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**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 http://www.springer.com/series/5859 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 This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, **and** transmission or information storage **and** retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws **and** regulations **and** therefore free for general use The publisher, the authors **and** the editors are safe to assume that the advice **and** information in this book are believed to be true **and** accurate at the date of publication Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer-Verlag GmbH Berlin Heidelberg 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 • 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 Introduction 1.1 Institutions, Models **and** Empirics 1.2 **Dynamic** Programming as Solution Method 1.3 Previous Work 1.4 Outline **and** Results 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 14 17 **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 **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 174 **Asset** **Accumulation** **and** **Portfolio** **Decisions** Under Inflation Risk Using the definition of 0D r in (8.18) we rewrite the equation above as d Brr / C Brr /Är rt d d 2 C Ar / Brr /.Är r r gr / C gr Brr / : d (8.72) Since rt is a stochastic process, the equation above holds if **and** only if d Ar / d d Brr / C Brr /Är D ; d 2 Brr /.Är r r gr / C gr Brr / D : (8.73) (8.74) Then, Brr / is solved as (8.29) **and** Ar / is solved as (8.30) The first part the model is of a multi-factor Gaussian model The solution is similar to the second part The solution process can be found, for example, in Brigo **and** Mercurio (2001) t u Proof of Property 8.2 Without loss of generality we set Q D Let O t be the factor when its mean is set to be zero The shifted factor t is obtained by t D O t C Let An / Bnr /, **and** Bn / be the coefficients in (8.28), (8.26) **and** (8.27) based on the factor O t of zero mean Let An /, Bnr /, **and** Bn / be those coefficients based on the factor t with the mean The shift change does not change the mean-reverting parameters (Är **and** Ä ) Also the market prices of risk rt **and** t remain the same Let tO D lO0 C lO1 O t be the market price of risk for Wt Because tO D t D l0 C l1 t , we have lO1 D l1 **and** lO0 D l0 C l1 Therefore, Bnr / D Bnr /, Bn / D Bn / due to the formulas (8.26) **and** (8.27) **And** An / An / D ÄQ D Bn / g l1 C Ä ÄQ Bn / C 0; /C 0 due to the formula (8.28), where the relation ÄQ D Ä C g l1 is used Using the transformations above, the nominal bond yield based on the shifted factor t is calculated by Yn t; T/ D D D An / An / An / C C C Bnr / rt C Bn / Bnr / rt C Bn / C Bn / t Ot C C Bnr / C rt C : Bn / Ot C Appendix 175 If the last bracket is equal to zero, meaning C D 0 ; then the nominal yield based on the shifted factor factor O t t is equivalent to that on the t u Proof of Property 8.3 We aim to prove rt D limT#t Yr t; T/ Taking t D T in (8.20) we have I t; 0/ D r t; 0/ C t Subtracting (8.23) at D by (8.24) we have I t; 0/ D rt C t Therefore we obtain r t; 0/ D rt Taking D in (8.18) we know r t; 0/ D A0r 0/ C B0rr 0/rt Therefore lim Yr t; T/ D A0r 0/ C B0rr 0/rt D rt T#t t u holds Proof of Property 8.5 The proof utilizes the technique of solving HLB equation (8) in Liu (2007) to solve the HJB equation (8.52) in this paper We map the terms in the HJB equations as specified in the following table The key point that we can apply Liu’s solution technique here is that with the addition terms due to the price uncertainty the HJB equation (8.52) still falls in Liu’s quadratic **asset** return class We map the terms in our HJB equation (8.52) to Liu’s HJB equation (8) **and** show that all terms satisfy Liu’s conditions defined through his Eqs (9)–(11) **and** (13)–(17) in Liu (2007) Terms in Liu’s equation (8) fO (a) (b) (c) ˙X˙X > X ˙X ˙ r/ Terms in Eq (8.52) ˚ Liu’s condition Gt RXX G> t Ft Gt RXA RAA ˙t Liu’s equation (10) Liu’s equation (9) Liu’s equation (15) /2 (d) (e) ˙X˙X 2 > > ˙X >˙X r/> ˙˙ > / r/ 1 1 r t Rt 1/ Gt RXA RAA RAI I /Gt RXI I RXX RXA RAA RAX t Rt 1/> ˙t RAA ˙t> / 2 C (f) 1 2 Rt 1/ Liu’s equation (16) Liu’s equation (14) /3 I RIA RAA RAI >˙ > 1R 1R R 1/ t t AI I AA t 2 I C I /2 Rt t) Liu’s equation (13) Our term in the mapping (a) satisfies Liu’s condition equation (10) where Â h0 D g2r gr g r gr g g2 Ã r ; h D ; h D ; Á D I2 : Our term in the mapping (b) satisfies Liu’s condition equation (9) where (8.75) 176 **Asset** **Accumulation** **and** **Portfolio** **Decisions** Under Inflation Risk Â Är r kD Ä Ã Â Är ; K2 D ; Á D I2 ; KO D K : ; KD Ä Ã (8.76) O h1 **and** h2 in Eqs (8.76) **and** (8.75) satisfy Liu’s condition equation The K, Á, K, (11) Our term in the mapping (c) is rewritten into Gt D Gt RXA RAA ˙t C rt C 1000 B B l0 C l1 t C B C A 0100 @ I ! /2 Rt 1/ t RXA RAA RAI rI ! C 1000 B B IC B C 0100 @1A /2 S D Gt D C rt l0 C l1 t 1 gr gr gr g l0 ! ! C I I ! I / rI I I sI ! / C I rI // I I rI I /RXI I gr 0 g l1 ! I ! rt : t Map it to Liu’s equation (15) we have Â gr g0 D g l0 gr gr Ã I rI I I Â Ã gr ; g1 D ; Á D I2 ; gO D g1 ; g2 D 0: g l1 (8.77) For mapping (d) our term RXX RXA RAA1 RAX is equal zero It is because the risks Wt in the **asset** returns in (8.39) include the risks WXt in the factor dynamics (8.36) Map it to Liu’s equation (16) we have l0 D ; l1 D ; l2 D ; Á D I2 : (8.78) For mapping (e) we rewrite our term first with ˙t t Rt 1/ D C 1X ; X WD Â Ã rt t based on Eq (8.40) **and** /3 2 I RIA RAA RAI 2 I C I D 2 I : 8.6 Conclusions 177 Now rewriting the term in mapping (3) using the results above D C 2 > 1 X/ RAA C > RAA 2 > RAA C 1X C C X/ 2 2 I /X : /2 C > 1 X/ RAA RAI I /2 I C X> I I > 1 RAA Map it to Liu’s equation (14) we have H0 D H1 D > RAA > RAA C I 2.1 ; H2 D / (8.79) I I > 1 RAA ; Á D I2 : Using Eq (8.32) to rewrite our term in mapping (f) **and** mapping it to Liu’s equation (13), we have ı0 D I I ; ı1 D Â Ã ; ı2 D ; Á D I2 : (8.80) The g1 , Á, gO , g2 , l1 **and** l2 in Eqs (8.77)–(8.79) satisfy the restriction Liu’s restriction in Eq (17) Then we apply Eqs (18)–(20) in Liu (2007) with D T t then we obtain the results (8.54)–(8.56) t u Proof of Property 8.7 For proving this property we just need to insert the following results: Bn / Bnr / Brr /Bnr / gr D g D ID B Bn / C Bnr / Brr /Bnr / B 0C > g D g D D B C r I ˙t / D B C 0 @ A I 0 S with Â Bnr / Bnr / D WD det Bn / Bn / 10 B0 1C C RAA1 RAX D B @0 0A ; 00 Ã ; B 0C C RAA1 RAI D B @1A ; (8.81) **and** the result of the factor elasticity in Property 8.6 into Eq (8.51) t u Chapter Concluding Remarks In this book we have combined theoretical **and** empirical work to study the issue of **sustainable** **asset** **accumulation** **and** **dynamic** **portfolio** **decisions** We mostly considered **asset** income but frequently included labor income in **portfolio** modeling in order to explore important issues regarding pension **and** retirement funds as well as wealth disparities Empirically, most chapters used actual US time series data to estimate the low frequency components of **asset** returns **and** labor income After fitting the US time series data to low frequency components in returns, our numerical procedure was used to solve for **dynamic** consumption **and** **asset** allocation **decisions** To some extent, we followed Campbell **and** Viceira (2002) **and** explored **dynamic** **asset** **accumulation** **and** allocation **decisions** for varying return, varying risk aversion, varying time horizon across investors, **and** for varying initial conditions As discussed in this book, our method appears to be more accurate than the method proposed by Campbell **and** Viceira (2002) The optimal saving decisions, the allocation of assets, investors’ wealth **and** welfare can be explored without linearization techniques The impact of varying return, risk aversion, time horizons **and** initial wealth differentials across investors on the dynamics of **asset** allocation **and** the evolution of wealth were traced We observed that there are cyclical movements in wealth **accumulation** as well as well upward **and** downward trends, depending on risk aversion, discount rates, the size of labor income, the saving rate **and** the size of the **asset** returns, as well as initial conditions This allowed us to study the extent to which financial markets could contribute to a secular rise in wealth disparity Wealth disparity, at least in terms of financial wealth, seems to arise not only from higher **asset** returns, lower risk aversion **and** discount rates, but also higher labor income, higher saving rates, **and** better access to leveraging The **dynamic** **portfolio** decision models presented here also allow for on-line decision updating as information on low frequency **asset** returns **and** labor income © 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_9 179 180 Concluding Remarks data evolves This suggests a practical method to rebalance portfolios as new information on returns **and** labor income are available The method is set up in such a way so as to help various types of institutions **and** investors to make optimal pension **and** retirement fund **decisions** As mentioned, more should be done on behavioral models of **asset** **accumulation** **and** allocation, though other authors have already begun to undertake this work.1 Specifically, more work should be pursued on socially **and** environmentally responsible investments To this end, Chap provides a method to undertake investment **decisions** with restrictions such as social, ethical **and** environmental guidelines, rules, **and** constraints concerning risk taking or safeness of investments Those rules **and** guidelines are likely to have considerable impact on not only a static mean-variance portfolio, but also on an intertemporal **portfolio** model through, for example, risk aversion or other behavioral parameters Most chapters employed models built on an infinite horizon decision making framework, as is common in this area But the limits of intertemporal modeling **and** the practicality of it have been laid out carefully Moreover, modeling **and** solution procedures could be more flexible **and** consider finite time **decisions** problems **and** solve such models for shorter time horizons Examples of this are given in Chappe **and** Semmler (2016), in which finite horizon model variants with changing behavior are discussed.2 Such models are open to wide behavioral interpretation since neither the objective functions nor the investment decision reaching it need to be fixed They be fixed only over a short planning horizon We can allow agents to switch objective function **and** investment strategies over the longer horizon as new challenges emerge In particular, as shown in Chappe **and** Semmler (2016), the NMPC solution method based on Grüne et al (2015), is quite flexible to permit such studies It allows for learning when knowledge of the agents is insufficient **and** needs to be improved along the paths Also, it might be that trajectories can become unstable, leading to deteriorations Agents can then switch behavior **and** one can allow for self-corrections so that after switching the behavior becomes more stabilizing and/or the targets are achieved through other means Finally, we also want to note that there can be extensive positive **and** negative externalities from the social behavior of agents in financial markets Social interactions can manifest itself as swarm **and** synchronized behavior, but also as behavior taking into account social interactions in preferences, **and** the impact of social norms, band wagon effects, imitations, learning **and** macro-level effects Many of those important aspects of behavior in **asset** markets could be built into such models, as proposed here, which should be the subject of future research See Grüne **and** Semmler (2008), **and** Zhang **and** Semmler (2009) In this work the nonlinear model predictive control (NMPC) technique is used Appendix A **Dynamic** Programming as Solution Method We here briefly describe the **dynamic** programming algorithm as applied in Grüne **and** Semmler (2004) that enables us to numerically solve our **dynamic** model variants The feature of the **dynamic** programming algorithm is an adaptive discretization of the state space which leads to high numerical accuracy with moderate use of memory Such algorithm is applied to discounted infinite horizon optimal control problems of the type introduced for the study of the global dynamics In our model variants we have to numerically compute V.x/ for Z V.x/ D max u e Ât f x; u/dt s.t xP D g.x; u/ where u represents a vector of control variables **and** x a vector of state variables which represents, in our case, wealth **accumulation** **and** a time index In the first step, the continuous time optimal control problem has to be replaced by a first order discrete time approximation given by Vh x/ D max Jh x; u/; Jh x; u/ D h j X C Âh/O{ f xh i/; ui / (A.1) iD0 where xu is defined by the discrete dynamics xh 0/ D x; xh i C 1/ D xh i/ C hg.xi ; ui / (A.2) **and** h > is the discretization time step Note that j D ji /i2N0 here denotes a discrete control sequence © 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 10.1007/978-3-662-49229-1 181 182 A **Dynamic** Programming as Solution Method The optimal value function is the unique solution of a discrete Hamilton-JacobiBellman equation such as Vh x/ D maxfhf x; uo / C C Âh/Vh xh 1//g j (A.3) where xh 1/ denotes the discrete solution corresponding to the control **and** initial value x after one time step h Abbreviating Th Vh /.x/ D maxfhf x; uo / C C Âh/Vh xh 1//g j (A.4) the second step of the algorithm now approximates the solution on a grid covering a compact subset of the state space, i.e a compact interval Œ0; K in our setup Denoting the nodes of by xi ; i D 1; : : : ; P, we are now looking for an approximation Vh satisfying Vh xi / D Th Vh /.xi / (A.5) for each node xi of the grid, where the value of Vh for points x which are not grid points (these are needed for the evaluation of Th ) is determined by linear interpolation We refer to the paper cited above for the description of iterative methods for the solution of (A.5) Note that an approximately optimal control law (in feedback form for the discrete dynamics) can be obtained from this approximation by taking the value j x/ D j for j realizing the maximum in (A.3), where Vh is replaced by Vh This procedure in particular 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Economics Letters, 108, 184–186 ... market, wealth accumulation and disparity See Ghilarducci (2008) © Springer-Verlag Berlin Heidelberg 2016 C Chiarella et al., Sustainable Asset Accumulation and Dynamic Portfolio Decisions, Dynamic. .. introduced in asset accumulation and allocation decisions Savings and asset allocation decisions are long-horizon decisions and expected inflation rates have to be properly taken into account in dynamic. .. 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_2 10 Forecasting and Low Frequency

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Xem thêm: Sustainable asset accumulation and dynamic portfolio decisions , Sustainable asset accumulation and dynamic portfolio decisions , 1 Institutions, Models and Empirics, 3 Dynamic Consumption and Portfolio Decisions: Two Assets and Time Varying Returns, 3 Business Cycles, Asset Returns and Labor Income