Risk management in finance and logistics

Chapter 3 presents optimization models forrisk control in investment decisions and the methods for solving these models,which are the main tools for risk management in investments.. We i

Translational Systems Sciences 14

Chunhui Xu · Takayuki Shiina

Risk

Management

in Finance and Logistics

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

of science He describes that it hopes to develop something like a “spectrum” of theories—asystem 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 thegaps 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 principleswithin 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 Societyfor the Systems Sciences) was founded One of the concerns of the founders, even then, wasthe state of the human condition, and what science could do 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 enhancinghuman health and well-being Translational medical research is often labeled as “Bench toBedside.” It places emphasis on translating the findings in basic research (at bench) morequickly and efficiently into medical practice (at bedside) At the same time, needs anddemands from practice drive the development of new and innovative ideas and concepts Inthis 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 systemsconcepts, 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 createcommon platforms for language to activate the “bench to bedside” cycle

In order to create a resilient and sustainable society in the twenty-first century, weunquestionably need open social innovation through which we create new social values, andrealize them in society by connecting diverse ideas and developing new solutions We assumethree 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, andmanagement 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 bydrawing on a range of disciplines in trans-disciplinary and cross-cultural ways They mayinclude social systems theory, sociology, business administration, management informationscience, organization science, computational mathematical organization theory, economics,evolutionary economics, international political science, jurisprudence, policy science, socio-information studies, cognitive science, artificial intelligence, complex adaptive systemstheory, philosophy of science, and other related disciplines In addition, this series willpromote translational systems science as a means of scientific research that facilitates thetranslation 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 andapplication, for twenty-first-century socioeconomic systems in a translational and trans-disciplinary context

More information about this series athttp://www.springer.com/series/11213

Risk Management in Finance and Logistics

123

Department of Risk Science in Finance

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

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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.

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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.

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Yoko, Tatsuro, Moeko

Risk has been recognized as an important factor that has to be considered inmanagerial decision-making in many disciplines, such as investments in financialmarkets 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 Hencerisk is a notion that spans multiple disciplines, and people’s diverse subjectiveperceptions should be considered in addressing issues regarding risk management.These features make this book an appropriate part of the Translational SystemsSciences series, which aims to cultivate a new frontier of systems sciences thatcontribute to the need for beneficial and practical applications

This book is intended to provide an introduction to the central concepts andquantitative tools for risk management, and simultaneously present some up-to-date research results on risk management in the fields of financial investments andlogistics planning To the best of our knowledge, this is the first book coveringdiverse 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 uate or graduate level for students who have a technical background in engineering,mathematics, or science This book aims to be informative for researchers whoseinterests are related to risk management, especially for those in the fields of financeand logistics The prerequisites for reading this book are relatively modest; theprime requirement is some familiarity with introductory elements of probabilitytheory and optimization techniques Certain sections do assume some knowledge ofmore advanced concepts of probability theory and optimization, such as stochasticprocesses and the branch and bound algorithm, but the text is structured so thatthe mainstream can be faithfully pursued without reliance on these advancedbackground concepts

undergrad-vii

This book is composed of two separate parts, each part is relatively independentand self-contained, readers interested in risk management in logistics can skip PartI,and those interested only in financial investments may only read PartI.

PartI is on risk management in finance, which is the most developed branch

in risk management The first three chapters in PartIintroduce the key conceptsand quantitative tools for risk management in financial investments Chapter 1is

an introduction to investments in financial markets, the associated risks, and riskmanagement in investments Chapter2presents 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 methodsfor estimating these risk measures Chapter 3 presents optimization models forrisk control in investment decisions and the methods for solving these models,which are the main tools for risk management in investments While the last twochapters of PartIinclude some up-to-date research results on financial investments.Chapter4 addresses a more general situation of investment known as the flexibleinvestment, wherein the start and/or exit time of investments are not fixed butflexible within certain time intervals We introduce the return and risk measuresfor flexible investments, some new indices for market risk are included, such as theperiod value at risk, and the methods for estimating these risk measures Chapter5

includes optimization models for risk control in flexible investments and methodsfor solving these models

Part II is on risk management in logistics Chapter 6 presents the basic ory of stochastic programming which originates from the linear programmingproblem developed by Dantzig In particular, the solution method is described indetail regarding the stochastic programming problem with recourse This model

the-is extended to models that include multi-stage planning and integer constraints

In addition, we show the basic concepts regarding a model having probabilisticconstraints and also including a variance term in the objective function In Chap.7,stochastic programming problem for inventory distribution is formulated withdemand as a stochastic variable and the effectiveness of the policy of usingboth preventive and emergency lateral transshipment is examined In Chap.8, thestochastic programming model for the logistics network reorganization problem andthe efficient solution method are shown

Finally, the first author wishes to thank his doctoral research supervisor ProfessorKyoiichi Kijima (Tokyo Institute of Technology), for his insightful guidance inPh.D research, and the author’s postdoctoral research supervisor Professor Yu-ChiLarry Ho (Harvard University), for his ingenious ideas regarding ordinal optimiza-tion from which the author benefited a lot He would like to thank many researchcollaborators for their helpful discussions, particularly Masakazu Ando_(ChibaInstitute of Technology, Japan), Shuning Wang_(Tsinghua University, China),Min Huang_(Northeastern University, China), Xiao Luo_(National University ofSingapore), Xiaolin Huang_(Shanghai Jiaotong University, China), and Utomo S.Putro and Santi Novani_(Institut Teknologi Bandung, Indonesia) He is also thankful

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

March 2018

Part I Risk Management in Finance

1 Financial Investment, Financial Risk and Risk Management 3

1.1 Financial Markets and Financial Investment 3

1.2 Main Risks in Financial Markets 5

1.3 Risk Countermeasures: Hedging and Diversifying 6

1.4 Risk Management by Diversification 7

1.5 Outline of Part I 9

2 Market Risk Measures in Financial Investments 13

2.1 Market Risk and Its Measurement 14

2.2 Variance: Fluctuation Is Taken as Risk 14

2.2.1 Definition of Variance 14

2.2.2 Estimation of Variance 15

2.3 Value at Risk: A Likely Loss Is Taken as Risk 17

2.3.1 Definition of Value at Risk 17

2.3.2 Estimation of VaR: Three Methods 18

2.4 Conditional VaR: Expected Loss Behind VaR Is Taken as Risk 27

2.4.1 Definition of Conditional VaR 27

2.4.2 Estimation of CVaR 27

2.5 Other Risk Measures: Failure Is Taken as Risk 31

2.6 Summary 34

3 Market Risk Control in Investment Decisions 35

3.1 Portfolio Selection and Its Models 36

3.2 MV Model and Its Variations 37

3.2.1 The Base MV Model and Its Two Variations 37

3.2.2 Solving Methods for MV Based Models 38

3.2.3 Two MV Based Models with Computational Advantages 39

xi

3.3 M-VaR Model and Its Solving Method 42

3.3.1 Methods for Solving M-VaR Models 43

3.3.2 Solving M-VaR Model Using the Soft Optimization Approach 45

3.4 M-CVaR Model and Its Solving Method 53

3.5 Other M-Risk Models and Solving Methods 55

3.6 Summary 57

4 Market Risk Measures for Flexible Investments 59

4.1 Flexible Investments 60

4.2 Risk Measures for Investments with Uncertain Exit Time 60

4.2.1 Period Value at Risk 61

4.2.2 Risk Measures Based on Average Loss in Time Axis 62

4.3 Estimation of PVaR with Scenario Simulation 64

4.3.1 Monte Carlo Simulation Method for Estimating PVaR 65

4.3.2 Historical Simulation Method for Estimating PVaR 67

4.4 Estimation of Risk Measures Based on Average Loss 69

4.4.1 Estimation of Risk Measures Under Complete Information About the Probabilities of Exit Time 69

4.4.2 Estimation of Risk Measures Under Partial Information About the Probabilities of Exit Time 75

4.5 Summary 78

5 Market Risk Control in Flexible Investment Decisions 79

5.1 Evaluation of Investments with Flexible Investment Term 80

5.2 Two Kinds of Model for Flexible Investment Decisions 81

5.3 M-PVaR Model and Solving Methods 82

5.3.1 Solving PVaR Minimization Model by Solving a Mixed Integer Linear Programming 82

5.3.2 Solving PVaR Minimization Model Using the Soft Optimization Approach 88

5.4 M-Risk Models and Solving Methods 89

5.4.1 M-Risk Models with Complete Probability Information Regarding Exit Time 90

5.4.2 M-Risk Models with Partial Probability Information Regarding Exit Time 91

5.5 M-Risk(t) Models and Their Solving Methods 93

5.6 Summary 100

References 101

Part II Risk Management in Logistics 6 Basic Results on Stochastic Programming 107

6.1 Stochastic Programming with Recourse 108

6.2 Multi-stage Stochastic Programming Model 110

6.3 Stochastic Integer Programming 115

6.4 Chance-Constrained Programming 125

6.5 Two-Stage Model Taking Variance into Account 130

7 Inventory Distribution Problem 137

7.1 Introduction 138

7.2 Lateral Transshipments 139

7.3 Numerical Experiments 144

7.4 Summary 149

8 Reorganization of Logistics Network 151

8.1 Background 152

8.2 Formulation 154

8.2.1 Sets 154

8.2.2 Parameters 154

8.2.3 Variables 154

8.3 Solution Algorithm 156

8.4 CVaR Optimization Model 161

8.5 Numerical Experiments 162

8.5.1 Node 162

8.5.2 Costs 162

8.5.3 Customer Demand 163

8.6 Conclusions 168

References 169

Appendix A Notations in Part I 173

Appendix B Historical Prices and Monthly Profit Rates of IBM and INTC: Data Used in Chap 2 175

Appendix C Historical Prices of 10 Components of the DJIA: Data Used in Chap 5 181

Index 183

Risk Management in Finance

Various kinds of financial markets have been developed to meet the need toraise capital, these markets simultaneously provide opportunities for investors tomake a profit but they may also suffer a loss from investments, which is known

as investment risk PartI is devoted to the issue of risk management in financialinvestments, building on the view point of investors

Part I aims to introduce the central concepts and quantitative tools for riskmanagement in the framework of modern portfolio theory, which are contents ofChaps.1,2and3, while Chaps.4and5present some up-to-date research results onrisk management in financial investments

Chapter1is an introduction to the background knowledge of financial marketsand risk management in such markets We include the features of major financialmarkets, the three main risks in financial markets, and the two countermeasuresfor market risk We do this in order to focus on and explain the structure of riskmanagement through diversification

Chapter2outlines the basics of risk management in investments It summarizesthe key indices for market risk and the practical methods used to estimate them.Many risk measures have been proposed since risk is a notion without a universallyaccepted definition; this chapter focuses on the three popular risk indices: theVariance, the Value at Risk, and the Conditional Value at Risk Several other riskmeasures are also included

Chapter 3 presents operational tools for risk management in investments Wepresent the main optimization models for making investment decisions, wherein therisk consideration is properly incorporated in these models, and the solving methodsfor these investment decision models, from conventional optimization methods toheuristic optimization algorithms

The next two chapters are recent research on risk management in a more generalsituation known as the flexible investment in which the timings of investments areflexible, we focus on the situation with a flexible exit time

Chapter4presents risk indices for market risk of flexible investments Becausethe investment term is flexible, risk measures proposed in traditional investmentcontexts do not apply to flexible investments This chapter first introduces several

new concepts for measuring market risk such as the Period Value at Risk(PVaR)and the Value at Risk in the worst case, which are needed in addressing risk control

of flexible investments It then introduces the methods for estimating these riskmeasures, including analytical methods based on the normal distribution assumptionand scenario simulation based methods

Chapter 5 introduces tools for risk management in flexible investments Wepresent two kinds of decision models for flexible investments wherein risk consid-eration is incorporated in the models, and introduce the methods for solving theseoptimization models

Financial Investment, Financial Risk

and Risk Management

Abstract This chapter introduces the background knowledge for risk management

in finance and outlines the contents of PartI

Section1.1briefly introduces the main financial markets, including the capitalmarkets, the money markets and the foreign exchange market, and investments infinancial markets We then explain the purpose of risk management from the viewpoint of investors

Section1.2 explains the three main risks in financial markets Investments infinancial markets are risky because of uncertainties in the markets; credit risk iscaused by uncertainty regarding the issuers of securities, market risk is due touncertainty in markets, and operational risk is caused by uncertainty in tradingsystems PartIwill focus on market risk

Section1.3summarizes the risk countermeasures available for use Risk termeasures are developed in order to cope with risks in financial markets, theyare used for hedging and/or reducing risk Derivatives are developed from hedginganticipated risks, while diversification of an investment is to reduce risk PartIwilladdress the issue of risk management through diversification

coun-Section 1.4 presents the framework for addressing risk management throughdiversification Modern portfolio theory (MPT) is the main stream theory forfinancial investments, we follow the framework MPT to address risk management

by diversification

Section1.5outlines the contents and structure of PartI

Keywords Financial market · Financial investment · Financial risks · Risk

hedge · Risk diversification · Risk management

1.1 Financial Markets and Financial Investment

Various financial markets have been developed to meet different needs for finance,they can be classified from the perspective of raising finance Markets for long-termfinance are called capital markets, while markets for short-term finance are calledmoney markets

© Springer Nature Singapore Pte Ltd 2018

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Translational Systems Sciences 14, https://doi.org/10.1007/978-981-13-0317-3_1

3

Foreign ExchangeMarket

Commodity Market

Fig 1.1 A classification of financial markets

Stock markets bond markets are two representatives of the capital markets Stockmarkets enable investors to buy and sell shares of publicly traded companies Theprimary stock market is where new issues of stocks are first offered, subsequenttrading of stock securities occurs in the secondary market Bond markets enableinvestors to buy and sell bonds issued by corporations, municipalities, states, andfederal governments from around the world

A money market trades highly liquid and short-term securities with a maturityshorter than one year, securities traded in this market include certificates of deposit,banker’s acceptances, certain bills, notes and commercial papers

In addition to money markets and capital markets, there are several other financialmarkets such as the derivative markets, the foreign exchange markets, and thecommodity markets A classification of financial markets is shown in Fig.1.1.The derivatives market trades securities that derive their values from theirunderlying assets, traded derivatives include forward contracts, futures, options, andswaps Derivatives are instruments developed to hedge financial risks, although theycan also be used for speculation

The foreign exchange market is a financial market where currencies are traded, it

is also referred to as the Forex market The inter-bank market is the financial systemthat trades currency between banks

The commodity markets facilitate the trading of commodities, such as preciousmetals or agricultural products

Refer to Valdez and Molyneux (2015) and Pilbeam (2010) for a conciseand insightful introduction to the workings of financial markets, products, andinstitutions

Financial markets are places where people raise capital by selling financialinstruments and these markets simultaneously provide opportunities to increase thevalue of capital One side receives capital from the market by offering financial

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 mainissue in short term finance since securities in the money markets are considered low-risk However, financial securities in capital markets are diversified, many high-risksecurities are traded on capital 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 forfinancial investments is to keep financial risk under control, which is different fromrisk management in other fields

This book will focus on risk management in capital markets from the viewpoints

of investors

1.2 Main Risks in Financial Markets

There are various uncertainties in financial markets which cannot be accuratelyanticipated, investments in financial markets is often risky due to some unforeseenevents in the future The main risks in capital markets are grouped into three types,

as explained below

Credit risk: Credit risk refers to the risk that a borrower may not repay a loan and

that the lender may therefore lose the principal of the loan or the interest associatedwith it

When lenders offer borrowers mortgages or other types of loans, there is always

an element of risk that the borrower may default on the loan Similarly, if a companyoffers credit to its client, there is a risk that they may not pay their invoices Creditrisk also describes the risk of a bond issuer failing to make payment when requested

or that an insurance company will not be able to make a claim

Credit risk arises because borrowers expect to use future cash flows to pay currentdebts; it is almost never possible to ensure that borrowers will definitely have thefunds to repay their debts.

Market risk: Market risk is the risk of losses in positions arising from movements

in market prices

The most commonly used types of market risk include the equity risk, which isthe risk that stock prices or their implied volatility will change, the interest rate risk,which is the risk that interest rates or their implied volatility will change, and thecurrency risk, which is the risk that foreign exchange rates or their implied volatilitywill change

Credit risk and market risk are two types of risks that are inherent in financialinvestments and are usually revenue driven, meaning that, controlling instead ofeliminating these risks is the main concern in investment

Operational risk: Operational risk is the prospect of loss resulting from

inade-quate or failed procedures, systems or policies It may be caused by employee errors,systems failures, fraud or other criminal activity, and any event that disrupts businessprocesses

Contrary to credit risk and market risk, operational risks are usually not ately incurred nor are they revenue driven Moreover, they are not diversifiable andlayoff cannot be used to temper such risk, meaning that, as long as people, systems,and processes remain imperfect, operational risk cannot be fully eliminated.Nonetheless, operational risk is manageable as by balancing the costs of improve-ment against the expected benefits, losses can be kept within some level of risktolerance

deliber-Many countermeasures have been developed to control these risks, they varyaccording to the type of risk and this book will only address the management ofmarket risk The reader may easily find resources on the management of credit riskand operational risk on the internet

With references to the management of credit risk, Renault and De gny (2004) provides a guideline for the understanding and controlling of creditrisk, while Duffie and Singleton (2003) provides an integrated treatment of theconceptual, practical, and empirical foundations of credit risk pricing, and riskmeasurement For the management of operational risk, Birindelli and Ferretti (2017)addresses several topical issues related to operational risk management in bank:regulation, organization, and strategy

Servi-1.3 Risk Countermeasures: Hedging and Diversifying

To cope with risks in investments, many countermeasures have been developed andthese can be regarded as having developed from two schools of thought

One approach to risk is the concept of hedging, which suggests that investors pare for anticipated risks Various derivatives, such as future contracts and options,

pre-are developed for hedging risks in financial markets, they provide countermeasures

to hedge against certain anticipated risks

For instance, a put option can be used to hedge the risk of holding the underlyingasset, while a call option can be employed to hedge the risk of a price rise of theunderlying asset in the future A future contract can fix a trade in the future, thuseliminating investors’ concern about the undesirable situation in the future.However, futures contracts may result in a loss when the market moves in theopposite direction, they can only hedge the risk in one direction, and using optionsalways incurs a cost, hence it is difficult to say that derivatives provide a perfectsolution to financial risks This book will not address risk hedging further, interestedreaders may refer to Hull and Basu (2016) and Durbin (2010) for derivatives and riskhedging in finance

The other approach to risk is the concept of diversifying, which suggestsinvestors diversify their investments in order to reduce market risk The idea behindthis approach is simple, as the old saying goes, do not put all your eggs in one basket.The idea of diversification is very old, but its practice in financial investmentwas initiated in the 1950s by the work of Markowitz (1952), risk diversificationencapsulates the main contents of the modern portfolio theory (MPT), which is themajor theory on investment with rich contents

MPT advocates that investors should consider both return and risk, and ment decisions should be made by trading off risk and return Refer to Elton et al.(2009) for more about MPT

invest-The approach of MPT has been widely accepted and the risk-return trade offbecomes the major framework for investment decisions This part will follow theframework of MPT to address the issue of risk management through diversification

1.4 Risk Management by Diversification

To explain the key concepts of risk management by diversification, we employ somemathematical notations in this section

Let W0be the value of an investment fund, an investor would like to allocate this

fund to n securities available in financial markets, and x i be the investment ratio in

security i, then vector x = (x1 , · · · , x n ) is an allocation of the fund, we call x an

investment or a portfolio in this book

Since the prices of securities change over time, let the price of security i at a future time t be ξ i (t ) , and ξ(t) = (ξ1 (t ), · · · , ξ n (t )), then the value of total holding

securities at time t is a function of ξ(t) and x, denote it by W (x, ξ(t)).

The evaluation of investment x is conducted from the aspects of return and risk,

which are usually derived from the profit rate of the investment The profit rate of

an investment provides the base for evaluation of the investment, we first define thenotion of profit rate

Fig 1.3 Risk management through diversification

Definition 1.1 The profit rate of an investment is the profit per unit invested Denote

the profit rate of investment x by R(x, ξ(t)), then it is calculated as follows:

R(x, ξ(t ))= W (x, ξ(t )) − W0

The profit rate of a security can be similarly defined, denote the profit rate of

security i by R i (ξ i (t )) , it is easy to show that the profit rate of investment x is a

weighted sum of the profit rates of the securities invested, that is,

In other words, the investment decision problem can be formulated with a criterion optimization indicated as follows:

bi-opt (x) {Return(x, ξ(t)), Risk(x, ξ(t))} (1.4)

where opt (x) stands for optimization with respect to x.

We follow this framework in addressing the issue of risk management bydiversification There are two popular ways to employ model (1.5) for risk control

by diversification, one way changes it to the risk minimization model, and the otherway changes it to the return maximization model We mainly use the following riskminimization model in this book,

Min (x) Risk(x, ξ(t )) : Return(x, ξ(t)) ≥ r0 (1.5)

where r0 is the minimum return investors ask from the investment This modelimplies that investors will make the risk of their investments as low as possiblewhile the return is kept above the required level

Model (1.5) is a conceptual model for risk control in investment decisions, in thenext chapters we will embody the return and risk for various investment situations

so as to make this model operational

1.5 Outline of Part I

PartIaddresses risk management in investment decisions following the framework

of MPT, investors are supposed to make investment decisions by trading off returnand risk as described in Model (1.4)

The notation t in model (1.4) is the time investors liquidate their investments, it isreferred to as exit time in this book Exit time is an important factor for investmentsbecause the return and risk of an investment are related to this time PartI willintroduce risk management in two typical situations regarding the exit time.One situation is the case of a fixed exit time This is the situation when investorswill use their investment fund for some other purposes at a future time which isknown and fixed

The conceptual model for investment decisions in this situation is as follows,

opt x ∈X f {Return(x, ξ(t)), Risk(x, ξ(t))}| t =t1, (1.6)

where t1 is a future time, and X f is a set of feasible investments that meet theinvestor’s other investment requirements

Because t is fixed in Model (1.6), we will not express it in our model and use thefollowing model for simplicity of notation

opt x ∈X f {Return(x, ξ), Risk(x, ξ)}. (1.7)The other situation is the case of a flexible exit time, the exit time is flexiblewithin some bounded interval Most investors in stock markets are believed to be insuch a situation because most stock holders are flexible regarding the time they selltheir stocks.

The conceptual model for the investment decision in this situation is as follows,

opt x ∈X f {Return(x, ξ(t)), Risk(x, ξ(t))}| t ∈T E (1.8)

where T Eis a bounded interval in future

We will introduce risk management in investment decision basing on model (1.7)

in Chaps.2and3, and present the contents based on model (1.8) in Chaps.4and5

As the base of quantitative risk management, Chap.2 introduces the majorindices for market risk and methods to estimate these risk indices Chapter 3 is

on risk control in investment It contains the main models for financial investmentdecisions, and methods to solve these investment decision models

When the exit time is flexible, risk measures for the cases with a fixed exit timecan no longer apply Chapter 4 is on new risk measures for investments with aflexible exit time We present several new risk measures, and methods to estimatethese new risk indices Chapter5is on risk control in investments with a flexibleexit time It includes decision models for investments with a flexible exit time, andmethods for solving these models

The contents of PartIcan be viewed in the following way: Chaps.2and3presentthe central concepts and quantitative tools for risk management in the framework ofmodern portfolio theory, while Chaps.4 and 5 contain some up-to-date researchresults on risk management in financial investments.

The structure of PartIis shown in Fig.1.4

Market Risk Measures in Financial

Investments

Abstract To precisely control the market risk of an investment, it is necessary to

quantitatively measure market risk This chapter introduces indices for market risk

of an investment A risk index is necessary to express market risk quantitatively, it

is the base for risk management in investments

As pointed out in previous chapter, risk is a notion without a consensus on itsdefinition, different people may have different perceptions on market risk Thefirst index for market risk was proposed in the pioneering work of Markowitz (JFinanc 7(1):77–91, 1952), various other indices have been proposed since then.This chapter introduces the popular indices for market risk with a focus on the threemajor indices: Variance (Sect.2.2), Value at Risk (Sect.2.3), and Conditional Value

at Risk (Sect.2.4)

Following the definition of a risk index, we present methods for estimating theindex To estimate the market risk of an investment, some kind of information aboutthe future market is necessary There are three ways to get such information, oneway is making certain assumption about the risk factors in future market, such asthe normal distribution assumption about the profit rate of an asset, the secondway is using data in the past to produce scenarios for risk factors in the future,implicitly assuming that history will repeat itself, while the third way is usingcomputer simulation to generate scenarios for risk factors in the future, which isusually based on some assumption about risk factors We will present estimatingmethods for market risk indices under some or all of these ways Each method will

be illustrated with a numerical example for better understanding

Beside of the three major risk indices, two other risk indices are also presented

in the last section of this chapter (Sect.2.5), they take failure as risk

Keywords Market risk · Risk measure · Variance · Value at risk · Conditional

value at risk · Scenario simulation

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Translational Systems Sciences 14, https://doi.org/10.1007/978-981-13-0317-3_2

13

2.1 Market Risk and Its Measurement

Market risk of investments is caused by uncertain factors that affect the value ofthe securities invested, such as the change in the exchange rates that may affect thestock prices of exporting companies We refer to all these related uncertain factors

as the risk factors of an investment Due to of the existence of risk factors, investorsfeel that they may lose money on their investments, this is the risk in investments.However, a definition of the notion of risk which is universally accepteddoes not exist, different investors may have a different understanding of marketrisk.This chapter will introduce three common perceptions of market risk, and thecorresponding risk indices

2.2 Variance: Fluctuation Is Taken as Risk

Fluctuation of the value of risky assets is viewed as risk, this idea was first suggested

in 1950s by Markowitz (1952) Viewing fluctuation as risk is consistent withpeople’s feelings about risk, since investors do not feel risk if the value of theirinvestment will not change,

This idea has been widely accepted since 1950s, it initiated a new field nownamed modern portfolio theory (MPT), refer to Elton et al (2009) for details aboutMPT Although variance is not a good measure from the perspective of risk control,

it is still the most popular measure of risk used in financial academia

2.2.1 Definition of Variance

Variance is a concept from statistics, it is the expectation of the squared deviation of

a random variable from its mean and it is an index that reflects the fluctuation of aninvestment

We use the profit rate as the base in defining market risk, let the profit rate of

investment x be R(x, ξ ) where ξ stands for the risk factors that affect the values of risky assets in the investment, and denote the variance of R(x, ξ ) by V ar(x), which

is defined as follows:

where r(x) = E ξ (R(x, ξ )) is the mean of R(x, ξ ).

Let R i (ξ i ) be the profit rate of risky asset i in portfolio x, it is easy to show that the profit rate of x is the weighted sum of the profit rates of its components,

Let r i = E ξ i (R i (ξ i )) , the covariance between R i (ξ i ) and R j (ξ j )is defined asfollows:

σ ij = E (ξ i ,ξ j ) [(R i (ξ i ) − r i )(R j (ξ j ) − r j )] (2.3)

Let the covariance matrix of profit rates R1 (ξ1), · · · , R n (ξ n ) by Σ = (σ ij ) n ×n,

it is easy to show that the variance of R(x, ξ ) can be calculated as follows:

Two methods are popular to obtain sample data of R i One is to use historicaldata For instance, sample data on the profit rate of a stock can be calculated usingpast price data Another method is to use simulated data generated from MonteCarlo simulation For instance, the future price data of a stock can be simulatedwith some assumption regarding the stock price, and then sample profit rate datacan be obtained using the simulated price data

Suppose m sample data of the profit rate of each component are obtained, denote them by R i (j ), i = 1 ∼ n, j = 1 ∼ m, as shown in Table2.1below

Then the mean and the variance of asset i’s profit rate are calculated as follows:

Table 2.1 Sample data of

profit rates Profit rates Samples of each profit rate Mean of eachprofit rate

R1 R1( 1), R1( 2), · · · , R1(m) r1

R2 R2( 1), R2( 2), · · · , R2(m) r2

R n R n ( 1), R n ( 2), · · · , R n (m) r n

The covariance between two profit rates is calculated as follows:

Hence the variance of portfolio x can be obtained from formula (2.4)

Example 2.1 Consider the risk of an investment in IBM and Intel stocks using

variance as a risk indicator, the length of investment is one month

The historical price data of the two stocks in the past one year (2016.8.1 ∼2017.7.31) are available from Yahoo!Finance, the daily adjusted close prices aresummarized in AppendixB

We first calculate the scenario data of the monthly profit rate of each stock usingthe stock prices in this period, take 22 trading days as one month, the monthly profitrate scenario can be calculated as follows:

R i (k)= p i (k + 22)

p i (k) − 1, i = IBM, INT C, k = 1, 2, · · · , 230. (2.7)

where p i ( 1) is the price on August 1, 2016, and p i ( 2) is the price on the next trading

day, and so forth

Hence we obtain 230 scenario data for each stock’s profit rate, which are alsoincluded in AppendixB

The co-variance matrix is then calculated using formula (2.5) and (2.6):

10-3 Variance of IBM-INTC portfolio

Fig 2.1 Variance of IBM-INTC portfolio

2.3 Value at Risk: A Likely Loss Is Taken as Risk

Investors do not feel risk if they are sure that they will not lose money, risk is aconcept that should be directly related to loss Since an investment leads to differentloss amounts with different possibilities, the largest loss that may occur in a certainprobability is taken as risk in this approach

2.3.1 Definition of Value at Risk

Value at Risk (VaR) is a concept proposed in 1990s by J.P Morgan, refer to Morgan(1997) and Schachter (2002) for details This approach suggests using the largestloss that may be incurred in a certain probability as an indicator of risk

The VaR of an investment is the worst loss expected over a given period of timewith a given probability The time period is known as the holding period, and theprobability is known as the confidence interval VaR is not an estimate of the worstpossible loss, but the largest likely loss, as illustrated in Fig.2.2below

For example, an investor might estimate VaR over 10 days to be $100 million,with a confidence interval of 95% This would mean there is a 5% chance of a losslarger than $100 million in the next 10 days

1-a

Expected Loss (EL) Unexpected Loss (UL)

Value-at-Risk (VaR)

Fig 2.2 Value at Risk: the largest likely loss

Definition 2.1 The Value at Risk (VaR) at confidence level α of investment x,

denoted by V aR α (x), is defined in the following formula:

V aR α (x) = inf {d ∈ R1|P (L(x, ξ)) ≤ d) ≥ α}, (2.10)

where L(x, ξ ) = −R(x, ξ) is the loss rate of investment x.

Hence V aR α (x)is the largest loss that may occur at a probability 1− α, in other words, the probability that the loss is larger than V aR α (x)will not be larger than

1− α.

When the loss rate L(x, ξ ) follows a continuous probability distribution function

f (L) , then the VaR of investment x can be expressed as follows:

2.3.2 Estimation of VaR: Three Methods

There are three basic methods that are used to compute Value at Risk The measurecan be computed analytically by making assumptions about return distributions formarket risk factors, and by using the variances in and covariances across these risk

factors It can also be estimated by running hypothetical portfolios through historicaldata or from Monte Carlo simulations.

2.3.2.1 Variance-Covariance Method

This method assumes that the profit rates of component assets follow the normal

distribution, then the profit rate of portfolio x also follows the normal distribution,

because the profit rate of the portfolio is a linear function of its component profitrates, as indicated in (2.2) The VaR of portfolio x can be calculated using this

property

We first introduce this method taking a portfolio consisting of two risky assets, Aand B, as an example Suppose the profit rates of assets A and B follow the normal

distributions N (r A , σ A2) and N (r B , σ B2), and the correlation coefficient between the

two profit rates is ρ AB , then we know that the profit rate of portfolio x = (x A , x B ) follows the normal distribution N (r(x), σ2(x)) , where r(x), σ2(x)are given by

hence the normalized variable L(x) +r(x)

σ (x) follows the standard normal distribution,

L(x) + r(x)

σ (x) ∼ N(0, 1) Let φ(z) and Φ(z) be the probability density function and the distribution

function of the normal distribution, respectively, that is,

as shown in Fig.2.3 z αcan be obtained from the standard normal distribution table

For instance, z0.95 = 1.645 and z0.99 = 2.33.

Fig 2.3 Feature of normal

distribution: P (Z ≥ z α )can

be decided by deviation away

from the mean

BecauseL(x) +r(x)

σ (x) follows the standard normal distribution, there is a unique z α

for a given α satisfying the following relation:

For a portfolio with n risky assets, its VaR can still be expressed with the formula

(2.16), but the mean and variance of portfolio x with n risky assets are given by the

Variance-ratio of Intel stock be x2 = 1 − x1.

Suppose the monthly profit rates of both stocks in the future will follow thenormal distribution, and the means and covariance of the profit rates can be

VaR of IBM-INTC portfolio

Fig 2.4 VaR of IBM-INTC portfolio

estimated using the historical price data used in Example2.1 Hence, we have

-END-2.3.2.2 Historical Simulation Method

The historical method simply re-organizes actual historical returns, putting them inorder from worst to best It then assumes that, from a risk perspective, history willrepeat itself

The fundamental assumption of the historical simulations method is that resultsare based on the past portfolio performance and make the assumption that the past

is a good indicator of the future

Suppose we have m scenario sample data for the profit rate of each component

asset, which are generated using their past prices, these scenario samples aresummarized in Table2.2

We obtain the scenarios of portfolio x’s profit rate using the following

and the scenarios of portfolio x’s loss rate L using relationship L(x, k) = −R(x, k).

We summarize these data in Table2.3

Then we organize the m scenarios of the loss rate from small to large, and denote the kth small loss rate by L (k) (x) , we arrange the m scenarios of the loss rate as

follows:

L ( 1) (x), L ( 2) (x), · · · , L (m) (x). (2.22)The historical simulation method selects one of the above scenario samples as

the VaR of portfolio x.

Table 2.2 Scenario sample

data of component profit rates Profit rate Scenarios samples of R i

R1 R1( 1), R1( 2), · · · , R1(m)

R2 R2( 1), R2( 2), · · · , R2(m)

· · · ·

R n R n ( 1), R n ( 2), · · · , R n (m)

Table 2.3 Estimation of VaR using the historical simulation method

Number of samples Samples of R i (k) Samples of R(x, k) Samples of L(x, k)

Let a be a real number,

Since the percentage of loss rate samples smaller than or equal to L ( (x)

is larger than or equal to α, but the percentage of loss rate samples smaller than or equal to L ( (x) is smaller than α, hence the historical simulation method uses

L ( (x) as V aR α (x)

For instance, let α

percentage of loss rate samples smaller than or equal to L ( 95) (x) is 0.95, but the percentage of loss rate samples smaller than or equal to L ( 94) (x) is 0.94, which is less than 0.95, hence V aR0.95 = L ( 95) (x)according to the definition of VaR

For the case α

percentage of loss rate samples smaller than or equal to L ( 96) (x)is 10196 = 0.9505, which is larger than 0.95, but the percentage of loss rate samples smaller than or equal to L ( 95) (x)is 10195 = 0.9406, which is less than 0.95, we see that V aR0.95 =

L ( 96) (x)according to the definition of VaR

Example 2.3 (Example 2.1 continued: VaR estimated using the Historical tion Method) We use the same historical price data of IBM and Intel stocks in

Simula-Example2.1to estimate the VaR of a portfolio comprised of these two stocks

We first calculate the sample data of the loss rate of the portfolio x = (x1 ,1−x1 )

using the 230 sample data of each component asset:

VaR of IBM-INTC portfolio

Fig 2.5 VaR of IBM-INTC portfolio estimated using historical simulation

2.3.2.3 Monte Carlo Simulation Method

The Monte Carlo simulation method randomly generates possible future scenarios(or market conditions) for risk factors The value of the portfolio being assessed isthen calculated for each set of market conditions generated The results are plotted

on a histogram (graph) from which the VaR can be determined

The main steps of this method are as follows

• Step 1: Determine distributions for related risk factors

To generate scenarios for risk factors using a computer program, their butions are required You may assume normal distributions, you may also usealternative distributions and subjective judgments to modify these distributions

distri-• Step 2: Generate future scenarios for risk factors and compute the correspondingprofit rates

Once the distributions are specified, the simulation process starts In each run,the risk factors take values according to their distributions, and the value andprofit rate of the portfolio reflect the outcomes

• Step 3: Estimate VaR using data obtained from the simulation

Fig 2.6 Computing VaR using the Monte Carlo simulation method

After a repeated series of runs, numbering usually in the thousands, you willhave a distribution of portfolio profit rate that can be used to assess Value at Risk.For instance, assume that you run a series of 10,000 simulations and derivecorresponding profit rates for the portfolio These values can be ranked fromhighest to lowest, and the 95% percentile VaR will correspond to the 500th lowestvalue and the 99th percentile to the 100th lowest value

As illustrated in Fig.2.6

Although this method of calculating VaR requires significant computationalcapability, it is a flexible method of measurement, you can predetermine the param-eters or distributions that will determine the possible future scenarios generated bythe computer

Example 2.4 (Example 2.1 continued: VaR estimated using the MS method) We first

suppose that the monthly profit rate of IBM and Intel stocks follow the normaldistribution for the sake of simplicity, and means and variances of the distributionsare estimated using historical prices of the two stocks as in Example2.2, that is,

VaR of IBM-INTC portfolio

Fig 2.7 VaR of IBM-INTC portfolio estimated using Monte Carlo simulation

The sample data of the loss rate of the portfolio x = (x1 ,1− x1 )is obtainedusing the following formula:

L(x, k) = −(x1 R1(k) + (1 − x1 )R2(k)), k = 1, · · · , m.

where R1 (k) and R2 (k)are simulated following the assumption (2.25)

Then the VaR of portfolio x at confidence level α is the α percentile of the sample data of L(x).

We generate 1000 samples for each the profit rate of each component, that is,

m = 1000, the VaR of portfolio x at confidence level 95% is obtained and illustrated

The historical simulation approach requires no assumptions about the returndistributions but implicitly assumes that the data used in the simulation is arepresentative sample of the risks looking forward.

The Monte Carlo simulation approach allows for the most flexibility in terms ofchoosing distributions for returns and bringing in subjective judgments and externaldata, but is the most demanding from a computational standpoint How to generatescenarios for uncertain factors is the key in using Monte Carlo simulation, refer toWang (2012) and Glasserman (2013) for more about using Monte Carlo simulations

of tail risk than VaR

2.4.1 Definition of Conditional VaR

CVaR is also referred to as the Expected Shortfall (ES) or the Expected Tail Loss(ETL), and is an interpretation of the expected loss given that the loss exceeds theVaR, it is defined as follows

Definition 2.2 Let L(x, ξ ) be the loss rate of portfolio x, and α ∈ (0, 1), the CVaR

of x at confidence level α, denoted by CV aR α (x), is

CV aR α (x) = E ξ [L(x, ξ)|L(x, ξ) ≥ V aR α (x)] (2.26)The relationship between VaR and CVaR is illustrated in Fig.2.8

Refer to Dowd (2003) and Rockafellar and Uryasev (2002) for more discussion

of conditional Value at Risk

Fig 2.8 Conditional Value at Risk: the expected loss larger than VaR

Table 2.4 An illustrative example for computing CVaR from its definition

Example 2.5 Suppose the loss of investment x is k− 50 with probability 1% for

k= 1 ∼ 100, as listed in Table2.4below

Then V aR0.95 (x)= 46 since the loss is 46 or more with probability 5%, and

CV aR 0.95 (x)= 1

0.05 (46+ 47 + 48 + 49 + 50) ∗ 1% = 48.

CVaR of IBM-INTC portfolio

When the scenario data about the component assets are available, we can estimateCVaR of a portfolio using these scenario data, as illustrated in the followingexample

Example 2.6 (Example 2.1 continued: CVaR estimated with historical data) We

calculate the CVaR of the portfolio consisting of IBM and Intel stocks using thesame historical price data as in Example2.1

For the case x = (0.5, 05), we know V aR0.95 (x) = 4.84% from Example2.3,

we assume that all scenarios will happen with the same probability, which is 2301since there are 230 scenarios for each component assets, hence the CVaR of portfolio

The CVaR of other portfolio can be estimated in the same way, Fig.2.9shows

the CV aR0.95 (x)of IBM-INTC portfolios at different investment ratios