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Table of contents
Part I Risk Measurement Framework
Chapter 1. Introduction 3
1.1 An introduction to Value-at-Risk and RiskMetrics 6
1.2 A more advanced approach to Value-at-Risk using RiskMetrics 7
1.3 What RiskMetrics provides 16
Chapter 2. Historical perspective of VaR 19
2.1 From ALM to VaR 22
2.2 VaR in the framework of modern financial management 24
2.3 Alternative approaches to risk estimation 26
Chapter 3. Applying the risk measures 31
3.1 Market risk limits 33
3.2 Calibrating valuation and risk models 34
3.3 Performance evaluation 34
3.4 Regulatory reporting, capital requirement 36
Part II Statistics of Financial Market Returns
Chapter 4. Statistical and probability foundations 43
4.1 Definition of financial price changes and returns 45
4.2 Modeling financial prices and returns 49
4.3 Investigating the random-walk model 54
4.4 Summary of our findings 64
4.5 A review of historical observations of return distributions 64
4.6 RiskMetrics model of financial returns: A modified random walk 73
4.7 Summary 74
Chapter 5. Estimation and forecast 75
5.1 Forecasts from implied versus historical information 77
5.2 RiskMetrics forecasting methodology 78
5.3 Estimating the parameters of the RiskMetrics model 90
5.4 Summary and concluding remarks 100
Part III Risk Modeling of Financial Instruments
Chapter 6. Market risk methodology 105
6.1 Step 1—Identifying exposures and cash flows 107
6.2 Step 2—Mapping cash flows onto RiskMetrics vertices 117
6.3 Step 3—Computing Value-at-Risk 121
6.4 Examples 134
Chapter 7. Monte Carlo 149
7.1 Scenario generation 151
7.2 Portfolio valuation 155
7.3 Summary 157
7.4 Comments 159
viii Table of contents
RiskMetrics
Ô —Technical Document
Fourth Edition
Part IV RiskMetrics Data Sets
Chapter 8. Data and related statistical issues 163
8.1 Constructing RiskMetrics rates and prices 165
8.2 Filling in missing data 170
8.3 The properties of correlation (covariance) matrices and VaR 176
8.4 Rebasing RiskMetrics volatilities and correlations 183
8.5 Nonsynchronous data collection 184
Chapter 9. Time series sources 197
9.1 Foreign exchange 199
9.2 Money market rates 199
9.3 Government bond zero rates 200
9.4 Swap rates 202
9.5 Equity indices 203
9.6 Commodities 205
Chapter 10. RiskMetrics volatility and correlation files 207
10.1 Availability 209
10.2 File names 209
10.3 Data series naming standards 209
10.4 Format of volatility files 211
10.5 Format of correlation files 212
10.6 Data series order 214
10.7 Underlying price/rate availability 214
Part V Backtesting
Chapter 11. Performance assessment 217
11.1 Sample portfolio 219
11.2 Assessing the RiskMetrics model 220
11.3 Summary 223
Appendices
Appendix A. Tests of conditional normality 227
Appendix B. Relaxing the assumption of conditional normality 235
Appendix C. Methods for determining the optimal decay factor 243
Appendix D. Assessing the accuracy of the delta-gamma approach 247
Appendix E. Routines to simulate correlated normal random variables 253
Appendix F. BIS regulatory requirements 257
Appendix G. Using the RiskMetrics examples diskette 263
Appendix H. RiskMetrics on the Internet 267
Reference
Glossary of terms 271
Bibliography 275
ix
List of charts
Chart 1.1 VaR statistics 6
Chart 1.2 Simulated portfolio changes 9
Chart 1.3 Actual cash flows 9
Chart 1.4 Mapping actual cash flows onto RiskMetrics vertices 10
Chart 1.5 Value of put option on USD/DEM 14
Chart 1.6 Histogram and scattergram of rate distributions 15
Chart 1.7 Valuation of instruments in sample portfolio 15
Chart 1.8 Representation of VaR 16
Chart 1.9 Components of RiskMetrics 17
Chart 2.1 Asset liability management 22
Chart 2.2 Value-at-Risk management in trading 23
Chart 2.3 Comparing ALM to VaR management 24
Chart 2.4 Two steps beyond accounting 25
Chart 3.1 Hierarchical VaR limit structure 33
Chart 3.2 Ex post validation of risk models: DEaR vs. actual daily P&L 34
Chart 3.3 Performance evaluation triangle 35
Chart 3.4 Example: comparison of cumulative trading revenues 35
Chart 3.5 Example: applying the evaluation triangle 36
Chart 4.1 Absolute price change and log price change in U.S. 30-year government bond 47
Chart 4.2 Simulated stationary/mean-reverting time series 52
Chart 4.3 Simulated nonstationary time series 53
Chart 4.4 Observed stationary time series 53
Chart 4.5 Observed nonstationary time series 54
Chart 4.6 USD/DEM returns 55
Chart 4.7 USD/FRF returns 55
Chart 4.8 Sample autocorrelation coefficients for USD/DEM foreign exchange returns 57
Chart 4.9 Sample autocorrelation coefficients for USD S&P 500 returns 58
Chart 4.10 USD/DEM returns squared 60
Chart 4.11 S&P 500 returns squared 60
Chart 4.12 Sample autocorrelation coefficients of USD/DEM squared returns 61
Chart 4.13 Sample autocorrelation coefficients of S&P 500 squared returns 61
Chart 4.14 Cross product of USD/DEM and USD/FRF returns 63
Chart 4.15 Correlogram of the cross product of USD/DEM and USD/FRF returns 63
Chart 4.16 Leptokurtotic vs. normal distribution 65
Chart 4.17 Normal distribution with different means and variances 67
Chart 4.18 Selected percentile of standard normal distribution 69
Chart 4.19 One-tailed confidence interval 70
Chart 4.20 Two-tailed confidence interval 71
Chart 4.21 Lognormal probability density function 73
Chart 5.1 DEM/GBP exchange rate 79
Chart 5.2 Log price changes in GBP/DEM and VaR estimates (1.65
s) 80
Chart 5.3 NLG/DEM exchange rate and volatility 87
Chart 5.4 S&P 500 returns and VaR estimates (1.65
s) 88
Chart 5.5 GARCH(1,1)-normal and EWMA estimators 90
Chart 5.6 USD/DEM foreign exchange 92
Chart 5.7 Tolerance level and decay factor 94
Chart 5.8 Relationship between historical observations and decay factor 95
Chart 5.9 Exponential weights for T = 100 95
Chart 5.10 One-day volatility forecasts on USD/DEM returns 96
Chart 5.11 One-day correlation forecasts for returns on USD/DEM FX rate and on S&P500 96
Chart 5.12 Simulated returns from RiskMetrics model 101
Chart 6.1 French franc 10-year benchmark maps 109
x List of charts
RiskMetrics
Ô —Technical Document
Fourth Edition
Chart 6.2 Cash flow representation of a simple bond 109
Chart 6.3 Cash flow representation of a FRN 110
Chart 6.4 Estimated cash flows of a FRN 111
Chart 6.5 Cash flow representation of simple interest rate swap 111
Chart 6.6 Cash flow representation of forward starting swap 112
Chart 6.7 Cash flows of the floating payments in a forward starting swap 113
Chart 6.8 Cash flow representation of FRA 113
Chart 6.9 Replicating cash flows of 3-month vs. 6-month FRA 114
Chart 6.10 Cash flow representation of 3-month Eurodollar future 114
Chart 6.11 Replicating cash flows of a Eurodollar futures contract 114
Chart 6.12 FX forward to buy Deutsche marks with US dollars in 6 months 115
Chart 6.13 Replicating cash flows of an FX forward 115
Chart 6.14 Actual cash flows of currency swap 116
Chart 6.15 RiskMetrics cash flow mapping 118
Chart 6.16 Linear and nonlinear payoff functions 123
Chart 6.17 VaR horizon and maturity of money market deposit 128
Chart 6.18 Long and short option positions 131
Chart 6.19 DEM 3-year swaps in Q1-94 141
Chart 7.1 Frequency distributions for and 153
Chart 7.2 Frequency distribution for DEM bond price 154
Chart 7.3 Frequency distribution for USD/DEM exchange rate 154
Chart 7.4 Value of put option on USD/DEM 157
Chart 7.5 Distribution of portfolio returns 158
Chart 8.1 Constant maturity future: price calculation 170
Chart 8.2 Graphical representation 175
Chart 8.3 Number of variables used in EM and parameters required 176
Chart 8.4 Correlation forecasts vs. return interval 185
Chart 8.5 Time chart 188
Chart 8.6 10-year Australia/US government bond zero correlation 190
Chart 8.7 Adjusting 10-year USD/AUD bond zero correlation 194
Chart 8.8 10-year Japan/US government bond zero correlation 195
Chart 9.1 Volatility estimates: daily horizon 202
Chart 11.1 One-day Profit/Loss and VaR estimates 219
Chart 11.2 Histogram of standardized returns 221
Chart 11.3 Standardized lower-tail returns 222
Chart 11.4 Standardized upper-tail returns 222
Chart A.1 Standard normal distribution and histogram of returns on USD/DEM 227
Chart A.2 Quantile-quantile plot of USD/DEM 232
Chart A.3 Quantile-quantile plot of 3-month sterling 234
Chart B.1 Tails of normal mixture densities 238
Chart B.2 GED distribution 239
Chart B.3 Left tail of GED (
n) distribution 240
Chart D.1 Delta vs. time to expiration and underlying price 248
Chart D.2 Gamma vs. time to expiration and underlying price 249
xi
List of tables
Table 2.1 Two discriminating factors to review VaR models 29
Table 3.1 Comparing the Basel Committee proposal with RiskMetrics 39
Table 4.1 Absolute, relative and log price changes 46
Table 4.2 Return aggregation 49
Table 4.3 Box-Ljung test statistic 58
Table 4.4 Box-Ljung statistics 59
Table 4.5 Box-Ljung statistics on squared log price changes (cv = 25) 62
Table 4.6 Model classes 66
Table 4.7 VaR statistics based on RiskMetrics and BIS/Basel requirements 71
Table 5.1 Volatility estimators 78
Table 5.2 Calculating equally and exponentially weighted volatility 81
Table 5.3 Applying the recursive exponential weighting scheme to compute volatility 82
Table 5.4 Covariance estimators 83
Table 5.5 Recursive covariance and correlation predictor 84
Table 5.6 Mean, standard deviation and correlation calculations 91
Table 5.7 The number of historical observations used by the EWMA model 94
Table 5.8 Optimal decay factors based on volatility forecasts 99
Table 5.9 Summary of RiskMetrics volatility and correlation forecasts 100
Table 6.1 Data provided in the daily RiskMetrics data set 121
Table 6.2 Data calculated from the daily RiskMetrics data set 121
Table 6.3 Relationship between instrument and underlying price/rate 123
Table 6.4 Statistical features of an option and its underlying return 130
Table 6.5 RiskMetrics data for 27, March 1995 134
Table 6.6 RiskMetrics map of single cash flow 134
Table 6.7 RiskMetrics map for multiple cash flows 135
Table 6.8 Mapping a 6x12 short FRF FRA at inception 137
Table 6.9 Mapping a 6x12 short FRF FRA held for one month 137
Table 6.10 Structured note specification 139
Table 6.11 Actual cash flows of a structured note 139
Table 6.12 VaR calculation of structured note 140
Table 6.13 VaR calculation on structured note 140
Table 6.14 Cash flow mapping and VaR of interest-rate swap 142
Table 6.15 VaR on foreign exchange forward 143
Table 6.16 Market data and RiskMetrics estimates as of trade date July 1, 1994 145
Table 6.17 Cash flow mapping and VaR of commodity futures contract 145
Table 6.18 Portfolio specification 147
Table 6.19 Portfolio statistics 148
Table 6.20 Value-at-Risk estimates (USD) 148
Table 7.1 Monte Carlo scenarios 155
Table 7.2 Monte Carlo scenarios—valuation of option 156
Table 7.3 Value-at-Risk for example portfolio 158
Table 8.1 Construction of rolling nearby futures prices for Light Sweet Crude (WTI) 168
Table 8.2 Price calculation for 1-month CMF NY Harbor #2 Heating Oil 169
Table 8.3 Belgian franc 10-year zero coupon rate 175
Table 8.4 Singular values for USD yield curve data matrix 182
Table 8.5 Singular values for equity indices returns 182
Table 8.6 Correlations of daily percentage changes with USD 10-year 184
Table 8.7 Schedule of data collection 186
Table 8.7 Schedule of data collection 187
Table 8.8 RiskMetrics closing prices 191
Table 8.9 Sample statistics on RiskMetrics daily covariance forecasts 191
Table 8.10 RiskMetrics daily covariance forecasts 192
xii List of tables
RiskMetrics
Ô —Technical Document
Fourth Edition
Table 8.11 Relationship between lagged returns and applied weights 193
Table 8.12 Original and adjusted correlation forecasts 193
Table 8.13 Correlations between US and foreign instruments 196
Table 9.1 Foreign exchange 199
Table 9.2 Money market rates: sources and term structures 200
Table 9.3 Government bond zero rates: sources and term structures 201
Table 9.4 Swap zero rates: sources and term structures 203
Table 9.5 Equity indices: sources 204
Table 9.6 Commodities: sources and term structures 205
Table 9.7 Energy maturities 205
Table 9.8 Base metal maturities 206
Table 10.1 RiskMetrics file names 209
Table 10.2 Currency and commodity identifiers 210
Table 10.3 Maturity and asset class identifiers 210
Table 10.4 Sample volatility file 211
Table 10.5 Data columns and format in volatility files 212
Table 10.6 Sample correlation file 213
Table 10.7 Data columns and format in correlation files 213
Table 11.1 Realized percentages of VaR violations 220
Table 11.2 Realized “tail return” averages 221
Table A.1 Sample mean and standard deviation estimates for USD/DEM FX 228
Table A.2 Testing for univariate conditional normality 230
Table B.1 Parameter estimates for the South African rand 240
Table B.2 Sample statistics on standardized returns 241
Table B.3 VaR statistics (in %) for the 1st and 99th percentiles 242
Table D.1 Parameters used in option valuation 249
Table D.2 MAPE (%) for call, 1-day forecast horizon 251
Table D.3 ME (%) for call, 1-day forecast horizons 251 |
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