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 dont 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 dont 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