CAViaR :
Conditional Value at Risk By
Regression Quantiles
Robert Engle and Simone Manganelli
U.C.S.D.
July 1999
Value at Risk
is a single measure of
market risk of a firm, portfolio, trading
desk, or other economic entity.
It is defined by a significance level and a
horizon. For convenience consider 5%
and 1 day.
Any loss tomorrow will be less than the
Value at Risk with 95% certainty
2
HISTOGRAM OF TOMORROW’S
VALUE - BASED ON PAST RETURNS
K e r n e l D e n s ity ( N o r m a l, h =
0 .1 1 4 5 )
0 .8
0 .6
0 .4
0 .2
0 .0
-20
-15
-10
S & P 500
-5
%
0
5
R E T U R N S
3
CUMULATIVE DISTRIBUTION
Empirical CDF of S&P500 RETURNS
1.0
0.8
0.6
0.4
0.2
0.0
-20
-10
0
10
4
Weakness of this measure
• The amount we exceed VaR is important
• There is no utility function associated with
this measure
• The measure assumes assets can be sold
at their market price - no consideration for
liquidity
• But it is simple to understand and very
widely used.
5
THE PROBLEM
• FORECAST QUANTILE OF FUTURE RETURNS
• MUST ACCOMMODATE TIME VARYING
DISTRIBUTIONS
• MUST HAVE METHOD FOR EVALUATION
• MUST HAVE METHOD FOR PICKING UNKNOWN
PARAMETERS
6
TWO GENERAL APPROACHES
• FACTOR MODELS--- AS IN RISKMETRICS
• PORTFOLIO MODELS--- AS IN ROLLING
HISTORICAL QUANTILES
7
FACTOR MODELS
– Volatilities and correlations between factors
are estimated
– These volatilities and correlations are updated
daily
– Portfolio standard deviations are calculated
from portfolio weights and covariance matrix
– Value at Risk computed assuming normality
8
PORTFOLIO MODELS
• Historical performance of fixed weight
portfolio is calculated from data bank
• Model for quantile is estimated
• VaR is forecast
9
COMPLICATIONS
• Some assets didn’t trade in the pastapproximate by deltas or betas
• Some assets were traded at different times
of the day - asynchronous pricessynchronize these
• Derivatives may require special
assumptions - volatility models and greeks.
10
PORTFOLIO MODELS - EXAMPLES
• Rolling Historical : e.g. find the 5% point of
the last 250 days
• GARCH : e.g. build a GARCH model to
forecast volatility and use standardized
residuals to find 5% point
• Hybrid model: use rolling historical but
weight most recent data more heavily with
exponentially declining weights.
11
THE CAViaR STRATEGY
• Define a quantile model with some
unknown parameters
• Construct the quantile criterion function
• Optimize this criterion over the historical
period
• Formulate diagnostic checks for model
adequacy
• Try it out!
12
Mathematical Formulation
Find VaR satisfying
P( y  VaR  )  
t
t 1
where y are returns and  is probability
Must be able to calculate VaR one day in
advance and to estimate unknown
parameters.
13
SPECIFICATIONS FOR VaR
• VaR is a function of observables in t-1
• VaR=f(VaR(t-1), y(t-1), parameters)
• For example - the Adaptive Model
VaR  VaR   (hit )
t 1
t
t 1
hit  I ( y  VaR )  
t
t
t
14
How to compute VaR
If beta is known, then VaR can be
calculated for the adaptive model from a
starting value.
Let VaR(1)  1.65
  * .95 if hit in 1
VaR(2)  VaR(1)  
  * (-.05) if no hit
VaR (3)  .....
15
CAViaR News Impact Curve
16
More Specifications
• Proportional Symmetric Adaptive
VaRt  VaRt 1   1( yt 1  VaRt 1 )   2 ( yt 1  VaRt 1 )
• Symmetric Absolute Value:
VaRt 1   0   1VaRt 1   2 yt 1
• Asymmetric Absolute Value:
VaRt 1   0   1VaRt 1   2 yt 1   3
17
• Asymmetric Slope

VaRt   0   1VaRt 1   2 yt 1   3 yt 1

• Indirect GARCH


VaR


2
t

1
VaRt  k   0   1 
  2 yt 1 



k




2
1/ 2
18
19
Koenker and Bassett(1978) maximize
Q(  )   yt  f t      I  y  f t (  )  0 1  2 
  hitt (  ) yt  f t (  ) 
Where f is the quantile which depends on
past information and parameters beta
The criterion minimizes absolute errors
where positive and negative errors are
weighted differently
20
Quantile Objective Function
21
Even though the quantile function is nondifferentiable at some points, the first
order conditions must be satisfied with
probability one.
 hitt ( ˆ )ft /   0
X t (  )  f t / 
hit ' X  0
Hits should be unpredictable and are
uncorrelated with regressors at an
optimum
22
Adaptive Criterion
-0.108
-0.110
-0.112
-0.114
-0.116
-0.118
500
1000
1500
2000
2500
3000
Quantile Criterion - Adaptive
23
Asymmetric Criterion
-0.12
-0.13
-0.14
-0.15
-0.16
-0.17
500
1000
1500
2000
2500
3000
Likelihood from CAViaR1
24
Optimization by Genetic
Algorithm
• DIFFERENTIAL EVOLUTIONARY GENETIC
ALGORITHM - Price and Storn(1997)
• Start with initial population of trial values
• Reproduction based on fitness
• Crossover to find next generation
• Mutation - random new elements
• Stopping Criterion
25
Testing the Model
• Should have the right proportion of hits
• Should have no autocorrelation
• Probability of exceeding VaR should be
independent of VaR (no measurement
error)
• Should be testable both in-sample and outof-sample
26
27
Tests
• Cowles and Jones (1937)
• Runs - Mood (1940)
• Ljung Box on hits (1979)
• Dynamic Quantile Test
28
Dynamic Quantile Test
To test that hits have the same distribution
regardless of past observables
Regress hit on
–
–
–
–
–
constant
lagged hits
Value at Risk
lagged returns
other variables such as year dummies
29
Distribution Theory
• If out of sample test , or
• If all parameters are known
• Then TR02 will be asymptotically Chi
Squared and F version is also available
• But the distribution is slightly different
otherwise
30
Mathematical Statistics
References
• Koenker and Bassett(1978) no dynamics
• Weiss(1991) least absolute deviation
• Newey and McFadden(1994)
31
Mathematical Statistics

Q     I  yt  f t       yt  f t   
n
ˆ  arg max Q(  )n

arg max
subject to r(  )0
Q(  )
Then


1
1
1 1 1 1
 LM  n hit ' XD R' RD AD R' R D X ' hit
d
1
1
ˆ
n    0   N 0, (1   ) D AD
32
Mathematical Assumptions
(1) E Q (  )   Q0  , is uniquely maximized at  0  
p
( 2) r (  )  R
(3) Q  
1
hit t (  ) X t (  ), with probabilit y one

n
( 4)  2 E Q   D, non singular in a neighborho od of  0
1
p
(5)
h 0 X t X t '  D, where h is the density of (y-f)

n
( 6)
d
n Q (  0 ) 
 N (0, A)
p
(7) X ' X / n  A, a non - singular matrix
(8) Q    Q  0   Q    0      0 D   0  / 2
 remainder , and remainder/ 1 
n    0   0
p
33
Estimating Standard Errors
• To calculate standard errors-must estimate D
• D weights X by the height of the conditional
density of returns at the estimated quantile
• Should estimate this without making
assumptions on the shape of the density
1
p
ht  0  X t X t '  D,

n
where ht is the conditiona l density of  yt -ft 
34
A Picture Gives Intuition
STANDARD DEVIATION OF Y1 IS TWICE Y2
BOTH ARE GAUSSIAN
1.0
0.8
0.6
0.4
0.2
0.0
200
400
600
Y1
800
1000
Y2
35
1% QUANTILE POINTS
1.0
0.8
0.6
0.4
0.2
0.0
200
400
Y1
Y2
600
800
1000
Q1_1/10
Q2_1/10
36
1% QUANTILE POINTS
0.25
0.20
0.15
0.10
0.05
0.00
220
240
260
280
300
Y1
Y2
320
340
360
380
400
Q1_1/10
Q2_1/10
37
5% QUANTILE
0.5
0.4
0.3
0.2
0.1
0.0
300
320
340
360
Y1
Y2
380
400
420
440
Q1_5/5
Q2_5/5
38
Density of Y2 divided by its standard deviation
0.5
0.4
0.3
0.2
0.1
0.0
200
400
Y1
Y2/2
600
800
1000
Q1_5/10
Q2_5/10
39
Assumption
• Define
t 
yt  f t
~ gt  
ft
gt    ht   ft
• Therefore
ht (0)  gt (0) / ft
• And
• NOW ASSUME:
gt (u )  g (u )
for all t for a Neighborhood of 0.
40
Estimate g Non-parametrically:
1
gˆ 0  cn n 
 k  t / cn 
• where k is a uniform kernel accepting
points between -1 and 1
• and for 2900 observations empirically we
chose cn=.05
hˆt (0)  gˆ (0) / fˆt
41
42
A little Monte Carlo
• 100 samples of 2000 observations of
GARCH(1,1) with parameters (.3, .05, .90)
• Estimate with Indirect GARCH CAViaR model
• Mean parameters are (.42, .05, .88)
• Some are far off showing no persistence
• Trimming 10 extremes, means become
(.31,.05,.90 )
43
Table 1 - Summary statistics of
the Monte Carlo experiment
0.1%
True mean
GAMMA1
GAMMA2
GAMMA3
4.15
0.90
0.69
7.16
0.80
0.67
t-statistic
8.54
-13.60
-0.95
Median
2.90
0.89
0.53
125.16
-2.45
2.60
-2.45
0.05
-0.07
2.60
-0.07
0.32
Mean
Var-Cov matrix
1%
True mean
GAMMA1
GAMMA2
GAMMA3
1.62
0.90
0.27
2.28
0.87
0.30
t-statistic
7.59
-7.91
6.01
Median
1.57
0.90
0.27
7.79
-0.28
0.19
-0.28
0.01
-0.01
0.19
-0.01
0.02
Mean
Var-Cov matrix
44
5%
True mean
Trimmed Mean
Trimmed Median
Trimmed Var-Cov matrix
25%
True mean
Trimmed Mean
Trimmed Median
Trimmed Var-Cov matrix
GAMMA1
0.81
0.99
0.81
0.49
-0.04
0.02
GAMMA2
0.90
0.89
0.90
-0.04
0.00
0.00
GAMMA3
0.135
0.14
0.14
0.02
0.00
0.00
GAMMA1
0.13
0.18
0.13
0.03
-0.01
0.00
GAMMA2
0.90
0.88
0.90
-0.01
0.01
0.00
GAMMA3
0.027
0.03
0.02
0.00
0.00
0.00
45
Table 2 - Monte Carlo summary statistics after
trimming the samples with GAMMA2<0.5
0.1%
True mean
Trimmed Mean
Trimmed Median
Trimmed Var-Cov matrix
1%
True mean
Trimmed Mean
Trimmed Median
Trimmed Var-Cov matrix
GAMMA1
4.15
4.04
2.49
18.36
-0.40
0.81
GAMMA2
0.90
0.87
0.90
-0.40
0.01
-0.03
GAMMA3
0.69
0.60
0.50
0.81
-0.03
0.23
GAMMA1
1.62
2.02
1.55
2.72
-0.11
0.12
GAMMA2
0.90
0.88
0.90
-0.11
0.00
-0.01
GAMMA3
0.27
0.29
0.27
0.12
-0.01
0.02
46
Applications
• Daily data from April 7, 1986 to April 7, 1999 3392 observations
• Save the last 500 for out- of- sample tests
• GM, IBM, S&P500
• Fit all 6 models for 5% ,1% , .1% and 25% VaR.
47
20
10
0
- 10
- 20
- 30
500
1000
1500
2000
2500
3000
GM
20
10
0
- 10
- 20
- 30
500
1000
1500
2000
2500
3000
IBM
10
0
- 10
- 20
- 30
500
1000
1500
2000
2500
3000
S&P500
48
News Impact Curve - 1% SP
12
10
8
6
4
2
200
400
VAR_ADAPTIVE
VAR_ASY_ABS
VAR_ASY_SLOP
600
800
1000
VAR_IND_GARCH
VAR_PRO_SYM_ADA
VAR_SYM_ABS
49
Caviar News Impact Curves
SP500 at 5%
5
5
5
4
4
4
3
3
3
2
2
2
1
1
1
0
0
50
100
150
200
250
300
350
400
0
50
100
150
A
200
250
300
350
400
50
5
4
4
4
3
3
3
2
2
2
1
1
1
0
0
150
200
AAV
250
300
350
400
200
250
300
350
400
250
300
350
400
SAV
5
100
150
PSA
5
50
100
0
50
100
150
200
AS
250
300
350
400
50
100
150
200
G
50
1% and 5% News Impact Curves
5
4
3
2
1
200
400
VAR_AS_5
600
800
1000
VAR_ASY_SLOP
51
Table 3 - Parameter estimates Statistics for the Adaptive model
ADAPTIVE *** 5%
Gamma 1
Standard Errors
P-values
RQ in sample
RQ out of sample
Hits in sample (%)
Hits out of sample (%)
DQ in sample (p-values)
1) [c, hit(-1 to -5)]
2) [VaR]
3) [c, hit(-1), VaR]
4) [c, hit(-1 to -5), VaR]
DQ out of sample (p-values)
1) [c, hit(-1 to -5)]
2) [VaR]
3) [c, hit(-1), VaR]
4) [c, hit(-1 to -5), VaR]
GM
0.22
0.03
0.00
553.26
100.84
4.91
6.40
IBM S&P 500
0.44
0.23
0.05
0.02
0.00
0.00
527.45 312.65
120.20
72.41
5.01
5.08
5.20
5.00
0.31
0.52
0.12
0.06
0.47
0.34
0.01
0.01
0.46
0.48
0.10
0.07
0.40
0.26
0.40
0.45
0.98
0.90
0.21
0.56
0.01
0.80
0.55
0.01
ADAPTIVE *** 25%
Gamma 1
Standard Errors
P-values
RQ in sample
RQ out of sample
Hits in sample (%)
Hits out of sample (%)
DQ in sample (p-values)
1) [c, hit(-1 to -5)]
2) [VaR]
3) [c, hit(-1), VaR]
4) [c, hit(-1 to -5), VaR]
DQ out of sample (p-values)
1) [c, hit(-1 to -5)]
2) [VaR]
3) [c, hit(-1), VaR]
4) [c, hit(-1 to -5), VaR]
GM
0.021
0.004
0.000
1507
291.62
24.86
27.00
IBM S&P 500
0.012 0.017
0.003 0.003
0.000 0.000
1368
752
312.44
184
25.31 25.07
24.80 27.40
0.94
0.73
0.58
0.81
0.25
0.80
0.64
0.23
0.59
0.68
0.30
0.32
0.57
0.43
0.23
0.64
0.67
0.97
0.28
0.30
0.45
0.29
0.39
0.41
52
ASYM SLOPE *** 0.1%
Gamma 1
Standard Errors
P-values
Gamma 2
Standard Errors
P-values
Gamma 3
Standard Errors
P-values
Gamma 4
Standard Errors
P-values
RQ in sample
RQ out of sample
Hits in sample (%)
Hits out of sample (%)
DQ in sample (p-values)
1) [c, hit(-1 to -5)]
2) [VaR]
3) [c, hit(-1), VaR]
4) [c, hit(-1 to -5), VaR]
DQ out of sample (p-values)
1) [c, hit(-1 to -5)]
2) [VaR]
3) [c, hit(-1), VaR]
4) [c, hit(-1 to -5), VaR]
LM test for VaR(t-2)
GM
2.7753
0.4342
0.6130
2.0416
25.01
4.15
0.10
0.00
IBM S&P 500
1.0863
0.4325
0.6587
0.6871
1.1402
1.8655
2.8743
2.2849
29.27
18.15
5.93
3.65
0.10
0.14
0.00
0.00
-
-
-
-
-
-
ASYM SLOPE *** 1%
Gamma 1
Standard Errors
P-values
Gamma 2
Standard Errors
P-values
Gamma 3
Standard Errors
P-values
Gamma 4
Standard Errors
P-values
RQ in sample
RQ out of sample
Hits in sample (%)
Hits out of sample (%)
DQ in sample (p-values)
1) [c, hit(-1 to -5)]
2) [VaR]
3) [c, hit(-1), VaR]
4) [c, hit(-1 to -5), VaR]
DQ out of sample (p-values)
1) [c, hit(-1 to -5)]
2) [VaR]
3) [c, hit(-1), VaR]
4) [c, hit(-1 to -5), VaR]
LM test for VaR(t-2)
GM
0.3928
0.2216
0.0381
0.7983
0.0676
0.0000
0.2725
0.1148
0.0088
0.4437
0.1589
0.0026
169.30
28.48
1.00
1.40
IBM S&P 500
0.0572 0.1473
0.0580 0.0833
0.1623 0.0385
0.9427 0.8699
0.0227 0.0484
0.0000 0.0000
0.0512 0.0001
0.0616 0.1168
0.2029 0.4997
0.2474 0.5045
0.1006 0.2403
0.0070 0.0179
179.54 105.84
40.54 22.69
0.97
0.97
1.60
1.60
0.60
0.98
0.96
0.71
0.81
0.89
0.96
0.88
0.56
0.96
0.94
0.68
0.96
0.46
0.67
0.97
0.92
0.05
0.21
0.53
0.07
0.92
0.05
0.13
0.45
0.07
0.96
53
ASYM SLOPE *** 5%
Gamma 1
Standard Errors
P-values
Gamma 2
Standard Errors
P-values
Gamma 3
Standard Errors
P-values
Gamma 4
Standard Errors
P-values
RQ in sample
RQ out of sample
Hits in sample (%)
Hits out of sample (%)
DQ in sample (p-values)
1) [c, hit(-1 to -5)]
2) [VaR]
3) [c, hit(-1), VaR]
4) [c, hit(-1 to -5), VaR]
DQ out of sample (p-values)
1) [c, hit(-1 to -5)]
2) [VaR]
3) [c, hit(-1), VaR]
4) [c, hit(-1 to -5), VaR]
LM test for VaR(t-2)
GM
0.0704
0.0425
0.0488
0.9353
0.0222
0.0000
0.0411
0.0285
0.0745
0.1182
0.0399
0.0015
548.63
99.20
4.98
5.20
IBM S&P 500
0.0951
0.0410
0.0444
0.0221
0.0161
0.0316
0.8916
0.9026
0.0272
0.0239
0.0000
0.0000
0.0597
0.0307
0.0335
0.0469
0.0372
0.2565
0.2110
0.2841
0.0558
0.0895
0.0001
0.0008
515.72
300.76
121.05
72.05
4.91
4.98
7.40
6.80
0.83
0.98
0.97
0.89
0.74
0.87
0.97
0.82
0.69
0.94
0.64
0.74
0.92
0.97
0.96
0.95
0.96
0.03
0.06
0.00
0.01
0.77
0.00
0.20
0.13
0.00
0.94
ASYM SLOPE *** 25%
Gamma 1
Standard Errors
P-values
Gamma 2
Standard Errors
P-values
Gamma 3
Standard Errors
P-values
Gamma 4
Standard Errors
P-values
RQ in sample
RQ out of sample
Hits in sample (%)
Hits out of sample (%)
DQ in sample (p-values)
1) [c, hit(-1 to -5)]
2) [VaR]
3) [c, hit(-1), VaR]
4) [c, hit(-1 to -5), VaR]
DQ out of sample (p-values)
1) [c, hit(-1 to -5)]
2) [VaR]
3) [c, hit(-1), VaR]
4) [c, hit(-1 to -5), VaR]
LM test for VaR(t-2)
GM
0.0404
0.0298
0.0877
0.9132
0.0393
0.0000
0.0415
0.0193
0.0157
0.0290
0.0170
0.0441
1500.88
289.41
25.00
25.60
IBM S&P 500
0.0125 0.0014
0.0104 0.0047
0.1151 0.3820
0.9605 0.9481
0.0169 0.0212
0.0000 0.0000
0.0108 0.0288
0.0098 0.0192
0.1349 0.0664
0.0297 0.0288
0.0127 0.0175
0.0097 0.0502
1360.53 746.90
311.51 183.48
25.14
24.93
23.40
25.80
0.69
0.97
0.97
0.79
0.83
0.85
0.90
0.90
0.49
0.93
0.99
0.60
0.88
0.88
0.67
0.94
0.99
0.64
0.45
0.79
0.70
0.89
0.29
0.77
0.67
0.32
0.60
54
Value at Risk for GM
12
12
10
10
8
8
6
6
4
4
2
2
0
0
500
1000
1500
2000
2500
3000
500
1000
A _V A R
1500
2000
2500
3000
2500
3000
2500
3000
A A V _V A R
12
12
10
10
8
8
6
6
4
4
2
2
0
0
500
1000
1500
2000
2500
3000
500
1000
A S _V A R
1500
2000
G _V A R
12
12
10
10
8
8
6
6
4
4
2
2
0
0
500
1000
1500
2000
P S A _V A R
2500
3000
500
1000
1500
2000
S A V _V A R
55
Value at Risk for SP
14
14
12
12
10
10
8
8
6
6
4
4
2
2
0
-2
0
500
1000
1500
2000
2500
3000
500
1000
A _V A R
1500
2000
2500
3000
2500
3000
2500
3000
A A V _V A R
14
14
12
12
10
10
8
8
6
6
4
4
2
2
0
0
500
1000
1500
2000
2500
3000
500
1000
A S _V A R
1500
2000
G _V A R
14
14
12
12
10
10
8
8
6
6
4
4
2
2
0
0
500
1000
1500
2000
P S A _V A R
2500
3000
500
1000
1500
2000
S A V _V A R
56
57
Dynamic Quantile Test -SP
Dependent Variable: SAV_HIT
Sample: 5 2892
Included observations: 2888
Variable
Coefficient Std. Error t-Statistic
Prob.
C
SAV_HIT(-1)
SAV_HIT(-2)
SAV_HIT(-3)
SAV_HIT(-4)
SAV_VAR
0.0051
0.0397
0.0244
0.0252
-0.0044
-0.0034
0.5977
0.0334
0.1920
0.1781
0.8127
0.6002
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
0.0029
0.0012
0.2190
138.2105
291.2040
1.9999
0.0096
0.0187
0.0187
0.0187
0.0187
0.0066
0.5277
2.1277
1.3051
1.3468
-0.2370
-0.5241
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic
Prob(F-statistic)
0.0006
0.2191
-0.1975
-0.1851
1.7043
0.1301
58
In-sample Dynamic Quantile Test
SAV_OUT
PSA_OUT
G_OUT
AS_OUT
AAV_OUT
GM
IBM
SP500
A_OUT
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
59
In-sample 1% Dynamic Quantile
Test
SAV_OUT
PSA_OUT
G_OUT
AS_OUT
AAV_OUT
GM
IBM
SP500
A_OUT
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
60
Out of Sample DQ Test
SAV_OUT
PSA_OUT
G_OUT
AS_OUT
AAV_OUT
GM
IBM
SP500
A_OUT
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
61
Out of Sample 1% DQ Test
SAV_OUT
PSA_OUT
G_OUT
AS_OUT
AAV_OUT
GM
IBM
SP500
A_OUT
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
62
TRADITIONAL GARCH(1,1) : IBM
C
0.133384 0.016911
ARCH(1) 0.112194 0.005075
GARCH(1) 0.851960 0.009923
VaR=1.65*standard deviation
63
DQ TESTS FOR NORMAL GARCH
0.25
0.2
0.15
GM
IBM
SP500
0.1
0.05
0
IN-SAMPLE
IN - TEST2
OUT
OUT-TEST2
64
TRADITIONAL GARCH(1,1) : IBM
C
0.133384 0.016911
ARCH(1) 0.112194 0.005075
GARCH(1) 0.851960 0.009923
5% POINT OF STANDARDIZED RESIDUALS = 1.48
FOR GM THIS POINT IS 1.56
FOR S&P THIS POINT IS 1.64
65
DQ TESTS FOR TRADITIONAL GARCH
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
GM
IBM
SP500
IN-SAMPLE
IN - TEST2
OUT
OUT-TEST2
66
Value at Risk for GM Asymmetric
20
10
0
-10
-20
-30
500
1000
1500
GM
2000
2500
3000
-AS_VAR
67
Value at Risk for IBM Adaptive
20
10
0
-10
-20
-30
500
1000
1500
IBM
2000
2500
3000
-A_VAR
68
Value at Risk for SP Implicit
GARCH
10
0
-10
-20
-30
500
1000
1500
SP500
2000
2500
3000
-G_VAR
69
Some Extensions
• Are there economic variables which can
predict tail shapes?
• Would option market variables have
predictability for the tails?
• Would variables such as credit spreads
prove predictive?
• Can we estimate the expected value of the
tail?
70
CONCLUSIONS-Contributions?
• Estimation strategy for VaR Models
• New Dynamic Specifications of Quantiles
• Estimation of VaR without estimating
volatility
• Test for VaR accuracy both in and out of
sample
• Promising empirical evidence on some
specifications
71