional rules without necessarily being optimal in the technical sense. Those
rules might then have some impact on the choice of discount rates, risk aversion
parameters, and dynamic savings and asset allocation. In this book we will not deal
extensively with the issue of social, ethical, environmental and ecological guidelines
for investments, or divestment for that matter,3 but some of our modeling procedures
are applicable to those problems.
A third perspective is that we want to move more toward dynamic decision
making and dynamic rebalancing of portfolios. Portfolio decisions are frequently
modeled as static decisions problems and, as a rule, they are supposed be optimal.
Yet, how should investors respond to changes in expected returns or respond
to how inflation rates affect the real value of their assets? Proper dynamic savings
and asset allocation decisions are essential in this context. In standard literature,
issues of savings and wealth accumulation are often separated from asset allocation
decisions. This book pursues a simultaneous and dynamic treatment of both savings
behavior and portfolio decision making.

1.1 Institutions, Models and Empirics
We do not go into the details of institutional arrangements of saving and investment
decisions. Rather we want to provide a more general framework within which
detailed institutional arrangements can be discussed. Many national governments,
the World Bank and the OECD are discussing institutional issues related to savings
and wealth accumulation, especially with regard to retirement funds operation.
There are many different operational structures for wealth accumulation, retirement
income and pensions, ranging from the purely public-managed pay-as-you-go
systems to fully funded systems to private pension fund systems. In many countries
there are hybrid systems that adopt aspects of pay-as-you-go and fully funded

systems. Both traditional mutual funds and publicly supported insurance schemes
(including pension funds) are management vehicles that can help build up wealth
for the future and reduce wealth disparities and inequality.

3

This has come up in recent discussions of climate change, where funds often set guidelines and
restrictions on investments into fossil energy, preferring instead to invest in renewable energy.


1.1 Institutions, Models and Empirics

3

Whatever institutional form is chosen, one must take account of labor income in
wealth accumulation and portfolio models. The suitable design and management of
portfolios that guarantee a sufficient retirement income for households with labor
income has also been at the center of recent debates on pension funds. Yet, portfolio
studies that include labor income are still rare. In their seminal work, Campbell and
Viceira (2002) devote two chapters to this issue, and this serves as an important
starting point for our study. In this direction we extend the modeling approach
to include not only asset income but also labor income in the dynamic decision
problem of asset accumulation and allocation.
In order to model the heterogeneity between generations, many researchers have
suggested overlapping generations models working with two periods4 : the first
period involves active labor market participation; the later period is for retirement.
Three generations models have also been used (see, e.g., Eggertsson and Mehrotra
2014). Those actually lead to life cycle models, which we will leave aside. However,
we could address those issues in the context of a dynamic decision approach in a
further step. For details of an overlapping generations model and its implication for

fund management, see Campbell and Viceira (2002, Chap. 7).
We follow here a procedure by Blanchard (1985) to convert an overlapping
generations model into a continuous time model. We stick to a continuous time
approach to avoid discrete time, two or three period models. This requires us to deal
with different time horizons at the different stages of agents’ lives: the time period
with primarily labor income and the period with primarily retirement income. To
deal with this problem of two time horizons, we employ a model with different
discount rates for the two periods as in Blanchard (1985).
We do not refer here to a saving and portfolio model for an individual investor.
If we had appropriate data for individual investors, we could also pursue, with our
method, an individual decision model, or life cycle model for an individual agent.
But this is not attempted in the first step of the research undertaken in this project.
We follow, to some extent, Merton (1971, 1973), Campbell and Viceira (2002,
Chap. 6) and Viciera (2001), but we depart from their assumptions that the expected
equity premium is a constant. In our model the equity premium will be time varying
and we also assume a time dependent risk free interest rate.
We start with an econometric harmonic fit of asset and labor incomes by
using spectral analysis. We use a Fourier transformation to decompose a function
(represented by time series data) into low frequency movements and residuals. We
employ actual time series data and estimate time variation of the data using the
harmonic fitting technique.5 We use US data, but financial and income data from
other countries could be employed as well. We employ low frequency movements
in asset and labor income in our dynamic decision approach to solve various model
specifications with Dynamic Programming.

4
5

See Kotlikoff and Burns (2005).
See Hsiao and Semmler (2009).

4

1 Introduction

1.2 Dynamic Programming as Solution Method
The Dynamic Programming (DP) algorithm has been employed in many areas of
economics and finance,6 using DP in our context has several advantages over other
methods. DP not only solves the dynamic decision model globally, it also lends itself
to extensions in which new market information becomes available. DP also helps to
simultaneously study the issue of accumulation and allocation of financial funds.
Other authors have already demonstrated the usefulness of dynamic programming
for dynamic decision making.7 The use of DP invokes the discussion on forwardlooking behavior of economic agents. This behavior accords with an individual who
invests current funds for some expected future outcome but, because of the long
time horizon, actually realized outcomes are uncertain. Similar forward-looking
decisions problems are present in the traditional static portfolio model. Regarding
the traditional model, Markowitz (2010) makes the following statement:
Judgment plays an essential role in the proper application of risk-return analysis for
individual and institutional portfolios. For example, the estimates of mean, variance, and
covariance of a mean variance analysis should be forward-looking rather than purely
historical. (Markowitz 2010: 7)

It is worth stressing that the use of DP to model forward-looking behavior of
individuals, households, and institutions, requires some methodological discussions.
The typical assumptions and postulates of DP are as follows:
• Marginal conditions, such as describing the balance between current costs and
future benefits, are instantaneously established (for example the Euler equation
in consumption and saving decisions)
• Information sets are a priori given for long time horizons, freely available and

fully used
• The decision maker can make smooth and continuous adjustments as the
environment changes
• The decisions are made under no income, liquidity, credit or other market
constraint
• The spillovers, externalities and contagion effects are negligible
• There are negligible macroeconomic feedback effects or propagation effects that
can significantly disturb the intertemporal arbitrage decision
• The decisions—responding to the realization of the state variables—can then be
made in nonlinear form at grid points of the state variables

6

See Grüne and Semmler (2004).
The many examples include dynamic choices over savings, occupation and job search, choices on
education and skills, investment in housing, health care choices, and insurance decisions. See Hall
(2010) and also the many examples in Grüne and Semmler (2004).

7


1.3 Previous Work

5

The use of the Dynamic Programming method thus presumes that none of the above
problems will significantly disturb dynamic decision making. Though in principle
one could claim that dynamic and forward-looking decision making is involved
in human behavior, particularly in economic decision making, but one should be
careful assuming away the above mentioned issues.

In our treatment of savings and asset allocation we will pay explicit attention to
the presumptions of the DP solution methodology. We will show that DP still gives
helpful answers to interesting questions of savings and portfolio decisions, such as
the role of risk aversion, discounting future outcomes, the role of initial condition on
wealth, constraints on the state and decision variables, and the evolution of income
and wealth arising from such decisions. This set up allows one also to study issues
of investor heterogeneity with respect to risk aversion, discount rates, initial wealth,
informational constraints and time horizons lengths on the paths of wealth and
inequality.8

1.3 Previous Work
Studying dynamic decision making in finance started with Merton (1971, 1973,
1990). More recently, seminal work has been undertaken to model dynamic
consumption and portfolio decisions. Originally, Merton (1971, 1973) provided a
general intertemporal framework for studying the decision problem of a long-term
investor who not only has to decide about savings but also of how to allocate
funds to different assets such as equity, bonds and cash. It is now increasingly
recognized that the static mean-variance framework of Markowitz needs to be
improved upon by extending it to a dynamic context that takes into account new
investment opportunities, different initial conditions, different risk aversion among
investors, different time horizons, and so on.
Much effort has been put forth to show that, under certain restrictive conditions,
the dynamic decision problem is the same as the static decision problem.9 Yet,
it is now well recognized that a more general dynamic framework is preferable.
However, there are many difficulties involved in obtaining closed-form solutions for
more general models. One must therefore employ numerical solution techniques to
solve for the consumption or saving paths and the dynamic asset allocation problem.
Important work on those issues has been presented by Campbell and Viceira
(1999, 2002). They use the assumption of log-normal distributions in consumption
and asset prices with the implication that the optimal consumption-wealth ratio—

or, equivalently, the saving-wealth ratio—does not vary too much. Using log-linear

8

A recent modification of the DP algorithm, making it useable for more complex decision making
problems, allows us to study those issues on a finite time horizon with informationally constrained
agents, (see Grüne et al. 2015 and Chap. 6 of this book).
9
See Campbell and Viceira (2002, Chap. 2).


6

1 Introduction

expansion of the consumption-wealth ratio around the mean, they show a link
between the myopic static decision problem and the dynamic decision problem (see
Campbell and Viceira, 2002, Chaps. 3–5). They solve a simplified model with time
varying bond returns but with a constant expected equity premium.10 In general,
models with time varying returns are difficult to solve analytically, and linearization
techniques as a solution method may not be quite accurate.11 This is likely to be the
case if returns and consumption-wealth ratios are too variable.12
If there is a predictable structure in equity (and bond) returns, and thus there
are time varying expected returns, then the dynamic decisions with respect to
consumption and portfolio weights need to respond to the time varying expected
returns.13 In some of our model variants we approximate the time varying expected
asset returns by the low frequency component of the returns.14 A further discussion
of these empirical issues is undertaken in Chap. 2.
Given such time varying returns, a buy and hold strategy for portfolio decisions
is surely not sufficient. Dynamic saving decisions, as well as a dynamic rebalancing

of a portfolio (following low frequency movements in investment opportunities
and returns), is needed in order to capture persistent changes in returns and to
avoid wealth and welfare losses. DP can be usefully applied here. It works with
flexible grid size, operates globally, and can solve for any point in the state space
simultaneously for both the consumption or saving decisions, as well as for time
varying portfolio weights. As our more solution technique shows, the consumptionwealth ratio can vary greatly but our solution remains sufficiently correct.15

10

See Campbell and Viceira (2002, Chap. 3).
For example, the log-linear expansion about the equilibrium consumption-wealth ratio is as
undertaken by Campbell and Viceira (2002, Chaps. 2–4).
12
In order to obtain an approximate solution of the model, Campbell and Viceira (2002:51)
presume that the consumption-wealth ratio is “not too” variable. However, they show that their
procedure loses serious accuracy with a parameter of risk aversion > 1. Moreover, Campbell
and Viceira use a model with a constant interest rate, see also Campbell (1993), Campbell and
Viceira (1999) and Campbell and Koo (1997). On the issue of the accuracy of first and second
order approximations in dynamic decision models, see Grüne and Semmler (2007).
13
There is much empirical evidence on time varying expected returns. For earlier work see
Campbell and Shiller (1989); for recent surveys, see Campbell and Viceira (1999) and Cochrane
(2006).
14
Recent theoretical research on asset pricing using loss aversion theory can give a sufficient
motivation for such an assumption on time varying expected asset returns following a low
frequency movement, for details see Grüne and Semmler (2008).
15
In Becker et al. (2007) the out-of-steady-state dynamics of second order approximations and
dynamic programming are compared. The errors from dynamic programming are much smaller do

not depend on the distance to the steady state.
11