LongRun Technical Document Jongwoo Kim, Allan M. Malz, Jorge Mina RiskMetrics Group http://www.riskmetrics.com The development of LongRun LongRun was created by the RiskMetrics Group in conjunction with a variety of groups within J.P. Morgan, including risk management services, corporate finance, advisory, foreign exchange, and capital markets. All data used in the production of the RiskMetrics Group's LongRun forecasts are provided by Reuters. LongRun First Edition (version 1.1, April 1999) Copyright 1999 RiskMetrics Group. All rights reserved. CorporateMetrics™ and CorporateManager™ are trademarks or registered trademarks of the Risk- Metrics Group in the United States and in other countries. They are written with the symbol ™ or ® on their first occurrence in this publication, and as CorporateMetrics and as CorporateManager here- after. RiskMetrics® and CreditMetrics® are registered trademarks of J.P. Morgan and Co, Inc. in the United States and in other countries. The RiskMetrics® and CreditMetrics® methodologies, data sets, and related software applications are exclusively managed by the RiskMetrics Group, LLC. Windows NT, Windows 98, and Windows 95 are trademarks of the Microsoft Corporation. RiskMetrics Group does not warrant any results obtained from the use of the LongRun data, methodology, documentation or any information derived from the data (collectively the “Data”) and does not guarantee its sequence, timeliness, accuracy, completeness or continued availability. The Data is calculated on the basis of historical observations and should not be relied upon to predict future market movements. The Data addresses market risk measurement and should not be relied upon to measure all of a company’s other risk types, such as credit, operational, business, legal, or reputational risks. Additional information is available on request. The information contained in this document is believed to be reliable but the RiskMetrics Group does not guarantee its completeness or accuracy. Opinions and estimates constitute our judgment and are subject to change without notice. Copyright 1999 RiskMetrics Group. iii Table of contents Preface v Chapter 1. Introduction to LongRun 1 1.1 What is LongRun?2 1.2 Scenario generation 5 1.2.1 Forecasting 5 1.2.2 Scenario simulation 10 1.3 Summary 12 Chapter 2. Forecasts based on current market prices 15 2.1 Forecasts using futures, forwards, and options 15 2.1.1 Cash and derivatives contracts and markets 16 2.1.2 The performance of derivatives prices in forecasting cash prices 26 2.1.3 Efficient markets theory 31 2.1.4 Risk neutral forecasts from derivative asset prices 38 2.2 Forecasts of extreme moves in asset prices 51 2.2.1 Statistical behavior of asset prices 51 2.2.2 The volatility smile in option markets 54 2.2.3 Interpreting the volatility smile 57 2.2.4 Implied probability distributions using the volatility smile 61 Appendix 2.A Covered parity in the currency, interest rate, commodity and equity index markets 69 2.A.1 Cost-of-carry and the mechanics of forward prices 69 2.A.2 Foreign exchange 70 2.A.3 Equities and commodities 71 2.A.4 Interest rates 72 Appendix 2.B Why do option prices contain so much information? 76 Chapter 3. Forecasts based on economic structure 81 3.1 Introduction 81 3.2 Constructing an econometric ‘forecasting system’ 82 3.3 Econometric framework 84 3.3.1 Advantages of the vector autoregressive model 84 3.3.2 Introduction to the error correction model 85 3.4 VECM: The functional form of our chosen model 94 3.4.1 Three types of VARM 95 3.4.2 Two types of ECM 96 3.4.3 Backtesting 100 3.4.4 Summary 102 3.5 Specification of the VECM 103 3.5.1 Foreign exchange 104 3.5.2 Interest rate 105 3.5.3 Equity index 107 3.5.4 Commodity price 107 3.6 Economic regimes in forecasts 108 3.6.1 Accounting for economic regimes in forecasts 109 3.6.2 Applying structural break tests 111 3.6.3 Incorporating structural breaks 112 3.7 Estimating the VECM 114 3.7.1 Selecting order of lags and cointegration vectors 114 3.7.2 USD per ZAR exchange rate example 115 LongRun Technical Document RiskMetrics Group iv Table of contents 3.7.3 USD per JPY exchange rate example 116 3.8 Summary 116 Appendix 3.A Estimating and testing the VECM 118 3.A.1 Full information maximum likelihood (FIML) estimation of VECM 118 3.A.2 Johansen trace test 119 Appendix 3.B Historical time series data 121 Chapter 4. Backtesting LongRun’s forecasting models 123 4.1 Performance assessment framework 123 4.1.1 Sample period 123 4.1.2 Benchmarks 124 4.1.3 Accuracy measures 124 4.1.4 Non-overlapping backtesting samples 125 4.2 Assessing the accuracy of mean forecasts 127 4.2.1 An example 127 4.2.2 Results by asset class 128 4.2.3 Summary of results 132 4.3 Assessing the robustness of confidence interval forecasts 133 4.3.1 An example 133 4.3.2 Results for historical volatility 133 4.3.3 Results for implied volatility 135 4.4 Conclusion 135 Chapter 5. Scenario simulation 137 5.1 Level I simulation: Monthly prices 137 5.1.1 Single price series: Autocorrelation 138 5.1.2 Multiple price series: Cross correlation 140 5.2 Level II simulation: Daily prices 142 5.2.1 Overview 142 5.2.2 Brownian bridges and the Level II algorithm 143 5.2.3 Putting the steps together: Unconditional moments 146 5.3 Simulation example 147 5.4 Summary 149 Appendix 5.A Matrix perturbations 151 Chapter 6. Applications of LongRun 153 6.1 Long-term VaR for a portfolio manager 153 6.2 CFaR and EaR for corporations 154 6.3 VaR for pension plans and mutual funds 156 6.3.1 Defined contribution pension plans and mutual funds 157 6.3.2 Defined benefit plan 157 6.4 Summary 158 List of acronyms 159 Bibliography 161 LongRun Technical Document RiskMetrics Group v Preface This technical document details the long-term forecasting and scenario generation methodologies in LongRun. It contains two sets of techniques for computing forecast values and confidence inter- vals for asset prices and a procedure for generating scenarios for use in Monte Carlo. In some circles of the economics and finance professions, forecasting is not a highly regarded ac- tivity. For some, it evokes images of speculators, chart analysts and questionable investor newslet- ters; for others, memories of the grandiose econometric forecasting failures of the 1970’s. There is nonetheless a need for forecasting in risk management. A prudent corporate treasurer or fund man- ager must have some way of measuring the risk to earnings, cash flows or returns. Any measure of risk must incorporate some estimate of the probability distribution of the future asset prices on which financial performance depends. Forecasting is an indispensable element of prudent financial management. How should corporate treasurers and fund managers approach forecasting? Forecasting accuracy per se is not the object of the exercise: every currently known forecasting tool often falls wide of the mark. In a risk management context, the forecasts should rather be practical, based on objective techniques, it must be possible to examine how the methodologies would have performed had they been applied in the past, and it should be possible to articulate the techniques to shareholders, in- vestors, and regulators. It is also desirable to have available different, methodologically indepen- dent forecasting techniques. The risk manager can then compare the results with one another and with his own judgements about future asset prices. We believe that LongRun meets these criteria for forecasting techniques. The RiskMetrics Group’s policy is to make public its risk management methodologies. In doing so we aim to foster pubic discussion of our approach, to help our clients grasp the methodologies which underline our products, and more generally to promote public understanding of risk manage- ment issues. We hope that interested practitioners and scholars will examine the LongRun method- ology and look forward to studying their criticisms, alternative approaches, and suggested applications. The authors have enjoyed support and constructive comments from a number of colleagues. We would particularly like to thank Ethan Berman, Alvin Lee and Jim Ye of the RiskMetrics Group, Mark Everson of Ford Motor Company, and John Byma of the Procter & Gamble Company for their detailed comments on several drafts of this document. Christopher Finger of the RiskMetrics Group helped formulate LongRun’s simulation procedures and had many useful suggestions throughout. Peter Zangari was instrumental in the early stages of this project. We would also like to thank Tatiana Kolubayev for editing and producing this document. Jongwoo Kim [email protected] Allan M. Malz [email protected] Jorge Mina [email protected] LongRun Technical Document RioskMetrics Group vi 1 Chapter 1. Introduction to LongRun The implementation of a comprehensive risk management system has become an essential step for conducting business in a global and increasingly liberalized trade environment. Given the numer- ous risks faced by corporations and financial institutions, the growing number of hedging alterna- tives and financing strategies, and the risk information disclosure required by regulators and sophisticated investors, the need for a framework to implement an adequate risk management pro- cess is imminent. In the case of financial institutions, an additional incentive for developing a risk management system can be found in the use of internal risk models to estimate the regulatory cap- ital charges to cover market risks.1 The RiskMetrics® methodology, first introduced in 1994, provides an answer to the market risk measurement problem when the relevant horizon is short.