CAViaR : Conditional Value at Risk By Regression Quantiles

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CAViaR Conditional Value at Risk By Regression
Quantiles

• Robert Engle and Simone Manganelli
• U.C.S.D.
• July 1999

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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
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HISTOGRAM OF TOMORROWS VALUE - BASED ON PAST
RETURNS
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CUMULATIVE DISTRIBUTION
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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.

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THE PROBLEM

• FORECAST QUANTILE OF FUTURE RETURNS
• MUST ACCOMMODATE TIME VARYING DISTRIBUTIONS
• MUST HAVE METHOD FOR EVALUATION
• MUST HAVE METHOD FOR PICKING UNKNOWN PARAMETERS

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TWO GENERAL APPROACHES

• FACTOR MODELS--- AS IN RISKMETRICS
• PORTFOLIO MODELS--- AS IN ROLLING HISTORICAL
  QUANTILES

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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

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PORTFOLIO MODELS

• Historical performance of fixed weight portfolio
  is calculated from data bank
• Model for quantile is estimated
• VaR is forecast

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COMPLICATIONS

• Some assets didnt 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.

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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.

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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!

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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.

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SPECIFICATIONS FOR VaR

• VaR is a function of observables in t-1
• VaRf(VaR(t-1), y(t-1), parameters)
• For example - the Adaptive Model

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How to compute VaR

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

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CAViaR News Impact Curve
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More Specifications

• Proportional Symmetric Adaptive
• Symmetric Absolute Value
• Asymmetric Absolute Value

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• Asymmetric Slope
• Indirect GARCH

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• 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

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Quantile Objective Function
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• 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

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Adaptive Criterion
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Asymmetric Criterion
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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

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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

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Tests

• Cowles and Jones (1937)
• Runs - Mood (1940)
• Ljung Box on hits (1979)
• Dynamic Quantile Test

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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

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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

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Mathematical Statistics References

• Koenker and Bassett(1978) no dynamics
• Weiss(1991) least absolute deviation
• Newey and McFadden(1994)

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Mathematical Statistics
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Mathematical Assumptions
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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

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A Picture Gives Intuition
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Assumption

• Define
• Therefore
• And
• NOW ASSUME

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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

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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 )

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Table 1 - Summary statistics of the Monte Carlo
experiment
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Table 2 - Monte Carlo summary statistics after
trimming the samples with GAMMA2lt0.5
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Applications

• Daily data from April 7, 1986 to April 7, 1999 -
  3392 observations
• Save the last 500 for out- of- sample tests
• GM, IBM, SP500
• Fit all 6 models for 5 ,1 , .1 and 25 VaR.

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News Impact Curve - 1 SP
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Caviar News Impact Curves SP500 at 5
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1 and 5 News Impact Curves
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Table 3 - Parameter estimates -Statistics for the
Adaptive model
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Value at Risk for GM
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Value at Risk for SP
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Dynamic Quantile Test -SP

• Dependent Variable SAV_HIT
• Sample 5 2892
• Included observations 2888
• Variable Coefficient Std. Error t-Statistic Prob.
  
• C 0.0051 0.0096 0.5277 0.5977
• SAV_HIT(-1) 0.0397 0.0187 2.1277 0.0334
• SAV_HIT(-2) 0.0244 0.0187 1.3051 0.1920
• SAV_HIT(-3) 0.0252 0.0187 1.3468 0.1781
• SAV_HIT(-4) -0.0044 0.0187 -0.2370 0.8127
• SAV_VAR -0.0034 0.0066 -0.5241 0.6002
• R-squared 0.0029 Mean dependent var 0.0006
• Adjusted R-squared 0.0012 S.D. dependent
  var 0.2191
• S.E. of regression 0.2190 Akaike info
  criterion -0.1975
• Sum squared resid 138.2105 Schwarz
  criterion -0.1851
• Log likelihood 291.2040 F-statistic 1.7043
• Durbin-Watson stat 1.9999 Prob(F-statistic) 0
  .1301

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In-sample Dynamic Quantile Test
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In-sample 1 Dynamic Quantile Test
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Out of Sample DQ Test
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Out of Sample 1 DQ Test
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TRADITIONAL GARCH(1,1) IBM

• C 0.133384 0.016911
• ARCH(1) 0.112194 0.005075
• GARCH(1) 0.851960 0.009923
• VaR1.65standard deviation

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DQ TESTS FOR NORMAL GARCH
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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 SP THIS POINT IS 1.64

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DQ TESTS FOR TRADITIONAL GARCH
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Value at Risk for GM Asymmetric
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Value at Risk for IBM Adaptive
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Value at Risk for SP Implicit GARCH
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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?

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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