Index

A

Accounting 405–6

Accrual accounting 405

Actual P&L 337

Aggregation risk 323, 355, 403, 422–4

Alpha model 311

Alpha strategies 7

Antithetic variance reduction 207–9

Apple 183

Archimedean copulas 229

Asymmetric volatility clustering see GARCH model

At-the-money (ATM) volatility 252, 306

Auditing 408

Autocorrelation 61–2, 138, 218, 292, 319

B

Backtesting 312, 313, 322, 332–55

banking regulations 335–7, 404

basic methodology 332–5

coverage tests 337–40

ETL 344–5

regression-based 341–2

volatility clustering, effect on 342–4

Banking book 406

Banking regulations 6, 14

Basel Accords 404–5

incremental risk charge 412–16

internal models 408–11

standardized rules 411–12

stress testing 378–84

Bank insolvency 364–6

Bank for International Settlements 404

Banks

accounting framework 405–6

capital requirements 403–16

Citigroup 183

economic capital 416–33

market risk requirement 406–8

Northern Rock 421

Base case scenarios 360

Basel Accords 404–5

Bayesian VaR 375–8, 399

Bayes’ rule 375

Behavioural models 232

Beliefs 375

Benchmark VaR 34–5, 48

Bernoulli process 334

Bias 206, 366

Bias statistics 345–8

Bid-ask spread 392

Black–Scholes–Merton formula 265

Bootstrap simulation 345

C

Candidate portfolio 332

Capital allocation 316, 401–35

Capitalization 403, 417

Cash flow

historical VaR 176–9, 190–2

linear VaR 67–78

Cash-flow maps 29, 76

interest rate sensitive portfolios 119

model risk 313–23

PCA 81–5

Cash-flow portfolios 69, 202, 236–9

Cash positions 86–7

CDO (Collateralized debt obligations) xxx–xxxiv, 421

CDS (Credit default swap) xxvi, 135

Chicago Mercantile Exchange 357

Cholesky matrix 180, 213, 226–7

Cleaned P&L 337

Closeness (matrix norm) 389

Cognitive biases 366

Coherent risk metrics 3, 38–41

Collateralized debt obligations (CDO) xxx–xxxiv, 421

Commodity futures 158

normal linear VaR 55, 103–6, 139

scenario-based VaR 369–71

Commodity portfolios 72

Commodity VaR 32, 64

Common trend 231

Compound distribution scenario 360, 367, 374

Compound distribution scenario VaR 371–5

Conditional coverage tests 337

Conditional distributions 201, 367

Conditional marginals 242

Conditional probability 376

Conditional scenarios 364

Conditional VaR, see Expected tail loss

Conditional volatility, see GARCH model

Confidence level 13

Congruence 203

Cooke ratio 407

Copula VaR, 236–9

Copulas 1, 225, 229, 242

Monte Carlo VaR for cash-flow portfolios 236–9

Core capital 407

Cornish–Fisher approximation 144, 170–2, 199, 261, 322

Correlation clustering 202, 242–4, 245

Correlation matrix 229, 231, 389–90

Cost accounting 405

Cost of capital coefficient 427

Cost of risk capital 421

Covariance matrix 54, 68, 85, 119, 153, 231, 293

Covariance VaR model 41

Coverage tests 337–40

Crack spread futures 192–4

Crash market 240

Credit crunch xxx–xxxiv, 264

Credit default swap (CDS) xxvi, 135

Credit rating 420

Credit risk factors 359

Credit spreads 53, 55, 60, 135–40, 231

Credit spread VaR 64, 75–8, 135–8, 368–9

Cross-gamma effects 305

Crude oil futures 192, 280–2, 302, 307

Currency 215, 411

portfolios 60, 241–4

volatility 158–63, 235

D

Data frequency

Historical simulation 145

DAX 30 97, 150, 276

Default intensities 359

Delta approximation to option VaR 257–59

Delta effect 252–3

Delta–gamma approximation 250

Delta–gamma VaR approximation 261–2

Delta–gamma–vega approximation 252

Monte Carlo VaR 298–9

Delta–normal VaR 257–59, 308

Dimension reduction 79, 85, 237, 302–8

Discounted P&L 15, 29

Discounting effect on VaR 23–5

Distribution aggregation 420

Distribution forecasts 348–51

Distribution scenarios 358, 360, 361–2, 366–7, 379, 399

DJIA (Dow Jones Industrial Average) 36

Domestic stock portfolios

systematic linear VaR 87–90

systematic historical VaR 179–85

Dow Jones Industrial Average (DJIA) 36

Downside risk metrics 9–13

Drift adjustment 58

Dynamically hedged option portfolios 272–3

Dynamic VaR 21, 247, 254–7, 267–8, 309

E

Economic capital (EC) 8, 416–33

aggregation 422–4

allocation 416–33

banking 421

measurement 416–20

minimum capital requirement 419–20

model 402

optimal allocation 430–3

Economic value 405

Eligible capital 407

Elliptical copulas 229, 235

EM algorithm 114

Endogenous liquidity effects 359, 394–5, 399

Energy futures 231

Energy options VaR 280–2, 302–7

Energy portfolios 60

Epanechnikov kernel 144, 165–6, 199

Equity beta estimation 316–17

Equity factor model 88–90

Equity indices 231

Equity marginal VaR 66–7, 95–6, 99

Equity scenario VaR 239–41

Equity VaR 27–8, 31, 64, 185–90, 199, 316

ES (Expected shortfall) 3, 34, 35–7, 48

Estimation period in backtest 332

Estimation risk 310, 312, 324–32, 353, 354

ETL, see Expected tail loss

EWMA, see Exponentially weighted moving average

Exceedances 338

Exogenous liquidity effects 359, 381, 399

Expected return – effect on VaR 23–5

Expected shortfall (ES) 3, 34, 35–7, 48

Expected tail loss (ETL) (conditional VaR) 1, 3, 34, 35–7, 48, 56, 128–35, 139

backtesting 344–5

credit spread 135–40

historical 195–8

Monte Carlo algorithms 235

normal 129–30

normal mixture 132–3

student t 130–2

student t mixture 133–5

Exponentially weighted moving average (EWMA) 20, 56, 88, 121–4

covariance 123

formula 125

historical VaR 156–7

OLS 316

Portfolio VaR 124–6

RiskMetrics^™ 126–8

specific VaR 91–2

volatility clustering 218–23

Exponential smoothing constant 161

Exponential twisting 218

Exponential weighting of probabilities 153

Extreme value distributions 144, 167–70

F

Factor push stress testing 363, 382–3

Fair value accounting 405

Faure sequences 206

Filtered historical simulation (FHS) 143, 163–5, 199, 234, 397

Financial risk factors 359–67

Foreign exchange rates 26

Forex 26, 72, 93, 215

Forex VaR 32, 64, 95–7, 185–94, 200

Frobenius norm 389

FTSE 100 95, 101–3, 115, 125–6, 149, 185–90

Funded activities 428

Funding liquidity risk 392

G

Gamma effect 250–2, 255, 256, 268, 290–2, 299, 303, 308

GARCH model 54, 61, 159

filtered historical simulation 163–5

Monte Carlo VaR 218–23, 242–4, 294–6

option VaR 294–6

Gasoline futures VaR 192–4

Gaussian copula 229

Gaussian kernel 144, 168

Generalized autoregressive conditional heteroscedasticity, see GARCH model

Generalized extreme value (GEV) 167

Generalized Pareto distribution (GPD) 167, 199

General risk 434

General risk charge (GRC) 407, 409–11

GEV (Generalized extreme value) 167

Ghost features 20, 156

Global equity crash (1987) 361–2, 385–7

Gold 411

GPD (Generalized Pareto distribution) 167, 199

Gradient vector 33, 66–7, 73–4, 76–8, 95–6, 99, 190

GRC (General risk charge) 407, 409–10

Greeks approximation 248, 249, 273–8, 310

H

Heating oil futures VaR 192

Herd behaviour 381

Historical accounting 405

Historical ETL 195–8

delta-hedged option 269–71

disaggregation 197–8

Historical scenarios 358, 360, 361–2, 401

Historical simulation 141–4, 198–200

dynamic VaR estimation 248

multi-step 151–2

Historical VaR 3, 43–4, 46, 50, 51, 141–4, 175–94, 244, 309

accuracy 152–65

case study 153–6

cash flows 176–9

crack spread futures 192–4

data frequency 145–6

definition 144

delta-hedged option 269–71

energy options trading book 280–2

ETL 144, 195–8

exact revaluation 263–72

Greeks approximation 274–7

international stock portfolios 185–90

linear portfolios 175–94

marginal VaR 186, 190

option portfolios 262–82

overlapping data 323

path-dependent options 278–80

precision 165–75

sample size 145–6, 155

scaling 146–51

specific returns 184–5

square-root scaling, errors from 151

systematic VaR 175–94

volatility adjustment 153, 158–63, 180

Holdings 20, 57

Homogeneity 39

Horizontal disallowance 412

Hyperbolic sine function 172

Hypercube 209

Hypothetical covariance matrices 388–90

Hypothetical distribution scenario 364

Hypothetical scenarios 358, 359, 360, 362–6, 402

I

Idiosyncratic risk 7

Illiquidity 384, 392–6

Implied volatility smiles 232, 302, 307

Importance sampling 215, 217

Incremental risk charge (IRC) 403, 407, 412–16, 434

Incremental VaR 3, 32–3, 56, 139

approximating 70–2

cash flow 70–2

equity 66–7

Independence of exceedances 337

Independence test 339–40

In the money (ITM) 279

Insolvency 14, 418

Intelligent aggregation 420

Interest rates 53, 176–7, 231

PCA 79–85, 237

scenarios for 368–9

Interest rate sensitive securities 72

Interest rate swaps 429

Interest rate term structures 231

Interest rate VaR 32, 64, 85, 199

equity portfolios 100

forex exposure 100

international bond 190–2

vertex choice 314–16

Internal models 434

Internal MRC 414–16

International bond position 190–4

International commodity portfolios 72

International equity portfolios 101–3

International stock portfolios 185–90

Inverse marginal distribution function 229

Investor confidence 421

IRC (Incremental risk charge) 403, 407, 413–16, 434

ITM (In the money) 279

iTraxx Europe index 135–6, 140

J

Johnson SU distribution 144, 172–5, 199, 261, 300

Johnson SU VaR 174–5

Junk bonds 359

K

Kernel fitting 165–7

Kurtosis 106–7, 166, 242

L

Latin hypercube sampling 210

Leptokurtosis 106–7, 242

Leverage effect 160

LIBOR rates 53, 55, 67, 75

Likelihood ratio backtest 351

Linear congruential generator 203, 245

Linear portfolio 4

historical VaR 150–1, 175–94

historical ETL 195–8

mixture VaR 111–20

mixture ETL 132–5

Monte Carlo VaR 234–44

normal linear VaR 42, 56–106

normal linear ETL 129–30

Student t VaR 106–11

Student t ETL 130–2

Liquidity 14, 359, 381, 384, 399

Liquidity-adjusted VaR 359, 392–6

Liquidity horizon 413

Liquidity risk 392, 420

Local maximum 260

Lognormal kernel 166

Low discrepancy sequences 201, 204–6

Lower partial moment (LPM) 9–11

M

Marginal VaR 3, 32–3, 48, 54–5, 64, 66–7

approximating 70–2

cash flows 73–4

commodities 105–6

credit spread 77–8

equity 66–7, 95–7, 99

forex 96–7, 99

historical simulation 186–90

Margins 358

Market crash scenario VaR 372–3

Market depth 381

Market discipline 406

Market impact 392

Market regimes 111

Market risk regulatory framework 406–8

Market Risk Amendment (1996) 379, 388, 403

Market risk capital charge (MRC) 407, 414–16

Market spread 392

Marking to model xxx, xxxv, 403, 405

Markov switching GARCH 223

Matrix norm 389

Maturity ladder 412

Maximum likelihood estimation (MLE) 109

Maximum loss 401

MBS (Mortgage backed securities) 421

Mean reversion 201, 218–23, 248

Mersenne prime 204

Mesokurtosis 178

Method of moments 109, 114

Minimum capital requirement 419–20

Minimum solvency ratio 407, 434

Mixing law 114

Mixture distributions 111–12

Mixture linear VaR 113, 115–18

Mixture parameter estimation 114–15

MLE (Maximum likelihood estimation) 109

Model risk 135, 202, 312, 316–17, 354, 403

Model validation 332–53

backtesting guidelines 335–7

backtesting methodology 332–5

bias statistics 345–8

coverage tests 337–40

distribution forecasts 348–51

ETL forecasts 345

regression 340–4

Monotonic transformation 38, 186

Monte Carlo credit spread VaR 236–8

Monte Carlo interest rate VaR 238–9

Monte Carlo simulations

path dependency 292–6

static VaR 255

Monte Carlo VaR 3, 44–5, 46–7, 51, 141, 201–2, 214–15, 245, 309

behavioural models 232

copulas 229–30

delta–gamma–vega mapping 298–9

energy options 302–7

exact revaluation 283–6

linear portfolios 233–44

low discrepancy sequences 204–6

multivariate delta–gamma mapping 299–300

multivariate delta–gamma–vega mapping 300–1

multivariate distribution 213

multivariate normal 227–8

multivariate student t 228–9

non-linear, non-normal 289–90

option portfolios 282–307

path-dependent option 295–6

PCA 305

pseudo-random number generation 203–4

regime switching models 223–5

short strangle position 296–8

univariate distributions 211–13

variance reduction 206–10

volatility clustering 218–23

Mortgage backed securities (MBS) 421

Moving average models 54

MRC (Market risk capital charge) 407, 414–16

Multi-step historical simulation 278

Multi-step Monte Carlo 201, 215–18, 244, 293

asymmetric GARCH volatility 221–3

EWMA volatility 218–20

Multivariate delta–gamma approximation 250, 251

Multivariate delta–gamma–vega approximation 300

Multivariate distribution 213, 226–30

Multivariate normal distribution 1, 53, 226, 228

Multivariate normal mixture distribution 202, 235

Multivariate student t 227–8, 287

N

Natural gas futures VaR 104

Net asset value 417

Non-linearity in risk factor dependence 287–90

Non-normality in risk factors 287–90

Normal copula 229

Normal distribution 42, 56, 138, 227, 320, 367–71

Normal linear VaR 3, 41–2, 45–6, 51, 56–67

bias statistics 345–8

cash-flow maps 67–78

commodity futures 103–6

confidence intervals 325–8

credit spread 75–8, 135–8

equity 27–8, 85–103

ETL 129–30

formula 18–20, 56–60

interest rate 29–30, 67–8, 72–4

stock portfolios 85–92

Normal mixture ETL 133

Normal mixture risk factor VaR 119–20

Normal mixture scenario VaR 373–5

Normal mixture VaR 115–16

Northern Rock 421

NYMEX futures 103

O

Off balance sheet instruments 406

OLS (Ordinary least squares) 316

Operational risk factors 359

Option portfolios 307

delta effect 252–3

dynamically hedged 272–3

gamma effect 250–1

historical VaR 263–82

Monte Carlo VaR 282–307

P&L 260

rho effect 253–4

risk factors 250–7, 308

theta effect 253–4

VaR 247–308

vega effect 252–3

Option price 217

Option pricing theory 257

Ordinary least squares (OLS) 316

OTC trades 135, 403

Out-of-sample diagnostic analysis 312

Over-the-counter (OTC) trades 135, 403

Overlapping data 151–2, 323, 336

P

Parametric linear VaR 53–140, 121–8, 324–9

Parametric simulation 248

Pareto distribution 144, 167, 199

Path dependency 292–6

Path-dependent options 278–80

PCA (Principal component analysis) 55, 79, 81–5, 140, 230–1

crude oil futures 304–6

interest rates 235

stress tests 359, 390–2, 399

Portfolio management

diversification 89–90

risk adjusted performance measures 23–4

risk measurement 6–7

VaR 20–1, 48

Portfolio sensitivity analysis 380

Portfolio theory 430

Posterior distribution 375–8

Power law scale exponents 143, 146–7

Present value of basis point (PV01), see PV01

Price beta mapping 250–1

Price process 217

Price-quantity impact 395–6

Price risk 271

Price sensitivities 2

Price-volatility relationship 232, 233, 253, 295

Pricing model 403

Primitive roots 204

Principal component analysis (PCA) 55, 79, 81–5, 140, 230–1

crude oil futures 304–6

interest rates 235

stress tests 359, 390–2, 399

Principal component risk factors 202

Principal component stress tests 391–2

Prior density 376

Prior distribution 375

Probability 364, 375, 379, 399

Probability distributions 366–7

Probability weights 157

Profit and loss (P&L)

actual 336–7

cleaned 337

discounted 15, 29

funded activities 428

option portfolios 259–60

Taylor expansions 255

uncertainty 1

VaR estimation 13–20

Pseudo-random numbers 201, 203–4

Public disclosure 406

PV01 (Present value of basis point) 29, 53, 55, 57, 63, 65, 68–83, 100, 102, 176–7, 179

Q

Quantile risk metrics 11–13, 48

Quantiles 32, 165, 186

confidence levels 330–2

Quanto correlation 93

Quasi Monte Carlo methods 201

R

RAPM (Risk adjusted performance measure) 421, 424–30, 431

RAROC (Risk adjusted return on capital) 427, 430–1, 434

RORAC (Return on risk adjusted capital) 425–9

Rating agencies 421

Realized P&L 336–7

Real-time VaR calculations 249

Rebalancing assumption 3, 20, 150, 254–7

Reference volatility 252

Regression 340–4

Regulatory capital 401, 403–16

Regulatory matrix 126

Resolution methods 2, 311, 322–3

Return on risk adjusted capital (RORAC) 425–9

Reward to risk ratio 425

Rho effects 253–4

Risk adjusted performance measure (RAPM) 421, 424–30, 431

Risk adjusted return on capital (RAROC) 427, 430–1, 434

Risk aggregation 420, 422–4

Risk attribution 26

Risk budgeting 8, 421

Risk drivers 358, 364

Risk factor dependence 225–33

Risk factor mapping 26, 27, 43, 49, 182, 311, 314–19

Risk factor models 225–34, 287, 319–22

Risk factors 26, 202

dynamic 152, 215–25

iTraxx Europe index 135–6

multivariate distribution 311

option portfolios 250–7, 297

orthogonalizing 230–1

principal components as 79, 81–5, 140, 230–1

scale exponents 147–50

trust region 383

volatility as 252, 280–2, 287–307

Risk factor sensitivities 21, 300 see also Equity beta, PV01, Greeks

Risk factor volatility 6

Risk free condition 39

Risk horizon 7, 13–15, 420

Risk limits 5

Risk metrics 1, 38, 47–8

ETL defined 36, 345

downside 9–13

VaR defined 13–17

Quantiles 11–13

RiskMetrics^™ 126–8, 344

Risk–return space 430

Rolling windows 332–3

S

S&P 500 153, 186, 385

EWMA 123

filtered historical simulation VaR 163–5

GARCH parameters 160

historical VaR 154

moments 115

normal linear VaR 155

RiskMetrics^™ VaR and ETL 343–7

VaR comparison 45–7

volatility adjusted VaR 159–62, 166

volatility index 149, 288–9, 300

Sample likelihood 376

Sample size 145, 155–6, 316

Sampling error 312, 313

estimation risk 329, 355

Sampling methods 203–11, 245

Scale exponent 143, 146, 200

major risk factors 147–50

Scaling VaR

different risk horizons 323–4, 357

Scenario analysis 357–78, 410

Scenario VaR 367–78, 399

Semi-standard deviation 9–10

Semi-variance 9

Sensitivity analysis 358

SER (Standardized exceedance residuals) 345, 346

Sharpe ratio 430

Shocks 357

Short strangle position 296–7

Significance level 13

Silver futures VaR 105

Simulation errors 202, 206

Single case scenarios 358, 360

Six sigma event 382

Skewness 111–17, 137, 166, 242

Sobol sequences 206

Solvency condition 419

Solvency ratio 407

SPAN (Standard Portfolio Analysis of Risk) 358

Specific risk charge (SRC) 407, 411, 434

Specific VaR 27, 31, 48, 54

empirical estimation 90–1

EWMA 91–2

historical simulation of 199

stock portfolio 87–90, 179–84

volatility adjustment of 182

Square-root-of-time rule 54, 146, 152

SRC (Specific risk charge) 407, 411, 434

Stable distribution 146

Stand-alone VaR 32–3, 48, 54, 56, 139, 199

bond positions 190–2

commodity futures 103–6, 199

normal linear 64–6

stock portfolios 93–100, 185–90

see also – credit spread VaR, commodity VaR

equity VaR, forex VaR, interest rate VaR

Standardized exceedance residuals (SER) 345, 346

Standardized MRC 414–16

Standardized rules 411–12, 434

Standard normal density function 129

Standard normal distribution function 18

Standard Portfolio Analysis of Risk (SPAN) 358

Standard uniform distribution 209–10

Static portfolio 20–1

Static VaR 21, 247, 254–7, 267–8, 309

Statistical bootstrap 163

Stock portfolios

historical VaR 179–89

normal linear VaR 55, 85, 103

systematic VaR 88–90, 93–103, 179–85

Stratified sampling 209, 212

Stressed covariance matrices 384, 385–8

Stressed VaR 387–8, 393–4

Stress event 357

Stress scenario 357

Stress testing 27, 85, 357, 378–84, 390–2, 399

bias 366

factor push 363, 382–3

FHS 397–8

GARCH model 398

regulatory guidelines 379–81, 408

systemic risk 381

volatility clustering 397–9

worst case loss 381–4

Structured Monte Carlo 201

Student t copulas 229

Student t distributed ETL 131

Student t distributed portfolio returns 327–8

Student t distribution 53, 106, 138

Student t linear VaR 108–9

Student t mixture ETL 134

Sub-additivity 32, 39, 55

Sub-prime mortgage crisis xxx–xxxv, 264

Sub-supplementary capital 407

Supervisory review 406

Supplementary capital 407

Systematic return 63

Systematic risk 31

Systematic VaR 31–2, 48, 54

decomposition 93–103

domestic stock portfolios 87–90

equity factor model 88–90

historical simulation 179–92

normal linear 63–4

T

Tangency portfolio 430

Target return 10

Taylor approximation 308

Taylor expansions 248, 255

Theoretical P&L 337

Theta effect 254, 255, 256, 290–2, 308

Threshold return 9

Tilt component 231

Time aggregation 186, 200

Total capital charge 407

Total VaR 31

Tracking error 6, 35, 48

Trading book 406

Trust region 383

Tuenter's algorithm 261, 262

U

UK bonds 79, 140, 176

Uncertainty 39, 359

Unconditional coverage tests 337, 338–9

Unconditional distributions 201

Undiversifiable risk 8

Unexpected return 5, 220

Unfunded activities 428

Unit interval 209

Unlisted securities 367

Unrealized P&L 15, 337

V

Value at risk (VaR) 1, 2, 48, 59–60, 137–8, 324–32, 333, 350

aggregation 63–6, 182–6

confidence level 13, 169

decomposition 30–3, 93–100, 189–92

defining 13–17

disaggregation 72–5, 90–1, 187

discounting, effect on 23

equally weighted averages 121

gamma effects 250–2, 308

mathematical definition 15–17

resolution method 322–3

risk attribution 26

risk factor mapping 26

risk horizon 13

risk metrics associated with VaR 33–41

scaling 21–3

significance level 13

Van der Corput sequences 206

Variance reduction 206–10, 245

Variance of simulation error 206

VaR, see Value at risk

Vega effect 252–3, 256, 268, 290–1

Vega mapping 299, 306–7

Vertex choice 314–16

Vertical disallowance 412

Volatility 1, 20, 80–1, 245, 312

adjustment 153, 158–63, 179, 182, 199

annualized 136

conditional 121

EWMA estimates 122

historical 121

mapping 252, 307

mean reversion 220

price relationship 232–3, 253, 295

risk factors 271, 306, 319

weighting 159

Volatility clustering 54, 141, 165, 201, 354, 399

coverage tests 342–4

GARCH models 54, 61, 159, 162–5, 218–25, 242–4, 294–5

Monte Carlo simulations 218–23

path dependency 292–6

stress testing 397–8

Volatility indices 150, 185–9, 252, 288–90

Volatility surface 308

Volatility term structures 231

W

Weak stochastic dominance 38

Worst case loss 357, 380, 381–4

Worst case scenarios 360, 361–2, 390

Z

Zero-coupon yield curves 26, 67–85, 140, 176–9, 236–9