CAViaR :Conditional Value at Risk By Regression Quantiles

Robert Engle and Simone Manganelli

U.C.S.D.July 1999

2

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

3

HISTOGRAM OF TOMORROW’S VALUE - BASED ON PAST RETURNS

0 . 0

0 . 2

0 . 4

0 . 6

0 . 8

- 2 0 - 1 5 - 1 0 - 5 0 5

S & P 5 0 0 % R E T U R N S

K e r n e l D e n s i t y ( N o r m a l , h = 0 . 1 1 4 5 )

4

CUMULATIVE DISTRIBUTION

0.0

0.2

0.4

0.6

0.8

1.0

-20 -10 0 10

Empirical CDF of S&P500 RETURNS

5

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.

6

THE PROBLEM

• FORECAST QUANTILE OF FUTURE RETURNS

• MUST ACCOMMODATE TIME VARYING DISTRIBUTIONS

• MUST HAVE METHOD FOR EVALUATION

• MUST HAVE METHOD FOR PICKING UNKNOWN PARAMETERS

7

TWO GENERAL APPROACHES• FACTOR MODELS--- AS IN

RISKMETRICS

• PORTFOLIO MODELS--- AS IN ROLLING HISTORICAL QUANTILES

8

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

9

PORTFOLIO MODELS

• Historical performance of fixed weight portfolio is calculated from data bank

• Model for quantile is estimated

• VaR is forecast

10

COMPLICATIONS

• Some assets didn’t trade in the past- approximate by deltas or betas

• Some assets were traded at different times of the day - asynchronous prices-synchronize these

• Derivatives may require special assumptions - volatility models and greeks.

11

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.

12

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!

13

Mathematical Formulation

Find VaR satisfying

where y are returns and is probability

Must be able to calculate VaR one day in advance and to estimate unknown parameters.

)(

1ttVaRyP

14

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

)(

)(11

ttt

ttt

VaRyIhit

hitVaRVaR

15

How to compute VaR

If beta is known, then VaR can be calculated for the adaptive model from a starting value.

.....)3(

hit no if (-.05)*

1in hit if .95*VaR(1)VaR(2)

1.65VaR(1)Let

VaR

16

CAViaR News Impact Curve

17

More Specifications

• Proportional Symmetric Adaptive

• Symmetric Absolute Value:

• Asymmetric Absolute Value:

)VaRy()VaRy(VaRVaR 1t1t21t1t11tt

1t21t101t yVaRVaR

31t21t101t yVaRVaR

18

• Asymmetric Slope

• Indirect GARCH

1t31t21t10t yyVaRVaR

2/12

1t2

21t

10t yk

VaRkVaR

19

20

Koenker and Bassett(1978) maximize

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

)()(

210)()(

ttt

ttt

fyhit

fyIfyQ

21

Quantile Objective Function

22

Even though the quantile function is non-differentiable at some points, the first order conditions must be satisfied with probability one.

Hits should be unpredictable and are uncorrelated with regressors at an optimum

0'

/)(

0/)ˆ(

Xhit

fX

fhit

tt

tt

23

Adaptive Criterion

-0.118

-0.116

-0.114

-0.112

-0.110

-0.108

500 1000 1500 2000 2500 3000

Quantile Criterion - Adaptive

24

Asymmetric Criterion

-0.17

-0.16

-0.15

-0.14

-0.13

-0.12

500 1000 1500 2000 2500 3000

Likelihood from CAViaR1

25

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

26

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 out-of-sample

27

28

Tests

• Cowles and Jones (1937)

• Runs - Mood (1940)

• Ljung Box on hits (1979)

• Dynamic Quantile Test

29

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

30

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

31

Mathematical Statistics References• Koenker and Bassett(1978) no

dynamics

• Weiss(1991) least absolute deviation

• Newey and McFadden(1994)

32

Mathematical Statistics

)(maxarg

)(maxargˆ

0)r( to

Q

nQ

fyfyIn

Q

subject

tttt

hitXDRRADRDRXDhitn

ADDNn

Then

LM

d

''''

)1(,0ˆ

111111

110

33

Mathematical Assumptions

0n1/ and ,

2/Q )8(

matrixsingular -non a ,/' )7(

),0()(n )6(

ofdensity theish where,'n

1 )5(

of odneighborho ain singular non , )4(

oney probabilit with ),()(1

Q )3(

)( )2(

at maximizeduniquely is ,)( )1(

0

0000

0

0

02

00

p

p

d

ptt

tt

p

remainderremainder

DQQ

AnXX

ANQ

(y-f)DXXh

DQE

Xhitn

Rr

QQE

34

Estimating Standard Errors

ofdensity lconditiona theis where

,'n1

0

-fy h

DXXh

ttt

pttt

• 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

35

A Picture Gives Intuition

0.0

0.2

0.4

0.6

0.8

1.0

200 400 600 800 1000

Y1 Y2

STANDARD DEVIATION OF Y1 IS TWICE Y2BOTH ARE GAUSSIAN

36

0.0

0.2

0.4

0.6

0.8

1.0

200 400 600 800 1000

Y1Y2

Q1_1/10Q2_1/10

1% QUANTILE POINTS

37

0.00

0.05

0.10

0.15

0.20

0.25

220 240 260 280 300 320 340 360 380 400

Y1Y2

Q1_1/10Q2_1/10

1% QUANTILE POINTS

38

0.0

0.1

0.2

0.3

0.4

0.5

300 320 340 360 380 400 420 440

Y1Y2

Q1_5/5Q2_5/5

5% QUANTILE

39

0.0

0.1

0.2

0.3

0.4

0.5

200 400 600 800 1000

Y1Y2/2

Q1_5/10Q2_5/10

Density of Y2 divided by its standard deviation

40

Assumption

• Define

• Therefore• And• NOW ASSUME:

tt

ttt g

f

fy~

ttt fhg

ttt fgh /)0()0(

.f hborhood ofor a Neigfor all t ugugt 0)()(

41

Estimate g Non-parametrically:

• where k is a uniform kernel accepting points between -1 and 1

• and for 2900 observations empirically we chose cn=.05

tt fgh ˆ/)0(ˆ)0(ˆ

ntn ckncg /0ˆ 1

42

43

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 )

44

Table 1 - Summary statistics of the Monte Carlo experiment

0.1% GAMMA1 GAMMA2 GAMMA3

True mean 4.15 0.90 0.69

Mean 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

Var-Cov matrix -2.45 0.05 -0.07

2.60 -0.07 0.32

1% GAMMA1 GAMMA2 GAMMA3

True mean 1.62 0.90 0.27

Mean 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

Var-Cov matrix -0.28 0.01 -0.01

0.19 -0.01 0.02

45

5% GAMMA1 GAMMA2 GAMMA3True mean 0.81 0.90 0.135

Trimmed Mean 0.99 0.89 0.14Trimmed Median 0.81 0.90 0.14

0.49 -0.04 0.02Trimmed Var-Cov matrix -0.04 0.00 0.00

0.02 0.00 0.00

25% GAMMA1 GAMMA2 GAMMA3True mean 0.13 0.90 0.027

Trimmed Mean 0.18 0.88 0.03Trimmed Median 0.13 0.90 0.02

0.03 -0.01 0.00Trimmed Var-Cov matrix -0.01 0.01 0.00

0.00 0.00 0.00

46

Table 2 - Monte Carlo summary statistics after trimming the samples with GAMMA2<0.5

0.1% GAMMA1 GAMMA2 GAMMA3True mean 4.15 0.90 0.69

Trimmed Mean 4.04 0.87 0.60Trimmed Median 2.49 0.90 0.50

18.36 -0.40 0.81Trimmed Var-Cov matrix -0.40 0.01 -0.03

0.81 -0.03 0.23

1% GAMMA1 GAMMA2 GAMMA3True mean 1.62 0.90 0.27

Trimmed Mean 2.02 0.88 0.29Trimmed Median 1.55 0.90 0.27

2.72 -0.11 0.12Trimmed Var-Cov matrix -0.11 0.00 -0.01

0.12 -0.01 0.02

47

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.

48

-30

-20

-10

0

10

20

500 1000 1500 2000 2500 3000

GM

-30

-20

-10

0

10

20

500 1000 1500 2000 2500 3000

IBM

-30

-20

-10

0

10

500 1000 1500 2000 2500 3000

S&P500

49

News Impact Curve - 1% SP

2

4

6

8

10

12

200 400 600 800 1000

VAR_ADAPTIVEVAR_ASY_ABSVAR_ASY_SLOP

VAR_IND_GARCHVAR_PRO_SYM_ADAVAR_SYM_ABS

50

Caviar News Impact Curves SP500 at 5%

0

1

2

3

4

5

50 100 150 200 250 300 350 400

A

0

1

2

3

4

5

50 100 150 200 250 300 350 400

PSA

0

1

2

3

4

5

50 100 150 200 250 300 350 400

SAV

0

1

2

3

4

5

50 100 150 200 250 300 350 400

AAV

0

1

2

3

4

5

50 100 150 200 250 300 350 400

AS

0

1

2

3

4

5

50 100 150 200 250 300 350 400

G

51

1% and 5% News Impact Curves

1

2

3

4

5

200 400 600 800 1000

VAR_AS_5 VAR_ASY_SLOP

52

Table 3 - Parameter estimates -Statistics for the Adaptive model

ADAPTIVE *** 5% GM IBM S&P 500 ADAPTIVE *** 25% GM IBM S&P 500Gamma 1 0.22 0.44 0.23 Gamma 1 0.021 0.012 0.017Standard Errors 0.03 0.05 0.02 Standard Errors 0.004 0.003 0.003P-values 0.00 0.00 0.00 P-values 0.000 0.000 0.000RQ in sample 553.26 527.45 312.65 RQ in sample 1507 1368 752RQ out of sample 100.84 120.20 72.41 RQ out of sample 291.62 312.44 184Hits in sample (%) 4.91 5.01 5.08 Hits in sample (%) 24.86 25.31 25.07Hits out of sample (%) 6.40 5.20 5.00 Hits out of sample (%) 27.00 24.80 27.40DQ in sample (p-values) DQ in sample (p-values)1) [c, hit(-1 to -5)] 0.31 0.47 0.46 1) [c, hit(-1 to -5)] 0.94 0.25 0.592) [VaR] 0.52 0.34 0.48 2) [VaR] 0.73 0.80 0.683) [c, hit(-1), VaR] 0.12 0.01 0.10 3) [c, hit(-1), VaR] 0.58 0.64 0.304) [c, hit(-1 to -5), VaR] 0.06 0.01 0.07 4) [c, hit(-1 to -5), VaR] 0.81 0.23 0.32DQ out of sample (p-values) DQ out of sample (p-values)1) [c, hit(-1 to -5)] 0.40 0.98 0.01 1) [c, hit(-1 to -5)] 0.57 0.67 0.452) [VaR] 0.26 0.90 0.80 2) [VaR] 0.43 0.97 0.293) [c, hit(-1), VaR] 0.40 0.21 0.55 3) [c, hit(-1), VaR] 0.23 0.28 0.394) [c, hit(-1 to -5), VaR] 0.45 0.56 0.01 4) [c, hit(-1 to -5), VaR] 0.64 0.30 0.41

53

ASYM SLOPE *** 0.1% GM IBM S&P 500 ASYM SLOPE *** 1% GM IBM S&P 500Gamma 1 2.7753 1.0863 0.4325 Gamma 1 0.3928 0.0572 0.1473Standard Errors - - - Standard Errors 0.2216 0.0580 0.0833P-values - - - P-values 0.0381 0.1623 0.0385

Gamma 2 0.4342 0.6587 0.6871 Gamma 2 0.7983 0.9427 0.8699Standard Errors - - - Standard Errors 0.0676 0.0227 0.0484P-values - - - P-values 0.0000 0.0000 0.0000

Gamma 3 0.6130 1.1402 1.8655 Gamma 3 0.2725 0.0512 0.0001Standard Errors - - - Standard Errors 0.1148 0.0616 0.1168P-values - - - P-values 0.0088 0.2029 0.4997

Gamma 4 2.0416 2.8743 2.2849 Gamma 4 0.4437 0.2474 0.5045Standard Errors - - - Standard Errors 0.1589 0.1006 0.2403P-values - - - P-values 0.0026 0.0070 0.0179

RQ in sample 25.01 29.27 18.15 RQ in sample 169.30 179.54 105.84RQ out of sample 4.15 5.93 3.65 RQ out of sample 28.48 40.54 22.69

Hits in sample (%) 0.10 0.10 0.14 Hits in sample (%) 1.00 0.97 0.97Hits out of sample (%) 0.00 0.00 0.00 Hits out of sample (%) 1.40 1.60 1.60

DQ in sample (p-values) DQ in sample (p-values)1) [c, hit(-1 to -5)] - - - 1) [c, hit(-1 to -5)] 0.60 0.81 0.562) [VaR] - - - 2) [VaR] 0.98 0.89 0.963) [c, hit(-1), VaR] - - - 3) [c, hit(-1), VaR] 0.96 0.96 0.944) [c, hit(-1 to -5), VaR] - - - 4) [c, hit(-1 to -5), VaR] 0.71 0.88 0.68

DQ out of sample (p-values) DQ out of sample (p-values)1) [c, hit(-1 to -5)] - - - 1) [c, hit(-1 to -5)] 0.96 0.05 0.052) [VaR] - - - 2) [VaR] 0.46 0.21 0.133) [c, hit(-1), VaR] - - - 3) [c, hit(-1), VaR] 0.67 0.53 0.454) [c, hit(-1 to -5), VaR] - - - 4) [c, hit(-1 to -5), VaR] 0.97 0.07 0.07

LM test for VaR(t-2) - - - LM test for VaR(t-2) 0.92 0.92 0.96

54

ASYM SLOPE *** 5% GM IBM S&P 500 ASYM SLOPE *** 25% GM IBM S&P 500Gamma 1 0.0704 0.0951 0.0410 Gamma 1 0.0404 0.0125 0.0014Standard Errors 0.0425 0.0444 0.0221 Standard Errors 0.0298 0.0104 0.0047P-values 0.0488 0.0161 0.0316 P-values 0.0877 0.1151 0.3820

Gamma 2 0.9353 0.8916 0.9026 Gamma 2 0.9132 0.9605 0.9481Standard Errors 0.0222 0.0272 0.0239 Standard Errors 0.0393 0.0169 0.0212P-values 0.0000 0.0000 0.0000 P-values 0.0000 0.0000 0.0000

Gamma 3 0.0411 0.0597 0.0307 Gamma 3 0.0415 0.0108 0.0288Standard Errors 0.0285 0.0335 0.0469 Standard Errors 0.0193 0.0098 0.0192P-values 0.0745 0.0372 0.2565 P-values 0.0157 0.1349 0.0664

Gamma 4 0.1182 0.2110 0.2841 Gamma 4 0.0290 0.0297 0.0288Standard Errors 0.0399 0.0558 0.0895 Standard Errors 0.0170 0.0127 0.0175P-values 0.0015 0.0001 0.0008 P-values 0.0441 0.0097 0.0502

RQ in sample 548.63 515.72 300.76 RQ in sample 1500.88 1360.53 746.90RQ out of sample 99.20 121.05 72.05 RQ out of sample 289.41 311.51 183.48

Hits in sample (%) 4.98 4.91 4.98 Hits in sample (%) 25.00 25.14 24.93Hits out of sample (%) 5.20 7.40 6.80 Hits out of sample (%) 25.60 23.40 25.80

DQ in sample (p-values) DQ in sample (p-values)1) [c, hit(-1 to -5)] 0.83 0.74 0.69 1) [c, hit(-1 to -5)] 0.69 0.83 0.492) [VaR] 0.98 0.87 0.94 2) [VaR] 0.97 0.85 0.933) [c, hit(-1), VaR] 0.97 0.97 0.64 3) [c, hit(-1), VaR] 0.97 0.90 0.994) [c, hit(-1 to -5), VaR] 0.89 0.82 0.74 4) [c, hit(-1 to -5), VaR] 0.79 0.90 0.60

DQ out of sample (p-values) DQ out of sample (p-values)1) [c, hit(-1 to -5)] 0.92 0.03 0.00 1) [c, hit(-1 to -5)] 0.88 0.64 0.292) [VaR] 0.97 0.06 0.20 2) [VaR] 0.88 0.45 0.773) [c, hit(-1), VaR] 0.96 0.00 0.13 3) [c, hit(-1), VaR] 0.67 0.79 0.674) [c, hit(-1 to -5), VaR] 0.95 0.01 0.00 4) [c, hit(-1 to -5), VaR] 0.94 0.70 0.32

LM test for VaR(t-2) 0.96 0.77 0.94 LM test for VaR(t-2) 0.99 0.89 0.60

55

Value at Risk for GM

0

2

4

6

8

10

12

500 1000 1500 2000 2500 3000

A_VAR

0

2

4

6

8

10

12

500 1000 1500 2000 2500 3000

AAV_VAR

0

2

4

6

8

10

12

500 1000 1500 2000 2500 3000

AS_VAR

0

2

4

6

8

10

12

500 1000 1500 2000 2500 3000

G_VAR

0

2

4

6

8

10

12

500 1000 1500 2000 2500 3000

PSA_VAR

0

2

4

6

8

10

12

500 1000 1500 2000 2500 3000

SAV_VAR

56

Value at Risk for SP

-2

0

2

4

6

8

10

12

14

500 1000 1500 2000 2500 3000

A_VAR

0

2

4

6

8

10

12

14

500 1000 1500 2000 2500 3000

AAV_VAR

0

2

4

6

8

10

12

14

500 1000 1500 2000 2500 3000

AS_VAR

0

2

4

6

8

10

12

14

500 1000 1500 2000 2500 3000

G_VAR

0

2

4

6

8

10

12

14

500 1000 1500 2000 2500 3000

PSA_VAR

0

2

4

6

8

10

12

14

500 1000 1500 2000 2500 3000

SAV_VAR

57

58

Dynamic Quantile Test -SPDependent Variable: SAV_HITSample: 5 2892Included observations: 2888Variable Coefficient Std. Error t-Statistic Prob.

C 0.0051 0.0096 0.5277 0.5977SAV_HIT(-1) 0.0397 0.0187 2.1277 0.0334SAV_HIT(-2) 0.0244 0.0187 1.3051 0.1920SAV_HIT(-3) 0.0252 0.0187 1.3468 0.1781SAV_HIT(-4) -0.0044 0.0187 -0.2370 0.8127SAV_VAR -0.0034 0.0066 -0.5241 0.6002

R-squared 0.0029 Mean dependent var 0.0006Adjusted R-squared 0.0012 S.D. dependent var 0.2191S.E. of regression 0.2190 Akaike info criterion -0.1975Sum squared resid 138.2105 Schwarz criterion -0.1851Log likelihood 291.2040 F-statistic 1.7043Durbin-Watson stat 1.9999 Prob(F-statistic) 0.1301

59

In-sample Dynamic Quantile Test

00.10.20.30.40.50.60.70.80.9

1

A_OUT

AAV_

OUT

AS_OUT

G_OUT

PSA_

OUT

SAV_

OUT

GMIBMSP500

60

In-sample 1% Dynamic Quantile Test

00.10.20.30.40.50.60.70.80.9

A_OUT

AAV_

OUT

AS_OUT

G_OUT

PSA_

OUT

SAV_

OUT

GMIBMSP500

61

Out of Sample DQ Test

00.10.20.30.40.50.60.70.80.9

1

A_OUT

AAV_

OUT

AS_OUT

G_OUT

PSA_

OUT

SAV_

OUT

GMIBMSP500

62

Out of Sample 1% DQ Test

00.10.20.30.40.50.60.70.80.9

1

A_OUT

AAV_

OUT

AS_OUT

G_OUT

PSA_

OUT

SAV_

OUT

GMIBMSP500

63

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

64

DQ TESTS FOR NORMAL GARCH

0

0.05

0.1

0.15

0.2

0.25

IN-SAMPLE IN - TEST2 OUT OUT-TEST2

GMIBMSP500

65

TRADITIONAL GARCH(1,1) : IBM

C 0.133384 0.016911ARCH(1) 0.112194 0.005075GARCH(1) 0.851960 0.009923

5% POINT OF STANDARDIZED RESIDUALS = 1.48

FOR GM THIS POINT IS 1.56FOR S&P THIS POINT IS 1.64

66

DQ TESTS FOR TRADITIONAL GARCH

00.10.20.30.40.50.60.70.80.9

1

IN-SAMPLE IN - TEST2 OUT OUT-TEST2

GMIBMSP500

67

Value at Risk for GM Asymmetric

-30

-20

-10

0

10

20

500 1000 1500 2000 2500 3000

GM -AS_VAR

68

Value at Risk for IBM Adaptive

-30

-20

-10

0

10

20

500 1000 1500 2000 2500 3000

IBM -A_VAR

69

Value at Risk for SP Implicit GARCH

-30

-20

-10

0

10

500 1000 1500 2000 2500 3000

SP500 -G_VAR

70

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?

71

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