Abstract
The paper presents and tests Dynamic Value at Risk (VaR) estimation procedures for equity index returns. Volatility clustering and leptokurtosis are well-documented characteristics of such time series. An ARMA (1, 1)-GARCH (1, 1) approach models the inherent autocorrelation and dynamic volatility. Fattailed behavior is modeled in two ways. In the first approach, the ARMA-GARCH process is run assuming alternatively that the standardized residuals are distributed with Pearson Type IV, Johnson SU, Manly’s exponential transformation, normal and t-distributions. In the second approach, the ARMA-GARCH process is run with the pseudonormal assumption, the parameters calculated with the pseudo maximum likelihood procedure, and the standardized residuals are later alternatively modeled with Mixture of Normal distributions, Extreme Value Theory and other power transformations such as John-Draper, Bickel-Doksum, Manly, Yeo-Johnson and certain combinations of the above. The first approach yields five models, and the second ap-proach yields nine. These are tested with six equity index return time series using rolling windows. These models are compared by computing the 99%, 97.5% and 95% VaR violations and contrasting them with the expected number of violations.
Highlights
VALUE AT RISK (VaR) is a popular measure of risk in a portfolio of assets
The ARMA-GARCH process is run with the pseudo-normal assumption, the parameters calculated with the pseudo maximum likelihood procedure, and the standardized residuals are later alternatively modeled with Mixture of Normal distributions, Extreme Value Theory and other power transformations such as John-Draper, Bickel-Doksum, Manly, Yeo-Johnson and certain combinations of the above
In the first one, consisting of five models, ARMA-GARCH model parameters are calculated assuming that standardized residuals alternatively follow Pearson Type IV distribution, Johnson SU distribution, Manly’s exponential transformation, normal and Student t-distributions
Summary
VALUE AT RISK (VaR) is a popular measure of risk in a portfolio of assets. It represents a high quantile of loss distribution for a particular horizon, providing a loss threshold that is exceeded only a small percentage of the time. The historical simulation approach assumes constant volatility of stocks over an extended period of time It fails to account for the phenomenon of volatility clustering, when periods of high and low volatility occur together. Normal distribution to model asset returns, sparking a slew of papers addressing the issue of accurately modeling leptokurtic time series with volatility clustering. In the first one, consisting of five models, ARMA-GARCH model parameters are calculated assuming that standardized residuals alternatively follow Pearson Type IV distribution, Johnson SU distribution, Manly’s exponential transformation, normal and Student t-distributions. The ARMA-GARCH parameters are calculated using the pseudo-normal assumption, i.e., assuming that standardized residuals are normally distributed, and they are later modeled using the mixture of normal distributions, Extreme Value Theory, and other power transformations such as John-Draper, BickelDoksum, Manly, Yeo-Johnson and certain combinations of the above. The importance of creating an accurate measure of risk cannot be understated, given how the stock market crash of 2008 bankrupted firms and individuals alike, and sent the world spiraling into recession
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