Risk management in finance and logistics

Translational Systems Sciences 14 Chunhui Xu · Takayuki Shiina Risk Management in Finance and Logistics Translational Systems Sciences Volume 14 Editors in Chief Kyoichi Kijima, Tokyo, Japan Hiroshi Deguchi, Yokohama, Japan Editorial Board Shingo Takahashi, Tokyo, Japan Hajime Kita, Kyoto, Japan Toshiyuki Kaneda, Nagoya, Japan Akira Tokuyasu, Tokyo, Japan Koichiro Hioki, Tottori, Japan Yuji Aruka, Hachioiji, Japan Kenneth Bausch, Riverdale, GA, USA Jim Spohrer, San Jose, CA, USA Wolfgang Hofkirchner, Wien, Austria John Pourdehnad, Philadelphia, PA, USA Mike C Jackson, Hull, UK Gary S Metcalf, Atlanta, GA, USA Marja Toivonen, Helsinki, Finland Sachihiko Harashina, Ichikawa, Japan In 1956, Kenneth Boulding explained the concept of General Systems Theory as a skeleton of science He describes that it hopes to develop something like a “spectrum” of theories—a system of systems which may perform the function of a “gestalt” in theoretical construction Such “gestalts” in special fields have been of great value in directing research towards the gaps which they reveal There were, at that time, other important conceptual frameworks and theories, such as cybernetics Additional theories and applications developed later, including synergetics, cognitive science, complex adaptive systems, and many others Some focused on principles within specific domains of knowledge and others crossed areas of knowledge and practice, along the spectrum described by Boulding Also in 1956, the Society for General Systems Research (now the International Society for the Systems Sciences) was founded One of the concerns of the founders, even then, was the state of the human condition, and what science could about it The present Translational Systems Sciences book series aims at cultivating a new frontier of systems sciences for contributing to the need for practical applications that benefit people The concept of translational research originally comes from medical science for enhancing human health and well-being Translational medical research is often labeled as “Bench to Bedside.” It places emphasis on translating the findings in basic research (at bench) more quickly and efficiently into medical practice (at bedside) At the same time, needs and demands from practice drive the development of new and innovative ideas and concepts In this tightly coupled process it is essential to remove barriers to multi-disciplinary collaboration The present series attempts to bridge and integrate basic research founded in systems concepts, logic, theories and models with systems practices and methodologies, into a process of systems research Since both bench and bedside involve diverse stakeholder groups, including researchers, practitioners and users, translational systems science works to create common platforms for language to activate the “bench to bedside” cycle In order to create a resilient and sustainable society in the twenty-first century, we unquestionably need open social innovation through which we create new social values, and realize them in society by connecting diverse ideas and developing new solutions We assume three types of social values, namely: (1) values relevant to social infrastructure such as safety, security, and amenity; (2) values created by innovation in business, economics, and management practices; and, (3) values necessary for community sustainability brought about by conflict resolution and consensus building The series will first approach these social values from a systems science perspective by drawing on a range of disciplines in trans-disciplinary and cross-cultural ways They may include social systems theory, sociology, business administration, management information science, organization science, computational mathematical organization theory, economics, evolutionary economics, international political science, jurisprudence, policy science, socioinformation studies, cognitive science, artificial intelligence, complex adaptive systems theory, philosophy of science, and other related disciplines In addition, this series will promote translational systems science as a means of scientific research that facilitates the translation of findings from basic science to practical applications, and vice versa We believe that this book series should advance a new frontier in systems sciences by presenting theoretical and conceptual frameworks, as well as theories for design and application, for twenty-first-century socioeconomic systems in a translational and transdisciplinary context More information about this series at http://www.springer.com/series/11213 Chunhui Xu • Takayuki Shiina Risk Management in Finance and Logistics 123 Chunhui Xu Department of Risk Science in Finance and Management Chiba Institute of Technology Chiba, Japan Takayuki Shiina Department of Industrial and Management Systems Science Waseda University Tokyo, Japan ISSN 2197-8832 ISSN 2197-8840 (electronic) Translational Systems Sciences ISBN 978-981-13-0316-6 ISBN 978-981-13-0317-3 (eBook) https://doi.org/10.1007/978-981-13-0317-3 Library of Congress Control Number: 2018950837 © Springer Nature Singapore Pte Ltd 2018 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 The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore To Freddy, Kaylin, Joanna; Yoko, Tatsuro, Moeko Preface Risk has been recognized as an important factor that has to be considered in managerial decision-making in many disciplines, such as investments in financial markets and reforming business plans Despite its generality, there is no consensus on the understanding of this notion; different people may have different perceptions of the meanings and implications of risk, even within the same discipline Hence risk is a notion that spans multiple disciplines, and people’s diverse subjective perceptions should be considered in addressing issues regarding risk management These features make this book an appropriate part of the Translational Systems Sciences series, which aims to cultivate a new frontier of systems sciences that contribute to the need for beneficial and practical applications This book is intended to provide an introduction to the central concepts and quantitative tools for risk management, and simultaneously present some up-todate research results on risk management in the fields of financial investments and logistics planning To the best of our knowledge, this is the first book covering diverse definitions of risk and quantitative methods for risk management in the fields of finance and logistics It is designed for self-study by professionals or classroom work at the undergraduate or graduate level for students who have a technical background in engineering, mathematics, or science This book aims to be informative for researchers whose interests are related to risk management, especially for those in the fields of finance and logistics The prerequisites for reading this book are relatively modest; the prime requirement is some familiarity with introductory elements of probability theory and optimization techniques Certain sections assume some knowledge of more advanced concepts of probability theory and optimization, such as stochastic processes and the branch and bound algorithm, but the text is structured so that the mainstream can be faithfully pursued without reliance on these advanced background concepts vii viii Preface This book is composed of two separate parts, each part is relatively independent and self-contained, readers interested in risk management in logistics can skip Part I, and those interested only in financial investments may only read Part I Part I is on risk management in finance, which is the most developed branch in risk management The first three chapters in Part I introduce the key concepts and quantitative tools for risk management in financial investments Chapter is an introduction to investments in financial markets, the associated risks, and risk management in investments Chapter presents the popular indices for market risk, which form the basis of risk management We introduce the concepts of variance, value at risk, and conditional value at risk in measuring market risk, and the methods for estimating these risk measures Chapter presents optimization models for risk control in investment decisions and the methods for solving these models, which are the main tools for risk management in investments While the last two chapters of Part I include some up-to-date research results on financial investments Chapter addresses a more general situation of investment known as the flexible investment, wherein the start and/or exit time of investments are not fixed but flexible within certain time intervals We introduce the return and risk measures for flexible investments, some new indices for market risk are included, such as the period value at risk, and the methods for estimating these risk measures Chapter includes optimization models for risk control in flexible investments and methods for solving these models Part II is on risk management in logistics Chapter presents the basic theory of stochastic programming which originates from the linear programming problem developed by Dantzig In particular, the solution method is described in detail regarding the stochastic programming problem with recourse This model is extended to models that include multi-stage planning and integer constraints In addition, we show the basic concepts regarding a model having probabilistic constraints and also including a variance term in the objective function In Chap 7, stochastic programming problem for inventory distribution is formulated with demand as a stochastic variable and the effectiveness of the policy of using both preventive and emergency lateral transshipment is examined In Chap 8, the stochastic programming model for the logistics network reorganization problem and the efficient solution method are shown Finally, the first author wishes to thank his doctoral research supervisor Professor Kyoiichi Kijima (Tokyo Institute of Technology), for his insightful guidance in Ph.D research, and the author’s postdoctoral research supervisor Professor Yu-Chi Larry Ho (Harvard University), for his ingenious ideas regarding ordinal optimization from which the author benefited a lot He would like to thank many research collaborators for their helpful discussions, particularly Masakazu Ando_(Chiba Institute of Technology, Japan), Shuning Wang_(Tsinghua University, China), Min Huang_(Northeastern University, China), Xiao Luo_(National University of Singapore), Xiaolin Huang_(Shanghai Jiaotong University, China), and Utomo S Putro and Santi Novani_(Institut Teknologi Bandung, Indonesia) He is also thankful Preface ix to his former Ph.D students, Jie Wang, Perla Rocio Calidonio Aguilar,Yanli Huo, and Chao Gong, for their help in research, and Miss Linjing Zou for preparing some of the figures in this book Chiba, Japan Tokyo, Japan March 2018 Chunhui Xu Takayuki Shiina Contents Part I Risk Management in Finance Financial Investment, Financial Risk and Risk Management 1.1 Financial Markets and Financial Investment 1.2 Main Risks in Financial Markets 1.3 Risk Countermeasures: Hedging and Diversifying 1.4 Risk Management by Diversification 1.5 Outline of Part I 3 Market Risk Measures in Financial Investments 2.1 Market Risk and Its Measurement 2.2 Variance: Fluctuation Is Taken as Risk 2.2.1 Definition of Variance 2.2.2 Estimation of Variance 2.3 Value at Risk: A Likely Loss Is Taken as Risk 2.3.1 Definition of Value at Risk 2.3.2 Estimation of VaR: Three Methods 2.4 Conditional VaR: Expected Loss Behind VaR Is Taken as Risk 2.4.1 Definition of Conditional VaR 2.4.2 Estimation of CVaR 2.5 Other Risk Measures: Failure Is Taken as Risk 2.6 Summary 13 14 14 14 15 17 17 18 27 27 27 31 34 Market Risk Control in Investment Decisions 3.1 Portfolio Selection and Its Models 3.2 MV Model and Its Variations 3.2.1 The Base MV Model and Its Two Variations 3.2.2 Solving Methods for MV Based Models 3.2.3 Two MV Based Models with Computational Advantages 35 36 37 37 38 39 xi 168 Reorganization of Logistics Network 8.6 Conclusions In this research, a logistics network reorganization problem is presented as an expected cost minimization model and a CVaR minimization model, and the solution is obtained using the L-shaped method The results of the numerical experiments show that the algorithm proposed in this research is more effective than the branch and bound method when the number of scenarios increases We also showed that the expected cost for the CVaR minimization model is slightly higher than that for the expected cost minimization model, while the worst case scenario can reduce the cost Regarding the CVaR minimization model, more research is needed to understand how an efficient solution may be attained This topic will be examined in the future In this research, we considered the uncertainty of demand, but it is also necessary to consider the uncertainty of the transport network For instance, given the disruption to transport caused by disasters such as earthquakes and typhoons, the current transportation routes may not always be optimal It is also necessary to consider risks due to corporate globalization References Agrawal, V., Chao, X., & Seshadri, S (2004) Dynamic balancing of inventory in supply chains European Journal of Operational Research, 159, 296–317 Ahmed, S., Tawarmalani, M., & Sahinidis, N V (2005) A finite branch-and-bound algorithm for two-stage stochastic integer programs Mathematical Programming, 100, 355–377 Allen, S G (1958) Redistribution of total stock over several user locations Redistribution of Total Stock Over Several User Locations, 5, 51–59 Ballou, R H (2001) Unresolved issues in supply chain network design Information System Frontiers, 3, 417–426 Banerjee, A., Burton, J., & Banerjee, S (2003) A simulation study of lateral transshipments in a single supplier multiple buyers supply chain network International Journal of Production Economics, 81–82, 103–114 Benders, J F (1962) Partitioning procedures for solving mixed-variables programming problems Numerische Mathematik, 4, 238–252 Birge, J R (1985) Decomposition and partitioning methods for multistage stochastic linear programs Operations Research, 33, 989–1007 Birge, J R (1997) Stochastic programming computation and applications INFORMS Journal on Computing, 9, 111–133 Birge, J R., Donohue, C J., Holmes, D F., & Svintsitski, O G (1996) A parallel implementation of the nested decomposition algorithm for multistage stochastic linear programs Mathematical Programming, 75, 327–352 Birge, J R., & Louveaux, F (1997) Introduction to stochastic programming (Springer series in operations research) New York: Springer Carøe, C C., & Tind, J (1998) L-shaped decomposition of two-stage stochastic programs with integer recourse Mathematical Programming, 83, 451–464 Charnes, A., & Cooper, W W (1959) Chance constrained programming Management Science, 6, 73–79 Committee, J E P S (1987) Denryoku-jyuyou-soutei oyobi Denryoku-kyoukyuu-keikaku Santeihoushiki no Kaisetsu (in Japanese) Dantzig, G B (1955) Linear programming under uncertainty Management Science, 1, 197–206 Drezner, Z (1992) Computation of the multivariate normal integral ACM Transactions on Mathematical Software, 18, 470–480 Ermoliev, Y., & Wets, R J.-B eds (1988) Numerical techniques for stochastic optimization (Springer series in computational mathematics, Vol 10) New York: Springer Fourer, R., Gay, D M., & Kernighan, B W (1993) AMPL: A modeling language for mathematical programming San Francisco: Scientific Press © Springer Nature Singapore Pte Ltd 2018 C Xu, T Shiina, Risk Management in Finance and Logistics, Translational Systems Sciences 14, https://doi.org/10.1007/978-981-13-0317-3 169 170 References Geoffrion, A., & Graves, G W (1974) Multicommodity distribution system design by benders decomposition Management Science, 20, 822–844 Graham, R L (1972) An efficient algorithm for determining the convex hull of a finite planar set Information Processing Letters, 1, 132–133 Haneveld, W K., Stougie, L., & van der Vlerk, M (1996) An algorithm for the construction of convex hulls in simple integer recourse programming Annals of Operations Research, 64, 67– 81 Herer, Y., Tzur, M., & Yucesan, E (2002) Transshipments: An emerging inventory recourse to achieve supply chain leagility International Journal of Production Economics, 80, 201–212 Herer, Y., Tzur, M., & Yucesan, E (2006) The multi-location transshipment problem IIE Transactions, 38, 185–200 Isa, H., Shiina, T., Morito, S., & Imaizumi, J (2015) Reorganization of logistics network via stochastic network (in Japanese) Journal of Japan Society for Fuzzy Theory and Intelligent Informatics, 27, 589–598 Kall, P., & Wallace, S W (1994) Stochastic programming Chichester: Wiley Karmarkar, U S., & Patel, N (1977) The one-period n-location distribution problem Naval Research Logistics Quarterly, 24, 559–575 Kiya, F., & Davoudpour, H (2012) Stochastic programming approach to redesigning a warehouse network under uncertainly Transportation Research Part E, 48, 919–936 Klein Haneveld, W K., Stougie, L., & van der Vlerk, M H (1995) On the convex hull of the simple integer recourse objective function Annals of Operations Research, 56, 209–224 Krishnan, K., & Rao, V (1965) Inventory control in n warehouses Journal of Industrial Engineering, XVI, 212–215 Laporte, G., & Louveaux, F V (1993) The integer l-shaped method for stochastic integer programs with recourse Operations Research Letters, 13, 133–142 Louveaux, F V (1980) A solution method for multistage stochastic programs with recourse with application to an energy investment Operations Research, 28, 889–902 Louveaux, F V (1986) Multistage stochastic programs with block-separable recourse Mathematical Programming Study, 28, 48–62 Louveaux, F V., & van der Vlerk, M H (1993) Stochastic programming with simple integer recourse Mathematical Programming, 61, 301–325 Melachrinoudis, E., & Min, H (2000) The dynamic relocation and phase-out of a hybrid, two-echelon plant/warehousing facility: A multiple objective approach European Journal of Operational Research, 123, 1–15 Melachrinoudis, E., & Min, H (2007) Redesigning a warehouse network European Journal of Operational Research, 176, 210–229 Melo, M., Nickel, S., & da Gama, F S (2005) Dynamic multi-commodity capacitated facility location: A mathematical modeling framework for strategic supply chain planning Computers and Operations Research, 33, 181–208 Paterson, C., Kiesmüller, G., Teunter, R., & Glazebrook, K (2011) Inventory models with lateral transshipments: A review European Journal of Operational Research, 210, 125–136 Prékopa, A (1971) Logarithmic concave measures with application to stochastic programming Acta Scientiarum Mathematicarum, 32, 301–316 Prékopa, A (1973) On logarithmic concave measures and functions Acta Scientiarum Mathematicarum, 34, 335–343 Prékopa, A (1988) Numerical solution of probabilistic constrained problems In Yu Ermoliev & R.J.B Wets (Eds.), Numerical techniques for stochastic optimization (pp 123–129) New York: Springer Prékopa, A (1995) Stochastic programming Dordrecht/Boston/London: Kluwer Academic Preparata, F P., & Shamos, M I (1985) Computational geometry: An introduction New York: Springer Robinson, L W (1990) Optimal an approximate policies in multiperiod, multilocation inventory models with transshipments Operations Research, 38, 278–295 References 171 Rockafellar, R T., & Uryasev, S (2000) Optimization of conditional value-at-risk The Journal of Risk, 2, 21–41 Ruszczy´nski, A., & Shapiro, A eds (2003) Stochastic programming (Handbooks in operations research and management science, Vol 10) Amsterdam: Elsevier Shiina, T (1999) Numerical solution technique for joint chance-constrained programming problem – An application to electric power capacity expansion Journal of the Operations Research Society of Japan, 42, 128–140 Shiina, T (2000a) L–shaped decomposition method for multi-stage stochastic concentrator location problem Journal of the Operations Research Society of Japan, 43, 317–332 Shiina, T (2000b) Stochastic programming model for the design of computer network (in japanese) Transactions of the Japan Society for Industrial and Applied Mathematics, 10, 37– 50 Shiina, T., & Birge, J R (2003) Multistage stochastic programming model for electric power capacity expansion problem Japan Journal of Industrial and Applied Mathematics, 20, 379– 397 Shiina, T., Tagaya, Y., & Morito, S (2007) Stochastic programming with fixed charge recourse Journal of the Operations Research Society of Japan, 50, 299–314 Shiina, T., Tagaya, Y., & Morito, S (2010) Stochastic programming considering variance (in Japanese) Transactions of the Operations Research Society of Japan, 53, 114–132 Tagaras, G (1999) Pooling in multi-location periodic inventory distribution systems Omega, 27, 39–59 Van Slyke, R., & Wets, R J.-B (1969) L-shaped linear programs with applications to optimal control and stochastic linear programs SIAM Journal on Applied Mathematics, 17, 638–663 Wets, R J.-B (1989) Stochastic programming In G L Nemhauser, A H G R Kan, & M J Todd (Eds.), Optimization (Handbooks in operations research and management science, Vol 1, pp 573–629) Amsterdam: Elsevier Yamauchi, Z ed (1972) STATISTICAL TABLES and formulas with computer applications Tokyo: Japanese Standard Association Appendix A Notations in Part I Table A.1 List of notations used in Part I x = (x1 , · · · , xn ) Xf ξ R(x, ξ ) R1 , Rn Z P (·) p(·) μ ri σij rE , r0 R0 te , t ts TE TS Min Max M V VaR PVaR CVaR W-CVaR WVaR Decision variable, investment ratios Feasible set of x Random variable(s), Random process(es), risk factors like stock prices Profit rate of portfolio x Real(1-dimension real space), n-dimension real space Set of integers Probability Price Mean of profit rate of a portfolio Mean of profit rate of asset i Covariance of Ri and Rj Required return value Acceptable risk value Exit time, set of possible exit times Start time Set of the exit time of an investmemt Set of the start time of an investment Minimization Maximization Mean Variance Value at Risk Period Value at Risk Conditional VaR Conditional VaR in the worst case VaR in the worst case © Springer Nature Singapore Pte Ltd 2018 C Xu, T Shiina, Risk Management in Finance and Logistics, Translational Systems Sciences 14, https://doi.org/10.1007/978-981-13-0317-3 173 Appendix B Historical Prices and Monthly Profit Rates of IBM and INTC: Data Used in Chap • The historical prices included in this appendix were the adjusted daily close price, adjusted for both dividends and splits, of IBM (International Business Machines Corporation) & INTC (Intel Corporation) in the following period: August 1, 2016 ∼ July 31, 2017 • The profit rate data of each stock is the monthly profit rate(profit rate in 22 business days), they are calculated using the historical price data For example, the monthly profit rate of IBM stock on August 1st, 2016 is calculated as follows R(I BM)2016.8.1 = p(2016.8.31) − p(2016.8.1) 153.09 − 154.23 = = − 0.74% p(2016.8.1) 154.23 • The price data were downloaded from Yahoo!Finance in January 8, 2018 • The price data were used in Examples 2.1 ∼ 2.7 of Chap © Springer Nature Singapore Pte Ltd 2018 C Xu, T Shiina, Risk Management in Finance and Logistics, Translational Systems Sciences 14, https://doi.org/10.1007/978-981-13-0317-3 175 R(IBM) −0.74 0.21 0.16 0.11 −0.28 −1.88 −3.76 −2.34 −4.72 −4.88 −3.84 −4.27 −3.47 −4.28 −2.82 −2.43 −3.29 −3.19 −1.17 −0.02 −1.01 −0.35 −0.80 R(INTC) 3.76 5.01 5.34 5.75 4.23 4.00 1.49 4.49 2.68 3.04 4.73 6.99 6.11 6.21 6.27 6.19 5.06 4.27 5.96 6.18 4.98 5.74 4.93 Date 2016/9/29 2016/9/30 2016/10/3 2016/10/4 2016/10/5 2016/10/6 2016/10/7 2016/10/10 2016/10/11 2016/10/12 2016/10/13 2016/10/14 2016/10/17 2016/10/18 2016/10/19 2016/10/20 2016/10/21 2016/10/24 2016/10/25 2016/10/26 2016/10/27 2016/10/28 2016/10/31 IBM 152.35 153.06 151.86 150.76 151.35 151.16 149.99 151.30 149.15 148.66 148.12 148.82 149.13 145.23 145.75 146.00 144.17 145.08 145.38 146.28 147.76 147.05 148.09 INTC 36.23 36.65 36.56 36.45 36.88 36.96 36.99 36.91 36.18 36.05 35.89 36.36 36.20 36.65 34.48 34.40 34.13 34.23 34.08 33.90 33.80 33.73 33.85 R(IBM) −2.80 −3.81 −3.59 −2.61 −2.96 −0.74 0.58 −0.51 4.45 5.47 3.85 3.66 3.85 6.99 7.00 8.40 9.70 8.55 9.11 9.36 7.61 7.26 4.93 R(INTC) −6.56 −8.56 −8.13 −8.93 −10.86 −8.19 −8.13 −7.91 −6.73 −6.08 −6.03 −6.08 −5.86 −6.53 −0.83 −0.52 1.70 0.59 1.73 2.46 2.20 0.64 −2.45 Date 2016/11/29 2016/11/30 2016/12/1 2016/12/2 2016/12/5 2016/12/6 2016/12/7 2016/12/8 2016/12/9 2016/12/12 2016/12/13 2016/12/14 2016/12/15 2016/12/16 2016/12/19 2016/12/20 2016/12/21 2016/12/22 2016/12/23 2016/12/27 2016/12/28 2016/12/29 2016/12/30 IBM 159.00 157.72 155.39 155.58 155.41 155.91 160.22 160.78 161.90 160.91 163.63 163.84 163.36 162.11 162.06 162.95 162.69 162.43 162.09 162.51 161.58 161.98 161.39 INTC 34.54 33.94 33.03 33.42 33.64 33.96 34.73 34.92 34.98 35.19 36.00 35.75 35.99 35.52 36.09 36.40 36.18 36.13 36.17 36.26 35.83 35.86 35.48 R(IBM) 1.50 3.06 5.91 5.42 6.06 4.55 0.44 1.45 0.86 1.11 −0.24 −1.01 −0.72 2.29 2.61 4.95 6.55 6.94 6.35 5.18 5.01 4.62 5.18 R(INTC) 2.72 5.48 7.85 6.41 6.08 5.44 2.93 3.50 2.66 2.28 0.00 0.57 −0.60 1.74 −0.33 1.10 2.22 1.71 2.73 0.94 0.52 −0.38 1.13 Date 2016/8/1 2016/8/2 2016/8/3 2016/8/4 2016/8/5 2016/8/8 2016/8/9 2016/8/10 2016/8/11 2016/8/12 2016/8/15 2016/8/16 2016/8/17 2016/8/18 2016/8/19 2016/8/22 2016/8/23 2016/8/24 2016/8/25 2016/8/26 2016/8/29 2016/8/30 2016/8/31 INTC 33.58 33.30 33.25 33.57 33.96 34.02 33.90 33.52 33.67 33.56 33.89 34.18 34.00 33.95 34.21 34.33 34.37 34.13 34.07 34.23 34.51 34.66 34.85 Historical Prices and Monthly Profit Rates of IBM and INTC: Data Used in Chap IBM 154.23 153.40 153.49 154.33 156.19 156.13 155.87 156.17 157.57 156.05 155.98 154.84 154.59 155.48 154.21 154.17 154.42 153.25 152.85 152.55 153.90 153.59 153.09 B Table B.1 Prices of IBM and INTC stocks(US $) and the monthly profit rate(%) generated 176 2016/9/1 2016/9/2 2016/9/6 2016/9/7 2016/9/8 2016/9/9 2016/9/12 2016/9/13 2016/9/14 2016/9/15 2016/9/16 2016/9/19 2016/9/20 2016/9/21 2016/9/22 2016/9/23 2016/9/26 2016/9/27 2016/9/28 153.72 153.73 154.50 155.75 153.20 150.01 152.52 150.13 148.43 149.99 148.23 149.22 148.82 149.86 150.42 149.33 148.37 151.05 152.52 34.97 35.03 35.51 35.40 35.38 34.41 35.03 34.57 34.58 35.50 36.57 36.08 36.06 36.36 36.46 36.11 35.58 36.10 36.35 −1.93 −1.55 −2.16 −3.69 −1.25 −0.58 −2.53 −1.34 0.26 −0.57 −2.03 −2.33 −1.90 −3.79 −3.55 −2.65 −1.41 −2.18 −3.59 4.22 5.29 4.10 4.50 4.34 5.16 2.91 3.82 5.14 2.00 0.21 −4.44 −4.60 −6.14 −6.10 −5.62 −4.72 −6.37 −7.21 2016/11/1 2016/11/2 2016/11/3 2016/11/4 2016/11/7 2016/11/8 2016/11/9 2016/11/10 2016/11/11 2016/11/14 2016/11/15 2016/11/16 2016/11/17 2016/11/18 2016/11/21 2016/11/22 2016/11/23 2016/11/25 2016/11/28 147.22 146.41 146.81 146.87 150.04 150.87 150.52 155.78 156.80 153.83 154.27 154.88 155.37 155.94 158.26 158.16 157.49 158.62 159.96 33.51 33.59 33.19 32.88 33.94 33.98 33.99 33.75 33.86 33.73 34.15 34.08 34.26 34.19 34.22 34.71 34.43 34.67 34.74 5.68 6.15 6.19 9.09 7.15 7.31 6.91 5.04 4.49 6.20 5.08 4.64 4.88 4.33 2.64 2.48 3.19 1.87 1.26 −0.29 0.15 2.33 5.62 2.91 2.94 3.51 6.67 5.61 6.70 4.01 5.88 6.25 5.81 5.57 4.20 5.31 3.36 3.24 2017/1/3 2017/1/4 2017/1/5 2017/1/6 2017/1/9 2017/1/10 2017/1/11 2017/1/12 2017/1/13 2017/1/17 2017/1/18 2017/1/19 2017/1/20 2017/1/23 2017/1/24 2017/1/25 2017/1/26 2017/1/27 2017/1/30 162.56 164.57 164.02 164.83 163.00 160.93 163.10 163.30 162.70 163.24 162.18 162.19 165.82 166.29 171.02 173.35 173.71 172.39 170.93 35.80 35.62 35.56 35.69 35.81 35.74 36.15 35.91 35.99 36.00 35.96 35.77 36.14 35.97 36.80 36.98 36.74 37.15 36.61 5.16 3.90 5.79 4.74 6.54 8.80 7.77 8.10 9.43 8.92 9.17 8.92 7.06 7.05 3.91 1.42 1.45 3.43 3.50 0.49 0.33 0.71 0.44 −2.45 −2.59 −2.42 −1.43 −1.31 −0.35 −0.05 0.58 −1.66 −0.90 −2.20 −2.72 −2.93 −4.72 −3.35 B Historical Prices and Monthly Profit Rates of IBM and INTC: Data Used 177 Date 2017/1/31 2017/2/1 2017/2/2 2017/2/3 2017/2/6 2017/2/7 2017/2/8 2017/2/9 2017/2/10 2017/2/13 2017/2/14 2017/2/15 2017/2/16 2017/2/17 2017/2/21 2017/2/22 2017/2/23 2017/2/24 2017/2/27 IBM 169.68 169.46 169.74 170.95 170.99 173.51 172.64 173.66 175.10 175.77 176.52 178.04 177.80 177.05 176.65 177.52 178.01 177.72 175.81 INTC 36.02 35.73 35.88 35.98 35.73 35.81 35.84 34.94 34.82 35.27 35.40 35.52 35.87 35.94 35.98 35.54 35.65 35.99 35.97 R(IBM) 3.98 4.36 4.14 2.87 1.55 0.43 0.16 −0.84 −1.61 −1.18 −2.49 −3.29 −4.16 −3.26 −3.02 −4.04 −4.34 −3.77 −3.04 R(INTC) −1.80 −1.91 −1.70 −2.46 −1.24 −1.21 −3.35 −0.79 −0.68 −1.84 −1.84 −1.72 −3.76 −3.04 −3.42 −2.52 −2.18 −2.55 −2.57 Date 2017/3/31 2017/4/3 2017/4/4 2017/4/5 2017/4/6 2017/4/7 2017/4/10 2017/4/11 2017/4/12 2017/4/13 2017/4/17 2017/4/18 2017/4/19 2017/4/20 2017/4/21 2017/4/24 2017/4/25 2017/4/26 2017/4/27 IBM 170.65 171.01 171.02 169.42 169.00 168.69 167.77 167.16 167.24 166.13 167.67 166.64 158.45 159.05 157.17 157.53 157.18 156.85 157.11 INTC 35.54 35.63 35.74 35.68 35.50 35.50 35.27 35.21 35.10 34.73 34.96 35.24 35.38 35.65 35.78 36.21 36.33 36.38 36.88 R(IBM) −8.91 −8.85 −11.16 −10.62 −10.93 −11.28 −11.14 −10.99 −10.35 −8.46 −10.93 −10.47 −5.09 −5.03 −4.28 −4.20 −3.55 −3.80 −4.43 R(INTC) 3.29 2.67 2.24 1.63 1.69 0.69 0.43 0.15 0.74 2.37 −0.51 −0.81 −0.69 −0.40 −0.53 −0.98 −0.92 −1.08 −2.62 Date 2017/6/1 2017/6/2 2017/6/5 2017/6/6 2017/6/7 2017/6/8 2017/6/9 2017/6/12 2017/6/13 2017/6/14 2017/6/15 2017/6/16 2017/6/19 2017/6/20 2017/6/21 2017/6/22 2017/6/23 2017/6/26 2017/6/27 IBM 151.07 150.46 150.82 150.78 149.40 150.51 152.49 153.56 152.64 152.20 152.61 153.76 153.22 153.33 152.18 152.79 152.50 153.61 153.13 INTC 35.85 36.05 36.07 35.86 35.99 36.21 35.44 35.46 35.61 35.27 35.05 34.95 35.25 34.60 34.32 34.10 33.94 33.82 33.40 R(IBM) 1.91 1.07 −0.03 0.37 1.62 0.72 −0.26 −1.00 −0.01 −0.52 −0.14 −5.05 −4.64 −5.08 −5.07 −5.32 −5.68 −6.55 −6.76 R(INTC) −7.36 −5.45 −7.46 −6.23 −7.20 −7.02 −4.09 −4.17 −3.34 −2.98 −2.21 −1.85 −2.14 −0.37 −0.23 0.90 1.64 2.64 4.93 B Table B.2 Historical prices of IBM and INTC stocks and the monthly profit rate generated 178 Historical Prices and Monthly Profit Rates of IBM and INTC: Data Used in Chap 2017/2/28 2017/3/1 2017/3/2 2017/3/3 2017/3/6 2017/3/7 2017/3/8 2017/3/9 2017/3/10 2017/3/13 2017/3/14 2017/3/15 2017/3/16 2017/3/17 2017/3/20 2017/3/21 2017/3/22 2017/3/23 2017/3/24 2017/3/27 2017/3/28 2017/3/29 2017/3/30 176.22 178.31 176.91 176.44 176.86 176.77 175.86 173.63 174.27 172.93 172.20 172.29 173.69 172.13 172.18 170.40 171.28 171.32 170.35 170.29 171.01 170.46 170.38 35.66 35.40 35.38 35.37 35.04 35.27 35.09 35.29 35.38 34.64 34.66 34.58 34.62 34.75 34.91 34.52 34.85 34.75 34.64 34.87 35.07 35.04 35.22 −3.31 −4.29 −3.34 −3.07 −4.21 −4.40 −4.07 −3.38 −4.08 −3.29 −3.52 −2.68 −4.06 −7.95 −7.63 −7.76 −8.03 −8.25 −7.92 −7.74 −8.15 −8.68 −8.49 −1.24 0.39 0.70 1.06 1.83 0.64 1.15 −0.06 −0.47 1.34 0.20 1.08 1.79 1.81 2.12 3.65 3.90 4.54 5.03 5.76 1.54 2.08 3.41 2017/4/28 2017/5/1 2017/5/2 2017/5/3 2017/5/4 2017/5/5 2017/5/8 2017/5/9 2017/5/10 2017/5/11 2017/5/12 2017/5/15 2017/5/16 2017/5/17 2017/5/18 2017/5/19 2017/5/22 2017/5/23 2017/5/24 2017/5/25 2017/5/26 2017/5/30 2017/5/31 157.08 155.66 155.91 155.45 155.86 151.94 151.43 150.52 149.67 149.07 148.80 149.93 152.07 149.35 149.20 150.39 151.04 150.44 150.92 151.60 150.90 150.14 151.03 35.62 35.77 36.42 36.70 36.58 36.55 36.27 36.10 35.74 35.42 35.27 35.36 35.55 34.78 34.96 35.14 35.50 35.59 35.85 35.99 35.99 35.91 35.84 −3.85 −2.95 −3.50 −2.98 −3.26 −1.67 −0.61 1.31 2.60 2.39 2.29 1.79 1.11 2.59 2.77 1.19 1.15 1.37 1.78 1.01 1.86 1.58 0.79 0.63 0.22 −1.03 −1.73 −1.95 −1.52 −0.16 −1.81 −0.78 0.53 0.00 −0.90 −1.70 1.34 −1.02 −2.32 −3.94 −4.66 −5.68 −7.20 −5.68 −7.30 −6.56 2017/6/28 2017/6/29 2017/6/30 2017/7/3 2017/7/5 2017/7/6 2017/7/7 2017/7/10 2017/7/11 2017/7/12 2017/7/13 2017/7/14 2017/7/17 2017/7/18 2017/7/19 2017/7/20 2017/7/21 2017/7/24 2017/7/25 2017/7/26 2017/7/27 2017/7/28 2017/7/31 153.70 152.52 152.22 153.95 152.06 150.77 151.34 151.82 151.59 152.09 152.02 152.63 151.41 152.39 145.99 146.12 145.54 144.46 144.66 143.84 143.55 142.78 143.16 33.95 33.29 33.49 33.21 34.08 33.38 33.63 33.40 33.67 33.99 33.98 34.42 34.21 34.27 34.30 34.49 34.47 34.24 34.41 34.49 34.71 35.05 35.21 −6.86 3.71 B Historical Prices and Monthly Profit Rates of IBM and INTC: Data Used 179 Appendix C Historical Prices of 10 Components of the DJIA: Data Used in Chap • The historical prices included in this appendix were the adjusted daily close price, adjusted for both dividends and splits, of the following 10 component stocks of DJIA(Dow Jones Industrial Average) in the following period: January 3, 2012 ∼ December 29, 2017 Symbol GE APPL XOM WMT UTX JNJ INTC GS VZ MSFT Company Name General Electric Company Apple Inc Exxon Mobil Corporation Wal-Mart Stores, Inc United Technologies Corporation Johnson & Johnson Intel Corporation The Goldman Sachs Group, Inc Verizon Communications Inc Microsoft Corporation *Only partial price data were included in this appendix for saving space • The price data were downloaded from Yahoo!Finance on January 8, 2018 • The price data were used in Examples 5.1 ∼ 5.4 of Chap 5, in generating scenarios for the profit rates of component stocks © Springer Nature Singapore Pte Ltd 2018 C Xu, T Shiina, Risk Management in Finance and Logistics, Translational Systems Sciences 14, https://doi.org/10.1007/978-981-13-0317-3 181 182 C Historical Prices of 10 Components of the DJIA: Data Used in Chap Table C.1 Prices of the 10 component stocks of Dow Jones Industrial Average(US $) Date 2012/1/3 2012/1/4 2012/1/5 2012/1/6 2012/1/9 2012/1/10 2012/1/11 2012/1/12 2012/1/13 2012/1/17 2012/1/18 2012/1/19 2012/1/20 2012/1/23 2012/1/24 2012/1/25 2012/1/26 2012/1/27 GE APPL 15.01 52.66 15.18 52.95 15.17 53.53 15.25 54.09 15.42 54.01 15.31 54.20 15.44 54.11 15.48 53.96 15.41 53.76 15.32 54.39 15.55 54.95 15.66 54.78 15.66 53.82 15.49 54.73 15.41 53.84 15.64 57.20 15.59 56.94 15.56 57.28 XOM 71.42 71.44 71.22 70.69 71.01 71.19 70.66 70.38 70.49 71.17 71.80 72.28 72.66 72.64 72.40 72.44 72.06 71.28 WMT UTX JNJ 51.67 64.73 55.23 51.14 65.07 54.89 50.89 64.44 54.83 50.53 64.07 54.35 50.69 64.24 54.43 50.57 65.92 54.66 50.87 66.50 54.60 50.96 66.96 54.69 50.99 65.96 54.71 51.26 66.79 54.59 51.40 67.28 54.73 51.91 66.92 54.65 52.25 66.49 54.72 52.17 66.63 54.49 52.58 67.43 54.49 52.65 67.32 54.68 52.22 67.11 55.08 52.00 67.29 54.96 INTC 20.20 20.67 20.91 20.79 20.97 21.07 21.24 21.20 20.70 20.61 20.90 21.10 21.72 21.99 22.15 22.15 22.02 22.01 GS 87.71 87.14 86.99 85.93 87.10 90.44 91.76 93.09 91.02 89.85 95.94 99.04 100.02 99.51 100.14 99.59 99.85 102.81 VZ MSFT 29.85 22.80 29.46 23.33 29.26 23.57 29.18 23.94 29.21 23.62 29.36 23.71 29.61 23.60 29.63 23.84 29.63 24.06 29.70 24.06 29.69 24.04 29.69 23.94 29.66 25.30 29.23 25.32 28.77 24.98 28.69 25.17 28.42 25.12 28.32 24.89 2017/11/30 2017/12/1 2017/12/4 2017/12/5 2017/12/6 2017/12/7 2017/12/8 2017/12/11 2017/12/12 2017/12/13 2017/12/14 2017/12/15 2017/12/18 2017/12/19 2017/12/20 2017/12/21 2017/12/22 2017/12/26 2017/12/27 2017/12/28 2017/12/29 18.16 17.76 17.83 17.64 17.54 17.59 17.59 17.53 17.79 17.64 17.52 17.70 17.64 17.47 17.33 17.35 17.38 17.43 17.38 17.36 17.45 83.29 83.46 83.57 82.89 82.28 82.55 82.66 83.03 82.76 83.12 82.90 83.03 82.94 82.44 82.87 83.85 83.97 83.98 83.90 84.02 83.64 96.72 96.84 96.50 97.32 96.77 96.78 96.55 96.93 96.70 97.76 97.13 97.11 97.90 98.80 98.75 98.06 98.21 99.16 99.26 99.40 98.75 44.84 44.68 44.49 43.44 43.45 43.08 43.35 43.66 43.33 43.34 43.26 44.56 46.26 47.04 47.56 46.76 46.70 46.08 46.11 46.22 46.16 247.64 248.95 250.65 248.33 245.95 248.56 250.35 250.13 257.68 255.56 255.48 257.17 260.02 256.48 255.18 261.01 258.97 257.72 255.95 256.50 254.76 50.32 50.67 51.14 50.35 50.11 49.85 50.51 51.26 52.59 52.29 51.75 52.08 52.65 52.24 52.18 52.41 52.59 52.62 52.68 52.83 52.33 171.85 171.05 169.80 169.64 169.01 169.32 169.37 172.67 171.70 172.27 172.22 173.97 176.42 174.54 174.35 175.01 175.01 170.57 170.60 171.08 169.23 121.45 120.12 120.04 120.29 121.20 122.40 122.81 123.30 123.48 124.30 123.76 126.17 126.71 126.78 127.00 127.31 127.23 127.14 127.58 128.12 127.57 139.33 139.98 139.01 139.67 141.06 140.01 140.59 141.14 142.60 142.89 141.65 142.46 141.80 141.78 141.16 141.06 140.12 140.09 140.57 140.56 139.72 84.17 84.26 81.08 81.59 82.78 82.49 84.16 85.23 85.58 85.35 84.69 86.85 86.38 85.83 85.52 85.50 85.51 85.40 85.71 85.72 85.54 Index A Absolute deviation, 40, 55 Aggregate level decision, 112, 115 Algorithm, 35, 38, 42, 44, 46–48, 57, 90–93, 100, 108, 109, 111, 115, 116, 121, 124, 132, 136, 138, 141–145, 149, 156–160, 162, 163, 168 A Modeling Language for Mathematical Programming (AMPL), 39, 136, 145, 162 B Benders decomposition, 109, 115, 156 Beta distribution, 126 Block separable recourse, 111–115 Branch-and-cut, 121, 124, 125 C Chance-constrained programming, 125–130 Complete recourse, 142, 160 Conditional value at risk (CVaR), 27–31, 34, 44, 53–55, 57, 61–64, 70, 71, 73, 75, 77, 78, 80, 82, 90, 91, 95, 97–99, 153, 156, 161–162, 164–168, 173 Confidence level, 18, 21, 23, 26, 27, 50, 54, 61–64, 77, 87 Convex envelope, 116, 121, 122 Convex hull, 121 Correlative, 126, 129 Covariance, 15, 16, 18–20, 26, 34, 39, 71–74, 95, 173 Credit risk, 5, Cumulative, 114, 128, 165–167 Customer, 153, 154, 159, 160, 162, 163 CVaR in the worst case(W-CVaR), 63, 64, 77, 78, 82, 91, 92, 173 CVaR minimization, 44, 54, 55, 91, 92, 156, 162, 164–168 D Derivatives, 4, 6, Detailed level decisions, 112, 114, 115 Deterministic equivalent, 108, 112–114, 126, 128 Dirichlet distribution, 126 Distribution center, 139–141 Distribution function, 18, 19, 74, 125–128 Down-side risk, 32, 34 E Emergency lateral transshipment, 138, 141, 143, 144, 146, 147, 149 Exit time, 9, 10, 60–64, 68–70, 72, 75, 78, 80–83, 86, 89–91, 93–97, 99, 100, 173 F Factory, 153, 154, 162, 163 Feasibility cut, 109, 110, 159, 160 Financial investment, 3–11, 13–34, 37 Financial market, 3–7, 36 Financial risk, 3–11 First stage decision, 108, 115, 116, 157 Flexible investments, 59–100 Future, 4–7, 9, 10, 15, 20, 22, 24, 25, 34, 36, 61, 65, 67, 86, 140, 149, 168 © Springer Nature Singapore Pte Ltd 2018 C Xu, T Shiina, Risk Management in Finance and Logistics, Translational Systems Sciences 14, https://doi.org/10.1007/978-981-13-0317-3 183 184 G Gamma distribution, 126 Gauss quadrature, 129 Geometric Brownian motion (GBM), 65, 66, 67 H Heuristic, 43–45, 57, 82, 153 Historical simulation, 22–24, 27, 30, 34, 42, 51, 52, 62, 64, 67, 68, 70, 75, 77, 78, 83, 84, 86, 88, 90, 97 I IBM ILOG CPLEX, 136, 162, 163 Inventory, 137–149 Inventory distribution problem, 137–149 J Joint chance-constraint, 126 L Lateral transshipment problem, 139–144 Linear programming (LP), 30, 31, 40, 42, 44, 45, 53–55, 57, 71, 75, 82, 85, 86, 91, 98, 108, 111–114, 126, 140 Logarithmic concave probabilistic measure, 126 Logistics network, 151–168 Lower bound, 121–123, 125, 131–136, 143, 156, 162 Lower semicontinuous, 120, 121 L-shaped method, 109, 110, 145, 153, 156, 162–164, 168 M Market risk, 6, 7, 10, 13–57, 59–100 Master problem, 116, 142, 143, 156, 157, 159, 160, 162, 164 M-CVaR(t) model, 95, 99 Mean variance model (MV model), 37–42, 93, 96 MIP, 139, 142–145, 149 Mixed integer linear programming (MILP), 82, 84–87, 89, 100 Monte Carlo simulation, 19, 24–27, 62, 64, 65, 67, 77 M-PVaR model, 81–89 M-risk models, 55–57, 81, 89–93, 100 M-risk(t) models, 93–100 Index Multistage stochastic programming with recourse, 110 Multivariate normal distribution, 128 MV(t) model, 93, 94 N Nested decomposition, 110–112 Nonlinear programming (NLP), 38, 42, 57, 90–93, 100, 125, 129 Normal distribution, 19, 20, 24, 25, 34, 47, 67, 71, 73, 74, 76–78, 90–92, 100, 128, 144 O Operational risk, 6, Optimality cut, 110, 115, 123–125, 142–144, 156, 159, 162 Option, 4, 6, 7, 156 P Period Value at Risk (PVaR), 61–69, 78, 80–89, 100, 173 Piecewise linear, 45, 109 Portfolio, 7, 14, 16, 17, 19–27, 29–33, 36–40, 42, 44, 45, 50–55, 57, 62, 63, 67–73, 75–77, 80, 83, 84, 86–91, 95, 96, 98, 173 Portfolio selection, 36–37, 40, 42–44, 50, 51, 53, 54, 57, 60, 81, 82, 86–92 Preventive lateral transshipment, 138–141, 144, 146, 147, 149 Probabilistic constraint, 108 Probability distribution, 18, 28, 125, 126, 167 Probability space, 108, 111, 112, 125 Profit rate, 7, 8, 14–16, 19, 20, 22–26, 30–34, 36, 39, 40, 50, 51, 53–55, 66–72, 74, 76–78, 80, 83, 86–92, 94, 95, 97, 98, 173, 175, 178, 181 PVaR minimization, 82–89, 100 Q Quadratic programming (QP), 38, 56, 57, 95, 136 R Recourse function, 108, 109, 110, 113, 115–122, 124, 131, 133, 134, 141–143 Reformulation, 146 Relative error, 145, 146 Relatively complete recourse, 117 Index Return, 7–9, 18, 22, 27, 31, 36–38, 44, 50, 53, 54, 80, 82, 87, 94–96, 98–100, 132, 133, 143 measure, 96, 99 Risk countermeasures, 6–7 diversification, 7–9 hedging, 6–7 measure, 10, 13–34, 37, 40, 42, 44, 45, 53, 57, 59–78, 81, 82, 89–93, 95, 100, 161 S Safety-first criterion, 31, 32, 34 Scenario, 16, 22, 24, 25, 27, 29–32, 34, 39, 40, 44, 45, 50, 53, 54, 55, 57, 64, 67–70, 83, 84, 86, 88, 89, 91, 94, 95, 97, 98, 108, 109, 111–114, 117, 122, 123, 125, 141, 143, 144, 145, 147, 153, 154, 156, 157, 159, 160, 163–168, 181 Scenario simulation, 64–70, 78, 83, 94 Scenario tree, 111, 112, 114 Second stage decision, 108, 116 Semi-variance, 32, 33, 55, 120, 121 Soft optimization approach, 43, 45–46, 82, 88–90, 100 Stochastic integer programming, 115–125 Stochastic programming, 107–136, 138, 141, 149, 155, 156 Stochastic programming problem with recourse, 108, 110 Supply chain, 138–140 Support, 108, 111, 117, 124, 130, 131 185 T Tender, 117, 118, 132, 134 Transshipment, 140, 146, 147 Two-stage stochastic problem, 108, 114, 115 U Uncertain exit time, 60–64, 72, 81, 89 Uncorrelated, 126, 128 Uniform distribution, 118, 163 Uniformly sampling, 48, 88 V Value at risk (VaR), 17–28, 34, 42–46, 50–53, 57, 61–64, 70–78, 80, 82, 87, 88, 90, 98, 99, 154, 161, 167, 173 Variance, 14–18, 20, 21, 25, 34, 37, 39, 40, 55, 57, 66, 67, 71–73, 75, 76, 82, 95, 130–136, 144, 147, 148, 149, 173 Variance-covariance, 19–21, 26, 34, 72, 74 VaR in the worst case (WVaR), 63, 64, 75, 76, 77, 82, 91, 92, 173 VaR minimization, 42, 45, 46, 50, 57, 82, 88, 90 W Warehouse, 139, 152–154, 156, 158, 159, 162, 163 Wishart distribution, 126 ... markets Because risk is inherent in financial investments and is often linked with gains, eliminating risk is not the purpose of risk management in finance, its purpose for financial investments... independent and self-contained, readers interested in risk management in logistics can skip Part I, and those interested only in financial investments may only read Part I Part I is on risk management. .. financial investment Financial investments may result in losses rather than gains due to the uncertainty in financial markets, this is called financial risk in investments Risk is not the main