Comparing Value-at-Risk Methodologies

1
Comparing Value-at-Risk Methodologies

• Breno Néri
• New York University
• breno.neri_at_nyu.edu
• http//homepages.nyu.edu/bpn207
• With Luiz Lima
• Financial Economics Workshop November 12th, 2007

2
Market Risk Exposure
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application

• 1987 Black Monday - 23 drop in value
• 1995 Mexico
• 1997 Asia
• 1998 Russia and Latin America
• 1998 Long-Term Capital Management

Oliver Linton
3
Measures of Market Risk
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application

• Market Risk Exposure
• Efficiency
• 1996 amendment to the 1988 Basle Capital Accord
• 1998 adopted by U.S. bank regulatory agencies

Lopez (JR, 1999)
4
VaR(p)
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application
5
VaR(p)
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application
6
General Framework
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application
Giot and Laurent (JEF, 2004)
7
General Framework
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application

• Conditional Mean OLS
• Lags and/or other Conditioning Variables
• Information Criteria
• Akaike AIC
• Schwarz (Bayesian) BIC
• Shibata
• Hannan-Quinn

8
General Framework
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application

• RiskMetrics
• Gaussian GARCH
• Skewed-t APARCH
• ARCH Quantile

9
General Framework
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application
ARCH(p)

• RiskMetrics
• Gaussian GARCH
• Skewed-t APARCH
• ARCH Quantile

Engle (ECA, 1982)
10
General Framework
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application
GARCH(p,q)

• RiskMetrics
• Gaussian GARCH
• Skewed-t APARCH
• ARCH Quantile

Bollerslev (JE, 1986)
11
General Framework
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application
APARCH(p,q)

• RiskMetrics
• Gaussian GARCH
• Skewed-t APARCH
• ARCH Quantile

Ding, Granger and Engle (JEF, 1993) He and
Teräsvirta (1999a,b)
12
General Framework
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application
Skewed Student-t
Fernández and Steel (JASA,1998) Lambert and
Laurent (2001)
13
General Framework
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application

• RiskMetrics
• Gaussian GARCH
• Skewed-t APARCH
• ARCH Quantile

J.P. Morgan (1996)
14
General Framework
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application

• RiskMetrics
• Gaussian GARCH
• Skewed-t APARCH
• ARCH Quantile

15
General Framework
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application

• RiskMetrics
• Gaussian GARCH
• Skewed-t APARCH
• ARCH Quantile

16
General Framework
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application

• RiskMetrics
• Gaussian GARCH
• Skewed-t APARCH
• ARCH Quantile

17
General Framework
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application

• RiskMetrics
• Gaussian GARCH
• Skewed-t APARCH
• ARCH Quantile

Giot (JFM, 2003) Giot and Laurent (JAE, 2003)
18
General Framework
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application

• RiskMetrics
• Gaussian GARCH
• Skewed-t APARCH
• ARCH Quantile

19
Exponential Power Function
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application
20
Exponential Power Function
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application
21
Exponential Power Function
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application
22
Exponential Power Function
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application

• Skewed Exponential Power Function
• Skewed Gaussian
• Skewed Student-t

23
M-Estimators
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application

• Minimum/Maximum
• Extremum Estimators

Huber (1964, 1965, 1982, 1981) Wooldridge / Green
/ Davidson and Mackinnon
24
M-Estimators
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application
Wooldridge
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M-Estimators
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application

• Uniform Weak Law of Large Numbers

Wooldridge
26
M-Estimators
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application

• Uniform Weak Law of Large Numbers

Wooldridge
27
M-Estimators
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application
28
M-Estimators
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application
29
M-Estimators
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application
30
M-Estimators
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application
31
M-Estimators FOC
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application
32
M-Estimators Examples
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application
33
M-Estimators Examples
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application
34
M-Estimators Examples
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application
35
M-Estimator Quantile Regression
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application
36
M-Estimator Quantile Regression
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application
37
M-Estimator Quantile Regression
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application
38
M-Estimator Quantile Regression
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application
39
M-Estimator Quantile Regression
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application
40
Quantile Regression Equivariance
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application
Koenker and Portnoy (BSA, 1996)
41
Quantile Regression
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application
Koenker and Portnoy (BSA, 1996)
42
ARCH Quantile VaR
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application

• RiskMetrics
• Gaussian GARCH
• Skewed-t APARCH
• ARCH Quantile

Wu and Xiao (JR, 2002)
43
ARCH Quantile VaR
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application
Wu and Xiao (JR, 2002)
44
ARCH Quantile VaR
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application
45
More on Quantile Regression
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application

• Original Paper
• Koenker and Basset (Econometrica, 1978)
• Goodness of Fit
• Koenker and Machado (JASA, 1999)
• Inference on Quantile Regression Process
• Koenker and Xiao (Econometrica, 2002)
• Quantile AutoRegressive Model, QAR(p)
• Koenker and Xiao (2004a)
• Unit Root Test for each quantile in a QAR(p)
• Koenker and Xiao (JASA, 2004b)

46
Backtests
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application

• Unconditional Coverage
• Point Estimator for p
• Independence
• Conditional Coverage
• Dynamic Quantile
• Magnitude Loss Function
• Other Backtests
• Time Until First Failure
• Duration Based Approach
• Mixed Test
• CD-Test
• Scale CD-Method

47
Backtests
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application

• Unconditional Coverage
• Point Estimator for p
• Independence
• Conditional Coverage
• Dynamic Quantile
• Magnitude Loss Function
• Other Backtests
• Time Until First Failure
• Duration Based Approach
• Mixed Test
• CD-Test
• Scale CD-Method

Kupiec (JD, 1995)
48
Backtests
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application

• Unconditional Coverage
• Point Estimator for p
• Independence
• Conditional Coverage
• Dynamic Quantile
• Magnitude Loss Function
• Other Backtests
• Time Until First Failure
• Duration Based Approach
• Mixed Test
• CD-Test
• Scale CD-Method

Haas (2001)
49
Backtests
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application

• Unconditional Coverage
• Point Estimator for p
• Independence
• Conditional Coverage
• Dynamic Quantile
• Magnitude Loss Function
• Other Backtests
• Time Until First Failure
• Duration Based Approach
• Mixed Test
• CD-Test
• Scale CD-Method

Christoffersen (IER, 1998)
50
Backtests
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application

• Unconditional Coverage
• Point Estimator for p
• Independence
• Conditional Coverage
• Dynamic Quantile
• Magnitude Loss Function
• Other Backtests
• Time Until First Failure
• Duration Based Approach
• Mixed Test
• CD-Test
• Scale CD-Method

Christoffersen (IER, 1998)
51
Backtests
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application

• Unconditional Coverage
• Point Estimator for p
• Independence
• Conditional Coverage
• Dynamic Quantile
• Magnitude Loss Function
• Other Backtests
• Time Until First Failure
• Duration Based Approach
• Mixed Test
• CD-Test
• Scale CD-Method

Engle and Manganelli (2002)
52
Backtests
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application

• Unconditional Coverage
• Point Estimator for p
• Independence
• Conditional Coverage
• Dynamic Quantile
• Magnitude Loss Function
• Other Backtests
• Time Until First Failure
• Duration Based Approach
• Mixed Test
• CD-Test
• Scale CD-Method

Lopez (FED-ER, 1999b)
53
Backtests
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application

• Unconditional Coverage
• Point Estimator for p
• Independence
• Conditional Coverage
• Dynamic Quantile
• Magnitude Loss Function
• Other Backtests
• Time Until First Failure
• Duration Based Approach
• Mixed Test
• CD-Test
• Scale CD-Method

54
Regulatory Framework
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application

• Basle Capital Accord
• 1996 amendment
• 1 billion
• 10

Lopez (JR, 1999a)
55
Regulatory Framework
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application

• Basle Capital Accord
• 1996 amendment
• 1 billion
• 10

Lopez (JR, 1999a)
56
Regulatory Framework
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application

• Basle Capital Accord
• 1996 amendment
• 1 billion
• 10

Lopez (JR, 1999a)
57
Monte Carlo Genesis
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application

• Enrico Fermi (1930)
• Manhattan Project
• Stanislay Ulam (1946)
• Nicholas Metropolis
• Top 10 Algorithms
• ENIAC (1946)
• MAthematical and Numerical Integrator And
  Computer
• MANIAC (1952) and MANIAC II (1957)
• with Richard Feynman

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ENIAC
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application
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ENIAC
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application
60
ENIAC
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application
61
ENIAC UPenn
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application
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MAX (at NYU)
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application
63
MAX (at NYU)
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application
64
Non-Random QuickSort
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application
65
Monte Carlo Simulations
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application
Frery and Cribary-Neto(2005)
66
Monte Carlo Simulations
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application
Frery and Cribary-Neto(2005)
67
Monte Carlo in Statistics
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application
68
MC Specification
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application

• 10 DGPs
• N 1000
• T 1250
• Rolling Temporal Window Size 250
• 1 day-ahead-forecast VaR(1)

69
MC DGPs
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application
70
MC Computational Details
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application

• 40,000,000 of optimizations (1000x1000x10x4)
• R
• RNG Mersenne-Twister (Matsumoto and Nishimura,
  1998)
• Quantile Regression (implemented in Fortran by
  Koenker)
• Interpreted Slower
• Ox
• Likelihood Maximizations
• Compiled Faster
• Estimation does not vary dramatically over time
• Estimated parameters at t-1 are the initial guess
  at t
• Maximum number of iterations
• Convergence Criterion

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MC Computational Details
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application

• EPGE03
• 4 Intel Pentium IV Xeon 2.8 GHz
• 4 Gb RAM
• 100 GB SCSI HD
• OS Linux Debian
• Peak Performance lt10 Gflops

72
MC Computational Details
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application

• EPGE03
• 4 Intel Pentium IV Xeon 2.8 GHz
• 4 Gb RAM
• 100 GB SCSI HD
• OS Linux Debian
• Peak Performance lt10 Gflops

73
MC Results
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application

• Test Size at 1 Significance Level
• Unconditional Coverage (Ho p1), Kupiec (1995)

74
Histograms Number of Violations per
Trajectory(DGP 6)
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application
75
Histograms Number of Violations per
Trajectory(DGP 7)
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application
76
Histograms Number of Violations per
Trajectory(DGP 8)
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application
77
Histograms Number of Violations per
Trajectory(DGP 9)
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application
78
Histograms Number of Violations per
Trajectory(DGP 10)
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application
79
MC Results
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application

• Distribution of Violations mean

80
MC Results
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application

• Distribution of Violations Variance

81
MC Results
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application

• Distribution of Violations MSE

82
MC Results
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application

• Distribution of Violations Skewness

83
MC Results
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application

• Distribution of Violations Excess Kurtosis

84
MC Results
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application

• Distribution of Violations Min

85
MC Results
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application

• Distribution of Violations Max

86
MC Results
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application

• Distribution of Violations Range Max - Min

87
Empirical Exercise
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application

• Data Ibovespa Daily Return
• Period from 07/08/1996 to 03/24/2000
• Observations 920 (670 forecasts, from
  07/11/1997)
• VaR(1) ? 7 violations

88
Empirical Exercise
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application

• Data Ibovespa Daily Return
• Period from 07/08/1996 to 03/24/2000
• Observations 920 (670 forecasts, from
  07/11/1997)
• VaR(1) Þ 7 violations

89
Empirical Exercise
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application

• Data Ibovespa Daily Return
• Period from 07/08/1996 to 03/24/2000
• Observations 920 (670 forecasts, from
  07/11/1997)
• VaR(1) Þ 7 violations

90
Empirical Exercise
Value-at-Risk Quantile Regression
Backtest Monte Carlo
Empirical Application

• Data Ibovespa Daily Return
• Period from 07/08/1996 to 03/24/2000
• Observations 920 (670 forecasts, from
  07/11/1997)
• VaR(1) ? 7 violations

91
Thank you!

• Breno Néri
• New York University
• breno.neri_at_nyu.edu
• http//homepages.nyu.edu/bpn207
• With Luiz Lima
• Financial Economics Workshop November 12th, 2007