• Value at risk

Thanks to its wide diffusion in the industry, Value-at-Risk (VaR) manages to became a cornerstone in the growing and complex regulation of capital requirements (Basel Accords). For this reason, despite the theoretical limitations of VaR,
the study of how improve the performance of such risk measure is still fundamental.
This thesis concerns the parametric method used to estimate Value-at-Risk and the evaluation of such estimates. The accuracy in predicting future risks, strictly depends on how such measure is calculated. The chosen method for the calculation is the parametric approach based on various extensions of the ARCH-GARCH models, combined with different assumed distributions for the returns. The ARCHGARCH models should be able to fit time series which show a time-varying volatility (heteroskedasticity), while more leptokurtic distributions (such as Student’s t and GED) than the Normal one, and their relative skew version, should provide better tail forecast and hence better VaR estimates.
The primary objective of this work is the evaluation of the estimates obtained from the models described above. For this purposes, several backtesting methods were performed and their results compared. Backtesting is a statistical procedure
where actual profits and losses are systematically compared to corresponding VaR estimates.
Backtesting methods here considered can be broadly divide in two categories.
Those tests that evaluate only a single VaR level (i.e. 1% or 5%) and those tests that evaluate a multiple VaR levels (hence they evaluate the entire density forecast).
To the first group belong test such as: Kupiec’s Unconditional Coverage test, Christoffersen’s Conditional Coverage test, Mixed Kupiec test and Duration test.
While to the second group belongs the Crnkovic-Drachman test, the Q-test and the Berkowitz test.
The results are then compared in the light of the strengths and the weaknesses of each approach. It emerged a substantial heterogeneity among the outcomes of these tests, especially between backtesting methods base on a single VaR level and
those based on a multiple VaR levels.
This empirical work is built on the framework of Angelidis, Benos and Degiannakis (2003). However, different volatility models, distributions and backtesting methods were employed. For these reasons, a comparison between the results of the
two study is also provided.