Sustainable asset accumulation and dynamic portfolio decisions

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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 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  h0 D g2r gr g r gr g g2 à 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  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  gr g0 D g l0 gr gr à I rI I I  à 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  à 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  à ; 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  Bnr / Bnr / D WD det Bn / Bn / 10 B0 1C C RAA1 RAX D B @0 0A ; 00 à ; 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 allows the numerical computation of approximately optimal trajectories In order the distribute the nodes of the grid efficiently, we make use of a posteriori error estimation For each cell Cl of the grid we compute Ál WD max j Th Vh /.k/ k2cl Vh k/ j More precisely we approximate this value by evaluating the right hand side in a number of test points It can be shown that the error estimators Ál give upper and lower bounds for the real error 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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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