13 th AFIR Colloquium 2003 The estimation of Market VaR using Garch models and a heavy tail distributions The dynamic VaR and The Static VaR The Garch Models The Heavy tails distributions

The market VaR The principal components The volatility The probability distributions of returns The probability defined for the maximum loss to be accepted 13 th AFIR Colloquium 2003

Why we need a credible VaR 13 th AFIR Colloquium )Because when we calculate a VaR position we need to make a reserve outside the portfolio 2) Because the traders must believe in this VaR and constraint the portfolio in order to comply with the limits as a result of VaR estimation 3) Because when we make a reserve we reduce the dividends, and add additional costs for this frozen funds

The first component of VaR: The volatility 13 th AFIR Colloquium 2003 How it is presented the volatility in the market? 1) The volatility don’t follows the law of t 0.5 2) The volatility is presented in clusters. There are moments of great volatility followed by moments of tranquility 3) The volatility series is a predictable process

13 th AFIR Colloquium 2003

How to forecast the volatility 13 th AFIR Colloquium 2003 Regress the series returns on a constant and the model is: The constant is the mean of the series and the residuals,  t are the volatility or the difference between the value observed and the constant or the mean of the series.

The presence of Arch in the model 13 th AFIR Colloquium 2003 First step: Test the hypothesis H o :        k = 0 H 1 : some  k  0 Use the statistic:

13 th AFIR Colloquium 2003 Second step: The Arch LM Test H o : There are absence of Arch H 1 : There are presence of Arch Estimate the following auto regression model: And calculate the Observations * R 2 = TR 2 This coefficient TR 2    k

Some results of different series 13 th AFIR Colloquium 2003 SeriesQ (8) ProbT*R 2 kProb Dow Jones Bovespa MSCI T.bond 5 y IDP Merval

With the presence of Arch the forecast volatility may be done by nonlinear models 13 th AFIR Colloquium ) Garch models 2) RiskMetrics™ or EWMA 3) Asymmetric Garch models RiskMetrics is a trade mark of J.P.Morgan

The Garch model 13 th AFIR Colloquium 2003 “ a little of the error of my prediction of today plus a little of the prediction for today ” If the volatility for tomorrow is a result of: Then we are in presence of a Garch(1,1)

The beauty of Garch (1,1) model 13 th AFIR Colloquium 2003 The square error of an heteroscedasticity process seems an ARMA (1,1). The autoregressive root that governs the persistence of the shocks of volatility is the sum of ( 

Now we can estimate the volatility 13 th AFIR Colloquium 2003 For the day  For  days or the volatility between t and t+ 

Risk Metrics™ 13 th AFIR Colloquium 2003 The analysts have fruitfully applied the Garch methodology in assets pricing models and in the volatility forecast. Risk Metrics use a special Garch model when use the decay factor . The behavior of this model is similar to: Garch (1,1) with  and  ™] Risk Metrics is a trade mark of J. P. Morgan]

The limitations of Garch (1,1) 13 th AFIR Colloquium ) Garch models only are sensitive to the magnitude of the excess of returns and not to the sign of this excess of return. 2) The non negative constraints on  and  which are imposed to ensure that  2 t remains positive 3) The conditional moments, may explode when the process itself is strictly stationary and ergodic.

The solutions for the limitations of Garch (1,1) The asymmetric models 13 th AFIR Colloquium 2003 Egarch (p,q) Tarch (1,1)

How to detect the asymmetry and select the correct model 13 th AFIR Colloquium 2003 The asymmetry test Log likelihood A.I.C. S.C. 1 2

The cross correlation for the asymmetry test 13 th AFIR Colloquium 2003 Where:

The asymmetry test 13 th AFIR Colloquium 2003 We must do a cross correlation between the squared residuals of the Garch model and the standardized residuals of the same (  t /  t ) The result of this cross correlation will be a white noise if the model is symmetric or in other words the Garch model is correctly specified, and a black noise is the model is asymmetric.

The results applied to Tbond 5 y. 13 th AFIR Colloquium 2003 Garch (1.1)Tarch(1,1)Egarch(1,1) C    

The results applied to Tbond 5 y. 13 th AFIR Colloquium 2003 Garch (1,1) Tarch (1,1) Egarch (1,1) Log likelihood AIC SC

The tests to confirm the use of an asymmetry model for Treasury 5 years 13 th AFIR Colloquium 2003 The cross correlogram Limits to accept a white noise

The second component of VaR The probability distribution 13 th AFIR Colloquium 2003 It was demonstrated that the returns don’t follows a normal distribution, for that reason I include the Heavy tails distributions What probability distribution follows the returns?

The heavy tails distributions found in returns series 13 th AFIR Colloquium 2003 The Logistic Distribution

The heavy tails distributions found in returns series 13 th AFIR Colloquium 2003 The Weibull Distribution

The EVD 13 th AFIR Colloquium 2003 This distribution depends of three parameters:  = mode;  = location and  = shape  Gumbel Distribution  Frechet Distribution  Weibull Distribution Where z = (y –  ) / 

The PWM for estimate EVD parameters 13 th AFIR Colloquium 2003 Where U is a plotting position that follows a free distribution and k takes the probability as: p k,n = [(n-k)+0.5]/n.

The EVD  13 th AFIR Colloquium 2003 Weibull distribution with different values of      

The EVD  <0 13 th AFIR Colloquium 2003 Frechet distribution with different values of    

The Kupiec solution 13 th AFIR Colloquium 2003 Kupiec demonstrate that on base a normal distribution that it is possible to extend the tails of the distribution in form that contemplate the probability of a catastrophe. The value that takes the abscissa named z of a standardized normal distribution extended by Kupiec is:

An example of returns: Tbond 5 y. 13 th AFIR Colloquium 2003 Tbond 5y. daily returns The Goodness of fit test K/S AD

An example of returns: Bovespa 13 th AFIR Colloquium 2003 Bovespa daily returns The Goodness of fit test K/S AD

The Goodness of fit tests 13 th AFIR Colloquium ) Kolmogorov Smirnov The Kolmogorov Smirnov test is a test that is independent of any Gaussian distribution, and have the benefit that not need a great number of observations. There is one critical value that depends on the number of observations and the level of confidence

The Goodness of fit tests 13 th AFIR Colloquium ) Anderson Darling The Anderson Darling test is a refinement of KS test, specially studied for heavy tails distributions. There are several critical values for each distribution fitted and depends from the number of observations and the level of confidence

What type of distributions we found 13 th AFIR Colloquium 2003 SeriesObs First Dist Fit Second Dist Fit Test Goodness of Fit KSAD 1st2nd1st2nd Tbond755LogisticEVD D.Jones1056LogisticWeibull Bovespa1036LogisticWeibull Merval773LogisticEVD IDP445LogisticWeibull MSCI1395LogisticWeibull

Simulations 13 th AFIR Colloquium 2003 After we define the best probability distribution for the series returns we can simulate trials using two methods 1)Montecarlo 2)Latin Hypercube The objective is found the value of the first percentile to determine the worst loss possible

Simulation with daily returns of Tbond 5y 13 th AFIR Colloquium 2003

Simulation daily returns D.Jones 13 th AFIR Colloquium 2003 The Logistic Distribution The EVD Distribution

Some results 13 th AFIR Colloquium 2003 AssetModelOutliers % of outlies / Observations TbondEgarch(1,1)61.0 Dow JonesEgarch(1,1)101.0 BovespaTarch (1,1)101.0 MervalEgarch(1,1)131.1 IDPEgarch(1,1)30.9 MSCIGarch (1,1)100.8

The Market VaR of Tbond 13 th AFIR Colloquium 2003

The Market VaR of DJI 13 th AFIR Colloquium 2003

Conclusions 13 th AFIR Colloquium 2003 The asymmetric Garch models, like Tarch or Egarch model, not only fulfill with the movements of the volatility, as we can observe with the back testing presented, also it is not necessary to use the heavy tails distributions, because the negative impact or the negative returns are included by the model form and is the form of a dynamic VaR

Conclusions 13 th AFIR Colloquium 2003 The static VaR estimated with the heavy tails distribution don’t follows the volatility movements and create reserves in excess. The time series history, complies with the requirements of Basel II, to make the volatility forecast. It is easy to teach this model to the traders, but not for the actuaries. The traders and the shareholders only import the recent past